Uniqueness of entire solutions to quasilinear equations of $p$-Laplace type

1.   Introduction

We prove the uniqueness property for a class of reachable solutions to the equation

where $\sigma\ge 0$ is a locally finite Borel measure in $\mathbb{R}^n$ absolutely continuous with respect to the $p$-capacity, and $\Delta_p u = {\rm div} (\nabla u | \nabla u |^{p-2})$ ($1 < p < \infty$) is the $p$-Laplace operator.

More general $\mathcal{A}$-Laplace operators ${\rm div}\, \mathcal{A}(x, \nabla u)$ in place of $\Delta_p$, under standard growth and monotonicity assumptions of order $p$ on $\mathcal{A}(x, \xi)$ ($x, \xi \in \mathbb{R}^n$), are treated as well (see Section 2). All solutions $u$ of (1.1) are understood to be $\mathcal{A}$-superharmonic (or, equivalently, locally renormalized) solutions in $\mathbb{R}^n$ (see ^[19] and Section 3 below).

We often use bilateral global pointwise estimates of solutions to (1.1) obtained by Kilpeläinen and Malý ^[20,21] in terms of the Havin–Maz'ya–Wolff potentials (often called Wolff potentials) ${\rm \bf W}_{1, \, p}\sigma$. Criteria of existence of solutions to (1.1), which ensure that ${\rm \bf W}_{1, \, p}\sigma\not\equiv \infty$, can be found in ^[33] (see also Section 3 below).

We remark that existence and uniqueness results are known for certain classes of solutions to quasilinear equations with $\mathcal{A}$-Laplace operators similar to (1.1) in arbitrary domains $\Omega\subseteq \mathbb{R}^n$ (not necessarily bounded), but with various additional restrictions on $\mathcal{A}(x, \xi)$ and data $\sigma$. We refer to ^[2] for $\sigma \in L^1(\Omega)$, and ^[25] for measures $\sigma$ with finite total variation in $\Omega$. Notice that in general only local analogues of the Kilpeläinen and Malý pointwise estimates are known for solutions in domains $\Omega\subsetneqq \mathbb{R}^n$. For our purposes, we need global pointwise estimates, which at the moment are available only for $\Omega = \mathbb{R}^n$, or in the case $p = 2$ for linear operators ${\rm div}\, (\mathcal{A}(x) \nabla u)$ in terms of positive Green's functions in domains $\Omega\subsetneqq \mathbb{R}^n$.

In Section 4, we prove uniqueness of nontrivial reachable solutions to the problem

in the sub-natural growth case $0 < q < p-1$, where $\mu, \sigma$ are nonnegative locally finite measures in $\mathbb{R}^n$ absolutely continuous with respect to the $p$-capacity. We observe that such a uniqueness property generally fails in the case $q\ge p-1$.

When we treat the uniqueness problem for solutions of equations of type (1.2), but with more general $\mathcal{A}$-Laplace operators ${\rm div}\, \mathcal{A}(x, \nabla \cdot)$ in place of $\Delta_p$, we impose the additional homogeneity condition $\mathcal{A}(x, \lambda \xi) = \lambda^{p-1} \mathcal{A}(x, \xi)$, for all $\xi \in\mathbb{R}^n$ and $\lambda > 0$ (see Section 4). We emphasize that our proof of uniqueness for reachable solutions of such equations relies upon the homogeneity of order $p-1$ of the $\mathcal{A}$-Laplacian, as well as homogeneity of order $q$ of the term $\sigma u^q$, for $0 < q < p-1$. Our main tool in this proof is provided by bilateral pointwise estimates for all entire solutions obtained recently in ^[37], which do not require the homogeneity of the $\mathcal{A}$-Laplacian.

We observe that in the case $p = 2$ all superharmonic solutions of Eqs (1.1) and (1.2) are reachable, and hence unique. An analogue of this fact is true for more general equations with the linear uniformly elliptic $\mathcal{A}$-Laplace operator ${\rm div}\, (\mathcal{A}(x) \nabla u)$, with bounded measurable coefficients $\mathcal{A} \in L^\infty({\bf R}^n)^{n\times n}$, in place of $\Delta$. In other words, all entire $\mathcal{A}$-superharmonic solutions to such equations are unique. For similar problems in domains $\Omega \subseteq \mathbb{R}^n$ and linear operators with positive Green's function satisfying some additional properties (in particular, in uniform domains) the uniqueness property was obtained recently in ^[38].

The uniqueness of nontrivial bounded (superharmonic) solutions for (1.2) in the case $p = 2$ was proved earlier by Brezis and Kamin ^[7]. For solutions $u\in C(\overline{\Omega})$ in bounded smooth domains $\Omega\subset \mathbb{R}^n$ and $\mu, \sigma \in C(\overline{\Omega})$, along with some more general equations involving monotone increasing, concave nonlinearities on the right-hand side, the uniqueness property was originally established by Krasnoselskii [23,Theorem 7.14].

As shown below, for $p\not = 2$, all $p$-superharmonic solutions $u$ to (1.1) or (1.2) are reachable, and hence unique, if, for instance, the condition ${\liminf\limits_{|x|\rightarrow \infty}}\, u = 0$ in (1.1) or (1.2), respectively, is replaced with ${\lim\limits_{|x|\rightarrow \infty}}\, u = 0$. See Sections 3 and 4, where we discuss this and other conditions that ensure that all solutions are reachable.

Existence criteria and bilateral pointwise estimates for all $\mathcal{A}$-superharmonic solutions to (1.2) were obtained in ^[37]. (See also earlier results in ^[9] involving minimal solutions in the case $\mu = 0$.) In particular, it is known that the measure $\sigma$ is necessarily absolutely continuous with respect to the $p$-capacity provided there exists a nontrivial $u \ge 0$ such that $- \Delta_p u \ge \sigma u^q$ (^[9], Lemma 3.6).

We remark that the proofs of the main existence results in ^[9,37] for (1.2) in the case $\mu = 0$ used a version of the comparison principle (^[9], Lemma 5.2) that contained some inaccuracies. A corrected form of this comparison principle is provided in Lemma 4.1 below. The other parts of ^[9,37] are unaffected by this correction.

With regards to the existence problem, we prove additionally that we can always construct a reachable solution to either (1.1) or (1.2), whenever a solution to the corresponding equation exists (see Theorem 3.10 and Remark 4.7 below).

2.   $\mathcal{A}$-superharmonic functions

Let ${\Omega}\subseteq \mathbb{R}^n$, $n\geq 2$, be an open set. By ${\mathcal{M}}^+({\Omega})$ we denote the cone of nonnegative locally finite Borel measures in ${\Omega}$, and by ${\mathcal{M}}^+_b({\Omega})$ the subcone of finite measures in ${\mathcal{M}}^+({\Omega})$. For $\mu \in {\mathcal{M}}^+({\Omega})$, we set $\Vert \mu\Vert_{{\mathcal{M}}^+(\Omega)} = \mu(\Omega)$ even if $\mu(\Omega) = +\infty$. The space of finite signed Borel measures in ${\Omega}$ is denoted by ${\mathcal{M}}_b({\Omega})$. By $\Vert \mu\Vert_{{\mathcal{M}}_b({\Omega})}$ we denote the total variation of $\mu \in {\mathcal{M}}_b({\Omega})$.

Let $\mathcal{A}\colon \mathbb{R}^n \times \mathbb{R}^n\rightarrow \mathbb{R}^n$ be a Carathéodory function in the sense that the map $x \rightarrow \mathcal{A}(x, \xi)$ is measurable for all $\xi\in\mathbb{R}^n,$ and the map $\xi\rightarrow \mathcal{A}(x, \xi)$ is continuous for a.e. $x\in\mathbb{R}^n$. Throughout the paper, we assume that there are constants $0 < \alpha\leq\beta < \infty$ and $1 < p < n$ such that for a.e. $x$ in $\mathbb{R}^n$,

In the uniqueness results of Sec. 4, we assume additionally the homogeneity condition

Such homogeneity conditions are often used in the literature (see ^[18,21]).

For an open set ${\Omega}\subset\mathbb{R}^n$, it is well known that every weak solution $u\in W^{1, \, p}_{{\rm loc}}({\Omega})$ to the equation

has a continuous representative. Such continuous solutions are said to be $\mathcal{A}$-$harmonic$ in ${\Omega}$. If $u\in W_{{\rm loc}}^{1, \, p}({\Omega})$ and

for all nonnegative $\varphi\in C^{\infty}_{0}({\Omega})$, i.e., $-{\rm div}\mathcal{A}(x, \nabla u) \geq 0$ in the distributional sense, then $u$ is called a supersolution to (2.3) in ${\Omega}$.

A function $u\colon{\Omega}\rightarrow (-\infty, \infty]$ is called $\mathcal{A}$-$superharmonic$ if $u$ is not identically infinite in each connected component of ${\Omega}$, $u$ is lower semicontinuous, and for all open sets $D$ such that ${\overline D}\subset{\Omega}$, and all functions $h\in C(\overline{D})$, $\mathcal{A}$-harmonic in $D$, it follows that $h\leq u$ on $\partial D$ implies $h\leq u$ in $D$.

A typical example of $\mathcal{A}(x, \xi)$ is given by $\mathcal{A}(x, \xi) = |\xi|^{p-2}\xi$, which gives rise to the $p$-Laplacian $\Delta_p u = {\rm div}\, (|\nabla u|^{p-2}\nabla u)$. In this case, $\mathcal{A}$-superharmonic functions will be called $p$-superharmonic functions.

We recall here the fundamental connection between supersolutions of (2.3) and $\mathcal{A}$-superharmonic functions discussed in ^[18].

Proposition 2.1 (^[18]). (ⅰ) If $u\in W_{{\rm loc}}^{1, \, p}({\Omega})$ is such that

then there is an $\mathcal{A}$-superharmonic function $v$ such that $u = v$ a.e. Moreover,

(ⅱ) If v is $\mathcal{A}$-superharmonic, then (2.4) holds. Moreover, if $v\in W^{1, \, p}_{{\rm loc}}({\Omega})$, then

(ⅲ) If v is $\mathcal{A}$-superharmonic and locally bounded, then $v\in W^{1, \, p}_{{\rm loc}}({\Omega})$, and

Note that if $u$ is $\mathcal{A}$-superharmonic, then the gradient of $u$ may not exist in the sense of distributions in the case $1 < p\leq 2-1/n$. On the other hand, if $u$ is an $\mathcal{A}$-superharmonic function, then its truncation $u_k = \min\{u, k\}$ is $\mathcal{A}$-superharmonic as well, for any $k > 0$. Moreover, by Proposition 2.1(ⅲ) we have $u_k\in W^{1, \, p}_{{\rm loc}}({\Omega})$. Using this we define the very weak gradient

If either $u\in L^{\infty}({\Omega})$ or $u\in W^{1, \, 1}_{{\rm loc}}({\Omega})$, then $Du$ coincides with the regular distributional gradient of $u$. In general we have the following gradient estimates ^[20] (see also ^[18]).

Proposition 2.2 (^[20]). Suppose u is $\mathcal{A}$-superharmonic in ${\Omega}$ and $1\leq q < \frac{n}{n-1}$. Then both $|Du|^{p-1}$ and $\mathcal{A}(\cdot, Du)$ belong to $L^{q}_{{\rm loc}}({\Omega})$. Moreover, if $p > 2-\frac{1}{n}$, then $Du$ coincides with the distributional gradient of u.

Note that by Proposition 2.2 and the dominated convergence theorem, we have

whenever $\varphi\in C^{\infty}_{0}({\Omega})$, $\varphi\geq 0$, and $u$ is $\mathcal{A}$-superharmonic in ${\Omega}$. It follows from Riesz's representation theorem (see [18,Theorem 21.2]) that there exists a unique measure $\mu[u]\in {\mathcal{M}}^+({\Omega})$ called the Riesz measure of $u$ such that

3.   Quasilinear equations with locally finite measure data in the entire space

In this section, we investigate the problems of existence and uniqueness of $\mathcal{A}$-superharmonic solutions in the entire space $\mathbb{R}^n$ to the equation

with measures $\sigma \in {\mathcal{M}}^+(\mathbb{R}^n)$ (not necessarily finite).

There has been a lot of work addressing the existence and uniqueness problem for quasilinear equations of the form

in a bounded domain ${\Omega}\subset \mathbb{R}^n$, where $\sigma\in L^1({\Omega})$, or, more generally, $\sigma\in {\mathcal{M}}_b({\Omega})$; see, e.g., ^{[2,4,5,6,10,11,12,22,25]}. For arbitrary domains, including $\mathbb{R}^n$, we refer to the papers ^[2] (for $L^1$ data) and ^[25] (for data in ${\mathcal{M}}_b({\Omega})$). In these papers one can find the notions of { entropy solutions (see ^[2,6,22]), SOLA (solutions obtained as limits of approximations) for $L^1$ data (see ^[11]), reachable solutions (see ^[10]), and renormalized solutions (see ^[12]).

The current state of the art on the uniqueness problem for (3.2) is that most results require that $\sigma < < {\rm cap}_p$, i.e., $\sigma$ is absolutely continuous with respect to the $p$-capacity in the sense that $\sigma(K) = 0$ for any compact set $K\subset \Omega$ such that ${\rm cap}_p(K) = 0$. The $p$-capacity ${\rm cap}_p(\cdot)$ is a natural capacity associated with the $p$-Laplacian defined by

for any compact set $K\subset \Omega$.

For later use, we now recall the following equivalent definitions of a (global) renormalized solution to Eq (3.2) (see ^[12]). For our purposes, we shall restrict ourselves to the case $\sigma\in {\mathcal{M}}^+_b({\Omega})$ and nonnegative solutions. Recall that we may use the decomposition $\sigma = \sigma_0+\sigma_s$, where both $\sigma_0$ and $\sigma_s$ are nonnegative measures such that $\sigma_0 < < {\rm cap}_0$, and $\sigma_s$ is concentrated on a set of zero $p$-capacity.

Definition 3.1. Let $\sigma\in {\mathcal{M}}^+_b({\Omega})$, where ${\Omega}\subset\mathbb{R}^n$ is a bounded open set. Then $u\geq 0$ is said to be a renormalized solution of (3.2) if the following conditions hold:

(a) The function $u$ is measurable and finite almost everywhere, and $T_{k}(u)$ belongs to $W_{0}^{1, \, p}({\Omega})$ for every $k > 0$, where $T_{k}(s): = \min \{k, s\}$, $s\ge 0$.

(b) The gradient $Du$ of $u$ satisfies $\left| {Du} \right|^{p-1}\in L^{q}({\Omega})$ for all $q < \frac{n}{n-1}$.

(c) For any $h\in W^{1, \infty}(\mathbb{R})$ with compact support, and any $\varphi\in W^{1, p}({\Omega})\cap L^\infty({\Omega})$ such that $h(u) \, \varphi \in W^{1, p}_0({\Omega})$,

and for any $\varphi\in C^{0}_{\rm b}({\Omega})$ (the space of bounded and continuous functions in ${\Omega}$),

Definition 3.2. Let $\sigma\in {\mathcal{M}}^+_b({\Omega})$, where ${\Omega}\subset\mathbb{R}^n$ is a bounded open set. Then $u\geq 0$ is said to be a renormalized solution of (3.2) if $u$ satisfies (a) and (b) in Definition 3.1, and if the following condition holds:

(c) For any $h\in W^{1, \infty}(\mathbb{R})$ with such that $h'$ has compact support, and any $\varphi\in W^{1, r}({\Omega})\cap L^\infty({\Omega})$, $r > n$, such that $h(u)\varphi \in W^{1, p}_0({\Omega})$,

Here $h(+\infty): = \lim\limits_{s\rightarrow +\infty} h(s)$.

Definition 3.3. Let $\sigma\in {\mathcal{M}}^+_b({\Omega})$, where ${\Omega}\subset\mathbb{R}^n$ is a bounded open set. Then $u\geq 0$ is said to be a renormalized solution of (3.2) if $u$ satisfies (a) and (b) in Definition 3.1, and if the following conditions hold:

(c) For every $k > 0$, there exists $\lambda_k \in {\mathcal{M}}^+_b({\Omega})$ concentrated on the set $\{u = k\}$ such that $\lambda_k < < {\rm cap}_p$, and $\lambda_{k}\rightarrow \sigma_{s}$ in the narrow topology of measures in ${\Omega}$ as $k\rightarrow \infty$, i.e.,

(d) For every $k > 0$,

for all $\varphi \in {\rm W}_{0}^{1, \, p}({\Omega})\cap L^{\infty}({\Omega})$.

We shall also need the notion of a local renormalized (nonnegative) solution on a general open set ${\Omega}\subseteq\mathbb{R}^n$ (not necessarily bounded) associated with a measure $\sigma \in {\mathcal{M}}^+({\Omega})$ (not necessarily finite). We recall the following equivalent definitions (see ^[3]), adapted to the case of nonnegative solutions.

Definition 3.4. Let $\sigma \in {\mathcal{M}}^+({\Omega})$, where ${\Omega}\subseteq\mathbb{R}^n$ is an open set. Then a nonnegative function $u$ is said to be a local renormalized solution of the equation $-{\rm div} \mathcal{A}(x, \nabla u) = \sigma$, if the following conditions hold:

(a) The function $u$ is measurable and finite almost everywhere, and $T_{k}(u)$ belongs to $W_{\rm loc}^{1, \, p}({\Omega})$, for every $k > 0$, where $T_{k}(s): = \min \{k, s\}$, $s \ge 0$.

(b) The gradient $Du$ of $u$ satisfies $\left| {Du} \right|^{p-1}\in L^{q}_{\rm loc}({\Omega})$ for all $0 < q < \frac{n}{n-1}$, and $u^{p-1} \in L^s_{\rm loc}({\Omega})$ for all $0 < s < \frac{n}{n-p}$.

(c) For any $h\in W^{1, \infty}(\mathbb{R})$ with compact support, and any $\varphi\in W^{1, p}({\Omega})\cap L^\infty({\Omega})$ with compact support in ${\Omega}$ such that $h(u) \, \varphi \in W^{1, p}({\Omega})$,

and for any $\varphi\in C_{\rm b}^{0}({\Omega})$ with compact support in ${\Omega}$,

Definition 3.5. Let $\sigma \in {\mathcal{M}}^+({\Omega})$, where ${\Omega}\subseteq\mathbb{R}^n$ is an open set. Then a nonnegative function $u$ is said to be a local renormalized solution of the equation $-{\rm div} \mathcal{A}(x, \nabla u) = \sigma$, if $u$ satisfies (a) and (b) in Definition 3.4, and if the following conditions hold:

(c) For every $k > 0$, there exists a nonnegative measure $\lambda_k < < {\rm cap}_p$, concentrated on the sets $\{u = k\}$, such that $\lambda_{k}\rightarrow \sigma_{s}$ weakly as measures in ${\Omega}$ as $k\rightarrow \infty$, i.e.,

for all $\varphi\in C^0_{\rm b}({\Omega})$ with compact support in ${\Omega}$.

(d) For every $k > 0$,

for all $\varphi$ in ${\rm W}_{0}^{1, \, p}({\Omega})\cap L^{\infty}({\Omega})$ with compact support in ${\Omega}$.

We now discuss solutions of (3.1) for general measures $\sigma \in {\mathcal{M}}^+(\mathbb{R}^n)$. It is known that a necessary and sufficient condition for (3.1) to admit an $\mathcal{A}$-superharmonic solution is the finiteness condition

(see, e.g., ^[33,34]). Thus, it is possible to solve (3.1) for a wide and optimal class of measures $\sigma$ satisfying (3.3) that are not necessarily finite.

We mention that (3.3) is equivalent to the condition ${\bf W}_{1, p}\sigma(x) < +\infty$ for some $x\in\mathbb{R}^n$ (or equivalently quasi-everywhere in $\mathbb{R}^n$ with respect to the $p$-capacity), where

is the Havin–Maz'ya–Wolff potential of $\sigma$ (often called the Wolff potential); see ^[17,26].

By the fundamental result of Kilpeläinen and Malý ^[20,21], any $\mathcal{A}$-superharmonic solution $u$ to Eq (3.1) satisfies the following global pointwise estimates,

where $K > 0$ is a constant depending only on $n, p$ and the structural constants $\alpha$ and $\beta$ in (2.1).

Our main goal here is to introduce a new notion of a solution to (3.1) so that existence is obtained under the natural growth condition (3.3) for $\sigma$, and uniqueness is guaranteed as long as $\sigma < < {\rm cap}_p$ (see Definition 3.8 below).

We begin with the following result on the existence of a minimal solution to (3.1) in case the measure $\sigma$ is continuous with respect to the $p$-capacity.

Theorem 3.6. Let $\sigma \in {\mathcal{M}}^+(\mathbb{R}^n)$, where $\sigma < < {\rm cap}_p$. Suppose that (3.3) holds. Then there exists a minimal $\mathcal{A}$-superharmonic solution to Eq (3.1).

Proof. Condition (3.3) implies that

for all $x\in\mathbb{R}^n$. Thus,

This yields

Here we used the fact that, if $\mu\in {\mathcal{M}}^+_b(\mathbb{R}^n)$, then ${\rm cap}_p(\{{\bf W}_{1, p}\mu = \infty \}) = 0$ (see [1,Proposition 6.3.12]). It follows that $\sigma(\{{\bf W}_{1, p}\sigma = \infty\}) = 0$, since $\sigma < < {\rm cap}_p$.

Let $\sigma_k$ ($k = 1, 2, \dots$) be the restriction of $\sigma$ to the set $B_k(0)\cap\{ {\bf W}_{1, p}\sigma < k\}$. We then have that $\sigma_k$ weakly converges to $\sigma$, and

Hence, $\sigma_k\in W^{-1, p'}(B_k(0))$ ($1/p+1/p' = 1$), and for each $k > 0$, there exists a unique nonnegative solution $u_k\in W^{1, p}_0(B_k(0))$ to the problem

If we set $u_k = 0$ in $\mathbb{R}^n\setminus B_k(0)$, then the sequence $\{u_k\}$ is non-decreasing, and by [33,Theorem 2.1{ }],

By [20,Theorem 1.17], it follows that the function $u: = \lim_{k\to \infty} u_k$ is $\mathcal{A}$-superharmonic in $\mathbb{R}^n$. Moreover, $u \le K\, {\bf W}_{1, p}\sigma$, and consequently

Thus, $u$ is an $\mathcal{A}$-superharmonic solution of (3.1).

To show the minimality of $u$, let $v$ be another $\mathcal{A}$-superharmonic solution of (3.1). From the construction of $u$, it is enough to show that $u_k\leq v$ for any $k\geq 1$. To this end, let $\nu_j$, $j = 1, 2, \dots$, be the Riesz measure of ${\rm min}\{v, j\}$. Since $v$ is $\mathcal{A}$-superharmonic, it is also a local renormalized solution to $-{\rm div}\, \mathcal{A}(x, \nabla v) = \sigma$ in $\mathbb{R}^n$ (see ^[19]). Hence, by a result of ^[3,12] and the fact that $\sigma < < {\rm cap}_p$, we obtain

for $\alpha_j\in {\mathcal{M}}^+(\mathbb{R}^n)$ concentrated in the set $\{v = j\}$.

Using the estimate $v\leq K{\bf W}_{1, p}\sigma$, we deduce

provided $j/K > k$. Since $u_k\in W^{1, p}_0(B_k(0))$ and ${\rm min}\{v, j\}\in W^{1, p}(B_k(0))$, by the comparison principle (see [9,Lemma 5.1]), we estimate

provided $j\geq K k$. Thus, $u = \lim_{k \to \infty} u_k \le v$. This completes the proof of the theorem.

The proof of the minimality of $u$ above can be modified to obtain the following comparison principle.

Theorem 3.7 (Comparison Principle). Let $\sigma, \tilde{\sigma} \in {\mathcal{M}}^+(\mathbb{R}^n)$, where $\sigma \leq \tilde{\sigma}$ and $\sigma < < {\rm cap}_p$, $1 < p < n$. Then $u \le \tilde{u}$, where $u$ is the minimal $\mathcal{A}$-superharmonic solution of (3.1) and $\tilde{u}$ is any $\mathcal{A}$-superharmonic solution of (3.1) with datum $\tilde{\sigma}$ in place of $\sigma$.

Proof. Let $\sigma'_k$, $k = 1, 2, \dots$, be the restriction of $\sigma$ to the set $B_k(0)\cap\{ {\bf W}_{1, p}\tilde{\sigma} < k\}$. Since $\sigma < < {\rm cap}_p$ we have that $\sigma'_k$ weakly converges to $\sigma$. Moreover, as ${\bf W}_{1, p}\sigma'_k\leq {\bf W}_{1, p}\sigma\leq {\bf W}_{1, p}\tilde{\sigma} < k$ on the set $\{ {\bf W}_{1, p}\tilde{\sigma} < k\}$, it follows that

Hence, $\sigma'_k\in W^{-1, \frac{p}{p-1}}(B_k(0))$, and for each $k > 0$ there exists a unique nonnegative solution $u'_k\in W^{1, p}_0(B_k(0))$ to the problem

Letting $u'_k = 0$ in $\mathbb{R}^n\setminus B_k(0)$, we have that the sequence $\{u'_k\}$ is non-decreasing, and by [33,Theorem 2.1],

Then $u'_k$ converges pointwise to an $\mathcal{A}$-superharmonic solution $u'$ of (3.1) by [20,Theorem 1.17]. On the other hand, by the comparison principle of [9,Lemma 5.1], we have

where $u_k$ is defined in (3.5). Hence, letting $k\rightarrow \infty$, we get $u'\leq u$, which yields $u' = u$ by the minimality of $u$.

We now let $\tilde{\sigma}_j$ ($j = 1, 2, \dots$) be the Riesz measure of ${\rm min}\{\tilde{u}, j\}$. Recall that the Riesz measure $\tilde{\sigma}$ of $\tilde{u}$ can be decomposed as

where $\tilde{\sigma}_0\in {\mathcal{M}}^+(\mathbb{R}^n)$, $\tilde{\sigma}_0 < < {\rm cap}_p$, and $\tilde{\sigma}_s\in {\mathcal{M}}^+(\mathbb{R}^n)$ is concentrated on a set of zero $p$-capacity. Then by a result of ^[3,12], we have

where $\tilde{\alpha}_j \in {\mathcal{M}}^+(\mathbb{R}^n)$ is concentrated in the set $\{\tilde{u} = j\}$. On the other hand, since $\tilde{\sigma}_s(\{\tilde{u} < \infty\}) = 0$ (see [19,Lemma 2.9]), we can rewrite (3.6) as

Now using the estimate $\tilde{u}\leq K{\bf W}_{1, p}\tilde{\sigma}$ and (3.7), we have

provided $j/K > k$.

Since $u'_k\in W^{1, p}_0(B_k(0))$ and ${\rm min}\{\tilde{u}, j\}\in W^{1, p}(B_k(0))$, by the comparison principle of [9,Lemma 5.1] we find

provided we choose a $j$ such that $j\geq K k$. Letting $k\rightarrow \infty$, we obtain $u\leq \tilde{u}$ as desired.

Theorem 3.6 justifies the existence (and hence uniqueness) of the minimal $\mathcal{A}$-superharmonic solution to (3.1) provided condition (3.3) holds and $\sigma < < {\rm cap}_p$. It is not known if condition (3.3) alone is enough for the existence of the minimal solution. It is also not known if under condition (3.3) and $\sigma < < {\rm cap}_p$ all $\mathcal{A}$-superharmonic solutions to (3.1) coincide with the minimal solution. For a partial result in this direction, see Theorem 3.12 below.

We now introduce a new notion of a solution so that uniqueness is guaranteed for all nonnegative locally finite measures $\sigma$ such that $\sigma < < {\rm cap}_p$. Our definition is an adaptation of the notion of the reachable solution of [10,Definition 2.3].

Definition 3.8. Let $\sigma \in {\mathcal{M}}^+(\mathbb{R}^n)$. We say that a function $u: \mathbb{R}^n \rightarrow [0, +\infty]$ is an $\mathcal{A}$-superharmonic reachable solution to Eq (3.1) if $u$ is an $\mathcal{A}$-superharmonic solution of (3.1), and there exist two sequences $\{u_i\}$ and $\{\sigma_i\}$, $i = 1, 2, \dots$, such that

${\rm (ⅰ)}$ Each $\sigma_i\in {\mathcal{M}}^+(\mathbb{R}^n)$ is compactly supported in $\mathbb{R}^n$, and $\sigma_i \leq \sigma$;

${\rm (ⅱ)}$ Each $u_i$ is an $\mathcal{A}$-superharmonic solution of (3.1) with datum $\sigma_i$ in place of $\sigma$;

${\rm (ⅲ)}$ $u_i\rightarrow u$ a.e. in $\mathbb{R}^n$.

Remark 3.9. The notion of reachable solution was introduced in ^[10] for equations over bounded domains with finite measure data. It is also related to the notion of SOLA (Solution Obtained as Limit of Approximations) of ^[11] for $L^1$ data over bounded domains. By ${\rm (ⅲ)}$ and the weak continuity result of ^[36], we see that $\sigma_i \rightarrow \sigma$ weakly as measures in $\mathbb{R}^n$. The extra requirement $\sigma_i \leq \sigma$ in our definition plays an important role in the proof of uniqueness in the case when the datum $\sigma$ is absolutely continuous with respect to ${\rm cap}_p$.

Theorem 3.10. Suppose $\sigma\in {\mathcal{M}}^+(\mathbb{R}^n)$, and suppose (3.3) holds. Then there exists an $\mathcal{A}$-superharmonic reachable solution to (3.1). Moreover, if additionally $\sigma < < {\rm cap}_p$, then any $\mathcal{A}$-superharmonic reachable solution is unique and coincides with the minimal solution.

Proof. Existence: Suppose that (3.3) holds. Then ${\bf W}_{1, p}\sigma < +\infty$ quasi-everywhere and hence almost everywhere. For each $i = 1, 2, \dots$, let $u^{j}_{i}$ be an $\mathcal{A}$-superharmonic renormalized solution (see ^[12]) to

Note that $\sigma|_{B_{i}(0)}\leq \sigma$ and $\sigma|_{B_{i}(0)}\rightarrow \sigma$ weakly as measures in $\mathbb{R}^n$. Also, by ^[33], we have

Hence, by [20,Theorem 1.17], there exist an $\mathcal{A}$-superharmonic function $u_i$ in $\mathbb{R}^n$ with

and a subsequence $\{u_i^{j_k}\}_{k}$ such that $u_{i}^{j_k}\rightarrow u$ and $D u_{i}^{j_k} \rightarrow D u_{i}$ a.e. as $k\rightarrow \infty$. These estimates yield that the Riesz measure of $u_i$ is $\sigma|_{B_{i}(0)}$ and

Using again [20,Theorem 1.17] and (3.8), we find a subsequence of $\{u_i\}$ that converges a.e. to an $\mathcal{A}$-superharmonic reachable solution $u$ of (3.1).

Uniqueness: We now assume further that $\sigma < < {\rm cap}_p$. Let $u$ be an $\mathcal{A}$-superharmonic reachable solution in the sense of Definition 3.8 with approximating sequences $\{{u_i}\}$ and $\{\sigma_i\}$. Let us fix an $i\in \{1, 2, \dots\}$. Then there exists a positive integer $N = N(ⅰ)$ such that ${\rm supp}(\sigma_i)\subset B_N(0)$. Let $v$ be the minimal $\mathcal{A}$-superharmonic solution to (3.1). Also, let $v_N$ be the minimal $\mathcal{A}$-superharmonic solution to (3.1) with datum $\sigma|_{B_N(0)}$ in place of $\sigma$. We have, by Theorem 3.7,

Thus, as $u_j \rightarrow u$ a.e., it is enough to show that

Note that since $\sigma_i\leq \sigma$ and ${\rm supp}(\sigma_i)\subset B_N(0)$ we have that

For $R > 0$, let $0\leq \Theta = \Theta_R\leq 1$ be a cutoff function such that

For any $k > 0$, we set

Also, for any $m > 0$, we define the following Lipschitz function with compact support on $\mathbb{R}$:

As $u_i$ and $v_N$ are both local renormalized solutions (see ^[3,19]), we may use

as test functions and thus obtaining

and

Let

and

Then by (3.10) we have

On the other hand, we can write

Thus, in view of (3.11), it follows that

To estimate $|A_m|$, we observe that $|h_m(t)|\leq 1$ and $|h'_m(t)|\leq 1/m$. Hence,

On the other hand, using $T^{+}_{2m}(u_i) \, \Theta$ as a test function for the equation of $u_i$ and invoking condition (2.1), we estimate

Since $T^{+}_{2m}(u_i)/m\leq 2$, and $T^{+}_{2m}(u_i)/m$ converges to zero quasi-everywhere, we deduce

Similarly,

Hence,

A similar argument gives

To estimate $|C_m|$, we first use the pointwise bound (3.4) to obtain

where $A_R$ is the annulus

Note that for $R > 4N$ we have

for all $x\in A_R$. Thus,

where we used the Caccioppoli inequality and the weak Harnack inequality in the second bound. This gives

Since $h_m(u_i) \, h_m(v_N) \rightarrow 1$ a.e. as $m\rightarrow \infty$, and $\Theta(x)\rightarrow 1$ everywhere as $R\rightarrow \infty$, it follows from (3.12)–(3.13) and Fatou's lemma that

Letting $k\rightarrow \infty$, we deduce

Since the integrand is strictly positive whenever $D u_i \not = D v_N$, we infer that $D u_i = D v_N$ a.e. on the set $\{u_i-v_N > 0\}$.

We next claim that the function $T_k^{+}(u_i-v_N)$ belongs to $W^{1, p}_{\rm loc}(\mathbb{R}^n)$ for any $k > 0$. To see this, for any $m > k$, we compute

where $\chi_A$ is the characteristic function of a set $A$. Thus,

On the other hand, using $H_{m, k}(v_N)\Theta$ as a test function for the equation of $v_N$, where

we have

Thus, for each fixed $k > 0$, the sequence $\{T^{+}_k(T^{+}_m(u_i)-T^{+}_m(v_N))\}_{m}$ is uniformly bounded in $W^{1, p}_{\rm loc}(\mathbb{R}^n)$. Since $T^{+}_k(T^{+}_m(u_i)-T^{+}_m(v_N))\rightarrow T_k^{+}(u_i-v_N)$ a.e. as $m\rightarrow \infty$, we see that $T_k^{+}(u_i-v_N)\in W^{1, p}_{\rm loc}(\mathbb{R}^n)$.

We are now ready to complete the proof of the theorem. Since $T_k^{+}(u_i-v_N) = T^{+}_k(T^{+}_m(u_i)-T^{+}_m(v_N))$ a.e. on the set $\{ u_i < m, v_N < m\}$ and the two functions belong to $W^{1, p}_{\rm loc}(\mathbb{R}^n)$, by (3.14) we have

a.e. on the set $\{ u_i < m, v_N < m\}$ for any $m > 0$. Thus, $\nabla T_k^{+}(u_i-v_N) = 0$ a.e. in $\mathbb{R}^n$, which implies the existence of a constant $\kappa\geq 0$ such that

a.e. in the entire space $\mathbb{R}^n$. Note that if $\kappa\not = 0$, then $u_i = v_N+\kappa$ in $\mathbb{R}^n$, which violates the condition at infinity, $\liminf_{|x|\rightarrow \infty} u_i(x) = 0$. It follows that $\kappa = 0$, which yields (3.9), as desired.

The following version of the comparison principle in $\mathbb{R}^n$ is an immediate consequence of Theorems 3.7 and 3.10.

Corollary 3.11. Let $\sigma, \tilde{\sigma} \in {\mathcal{M}}^+(\mathbb{R}^n)$, where $\sigma \leq \tilde{\sigma}$ and $\sigma < < {\rm cap}_p$, $1 < p < n$. Let $u$ be an $\mathcal{A}$-superharmonic reachable solution of (3.1), and $\tilde{u}$ any $\mathcal{A}$-superharmonic solution of (3.1) with datum $\tilde{\sigma}$ in place of $\sigma$. Then $u \le \tilde{u}$ in $\mathbb{R}^n$.

For $\sigma\in {\mathcal{M}}^+(\mathbb{R}^n)$ such that that $\sigma < < {\rm cap}_p$, sometimes it is desirable to know when an $\mathcal{A}$-superharmonic solution to (3.1) is also the $\mathcal{A}$-superharmonic reachable solution to (3.1), and hence also the minimal $\mathcal{A}$-superharmonic solution to (3.1). The following theorem provides some sufficient conditions in terms of the weak integrability of the gradient of the solution, or in terms of the finiteness of the datum $\sigma$.

Theorem 3.12. Let $\sigma\in {\mathcal{M}}^+(\mathbb{R}^n)$, where $\sigma < < {\rm cap}_p$. Suppose that any one of the following conditions holds:

(ⅰ) $|D u| \in L^{\gamma, \infty}(\mathbb{R}^n)$ for some $(p-1)n/(n-1)\leq \gamma < p$, where $L^{\gamma, \infty}(\mathbb{R}^n)$ is the weak $L^\gamma$ space in $\mathbb{R}^n$;

(ⅱ) $|D u| \in L^{p}(\mathbb{R}^n)$;

(ⅲ) $\sigma \in {\mathcal{M}}^+_b(\mathbb{R}^n)$.

Then any $\mathcal{A}$-superharmonic solution $u$ to the Eq (3.1) coincides with the minimal $\mathcal{A}$-superharmonic solution.

Proof. Let $v$ be the minimal $\mathcal{A}$-superharmonic solution of (3.1). Our goal is to show that $u\leq v$ a.e. Let $\Theta(x) = \Theta_R(x), R > 0$, $T^{+}_k(t), k > 0$, and $h_m(t), m > 0$ be as in the proof of Theorem 3.10. Then arguing as in the proof of Theorem 3.10, with $u$ in place of $u_i$ and $v$ in place of $v_N$, we have

where now

and

As in the proof of Theorem 3.10, we have

As for $C_m$, we have

where, as above, $A_R$ is the annulus

Suppose now that condition (ⅰ) holds. Then $|D u|^{p-1} \in L^{\frac{q}{q-1}, \infty}(\mathbb{R}^n)$ for some $q\in (p, n]$. Set

and note that ${\bf W}_{1, p}\sigma \in L^{m, \infty}(\mathbb{R}^n).$ A proof of this fact in the 'sublinear' case $(p-1)q/(q-1)\leq 1$ can be found in ^[32].

We have that either $m\leq q$ or $m > q$. In the case $m\leq q$, for any $\epsilon > 0$ we find

Here we shall choose $\epsilon > 0$ such that

In the case $m > q$, we have

Note that, since $q < p$,

Hence, in both cases we have, for any fixed $k > 0$,

and likewise,

On the other hand, suppose now that condition (ⅱ) holds, i.e., $|D u| \in L^{p}(\mathbb{R}^n)$. Then ${\bf W}_{1, p}\sigma \in L^{\frac{np}{n-p}}(\mathbb{R}^n)$, and as in (3.16) we have

and likewise for $v$. Thus (3.17) and (3.18) also hold under condition (ⅱ).

Finally, suppose that ${\rm (ⅲ)}$ holds. For any $1 < r < \frac{n}{n-1}$ and $\epsilon\in(0, 1)$ such that ${\epsilon} \frac{r}{r-1} < \frac{n(p-1)}{n-p}$, we have

where we used the Caccioppoli inequality and the weak Harnack inequality in the last bound (see [18,Theorem 7.46]).

Hence, using [9,Lemma 3.1] we get

A similar inequality holds for $v$ in place of $u$. Thus, we see that (3.17) and (3.18) hold under condition (ⅲ) as well.

Now (3.17) and (3.18) yield that, for any $k > 0$, we have

Using (3.15) and (3.20), we deduce

for any $k > 0$. This implies $Du = Dv$ a.e. on the set $\{u-v > 0\}$ and, as in the proof of Theorem 3.10, in view of the condition at infinity, we deduce $u\leq v$ a.e. as desired.

We now provide a criterion for reachability by requiring only the finiteness of the approximating measures $\sigma_i$.

Corollary 3.13. Let $u$ be an $\mathcal{A}$-superharmonic solution of (3.1), where $\sigma \in {\mathcal{M}}^+(\mathbb{R}^n)$, and $\sigma < < {\rm cap}_p$. Suppose that there exist two sequences $\{u_i\}$ and $\{\sigma_i\}$, $i = 1, 2, \dots$, such that the following conditions hold:

${\rm (ⅰ)}$ each $\sigma_i \in {\mathcal{M}}^+_b(\mathbb{R}^n)$, and $\sigma_i \leq \sigma$;

${\rm (ⅱ)}$ each $u_i$ is an $\mathcal{A}$-superharmonic solution of (3.1) with datum $\sigma_i$ in place of $\sigma$;

${\rm (ⅲ)}$ $u_i\rightarrow u$ a.e. in $\mathbb{R}^n$.

Then $u$ is an $\mathcal{A}$-superharmonic reachable solution of (3.1), and thus coincides with the minimal solution.

Proof. By Theorem 3.12, each $u_i$ is a reachable solution. Thus by a diagonal process argument, we see that $u$ is also a reachable solution. Alternatively, this can also be proved by modifying the proof of the uniqueness part in Theorem 3.10, taking into account estimates of the form (3.19).

Theorem 3.12 formally holds under the condition $|D u| \in L^{\gamma, \infty}(\mathbb{R}^n)$ for $0 < \gamma < (p-1)n/(n-1)$ as in this case $\sigma = 0$. The proof of this fact, especially in the case $0 < \gamma\leq p-1$, requires some results obtained recently in ^[31].

Theorem 3.14. If $u$ is an $\mathcal{A}$-superharmonic function in $\mathbb{R}^n$ such that $|D u| \in L^{\gamma, \infty}(\mathbb{R}^n)$ for some $0 < \gamma < (p-1)n/(n-1)$, then $\sigma = 0$ where $\sigma$ is the Riesz measure of $u$.

Proof. Let $Q_r(x)$, $r > 0$, denote the open cube $Q_r(x): = x+ (-r, r)^n$ with center $x\in\mathbb{R}^n$ and side-length $2r$. Using $\Phi\in C_0^\infty (Q_r(x))$, $\Phi\geq 0$, $\Phi = 1$ on $Q_{r/2}(x)$, and $|\nabla \Phi|\leq C/r$, as a test function we have

Thus if $\gamma\in (p-1, (p-1)n/(n-1))$, for any $R > 0$ we use Hölder's inequality to get

Note that $n\frac{\gamma-p+1}{\gamma} < 1$ and thus letting $R\rightarrow \infty$ we get $\sigma = 0$.

We now consider the case $0 < \gamma\leq p-1$. Let $\gamma_1$ be a fixed number in $(p-1), (p-1)n/(n-1))$. By [31,Lemma 2.3], for any cube $Q_\rho(x)\subset\mathbb{R}^n$, we have

On the other hand, by [31,Corollary 1.3] we find

Note that [31,Corollary 1.3] is stated for $1 < p < 3/2$ but the argument there also works for all $1 < p\leq n$ after taking into account the comparison estimates of ^[14,27,28].

Hence, it follows that

where we used (3.21) with $r = 4 \rho$ in the last inequality. This allows us to employ a covering/iteration argument as in [16,Remark 6.12] to obtain that

for any $\epsilon > 0$.

Thus, if $0 < \gamma < p-1$, in view of (3.21), (3.22), and Hölder's inequality, we get

as $R\rightarrow \infty$. Hence, $\sigma = 0$. The case $\gamma = p-1$ is treated similarly, starting with the inequality

for a sufficiently small $\epsilon > 0$.

Due to the results of ^{[13,14,24,30]} (see also ^[15,29]), under some additional regularity conditions on the nonlinearity $\mathcal{A}(x, \xi)$, one has

provided $u$ is an $\mathcal{A}$-superharmonic solution to the Eq (3.1). This gradient estimate holds in particular for $\mathcal{A}(x, \xi) = |\xi|^{p-2}\xi$, i.e., the $p$-Laplacian $\Delta_p$, which yields the following corollary.

Corollary 3.15. Let $\sigma \in {\mathcal{M}}^+(\mathbb{R}^n)$. Suppose that one of the following conditions holds:

$\rm (ⅰ)$ $\sigma < < {\rm cap}_p$ and ${\bf I}_1 \sigma \in L^{s, \infty}(\mathbb{R}^n)$ for some $n/(n-1) < s < p/(p-1)$. This holds in particular if $\sigma = f\in L^{t, \infty}(\mathbb{R}^n)$ for some $1 < t < np/(np-n+p)$;

$\rm (ⅱ)$ ${\bf I}_1 \sigma \in L^{p/(p-1)}(\mathbb{R}^n)$, i.e., $\sigma$ is of finite energy.

Then any $p$-superharmonic solution $u$ to the equation

coincides with the minimal $p$-superharmonic solution.

Finally, we show that if the condition at infinity, ${\liminf\limits_{|x|\rightarrow \infty}}\, u = 0$ in (3.1), is replaced with the stronger one ${\lim\limits_{|x|\rightarrow \infty}}\, u = 0$, then all $\mathcal{A}$-superharmonic solutions are indeed reachable.

Theorem 3.16. Suppose that $u$ is an $\mathcal{A}$-superharmonic solution of the equation

where $\sigma \in {\mathcal{M}}^+(\mathbb{R}^n)$, and $\sigma < < {\rm cap}_p$. Then $u$ is the unique $\mathcal{A}$-superharmonic solution of (3.23), which coincides with the minimal $\mathcal{A}$-superharmonic reachable solution of (3.1).

Proof. First notice that the condition ${\lim\limits_{|x|\rightarrow \infty}}\, u = 0$ in (3.23) yields, in view of (3.4),

For any $\epsilon > 0$, let

and

Clearly, ${\Omega}_{\epsilon}$ is a bounded open set, $u_{\epsilon} = u-\epsilon$ on $\Omega_{\epsilon}$, and $u_{\epsilon} = 0$ in $\mathbb{R}^n\setminus {\Omega}_{\epsilon}$.

Let $v$ be the minimal solution of (3.1), which is also the minimal solution of (3.23), since $v \le K \, {\bf W}_{1, p}\sigma$, and hence ${\lim\limits_{|x|\rightarrow \infty}}\, v = 0$. It is enough to show that

in $\Omega_{\epsilon}$, as this will yield that $u\leq v$ in $\mathbb{R}^n$ after letting ${\epsilon}\rightarrow 0^{+}$.

Now by Lemma 4.1 below, to verify (3.24), it suffices to show that $u_{\epsilon}$ is a renormalized solution of

Note that, for any $k > 0$, $T_k(u_{\epsilon}) = T_{k+\epsilon}(u)-{\epsilon}$. We have $T_k(u_{\epsilon})\in W^{1, p}_{\rm loc} (\mathbb{R}^n)$, $T_k(u_{\epsilon})$ is quasi-continuous in $\mathbb{R}^n$, and $T_k(u_{\epsilon}) = 0$ everywhere in $\mathbb{R}^n\setminus{\Omega}_{\epsilon}$. Thus $T_k(u_{\epsilon})\in W^{1, p}_0({\Omega}_{\epsilon})$ (see [18,Theorem 4.5]).

As $u$ is a local renormalized solution in $\mathbb{R}^n$, for every $k > 0$ there exists a nonnegative measure $\lambda_{k+{\epsilon}} < < {\rm cap}_p$, concentrated on the sets $\{u = k+{\epsilon}\}$, such that $\lambda_{k+{\epsilon}}\rightarrow 0$ weakly as measures in $\mathbb{R}^n$ as $k\rightarrow \infty$. Since ${\Omega}_{\epsilon}$ is bounded, this implies that $\lambda_{k+{\epsilon}}\rightarrow 0$ in the narrow topology of measures in $\Omega_{\epsilon}$.

Moreover, for $k > 0$,

for every $\varphi\in{\rm W}_{0}^{1, \, p}(\mathbb{R}^n)\cap L^{\infty}(\mathbb{R}^n)$ with compact support in $\mathbb{R}^n$. In particular, we have

for every $\varphi\in{\rm W}_{0}^{1, \, p}({\Omega}_{\epsilon})\cap L^{\infty}({\Omega}_{\epsilon})$.

Thus, we conclude that $u_{\epsilon}$ is a renormalized solution in ${\Omega}_{\epsilon}$, as desired.

4.   Quasilinear equations with a sub-natural growth term in $\mathbb{R}^n$

In this section, we study solutions to the equation

in the sub-natural growth case $0 < q < p-1$, with $\mu, \sigma \in {\mathcal{M}}^+(\mathbb{R}^n)$.

We consider nontrivial $\mathcal{A}$-superharmonic solutions to (4.1) such that $0 < u < \infty$ $d \sigma$-a.e., which implies $u \in L^q_{{\rm loc}} (\mathbb{R}^n, \sigma)$, so that $\sigma u^q + \mu \in {\mathcal{M}}^+(\mathbb{R}^n)$ (see ^[37]).

As was noted in the Introduction, $\sigma < < {\rm cap}_p$ whenever there exists a nontrivial solution $u$ to (4.1), for any $\mu$ (in particular, $\mu = 0$).

The existence and uniqueness of nontrivial reachable $\mathcal{A}$-superharmonic solutions to (4.1), under the additional assumption $\mu < < {\rm cap}_p$, are proved below. Without this restriction on $\mu$, the existence of nontrivial solutions, not necessarily reachable, was obtained recently in ^[37], along with bilateral pointwise estimates of solutions in terms of nonlinear potentials.

We use this opportunity to make a correction in the proof of the existence property for (4.1) in the case $\mu = 0$ given in [9,Theorem 1.1], which used a version of the comparison principle ([9], Lemma 5.2). It was invoked in the proof of [37,Theorem 1.1] as well. Some inaccuracies in the statement of this comparison principle and its proof are fixed in the following lemma. The rest of the proofs of [9,Theorem 1.1] and [37,Theorem 1.1] remains valid with this correction. (See the proof of Theorem 4.2 below.)

Lemma 4.1. Let ${\Omega}$ be a bounded open set in $\mathbb{R}^n$. Suppose that $\mu, \nu \in {\mathcal{M}}^+_b({\Omega})$, where $\mu\leq \nu$ and $\mu < < {\rm cap}_p$. If $u\ge 0$ is a renormalized solution of

and if $v\ge 0$ is an $\mathcal{A}$-superharmonic function in ${\Omega}$ with Riesz measure $\nu$ such that $\min\{v, k\}\in W^{1, p}({\Omega})$ for any $k > 0$, then $u\leq v$ a.e.

Proof. Let $\nu_j$, $j > 0$, be the Riesz measure of $\min\{v, j\}$. Since $\min\{v, j\}\in W^{1, p}({\Omega})$ we see that $\nu_j$ belongs to the dual of $W^{1, p}_0({\Omega})$ (see [18,Theorem 21.6]). As in (3.7), we have

for a measure $\alpha_j\in {\mathcal{M}}^+_b({\Omega})$ concentrated in the set $\{v = j\}$. Thus the measure $\mu_j: = \mu|_{\{v < j\}}\leq \nu|_{\{v < j\}} \leq \nu_j$ for any $j > 0$. This implies that $\mu_j$ also belongs to the dual of $W^{1, p}_0({\Omega})$, and hence there exists a unique solution $u_j$ to the equation

Then by the comparison principle (see [9,Lemma 5.1]) we find

for any integer $j > 0$. Thus there is a function $\tilde{u}$ on ${\Omega}$ such that $0\leq \tilde{u}\leq v$ a.e. and $u_j\rightarrow \tilde{u}$ as $j\rightarrow \infty$. We now claim that $\tilde{u}$ is also a renormalized solution to Eq (4.2). If this is verified then, as $\mu < < {\rm cap}_p$, we must have that $\tilde{u} = u$ a.e. (see ^[12,25]) and thus $u\leq v$ a.e. as desired.

To show that $\tilde{u}$ is the renormalized solution of (4.2), we first use $T_k^{+}(u_j)$, $k > 0$, as a test function for (4.3) to obtain

Since $T_k^{+}(u_j)\rightarrow T_k^{+}(\tilde{u})$ a.e. as $j\rightarrow \infty$, we see that $T_k^{+}(\tilde{u})\in W^{1, p}_0({\Omega})$ and

for any $k > 0$. By [2,Lemmas 4.1 and 4.2], this yields

Moreover, arguing as in Step 4 of the proof of Theorem 3.4 in ^[12], we see that $\{\nabla u_j\}$ is a Cauchy sequence in measure which converges to $D\tilde{u}$ a.e. in ${\Omega}$. There is no need to take a subsequence here as the limit is independent of any subsequence.

Moreover, for any Lipschitz function $h: \mathbb{R} \rightarrow \mathbb{R}$ such that $h'$ has compact support and for any function $\varphi\in W^{1, r}({\Omega})\cap L^\infty({\Omega})$, $r > n$, such that $h(\tilde{u})\varphi\in W^{1, p}_0({\Omega})$, we have

Thus if the support of $h'$ is in $[-M, M]$, $M > 0$, then, using $0\leq u_j\leq \tilde{u}$, we can rewrite the above equality as

Note that by (4.4) and [2,Lemma 4.2], we have that $\nabla u_j$ is uniformly bounded in $L^{\frac{n(p-1)}{n-1}, \infty}({\Omega})$ and $\nabla T^{+}_{M}(u_j)$ is uniformly bounded in $L^{p}({\Omega})$. Thus by the Vitali Convergence Theorem, the left-hand side of the above equality converges to

On the other hand, by the Lebesgue Dominated Convergence Theorem we have

Thus, we get

which yields that $\tilde{u}$ is the renormalized solution of (4.2) (see Definition 3.2).

We recall that by $\varkappa = \varkappa (\mathbb{R}^n)$ we denote the least constant in the weighted norm inequality (see ^[9,37])

for all $\mathcal{A}$-superharmonic functions $\varphi\ge 0$ in $\mathbb{R}^n$ such that ${\liminf\limits_{|x| \to \infty}} \, \varphi(x) = 0$. Notice that by estimates (3.4), $K^{-1} \, \varphi \le {\bf W}_{1, p} \mu \le K \, \varphi$, where $\mu = - {\rm{div}}\, \mathcal{A}(x, \nabla \varphi)\in {\mathcal{M}}^{+}(\mathbb{R}^n)$. Here we may assume without loss of generality that $\mu \in {\mathcal{M}}_b^{+}(\mathbb{R}^n)$, so that ${\bf W}_{1, p} \mu\not\equiv \infty$. Consequently, (4.5) is equivalent to the inequality

where $K^{-1} \, \varkappa\le \kappa \le K \, \varkappa$. In particular, one can replace ${\rm{div}}\, \mathcal{A}(x, \nabla \varphi)$ in (4.5) by $\Delta_p$, up to a constant which depends only on $K$.

By $\varkappa(B)$, where $B$ is a ball in $\mathbb{R}^n$, we denote the least constant in a similar localized weighted norm inequality with the measure $\sigma_B$ in place of $\sigma$, where $\sigma_B = \chi_B \, \sigma$ is the restriction of $\sigma$ to $B$.

The so-called intrinsic nonlinear potential $\bf{K}_{p, q} \sigma$, introduced in ^[9], is defined by

Here $B = B(x, t)$ is a ball in $\mathbb{R}^n$ of radius $t > 0$ centered at $x$. As was noticed in ^[9], $\bf{K}_{p, q} \sigma \not\equiv + \infty$ if and only if

By [9,Theorem 1.1], there exists a nontrivial $\mathcal{A}$-superharmonic solution to the homogeneous equation (4.1) in the case $\mu = 0$ if and only if $\bf{W}_{1, p} \sigma \not\equiv + \infty$ and $\bf{K}_{p, q} \sigma \not\equiv + \infty$, i.e., conditions (3.3) and (4.6) hold. The next theorem shows that this solution is actually reachable.

Theorem 4.2. Let $0 < q < p-1$, and let $\sigma\in {\mathcal{M}}^+(\mathbb{R}^n)$. Then the nontrivial minimal $\mathcal{A}$-superharmonic solution $u$ of

constructed in the proof of [9,Theorem 1.1] under the conditions (3.3) and (4.6), is an $\mathcal{A}$-superharmonic reachable solution.

Proof. We start with the same construction as in the proof of [9,Theorem 1.1] for the minimal $\mathcal{A}$-superharmonic solution $u$, but with datum $\sigma|_{B_m(0)}$ in place of $\sigma$ ($m = 1, 2, \dots$).

For a fixed $m$, let $v_m$ be the minimal $\mathcal{A}$-superharmonic solution to the equation

We recall from the construction in ^[9] that

where $v_{1, m}^k$ ($k = 0, 1, 2, \dots$) is the $\mathcal{A}$-superharmonic renormalized solution of

with $w_{0, m} = c_0 ({\bf W}_{1, p}(\sigma|_{B_m(0)}))^{\frac{p-1}{p-1-q}}$, and $v_{j, m}^k$ ($k = 0, 1, 2, \dots$, $j = 2, 3, \dots$) is the $\mathcal{A}$-superharmonic renormalized solution of

Here $c_0$ is a fixed constant such that

where $C$ is the constant in (3.9) of ^[9], and $\mathfrak{c}$ is the constant in (3.10) of ^[9] with $\alpha = 1$.

We also recall from ^[9] that

where $u_{j}^{k}$ are the $\mathcal{A}$-superharmonic renormalized solutions of the corresponding problems in $B_{2^k}(0)$ with $\sigma$ in place of $\sigma|_{B_m(0)}$. In particular, $\min (u_{j}^k, l) \in W^{1, p}_0 (B_{2^k}(0))$ and $\min (v_{j, m}^k, l) \in W^{1, p}_0 (B_{2^k}(0))$ for all $l > 0$.

Thus, by the above version of the comparison principle (Lemma 4.1) we see that

whenever $m_1 \le m_2$.

This yields

Letting now $m\rightarrow \infty$, we obtain an $\mathcal{A}$-superharmonic reachable solution

to (4.7) such that $v\leq u$ in $\mathbb{R}^n$. As $u$ is the minimal $\mathcal{A}$-superharmonic solution of (4.7), we see on the other hand that $u\leq v$, and thus $u = v$, which completes the proof.

Remark. In the proof of [9,Theorem 1.1], there is a misprint in the exponent in inequality (4.8) above for the constant $c_0$. This choice of $c_0$ ensures the minimality of the solution $u$ of (4.7) constructed in ^[9].

We recall that, by [37,Theorem 1.1] and [37,Remark 4.3], a nontrivial $\mathcal{A}$-superharmonic solution of (4.1) exists if and only if $\bf{W}_{1, p} \sigma\not \equiv \infty$, $\bf{K}_{p, q} \sigma\not \equiv \infty$, and $\bf{W}_{1, p} \mu \not \equiv \infty$, i.e., the following three conditions hold:

Theorem 4.3. Let $0 < q < p-1$, and let $\mu, \sigma\in {\mathcal{M}}^+(\mathbb{R}^n)$, where $\mu < < {\rm cap}_p$. Then, under the conditions (4.9), (4.10), and (4.11), there exists a nontrivial minimal reachable $\mathcal{A}$-superharmonic solution of (4.1).

Proof. Since the case $\mu = 0$ was treated in Theorem 4.2 above, without loss of generality we may assume that $\mu\not = 0$. We recall that in the proof of [37,Theorem 1.1], a nontrivial $\mathcal{A}$-superharmonic solution $u$ of (4.1), was constructed using the following iteration process. We set $u_0 = 0$, and for $j = 0, 1, 2, \ldots$ construct the iterations

where $u_j \in L^q_{{\rm loc}} (\mathbb{R}^n, d \sigma)$. We observe that, for each $j$, the solution $u_{j+1}$ was chosen in ^[37] so that $u_{j}\le u_{j+1}$ ($j = 0, 1, 2, \ldots$) by a version of the comparison principle (see [34,Lemma 3.7 and Lemma 3.9]). Then $u : = \lim_{j \to \infty} u_j$ is a nontrivial $\mathcal{A}$-superharmonic solution of (4.1).

We now modify this argument as follows to obtain a minimal nontrivial $\mathcal{A}$-superharmonic solution of (4.1). Notice that $\mu < < {\rm cap}_p$ by assumption, and, as mentioned above, $\sigma < < {\rm cap}_p$, since a solution exists. Hence, clearly the measure $\sigma u_j^q + \mu < < {\rm cap}_p$ as well. By Theorem 3.6, $u_{j+1}$ can be chosen as the minimal $\mathcal{A}$-superharmonic solution to (4.12).

It follows by induction that $u_{j}\le u_{j+1}$ ($j = 0, 1, 2, \ldots$). Indeed, this is trivial when $j = 0$, and then by the inductive step,

which is obvious when $j = 1$. From this, using Theorem 3.7 we deduce $u_{j}\le u_{j+1}$ for all $j = 1, 2, \ldots$.

Similarly, if $\tilde{u}$ is any $\mathcal{A}$-superharmonic solution of (4.1), then again arguing by induction and using Theorem 3.7, we deduce that $u_{j+1} \le \tilde{u}$ ($j = 0, 1, 2, \ldots$), since

Consequently, $u\le \tilde{u}$, i.e., $u$ is the minimal $\mathcal{A}$-superharmonic solution of (4.1).

We next show that $u$ is a reachable solution. Using a similar iteration process with $\sigma|_{B_m(0)}$ in place of $\sigma$ and $\mu|_{B_m(0)}$ in place of $\mu$ ($m = 1, 2, \dots$), we set $v_{0, m} = 0$ and define $v_{j, m}$ to be the minimal $\mathcal{A}$-superharmonic solution to the equation

where $v_{j, m} \le v_{j+1, m}$ for each $m = 1, 2, \ldots$.

As above, arguing by induction and using Theorem 3.7, we deduce

whenever $m_1\le m_2$. It follows that $v_m : = \lim_{j\to \infty} v_{j, m}\le u$ ($m = 1, 2, \ldots$) is an $\mathcal{A}$-superharmonic solution of the equation

where $v_{m_1} \le v_{m_2} \le u$ if $m_1\le m_2$.

Thus, letting $m\rightarrow \infty$, we obtain an $\mathcal{A}$-superharmonic reachable solution $v: = \lim\limits_{m\to \infty} v_{m}$ to (4.7) such that $v\leq u$. Since $u$ is the minimal $\mathcal{A}$-superharmonic solution of (4.7), we see that $u = v$, which completes the proof.

We now prove the uniqueness property for reachable solutions of (4.1).

Theorem 4.4. Let $0 < q < p-1$, and let $\mu, \sigma\in {\mathcal{M}}^+(\mathbb{R}^n)$, where $\mu < < {\rm cap}_p$. Suppose $\mathcal{A}$ satisfies conditions (2.1) and (2.2). Then nontrivial $\mathcal{A}$-superharmonic reachable solutions of (4.1) are unique.

Proof. Let $u, v$ be two nontrivial $\mathcal{A}$-superharmonic solutions of (4.1) in $\mathbb{R}^n$. Then by [37,Theorem 1.1] and [37,Remark 4.3], there exists a constant $C\ge 1$, depending only on $p$, $q$ and $n$, such that

Hence, clearly,

Notice that here by definition $u, v \in L^{q}_{{\rm loc}}(\mathbb{R}^n, \sigma)$. Suppose that $v$ is a reachable solution of (4.1) in $\mathbb{R}^n$. Then by Corollary 3.11 with $\sigma v^q + \mu$ in place of $\sigma$, and $\tilde{\sigma} = C^q \left(\sigma u^q + \mu \right)$, it follows that $v \le C^{\frac{q}{p-1}}u$.

By iterating this argument, we deduce

Since $0 < q < p-1$, letting $j \to \infty$ in the preceding inequality, we obtain $v \le u$ in $\mathbb{R}^n$. Interchanging the roles of $u$ and $v$, we see that actually $u = v$ in $\mathbb{R}^n$.

Corollary 4.5. Nontrivial $\mathcal{A}$-superharmonic solutions $u$ of (4.1) are unique under the assumptions of Theorem 4.4, provided any one of the following conditions holds:

$\rm (ⅰ)$ $u \in L^q (\mathbb{R}^n, d \sigma)$ and $\mu \in {\mathcal{M}}^+_b(\mathbb{R}^n)$, or equivalently $\varkappa (\mathbb{R}^n) < \infty$ and $\mu \in {\mathcal{M}}^+_b(\mathbb{R}^n)$;

$\rm (ⅱ)$ $\lim_{|x| \to \infty} u(x) = 0$;

$\rm (ⅲ)$ $|D u| \in L^{p}(\mathbb{R}^n)$, or $|D u| \in L^{\gamma, \infty}(\mathbb{R}^n)$ for some $(p-1)n/(n-1)\leq \gamma < p$.

Proof. Suppose first that (ⅰ) holds. By [9,Theorem 4.4], $\varkappa (\mathbb{R}^n) < \infty$ if and only if there exists a nontrivial $\mathcal{A}$-superharmonic solution $u \in L^q(\mathbb{R}^n, d\sigma)$ of (4.7). In particular, since by [37,Theorem 4.1],

it follows that

Next, we denote by $\varphi$ an $\mathcal{A}$-superharmonic solution to the equation

where $\mu$ is the Riesz measure of $\varphi$. Notice that $\varphi \ge K^{-1} \, {\bf W}_{1, p} \mu$ by the lower bound in inequality (3.4). Since $\mu \in {\mathcal{M}}_b(\mathbb{R}^n)$ and $\varkappa (\mathbb{R}^n) < \infty$, using $\varphi$ as a test function in inequality (4.5) yields ${\bf W}_{1, p} \mu \in L^q (\mathbb{R}^n, d \sigma)$.

Hence, by [37,Theorem 1.1] and [37,Remark 4.3], we deduce that there exists a nontrivial $\mathcal{A}$-superharmonic solution of (4.1) $u \in L^q (\mathbb{R}^n, d \sigma)$, and, for any such a solution, $\sigma u^q + \mu\in {\mathcal{M}}_b(\mathbb{R}^n)$. It follows that $u$ is a reachable $\mathcal{A}$-superharmonic solution of (4.1) by Theorem 3.10 and Theorem 3.12 (ⅲ).

In case (ⅱ), by Theorem 3.16 $u$ is a reachable solution of (4.1).

In case (ⅲ), $u$ is a reachable $\mathcal{A}$-superharmonic solution of (4.1) by Theorem 3.10 and Theorem 3.12 (ⅰ), (ⅱ).

In all these cases, reachable $\mathcal{A}$-superharmonic solutions are unique by Theorem 4.4.

Remark 4.6. Uniqueness of finite energy solutions $u$ to (4.1) such that $|D u| \in L^{p}(\mathbb{R}^n)$ in Corollary 4.5(ⅲ) was established in [35,Theorem 6.1] in the special case of the $p$-Laplace operator using a different method. (See also an earlier result [8,Theorem 5.1] in the case $\mu = 0$.) Solutions of finite energy to (4.1) exist if and only if ${\bf W}_{1, p} \sigma\in L^{\frac{(1+q)(p-1)}{p-1-q}}(\mathbb{R}^n, d \sigma)$ and ${\bf W}_{1, p} \mu \in L^{1}(\mathbb{R}^n, d \mu)$ ([35,Theorem 1.1]).

Remark 4.7. Under the assumptions of Theorem 4.3, but without the restriction $\mu < < {\rm cap}_p$, it is still possible to prove the existence of an $\mathcal{A}$-superharmonic reachable solution (not necessarily minimal) of (4.1). The construction of such a solution makes use of an extension of [33,Lemma 6.9] proved below.

Proof. To prove this claim, we shall construct first a nondecreasing sequence $\{u_m\}_{m\geq 1}$ of $\mathcal{A}$-superharmonic solutions of

Then by [37,Theorem 1.1 and Remark 4.3],

It follows from [20,Theorem 1.17] that $u_{m}\rightarrow u$ pointwise everywhere, where $u$ is an $\mathcal{A}$-superharmonic reachable solution of (4.1).

The construction of $\{u_m\}_{m\geq 1}$ can be done as follows. It suffices to demonstrate only how to construct $u_1$ and $u_2$ such that $u_2\geq u_1$, since the construction of $u_m$ for $m\ge 3$ is completely analogous. Let $v_1$ be an $\mathcal{A}$-superharmonic solution of

Here as above $v_1$ is an a.e. pointwise limit of a subsequence of $\{v_1^{(k)}\}_{k\geq 1}$, where each $v_1^{(k)}$ is a nonnegative $\mathcal{A}$-superharmonic renormalized solution of

Next, for any $j\geq1$, let $v_{j+1}$ be an $\mathcal{A}$-superharmonic solution of

Notice that $v_{j+1}$ is an a.e. pointwise limit of a subsequence of $\{v_{j+1}^{(k)}\}_{k\geq 1}$, where each $v_{j+1}^{(k)}$ is a nonnegative $\mathcal{A}$-superharmonic renormalized solution of

By [33,Lemma 6.9] we may assume that $v_{2}^{(k)}\geq v_1^{(k)}$ for all $k\geq 1$, and hence $v_2\geq v_1$. In the same way, by induction we deduce that $v_{j+1}^{(k)}\geq v_j^{(k)}$ for all $j, k\geq 1$. It follows that $v_{j+1}\geq v_j$, and

Then by [37,Theorem 4.1], for any $j \geq 1$, we obtain the bound

Thus, the nondecreasing sequence $\{v_j\}_{j\geq 1}$ converges to an $\mathcal{A}$-superharmonic solution $u_1$ of

To construct $u_2$ such that $u_2\geq u_1$, let $w_1$ be an $\mathcal{A}$-superharmonic solution of

Notice that $w_1$ is an a.e. pointwise limit of a subsequence of $\{w_1^{(k)}\}_{k\geq 1}$, where each $w_1^{(k)}$ is a nonnegative $\mathcal{A}$-superharmonic renormalized solution of

Again, by [33,Lemma 6.9] we may assume that $w_{1}^{(k)}\geq v_1^{(k)}$ for all $k\geq 1$, and hence $w_1\geq v_1$.

Next, for any $j\geq1$, let $w_{j+1}$ be an $\mathcal{A}$-superharmonic solution of

Notice that $w_{j+1}$ is an a.e. pointwise limit of a subsequence of $\{w_{j+1}^{(k)}\}_{k\geq 1}$, where each $w_{j+1}^{(k)}$ is a nonnegative $\mathcal{A}$-superharmonic renormalized solution of

We can ensure here that $w_{j+1}^{(k)}\geq \max\{v_{j+1}^{(k)}, w_{j}^{(k)}\}$ for all $j, k\geq 1$. Indeed, since $w_1\geq v_1$ and $w_1\geq 0$, by Lemma 4.8 below we may assume that $w_{2}^{(k)}\geq \max\{v_{2}^{(k)}, w_{1}^{(k)}\}$ for all $k\geq 1$, and hence $w_{2}\geq \max\{v_{2}, w_1\}$. Repeating this argument by induction we obtain $w_{j+1}^{(k)}\geq \max\{v_{j+1}^{(k)}, w_{j}^{(k)}\}$ for all $j, k\geq 1$.

It follows that $w_{j+1}\geq \max\{v_{j+1}, w_{j}\}$ for all $j\geq 1$. As in (4.13) we have

and hence the nondecreasing sequence $\{ w_j\}$ converges to an $\mathcal{A}$-superharmonic solution $u_2$ of

such that $u_2\geq u_1$, as desired.

The following lemma invoked in the argument presented above is an extension of [33,Lemma 6.9].

Lemma 4.8. Let ${\Omega}$ be a bounded open set in $\mathbb{R}^n$ and let $\mu_1, \mu_2 \in \mathcal{M}_{b}^+({\Omega})$. Suppose that $u_i$ $(i = 1, 2)$ is a renormalized solution of

Then for any measure $\nu \in \mathcal{M}_{b}^+({\Omega})$ such that $\nu\geq \mu_1$ and $\nu\geq \mu_2$, there is a renormalized solution $v$ of

such that $v\geq u_1$ and $v\geq u_2$ a.e.

Proof. For $i = 1, 2$, let $u_{i, k} = \min\{u_i, k\}$ ($k = 1, 2, \dots$). Then $u_{i, k}$ is the bounded renormalized solution of

Here $\mu_i = \mu_{i0}+\mu_{is}$ ($i = 1, 2$) is the decomposition of $\mu_i$ used in Section 3 above, where $\mu_{i0}, \mu_{is} \in \mathcal{M}_{b}^+(\Omega)$, $\mu_{i0} < < {\rm cap}_p$, and $\mu_{is}$ is concentrated on a set of zero $p$-capacity. Moreover, $\lambda_{i, k}\in\mathcal{M}_{b}^+(\Omega)$ and $\lambda_{i, k} \rightarrow \mu_{is}$ in the narrow topology of measures as $k\rightarrow \infty$ (see Definition 3.3).

Now let $v_k$ ($k = 1, 2, \dots$) be a renormalized solution of

Then by [33,Lemma 6.8] we deduce $v_k\geq \max\{u_{1, k}, u_{2, k}\}$ for all $k\geq 1$. Finally, we use the stability results of ^[12] to find a subsequence of $\{v_k\}$ that converges a.e. to a desired function $v$.

Acknowledgments

Nguyen Cong Phuc is supported in part by the Simons Foundation, award number 426071.

Conflict of interest

The authors declare no conflict of interest.