Nonlinear nonlocal equations involving subcritical or power nonlinearities and measure data

1.   Introduction

Let $\Omega\subset{\mathbb R}^N$ be an open bounded domain, $s\in(0, 1)$ and $1 < p < \frac{N}{s}.$ In this article we are concerned with the existence of very weak solutions to the quasilinear nonlocal problems

where $\mu$ is a bounded Radon measure on $\Omega$ and $g\in C({\mathbb R})$ is a nondecreasing function such that $g(0) = 0.$ Here, the nonlocal operator $L$ is defined by

where the symbol P.V. stands for the principle value integral and $K:{\mathbb R}^N\times{\mathbb R}^N\to {\mathbb R}$ is a measurable and symmetric (i.e., $K(x, y) = K(y, x)$) function. Note that if $K(x, y)\equiv|x-y|^{-N-sp}$ then $L$ coincides with the standard fractional $p$-Laplace operator $(- \Delta)^s_p.$

Throughout this work, we assume that there exists a positive constant $\Lambda_K\geq1$ such that the following ellipticity condition holds

In addition, we denote by $\mathfrak{M}_b(\Omega)$ the space of Radon measures on ${\mathbb R}^N$ such that $\mu({\mathbb R}^N\setminus \Omega) = 0,$ as well as by $\mathfrak{M}^+_b(\Omega)$ its positive cone.

Let

For $p = 2$ and $K(x, y) = C_{N, s}|x-y|^{-N-2s},$ operator $L$ reduces to the well-known fractional Laplace operator $(- \Delta)^s$ and the problem $P_+$ becomes

When $g$ satisfies the subcritical integral condition

Chen and Véron ^[9] showed that problem (1.1) admits a unique very weak solution for any $\mu\in \mathfrak{M}_b(\Omega).$ In addition they showed that problem (1.1) with $g(u) = |u|^{ \kappa-1}u\; (\kappa > 1)$ possesses a very weak solution if and only if $\mu$ is absolutely continuous with respect to Bessel capacity $C_{L_{2s, \kappa'}},$ i.e., $\mu$ vanishes on compact set $E$ of $\Omega$ satisfying $\mathrm{Cap}_{{2s, \kappa'}}(E) = 0$ (see (3.21) for the definition of the Bessel capacities). Their approach is based on the properties of the Green Kernel associated with fractional Laplace operator $(- \Delta)^s$ in $\Omega.$

In the local theory and more precisely when $Lu = - \Delta_p u = -\text{div}(|\nabla u|^{p-2}\nabla u),$ related problems have been studied in ^{[4,5,6,15,30,31,32]}. In particular, in the power case, i.e.,

Bidaut-Véron, Nguyen and Véron ^[5] established that if $\mu\in \mathfrak{M}_b(\Omega)$ is absolutely continuous with respect to the Bessel capacity $\mathrm{Cap}_{{p, \frac{ \kappa}{ \kappa-p+1}}}$, then there exists a renormalized solution to problem (1.2) with $\kappa > p-1.$ A main ingredient in the proof of this result is the pointwise estimates for $p$-superharmonic functions in $\Omega.$ These pointwise estimates are expressed in terms of the truncated Wolff potentials $W_{1, p}^{R}[ \mu]$ (see, e.g., ^{[17,19,20,31]}). We recall here that the truncated Wolff potential is given by

for any $R > 0$ and $\alpha\in (0, N)$ such that $p\in(1, \frac{N}{ \alpha}).$ Conversely, Bidaut-Véron ^[4] showed that if problem (1.2) with $\kappa > p-1$ admits a renormalized solution, then $\mu$ is absolutely continuous with respect to the Bessel capacity $\mathrm{Cap}_{{p, \frac{ \kappa}{ \kappa-p+1}+ \varepsilon}}$, for any $\varepsilon > 0.$

Phuc and Verbitsky ^[31] showed that if $\tau\in\mathfrak{M}_b^+(\Omega)$ has compact support in $\Omega,$ then the problem

admits a nonnegative renormalized solution for some $\rho > 0,$ if and only if, there exists a positive constant $C$ such that

for any compact $K\subset \Omega.$ Moreover, they showed that (1.5) is equivalent to

for some positive constant $C > 0.$

Recently, a great attention has been drawn to the study of the fractional $p$-Laplacian or more general nonlocal operators (see for example ^{[2,11,12,18,21,22,23,24,25,26,27,28,29]}). More precisely, Kuusi, Mingione and Sire ^[26] dealt with the problem

where $g\in W^{s, p}({\mathbb R}^N),$ $L_ \Phi$ is a nonlocal operator defined by

Here $\Phi:{\mathbb R}\to{\mathbb R}$ is a continuous function such that $\Phi(0) = 0$ and

When $2-\frac{s}{N} < p,$ they show the existence of a very weak solution to (1.6), which they called SOLA (Solutions obtained as limits of approximations). They also showed local pointwise estimates for SOLA to (1.6) in terms of the truncated Wolff Potential $W_{s, p}^{R}[ \mu].$ In the particular case $\Phi(t) = |t|^{p-2}t$ and $g = 0,$ the existence of very weak solutions was established in ^[2] for any $1 < p < \frac{N}{s}.$

The objective of this work is to determine the subcritical integral conditions on $g,$ which ensure the existence of very weak solutions to problems (P_±). In addition, in the power case, i.e., $g(u) = |u|^{ \kappa-1}u;$ $\kappa > p-1,$ we aim to find sufficient conditions, expressed in terms of Bessel capacities like above, for the solvability of (P_±).

Let us mention here that our work is inspired by the article ^[5] for problem ($P_+$) and by the articles ^[30,31] for problem ($P_-$) with $g(u) = |u|^{ \kappa-1}u; \; \kappa > p-1$. However, due to the presence of the nonlocal operator, new essential difficulties arise which complicate drastically the study of problems (P_±).

In order to state our main results, we need to introduce the notion of the very weak solutions.

Definition 1.1. Let $s\in(0, 1),$ $1 < p < \frac{N}{s},$ $\tilde g\in C({\mathbb R}),$ $\Omega\subset{\mathbb R}^N$ be an open bounded domain and $\mu\in \mathfrak{M}(\Omega).$ We will say that $u:{\mathbb R}^N\to{\mathbb R}$ is a very weak solution to the problem

if $\tilde g(u)\in L^1_{loc}(\Omega)$ and if the following conditions are valid:

(ⅰ) $u = 0$ a.e. in ${\mathbb R}^N\setminus \Omega$ and $u\in W^{h, q}({\mathbb R}^N)$ for any $0 < h < s$ and for any $0 < q < \frac{N(p-1)}{N-s}.$

(ⅱ) $T_k(u): = \max(-k, \min(u, k))\in W_0^{s, p}(\Omega)$ for any $k > 0.$

(ⅲ)

for any $\phi\in C_0^\infty(\Omega).$

We note here that if $2-\frac{s}{N} < p < \frac{N}{s},$ then the very weak solution $u$ belongs to the fractional Sobolev space $W^{h, q}({\mathbb R}^N)$ for any $q \in (1, \frac{N(p-1)}{N-s}).$ If $p\leq 2-\frac{s}{N},$ the space $W^{h, q}({\mathbb R}^N)$ in the above definition is no longer a fractional Sobolev space, however it is defined in the same way (see (2.1)).

In Section 2, we discuss the existence and main properties of the very weak solutions of problem (1.7) with $\tilde g\equiv0.$ Particularly, in the spirit of ^[26], we show the existence of a SOLA $u$ satisfying statements (ⅰ)–(ⅲ) of the above definition (see Proposition 2.8). The approximation sequence consists of solutions of (1.7) with $\tilde g\equiv0$ and smooth data. In addition, we prove that these solutions satisfy a priori estimates (2.8) and (2.11). As a result, we establish that the very weak solution satisfies (2.11) and

where $\left \|{\cdot}\right \|^{*}_{L^{\frac{N}{N-sp}}_w({\mathbb R}^N)}$ has been defined in (2.4) and is related to the Marcinkiewicz spaces. Finally, when $\mu\in \mathfrak{M}_b^+(\Omega),$ we construct this solution (see Propositions 2.9 and 2.10) such that $u\geq0$ and

where $d(x) = \mathrm{dist}\, (x, \partial \Omega).$ The lower estimate in the above display can be obtained as a consequence of [26, estimate (1.25)]. The upper estimate in the above display is an application of [21, Theorem 5.3] and (1.8).

Using the above properties of the very weak solutions and the fact that if $u, g$ satisfies (1.8) and (1.9) respectively then $g(u)\in L^1(\Omega),$ we obtain the following result.

Theorem 1.2. Let $s\in(0, 1),$ $1 < p < \frac{N}{s},$ $\mu\in \mathfrak{M}_b(\Omega).$ We assume that $g\in C({\mathbb R})$ is a nondecreasing function satisfying $g(0) = 0$ and

Then there exist a very weak solution $u$ to problem ($P_+$) satisfying (1.8) and

In addition, for any $q\in(0, \frac{N(p-1)}{N-s})$ and $h\in(0, s),$ there exists a positive constant $c = c(N, p, s, \Lambda_K, q, h, | \Omega|)$ such that

We note here that the integral conditions (1.9) and (1) coincide for $p = 2$. In addition, in the corresponding local case, the integral condition (1.9) with $s = 1$ ensures the existence of the associated renormalized solutions (see [32, Theorem 5.1.2 and (5.1.40)]).

Let us consider problem $(P_+)$ with a power absorption, i.e.,

We first notice that the function $g(t) = |t|^{ \kappa-1}t$ with $k > 0$ satisfies (1.9) if and only if $0 < \kappa < \frac{N(p-1)}{N-sp},$ hence problem (1.12) admits a very weak solution in this case. In the supercritical case $\kappa\geq\frac{N(p-1)}{N-sp},$ the sufficient condition for the solvability of problem (1.12) is expressed in terms of the Bessel capacity $\mathrm{Cap}_{{sp, \frac{ \kappa}{ \kappa-p+1}}}$ as follows.

Theorem 1.3. Let $s\in(0, 1),$ $1 < p < \frac{N}{s},$ $\kappa > p-1$ and $\mu\in \mathfrak{M}_b(\Omega).$ In addition we assume that $\mu$ is absolutely continuous with respect to the Bessel capacity $\mathrm{Cap}_{{sp, \frac{ \kappa}{ \kappa-p+1}}}$. Then there exists a very weak solution $u$ to problem (1.12) such that

In addition, for any $q\in(0, \frac{N(p-1)}{N-s})$ and $h\in(0, s),$ there exists a positive constant $c = c(N, p, s, \Lambda_K, q, h, | \Omega|)$ such that

In view of the discussion on the existence of solutions to problem (1.4), we expect that the existence phenomenon occurs for ($P_-$) only for measures $\mu\in \mathfrak{M}_b(\Omega)$ with small enough total mass. Indeed, using the Schauder fixed point theorem and sharp weak Lebesgue estimates, we prove the following existence result for any $\mu\in \mathfrak{M}_b(\Omega)$ with small enough total mass.

Theorem 1.4. Let $s\in(0, 1),$ $1 < p < \frac{N}{s}$ and $\tau\in\mathfrak{M}_b(\Omega)$ be such that $|\tau|(\Omega)\leq 1.$ Assume that $g \in C({\mathbb R})$ is a nondecreasing function satisfying (1.9) and

Then there exists a positive constant $\rho_0$ depending on $N, |{\Omega}|, \Lambda_g, {\Lambda}_K, a, s, p, d, | \Omega|$ such that for every $\rho \in (0, \rho_0)$ the following problem

admits a very weak solution $v$ satisfying

Here, $t_0 > 0$ depends on $N, |{\Omega}|, \Lambda_g, {\Lambda}_K, a, s, p, d, \rho_0.$ In addition, for any $q\in(0, \frac{N(p-1)}{N-s})$ and $h\in(0, s),$ there exists a positive constant $c$ depending only on $N, p, s, \Lambda_g, \Lambda_K, q, h, | \Omega|, a, d, \rho_0$ and $t_0,$ such that

In the linear case, i.e., $p = 2$, problem ($P_-$) with $L = (- \Delta)^s$ was thoroughly studied in ^[7]. More precisely, the authors in ^[7] showed that the same existence result occurs provided $g$ satisfies (1) and (1.15).

Problem ($P_-$) with $g(t) = |t|^{ \kappa-1}t$ and $\mu\in\mathfrak{M}_b^+(\Omega)$ becomes

When $p = 2$, problem ($P_-$) with $L = (- \Delta)^s$ and $\tau = \delta_0$ was studied in ^[8]. Here $\delta_0$ denotes the dirac measure concentrated at a point $x_0\in \Omega.$ In particular, the authors in ^[8] established that if $\kappa\geq\frac{N}{N-2s}$ and $u$ is a nonnegative solution of (1.19) then $\rho = 0.$ Concerning problem (1.19), conditions (1.9) and (1.15) are satisfied if $\kappa$ belongs to the subcritical range, that is when $p-1 < \kappa < \frac{N(p-1)}{N-sp}.$ In general, a sufficient condition for the solvability of (1.19) is the following.

Proposition 1.5. Let $s\in(0, 1),$ $1 < p < \frac{N}{s},$ $\kappa > p-1$ and $\tau\in\mathfrak{M}_b^+(\Omega)$ be such that

for some positive constant $M.$ Then problem (1.19) admits a nonnegative very weak solution $u$ for some $\rho > 0.$ Furthermore, there holds

where ${{\mathrm{d}}} \mu = u^ \kappa {{\mathrm{d}}} x+ \rho {\mathrm{d}}\tau$ and the positive constant $M$ depends only on $C, N, p, q, \Lambda_K.$

Finally, inspired from Phuc and Verbitsky's ideas in ^[30,31], we establish the following existence result in the whole range $\kappa > p-1.$

Theorem 1.6. Let $s\in(0, 1),$ $1 < p < \frac{N}{s},$ $\kappa > p-1$ and $\tau\in \mathfrak{M}_b^+(\Omega)$ with compact support in $\Omega.$ Then the following statements are equivalent.

(i) Problem (1.19) admits a nonnegative very weak solution $u_ \rho$ for some $\rho > 0$ such that

where ${{\mathrm{d}}} \mu = u^ \kappa {{\mathrm{d}}} x+ \rho {\mathrm{d}}\tau$ and for some constant $C_1 > 0.$ (ii) There exists a positive constant $C_2$ such that

for any Borel set $E\subset{\mathbb R}^N.$

(iii) There exists a positive constant $C_3$ such that

for any ball $B\subset{\mathbb R}^N,$ where ${{\mathrm{d}}}\tau_{\lfloor B} = \chi_B{{\mathrm{d}}} \tau.$

(iv) There exists a positive constant $C_4$ such that

We note here that if $p-1 < q < \frac{N(p-1)}{N-sp}$ then $\frac{spq}{q-p+1} > N$, this implies that $\mathrm{Cap}_{{sp, \frac{q}{q-p+1}}}(\{x\}) > 0$ for any $x\in{\mathbb R}^N$ (see [1, Section 2.6]). Hence, the statement (ⅱ) in the above theorem is always satisfied in the subcritical range.

Section 2 is devoted to the study of the very weak solutions to problem (1.7) with $\tilde g\equiv0.$ In Section 3, we discuss problem ($P_+$) as well as Theorems 1.2 and 1.3 are proved in Subsections 3.2 and 3.3 respectively. In section 4, we deal with problem ($P_-$). More precisely, we prove Theorem 1.4 in Subsection 4.1 and demonstrate Proposition 1.5 and Theorem 1.6 in Subsection 4.2.

2.   Very weak solutions

We start with the definition of the fractional spaces, which will be used frequently in this work. For any $s\in(0, 1)$ and $q > 0,$ we denote by $W^{s, q}({\mathbb R}^N)$ the fractional space

endowed with the quasinorm

When $q\geq 1,$ $W^{s, q}({\mathbb R}^N)$ is a Banach space and is called fractional Sobolev space. Finally, for any $p > 1,$ we denote by $W_0^{s, p}(\Omega)$ the closure of $C_0^\infty(\Omega)$ in the norm $\left \|{\cdot}\right \|_{W^{s, p}({\mathbb R}^N)}$ and by $(W^{s, p}_0(\Omega))^*$ its dual space.

2.1.   Weak solutions and a priori estimates

In this subsection, we introduce the notion of the weak solution of the following problem

where $\mu\in (W^{s, p}_0(\Omega))^*.$ In addition, when $\mu\in L^{p'}(\Omega),$ we establish a priori estimates, which will be used in the construction of the very weak solutions of the above problem with measure data.

Definition 2.1. Let $s\in(0, 1),$ $p > 1,$ and $\mu\in (W^{s, p}_0(\Omega))^*.$ We will say that $u\in W^{s, p}_0(\Omega)$ is a weak solution of (2.2), if it satisfies

Let us now give the definition of weak supersolutions of $L$ in $\Omega.$

Definition 2.2. Let $s\in(0, 1)$ and $p > 1.$ We will say that $u\in W^{s, p}({\mathbb R}^N)$ is a weak supersolution (resp. subsolution) of $L$ in $\Omega,$ if and only if satisfies

for any nonnegative $\phi\in W^{s, p}_0(\Omega).$

Next we state the comparison principle.

Proposition 2.3. ([23, Lemma 6]). Let $u\in W^{s, p}({\mathbb R}^N)$ be a weak supersolution of $L$ in $\Omega$ as well as let $v\in W^{s, p}({\mathbb R}^N)$ be a weak subsolution of $L$ in $\Omega$ such that $(v-u)_+\in W^{s, p}_0(\Omega).$ Then, $u\geq v$ a.e. in ${\mathbb R}^N.$

In view of the proof [11, Theorem 2.3], we may obtain the following existence result.

Proposition 2.4. For any $\mu\in (W^{s, p}_0(\Omega))^*$ there exists a unique weak solution of (2.2).

In order to state the first a priori estimate for the weak solution of (2.2), we need to give the definition and the main properties of Marcinkiewicz spaces. Let $D \subset {\mathbb R}^N$ be a domain. Denote $L^p_w(D)$, $1 \leq p < \infty,$ the weak $L^p$ space (or Marcinkiewicz space) defined as follows. A measurable function $f$ in $D$ belongs to this space if there exists a constant $c$ such that

The function ${\lambda}_f$ is called the distribution function of $f$. For $p \geq 1$, denote

The $\left \|{.}\right \|_{L^p_w(D)}^*$ is not a norm, but for $p > 1$, it is equivalent to the norm

More precisely,

Notice that,

From (2.4) and (2.6), one can derive the following estimate which is useful in the sequel.

Proposition 2.5. Let $1 < p < \frac{N}{s}$, $\mu\in L^{p'}(\Omega)$ and $u\in W^{s, p}_0(\Omega)$ be the unique weak solution of (2.2). Then there exists a positive constant $C = C(p, s, N, \Lambda_K)$ such that

Proof.

Let $k > 0.$ Taking $T_k(u)$ as test function and using the fact that

we obtain

Now, by the above inequality and the fractional Sobolev inequality we have

which implies the desired result. □

Proposition 2.6. Let $\mu\in L^{p'}(\Omega)$ and $u\in W^{s, p}_0({\mathbb R}^N)$ be the unique weak solution of (2.2). Then there exists a positive constant $C = C(p, s, N, \Lambda_K)$ such that

for any $\xi > 1$ and $d > 0.$

Proof. The proof is very similar to that of [26, Lemma 3.1] (see also [25, Lemma 8.4.1]). For the sake of convenience we give it below.

Set $\phi_\pm: = \pm(d^{1-\xi}-(d+u_\pm)^{1-\xi}).$ Using $\phi_\pm$ as test function we obtain

Now,

which implies

Combining all above we can easily reach the desired result. □

We conclude this subsection by the following a priori estimate for the weak solutions of (2.2) in the whole range $p > 1.$

Proposition 2.7. Let $\overline{q} = \min\{\frac{N(p-1)}{N-s}, p\}$ $\mu\in L^{p'}(\Omega)$ and $u\in W^{s, p}_0({\mathbb R}^N)$ be the unique weak solution of (2.2). For any $q\in(0, \overline{q})$ and $h\in(0, s),$ there exists a positive constant $c$ depending only on $N, s, p, \Lambda_K, q$ and $| \Omega|$ such that

Proof. The proof is an adaptation of the argument in [26, Lemma 3.2]. Let $R = \mathrm{diam}\, (\Omega)$ and $x_0\in \Omega.$ First, we note that

Taking into account that $u = 0$ a.e. in ${\mathbb R}^N\setminus \Omega,$ we can easily prove that

and

Here, we have also used the fact that $|x-y|\approx 1+|y-x_0|$ for any $(x, y)\in B_{R}(x_0)\times({\mathbb R}^N\setminus B_{2R}(x_0)),$ where the implicit constants in the last estimate depend only on $R.$ Similarly, we have

Combining all above, we have

Now, by Hölder inequality we obtain

Setting

and combining (2.10) and (2.13), we conclude

If $p > 2-\frac{s}{N},$ without loss of generality, we may assume that $q > 1.$ Therefore, we may apply the fractional Sobolev inequality to $d$ as in the proof of [26, Lemma 3.2] to obtain the desired result.

If $1 < p\leq 2-\frac{N}{s},$ we have that $0 < q\leq 1,$ therefore, we can not apply the fractional Sobolev inequality to $d$. To overcome this difficulty we use (2.8) instead of fractional Sobolev inequality. More precisely, let $1 < p < \frac{N}{s},$ then $0 < q < \frac{N(p-1)}{N-s} < p.$ Hence, we may choose $\xi > 1$ such that $1 < \gamma: = \frac{\xi q}{(p-1)(p-q)} < \frac{N}{N-sp}.$ Thus, by (2.6) and (2.8), we deduce

which in turn implies

The desired result follows by (2.12), (2.14) and the above inequality. □

2.2.   Existence and main properties

In this subsection, we construct a very weak solution to problem (2.2) which possesses several important properties, such as it satisfies pointwise estimates in terms of Wolff's potential. These estimates play an important role in the study of problems (P_±).

We start with the following existence result.

Proposition 2.8. Let $1 < p < \frac{N}{s}$ and $\mu\in \mathfrak{M}_b(\Omega).$ Then there exists a very weak solution to (2.2) satisfying

and

In addition, for any $q\in(0, \frac{N(p-1)}{N-s})$ and $h\in(0, s),$ there exist a positive constant $C_2 = C_2(N, p, s, \Lambda_K, q, h, | \Omega|)$ such that

Proof. Let $\{ \rho_n\}_n$ be a sequence of mollifiers and $\mu_n = \rho_n* \mu.$ Then $\mu_n\in C_0^\infty({\mathbb R}^N)$ and $\mu_n\rightharpoonup \mu$ weakly in $\mathfrak{{\mathbb R}}^N.$ We denote by $u_n$ the weak solution of (2.2) with $\mu = \mu_n.$

By (2.8), (2.9) and (2.11), there exist positive constants $C_1$ and $C_2$ such that

and

for any $n\in{\mathbb N},$ $q\in (0, \frac{N(p-1)}{N-s})$ and $h\in (0, s).$ In the spirit of the proof of [10, Theorem 3.4], we will show that the existence of a subsequence (still denoted by $\{u_n\}$) and a function $u:{\mathbb R}^N\to {\mathbb R}$ satisfying the following properties:

(ⅰ) $u\in W^{h, q}({\mathbb R}^N)$ for any $0 < q < \frac{N(p-1)}{N-s}$ and $0 < h < s.$

(ⅱ) $u_n\to u$ a.e. in ${\mathbb R}^N,$ $u = 0$ a.e. in ${\mathbb R}^N\setminus \Omega$ and $\left \|{u-u_n}\right \|_{W^{h, q}({\mathbb R}^N)}\to 0$ for any $q\in (0, \frac{N(p-1)}{N-s})$ and $h\in (0, s).$

(ⅲ) $T_k(u)\in W_0^{s, p}(\Omega)$ for any $k > 0.$

Step 1. There exists a subsequence, still denoted by $u_n,$ such that

Let $n, m\in {\mathbb N}$ and $\eta, \rho > 0.$ Then

By (2.18) and (2.7), there exists $k_0 > 0$ such that

By (2.19), the fractional Sobolev embedding theorem (see e.g., [13, Corollary 7.2]) and the fact that $W_0^{s, p}(\Omega)$ is a reflexive Banach space, we may prove the existence of a subsequence $T_{k_0}(u_{n_j})$ of $T_{k_0}(u_{n})$ such that $T_{k_0}(u_{n_j})\to v_{k_0}$ in $L^p({\mathbb R}^N)$ and a.e. in ${\mathbb R}^N$ as well as $T_{k_0}(u_{n_j})\rightharpoonup v_{k_0}$ in $W^{s, p}_0(\Omega).$ Hence,

The desired result follows by (2.21) and (2.22).

Step 2. Weak convergence of the truncates. Since $u_{n}\to u$ a.e. in ${\mathbb R}^N,$ we have that $T_k(u_{n})\to T_k(u)$ a.e. in ${\mathbb R}^N.$ Furthermore, by (2.19) and the fractional Sobolev embedding theorem, we can find a subsequence $\{T_k(u_{n_j})\}_{j = 1}^\infty$ such that $T_{k}(u_{n_j})\to v_{k}$ in $L^p({\mathbb R}^N)$ and $T_{k}(u_{n_j})\rightharpoonup v_{k}$ in $W^{s, p}_0(\Omega).$ Since $v_k = T_k(u)$ a.e. in ${\mathbb R}^N,$ we have that $T_k(u)\in W^{s, p}_0(\Omega).$ This implies that the limit does not depend on the subsequence. Hence, for the same subsequence $u_n$ of the Step 1, we have that

Furthermore, by (2.18)–(2.20) and Fatou's lemma, we have that

and

for any $q\in (0, \frac{N(p-1)}{N-s})$ and $h\in (0, s).$

By (2.20), (2.25) and the fact that $u_n\to u$ a.e. in ${\mathbb R}^N$, We can easily show that $\left \|{u-u_n}\right \|_{W^{h, q}({\mathbb R}^N)}\to 0$ for any $q\in (0, \frac{N(p-1)}{N-s})$ and $h\in (0, s).$ Let $\phi\in C_0^\infty(\Omega),$ $q\in(p-1, \frac{N(p-1)}{N-s})$ and $h\in (\max(\frac{sp-1}{p-1}, 0), s).$ For any bounded Borel set $E\subset{\mathbb R}^N,$ we have that

This, together with (ⅰ), (ⅱ) and the fact that $\frac{q(sp-h(p-1)-1)}{q-p+1} < 0$, implies that

The proof is complete. □

In the next theorem, we establish a priori pointwise estimates for a certain nonnegative very weak solution of problem (2.2) with $\mu\in \mathfrak{M}_b^+(\Omega).$

Proposition 2.9. Let $1 < p < \frac{N}{s}$ and $\mu\in \mathfrak{M}_b^+(\Omega)$. Then there exist a nonnegative very weak solution $u$ of (2.2) and a positive constant $C$ depending only on $N, s, p, \Lambda_K$ such that

for a.e. $x\in \Omega.$

Proof. Let $u$ be the solution constructed in Proposition 2.8 and $\{u_n\}$ be the sequence defined in Proposition 2.8 such that

(ⅰ) $u\in W^{h, q}({\mathbb R}^N)$ for any $0 < q < \frac{N(p-1)}{N-s}$ and $0 < h < s.$

(ⅱ) $u_n\to u$ a.e. in ${\mathbb R}^N,$ $u = 0$ a.e. in ${\mathbb R}^N\setminus \Omega$ and $\left \|{u-u_n}\right \|_{W^{h, q}({\mathbb R}^N)}\to 0$ for any $q\in (0, \frac{N(p-1)}{N-s})$ and $h\in (0, s).$

Since $\mu_n\in C_0^\infty({\mathbb R}^N)$ is a nonnegative function, by (2.3), we have that $u_n\geq0$ a.e. in ${\mathbb R}^N.$ Hence, by [23, Lemma 7], $u_{k, n} = \min(u_n, k)$ is a nonnegative weak supersolution. By properties (ⅰ) and (ⅱ), we may show that $u_{k} = \min(u, k)$ is a nonnegative weak supersolution. Hence, there exists a nonnegative Radon measure $\mu_k\in \mathfrak{M}^+(\Omega)$ such that

for any $\phi\in C_0^\infty(\Omega).$ Since $u_k\to u$ in ${\mathbb R}^N,$ we have that $\left \|{u-u_k}\right \|_{W^{h, q}({\mathbb R}^N)}\to 0$ for any $h\in (0, s)$ and $q\in(0, \frac{N(p-1)}{N-s}).$ This, together with (2.27), implies

Now, we remark that, in view of the proof of [26, Theorem 1.3], we may apply [26, estimate (1.25)] to $u_k.$ Hence,

Letting $k\to\infty$ in the above inequality and using some elementary manipulations, we may obtain the lower estimate in (2.26).

For the upper estimate in (2.26), by [23, Theorem 9], we have that

Hence, $v_k$ is a lower semicontinuous functions in $\Omega$ and a nonnegative weak supersolution. By [23, Theorem 12], $v_k$ is $(s, p)$-superharmonic function in $\Omega$ (see [23, Definition 1] for the definition of $(s, p)$-superharmonic function). This, together with [23, Lemma 12], implies that $v: = \lim_{k\to\infty} v_k$ is $(s, p)$-superharmonic function in $\Omega$ and $v = u$ a.e. in ${\mathbb R}^N.$ The desired result follows by applying [21, Theorem 5.3] to $v$ and the fact that $v = u$ a.e. in ${\mathbb R}^N.$ □

Proposition 2.10. Let $\mu\in \mathfrak{M}_b(\Omega).$ Then there exists a very weak solution $u$ of (2.2) and a positive constant $C$ depending only on $N, s, p$ and $\Lambda_K$ such that

Proof. Let $u$ be the solution constructed in Proposition 2.8 and $x_0\in \Omega.$ Set $R = \mathrm{diam}\, (\Omega),$ $\mu_n = \rho_n* \mu$ and $\mu_n^\oplus = \rho_n* \mu^+.$ We denote by $v_n^{\oplus}\in W^{s, p}_0(\Omega)$ the solution of

By Proposition 2.3, we have that $v_n^{\oplus}\geq 0$ and $v_n^{\oplus}\geq u_n,$ where $u_n\in W^{s, p}_0(\Omega)$ is the weak solution of (2.2) with $\mu = \mu_n.$ By statements (ⅰ)–(ⅲ) in the proof of Proposition 2.8, there exist subsequences $\{u_{n_k}, v_{n_k}^{\oplus}\}_{k = 1}^\infty$ such that $u_{n_k}\to u$ and $v_{n_k}^\oplus\to v^\oplus$ a.e. in ${\mathbb R}^N$ and

for any $h\in (0, s)$ and $q\in(0, \frac{N(p-1)}{N-s}).$ Combining all above, we may deduce that $u\leq v^\oplus$ a.e. in ${\mathbb R}^N$ and $v^\oplus$ is a nonnegative very weak solution to

In addition, in view of the proof of Proposition 2.9, there exists a positive constant $C = C(p, s, \Lambda_K, N)$ such that

where

By (2.15) and (2.6), we derive that

and

where the implicit constants in (2.31) and (2.32) depend only on $p, s, \Lambda_K, N.$ The inequalities in (2.32) follow by the fact that $v^\oplus = 0$ in ${\mathbb R}^N\setminus B_{2R}(x_0)$ and $\mu({\mathbb R}^N\setminus \Omega) = 0.$

Combining (2.30)–(2.32), we obtain the upper bound in (2.29).

The proof of the lower bound in (2.29) is similar and we omit it. □

3.   Nonlocal equations with absorption nonlinearities

3.1.   The variational problem

We assume that $g\in C({\mathbb R})$ and $rg(r)\geq0.$ Let $\Omega\subset{\mathbb R}^N$ be an open bounded domain and $\mu\in (W^{s, p}_0(\Omega))^*.$ Set $G(r) = \int_0^rg(s){{\mathrm{d}}} s,$

and

Theorem 3.1. Let $s\in(0, 1),$ $p > 1$ and $\mu\in (W^{s, p}_0(\Omega))^*.$ Then, there exists a minimizer $u_ \mu$ of $J$ in $\mathbf{X}_G(\Omega).$ Furthermore, $u_ \mu$ is a weak solution of $J,$ in the sense of

for any $\zeta\in W^{s, p}_0(\Omega)\cap L^\infty(\Omega).$

If $g$ is nondecreasing the solution $u_ \mu$ is unique and the mapping $\mu\mapsto u_ \mu$ is nondecreasing.

Proof. We adapt the argument used in the proof of [15, Theorem 5.1]. Let $\{v_n\}$ be a minimizing sequence. Taking in to account that $G(t)\geq0$ for any $t\in {\mathbb R}$ and the fractional Sobolev inequality, we can easily show the existence of a positive constant $C = C(p, \Omega, \Lambda_K)$ such that

This implies that $v_n$ is uniformly bounded in $W_0^{s, p}(\Omega).$ Thus, by the fractional Sobolev embedding theorem (see e.g., [13, Corollary 7.2]) and the fact that $W_0^{s, p}(\Omega)$ is a reflexive Banach space, we may prove the existence of a subsequence, still denoted by $\{v_{n}\}$ and a function $v\in W^{s, p}_0(\Omega)$ such that there hold:

(ⅰ) $v_{n}\to v$ a.e. in ${\mathbb R}^N.$

(ⅱ) $v_{n}\rightharpoonup v$ in $W_0^{s, p}(\Omega)$ and $v_{n}\rightarrow v$ in $W^{h, q}({\mathbb R}^N)$ for any $h\in(0, s)$ and $q\in(1, p).$

By Fatou's lemma, we obtain

Hence $v$ is a minimizer. If $g$ is nondecreasing, the uniqueness of the minimizer follows by the fact that $J$ is strictly convex.

We next show (3.1). Let $v_k$ be the minimizer of $J$ associated with $g_k = \max(-k, \min(g, k)).$ Then, in view of the proof of [11, Theorem 2.3], $v_k$ satisfies

for any $\zeta\in W^{s, p}_0(\Omega).$ Taking $v_k$ as test function, we have

which implies

By the above inequality, we may deduce that there exists a subsequence, still denoted by $\{v_{k}\}$ and a function $v\in W^{s, p}_0(\Omega)$ such that they satisfy statements (ⅰ) and (ⅱ).

Let $\zeta\in L^\infty(\Omega)$ with $\left \|{ \zeta}\right \|_{L^\infty(\Omega)} = N$ and $E\subset \Omega$ be a Borel set. Then, for any $\lambda > 0,$ we have

Also,

Let $\varepsilon > 0,$ $\lambda = \frac{2MN}{ \varepsilon}$ and $\delta = \frac{ \varepsilon}{2N\sup\{|g(t)|:|t|\leq\frac{2MN}{ \varepsilon}\}+1}$. Then for any Borel set $E\subset \Omega$ with $|E| < \delta$, we have

Thus, by Vitali's theorem, we conclude

Combining all above, we obtain that $v$ satisfies (3.1).

Now for any $u\in \mathbf{X}_G(\Omega),$ we have that $u\in \mathbf{X}_{G_{k}}(\Omega),$ $G_{k}(u)\leq G(u)$ and

where $G_{k}(r) = \int_0^rg_{k}(s){{\mathrm{d}}} s.$ By the above inequality and Fatou's Lemma, we deduce that $v$ is a minimizer of $J$ in $\mathbf{X}_G(\Omega).$

Let $g$ be nondecreasing and $u_ \nu$ be the minimizer of $J$ associated with $\nu\in (W^{s, p}_0(\Omega))^*,$ such that $\nu\leq \mu.$ Then, using $v_k = \min\{(u_ \nu-u_ \mu)_+, k\}$ as test function, we have that

Letting $k\to\infty$ in the above inequality and then proceeding as in the proof of [23, Lemma 6], we obtain that $u_ \nu\leq u_ \mu$ a.e. in ${\mathbb R}^N.$ □

When $\mu\in L^{p'}(\Omega),$ we derive the following result which will be useful in the next subsection.

Lemma 3.2. Let $\mu\in L^{p'}(\Omega),$ $g\in C({\mathbb R}^N)$ be a nondecreasing function with $g(0) = 0$ and $u\in W^{s, p}_0(\Omega)$ satisfy (3.1). Then there holds,

In addition, if we assume that $\mu\geq0,$ then $u\geq0$ a.e. in ${\mathbb R}^N.$

Proof. Let $k > 0.$ Using $\phi_k = \tanh(ku)$ as test function in (3.1), we obtain

If $\infty > u(x) > u(y) > -\infty,$ then there exists $\xi\in (u(y), u(x))$ such that

Combining the last two displays, we can easily obtain that

Since $g(u) \phi_k\geq 0$ a.e. in $\Omega,$ by Fatou's lemma and the above inequality, we can easily deduce (3.6). □

3.2.   Subcritical nonlinearities

In this subsection, we always assume that $s\in(0, 1),$ $1 < p < \frac{N}{s}$ and $g\in C({\mathbb R})$ is nondecreasing such that $g(0) = 0.$

Lemma 3.3. Let $g\in L^\infty({\mathbb R})$ and $\lambda_i\in \mathfrak{M}_b^+(\Omega)$ ($i = 1, 2$). Then there exist very weak solutions $u, u_i$ ($i = 1, 2$) to problems

and

such that there hold

In addition, for any $q\in(0, \frac{N(p-1)}{N-s})$ and $h\in(0, s),$ there exists a positive constant $c = c(N, p, s, \Lambda_K, q, h, | \Omega|)$ such that

and

Finally, there exist very weak solutions $v_i$ to (2.2) with $\mu = \lambda_i$ (i = 1, 2) such that

where $C_i$ is a positive constant depending only on $p, s, \Lambda_K$ and $N.$

Proof. Let $\{ \rho_n\}_{1}^\infty$ be a sequence of mollifiers and $\lambda_{n, i} = \rho_n* \lambda_i.$ Then $\lambda_{n, i}\in C_0^\infty({\mathbb R}^N).$ By Proposition 3.1, there exist unique solutions $u_n, u_{n, i}, v_{n, i}\in W^{s, p}_0(\Omega)$ to the following problems

and

such that there holds

By Lemma 3.2 and Proposition 2.7, for any $q\in(0, \frac{N(p-1)}{N-s})$ and $h\in(0, s),$ there exists a positive constant $c = c(N, p, s, \Lambda_K, q, h, | \Omega|)$ such that

and

Furthermore, in view of the proof of (2.16), we have that $T_k(u_n), T_k(u_{n, i}), T_k(v_{n, i})\in W^{s, p}_0(\Omega)$ and satisfy (2.19) with $\mu = \lambda_1+ \lambda_2.$

Since the sequences $\{ \lambda_{n, i}\}_{n}$ are uniformly bounded in $\mathfrak{M}_b(\Omega),$ as in the proof of Proposition 2.8, we may show that there exist subsequences, still denoted by the same index, such that $u_n\to u,$ $u_{n, i}\to u_i$ $v_{n, i}\to v_i$ in $W^{h, q}({\mathbb R}^N)$ and a.e. in ${\mathbb R}^N.$ In addition, we may prove that $T_k(u), T_k(u_{i}), T_k(v_{i})\in W^{s, p}_0(\Omega)$ for any $k > 0.$ Finally, by dominated convergence theorem, we deduce that $g(u_{n})\to g(u),$ $g(u_{n, 1})\to g(u_{1}),$ $g(-u_{n, 2})\to g(-u_{2})$ in $L^1(\Omega)$. Hence, combining all above, we can easily show that $u, u_i$ are very weak solutions of problems (3.7)–(3.9) respectively and $v_i$ are very weak solutions of problem (2.2) with $\mu = \lambda_i$ $(i = 1, 2).$ By proceeding as in the proof of Proposition 2.10 and using (3.14), we derive (3.13).

Estimates (3.11) and (3.12) follow by (3.15), (3.16) and Fatou's lemma. □

Lemma 3.4. Let $\lambda_i\in \mathfrak{M}_b^+(\Omega)$ for $i = 1, 2.$ We also assume that $g((-1)^{1+i} C W_{s, p}^{2R}[ \lambda_i])\in L^1(\Omega),$ where $C$ is the constant in Proposition 2.10. Then the conclusion of Lemma 3.3 holds true.

Proof. Let $T_n(t) = \max(-n, \min(t, n))$ for any $n\in{\mathbb N}.$ By Lemma 3.3, there exist very weak solutions $u_n, u_{n, i}, v_{n, i}\in W^{s, p}_0(\Omega)$ of the following problems

and

such that there holds

and for any $n\in {\mathbb N}.$ The rest of the proof can proceed similarly to the proof of Lemma 3.3 and we omit it. □

Proposition 3.5. Assume

for $\tilde q > 0.$ Let $v$ be a measurable function defined in $\Omega.$ For $s > 0$, set

Assume that there exists a positive constant $C_0$ such that

Then for any $s_0\geq1,$ there hold

Proof. The proof is very similar to the one of [16, Lemma 5.1] and we omit it. □

Proof of Theorem 1.2. Let $\lambda_1 = \mu^+$ and $\lambda_2 = \mu^-.$ By Lemma 3.3, there exist very weak solutions $u_n, v_{i}$ of the following problems

and

such that there holds

Also, taking into consideration that $g$ in nondecreasing with $g(0) = 0$, we may show that $T_k(u_n), T_k(v_i)$ satisfy (2.19) with $\mu = \lambda_1+ \lambda_2.$ In addition, by (2.15), there holds

By (2.7) and Proposition 3.5, we have that $|T_nog(u_n)|\leq g(v_1)-g(-v_2)$ and

where $\tilde q = \frac{N(p-1)}{N-sp}.$ The desired result follows by proceeding as in the proof of Lemma 3.3. □

3.3.   Power nonlinearities: proof of Theorem 1.3

In order to prove Theorem 1.3, we need to introduce some notations concerning the Bessel capacities, we refer the reader to ^[1] for more detail. For ${\alpha}\in{\mathbb R}$ we define the Bessel kernel of order ${\alpha}$ by $G_{{\alpha}}(\xi) = {{\mathcal F}}^{-1}(1+|.|^2)^{-\frac{{\alpha}}{2}}(\xi)$, where ${{\mathcal F}}$ is the Fourier transform of moderate distributions in ${\mathbb R}^N$. For any $\beta > 1,$ the Bessel space $L_{{\alpha}, \beta}({\mathbb R}^N)$ is given by

with norm

The Bessel capacity is defined as follows.

Definition 3.6. Let $\alpha > 0,$ $1 < \beta < \infty$ and $E\subset{\mathbb R}^N.$ Set

Then

If $\mathcal{S}_E = \emptyset,$ we set $\mathrm{Cap}_{{{\alpha}, \beta}}(E) = \infty.$

In the sequel, we denote by $L_{- \alpha, \beta'}({\mathbb R}^N)$ the dual of $L_{ \alpha, \beta}({\mathbb R}^N)$ and we set

Proof of Theorem 1.3. Since $\mu$ is absolutely continuous with respect to the capacity $\mathrm{Cap}_{{sp, \frac{ \kappa}{ \kappa-p+1}}},$ the measures $\mu^+, \mu^-$ have the same property. Thus, by [5, Theorem 2.5] (see also ^[3]), there are nondecreasing sequences $\{ \mu_{n}^\pm\}_n\subset L^{-sp, \frac{ \kappa}{p-1}}({\mathbb R}^N)\cap\mathfrak{M}^+_b({\mathbb R}^N)$ with compact support in $\Omega,$ such that they converge to $\mu^\pm$ in the narrow topology. Furthermore, by [5, Theorem 2.3] (see also [1, Corollary 3.6.3]),

By Lemma 3.4, there exist solutions $u_n, u_{n, i}, v_i$ to the problems

and

such that there holds

Furthermore, in view of the proof of Lemmas 3.3 and 3.4, the sequences $\{u_{n, i}\}, \{v_{n, i}\}$ satisfy (3.15)–(3.18) with $g(t) = |t|^ \kappa \mathrm{sign}(t),$ $\lambda_{n, 1} = \mu_n^+$ and $\lambda_{n, 2} = \mu_n^-,$ as well as they can be constructed such that

By (3.15) and (3.16) with $g(t) = |t|^ \kappa \mathrm{sign}(t),$ $\lambda_{n, 1} = \mu_n^+$ and $\lambda_{n, 2} = \mu_n^-,$ we have

By (3.15)–(3.18) with $g(t) = |t|^ \kappa \mathrm{sign}(t),$ $\lambda_{n, 1} = \mu_n^+$ and $\lambda_{n, 2} = \mu_n^-,$ there are subsequences, still denoted by the same index, such that $u_n\to u,$ $u_{n, i}\to u_i$ $v_{n, i}\to v$ in $W^{h, q}({\mathbb R}^N)$ and a.e. in ${\mathbb R}^N.$ In addition, $T_k(u), T_k(u_i), T_k(v_i)\in W^{s, p}_0({\mathbb R}^N)$ and

Therefore, by dominated convergence theorem, we obtain that $|u_{n}|^ \kappa\to |u|^ \kappa,$ $|u_{n, 1}|^ \kappa\to |u_{1}|^ \kappa,$ $|u_{n, 2}|^ \kappa\to |u_{2}|^ \kappa$ in $L^1(\Omega)$. This, implies that $u, u_i$ are very weak solutions of problems (3.7)–(3.9) respectively and $v_i$ are very weak solution of problem (2.2) with $\mu = \lambda_i,$ where $\lambda_1 = \mu^+$ and $\lambda_2 = \mu^-.$

Estimate (1.13) follows by (3.25) and (3.13). Estimate (1.14) follows by (3.15) with $g(t) = |t|^ \kappa \mathrm{sign}(t),$ $\lambda_{n, 1} = \mu_n^+,$ $\lambda_{n, 2} = \mu_n^-$ and Fatou's lemma. □

4.   Nonlocal equations with source nonlinearities

4.1.   Subcritical nonlinearities

In this subsection, we investigate the existence of solutions to the following problem

where $\rho > 0,$ $g\in C({\mathbb R})$ is a nondecreasing function and

Let us state the first existence result.

Lemma 4.1. Let $1 < p < \frac{N}{s}$ and $\tau \in C_0^\infty({\mathbb R}^N)$ be such that $\left \|{{\tau}}\right \|_{L^1({\mathbb R}^N)}\leq 1.$ Assume that $g \in L^\infty(\Omega) \cap C({\mathbb R})$ satisfies (3.19) for

In addition, we assume that $g$ is nondecreasing and satisfies (4.2).

Then there exists a positive constant $\rho_0$ depending on $N, \Omega, {\Lambda}_{g}, \Lambda_K, a, d, p, s$ such that for every $\rho \in (0, \rho_0),$ problem (4.1) admits a weak solution $v\in W^{s, p}_0(\Omega)$ satisfying

where $t_0 > 0$ depends on $N, \Omega, {\Lambda}_{g}, \Lambda_K, a, d, p, s$.

Proof. We shall use Schauder fixed point theorem to show the existence of a positive weak solution of (4.1).

Let $1 < \kappa < \min\{\frac{N}{N-sp}, \frac{d}{p-1}\}$ and $v\in L^1(\Omega).$ Since $g\in L^\infty(\Omega),$ we can easily show that the following problem

admits a unique weak solution ${\mathbb T}(v)\in W_0^{s, p}(\Omega).$ We define the operator ${\mathbb S}$ by

By (2.8), we obtain

Let $v \in L_w^{\frac{N}{N-sp}}({\Omega})$. For any $\lambda > 0$, we set $E_ \lambda: = \{x\in \Omega: |v(x)|^{\frac{1}{p-1}} > \lambda\}$ and $e(\lambda) = \int_{E_ \lambda}{{\mathrm{d}}} x$. By (2.4) and (2.6), we can easily show that

By the above inequality and Lemma 3.5 with $\lambda_0 = 1$ and $\tilde q = \frac{N(p-1)}{N-sp},$ we deduce

Let $\lambda = \left \|{v}\right \|_{L_w^{\frac{N}{N-sp}}({\Omega})}$. By (2.6), we have that

Combining all above, we may prove that

Therefore, if $\left \|{v}\right \|_{L_w^{\frac{N}{N-sp}}({\Omega})}\leq t$ then

Since $1 < \kappa < \frac{N}{N-sp}$, there exist $t_0 > 0$ and $\rho_0 > 0$ depending on $|{\Omega}|, {\Lambda}_g, p, \kappa, N, a$ such that for any $t\in(0, t_0]$ and $\rho \in (0, \rho_0),$ the following inequality holds

where $C$ is the constant in (4.7). Hence,

Next, we apply Schauder fixed point theorem to our setting.

We claim that ${\mathbb S}$ is continuous. First we assume that $v_n\rightarrow v$ in $L^1({\Omega})$ and ${\mathbb T}(v_n)\to {\mathbb T}(v)$ in $W^{1, p}_0(\Omega),$ then by fractional Sobolev inequality, we have

Let $k > 0$ and $\varepsilon > 0,$ then

By (4.6) and the fact that $g\in L^\infty({\mathbb R}),$ we have that ${\mathbb S}(v_n)\in L^{ \beta}(\Omega)$ and $\{{\mathbb S}(v_n)\}$ is uniformly bounded in $L^{ \beta}(\Omega)$ for any $\beta\in (1, \frac{N}{N-sp}).$ Hence, there exists $k_0\in{\mathbb N},$ such that

Now, we set

and $B_{ \delta, n} = \{x\in \Omega:\; |{\mathbb T}(v)(x)-{\mathbb T}(v_n)(x)|\leq \delta\}.$ Then, we have that

Since $h(t) = t^{p-1} \mathrm{sign}(t)$ is uniformly continuous in $[-k_0, k_0],$ there exists $\delta_0 > 0$ independent of $n$ such that

Moreover, by (4.9), there exists $n_0 = n_0(\delta_0, k_0, p)\in {\mathbb N}$ such that

Hence, combining (4.9)–(4.14), we obtain that ${\mathbb S}(v_n)\to{\mathbb S}(v)$ in $L^1(\Omega).$

Therefore, it is enough to show that ${\mathbb T}(v_n)\to {\mathbb T}(v)$ in $W^{s, p}_0(\Omega).$ In order to prove this, we will consider two cases.

Case 1. $1 < p < 2.$ Let $M: = \sup_{t\in{\mathbb R}}|g(t)|.$ We will show that ${\mathbb T}(v_n)\to {\mathbb T}(v)$ in $W^{s, p}_0(\Omega).$ Since ${\mathbb T}(v_n), {\mathbb T}(v)\in W_0^{s, p}(\Omega)$ are weak solutions of (4.4) with $v_n$ and $v$ respectively, we have

Therefore,

Using $\phi = {\mathbb T}(v_n)-{\mathbb T}(v)$ as test function, we have

We first treat $I.$ On one hand, since

for any $(a, b)\in{\mathbb R}^{2N}\setminus\{(0, 0)\}$ and $p\in(1, 2),$ we have

On the other hand, by Hölder inequality, we obtain

where $C_1$ is the constant in (4.16). Hence, by (4.18) and (4.19), we obtain

Next we treat $II.$ Let $r = \frac{N p}{N-sp},$ proceeding as in the proof of (4.15), we have

where in the last inequality we used the fractional Sobolev inequality.

Combining (4.17), (4.20) and (4.21), we obtain

Since $g\circ(|\cdot|^{p-1} \mathrm{sign}(\cdot))$ is uniformly continuous in ${\mathbb R},$ bounded and $v_n\to v$ in $L^1(\Omega),$ we obtain

which, together with (4.22), implies the desired result.

Case 2. $p\geq 2.$ We note here that

for any $(a, b)\in{\mathbb R}^{2N}$ and $p\geq2.$ Thus,

By using a similar argument to the one in Case 1, we may show that ${\mathbb T}(v_n)\to {\mathbb T}(v)$ in $W^{s, p}_0(\Omega).$

Next we claim that ${\mathbb S}$ is compact. Indeed, let $\{v_n\}$ be a sequence in $L^1({\Omega})$ then by (4.16), we obtain that ${\mathbb T}(v_n)$ is uniformly bounded in $W^{s, p}_0(\Omega).$ Hence there exists a subsequence still denoted by $\{{\mathbb T}(v_n)\}$ such that ${\mathbb T}(v_n)\rightharpoonup \psi$ in $W^{s, p}_0(\Omega)$ and ${\mathbb T}(v_n)\to \psi$ a.e. in ${\mathbb R}^N.$ Furthermore, in view of (4.6), we can easily show that ${\mathbb S}(v_n) = |{\mathbb T}(v_n)|^{p-1} \mathrm{sign}({\mathbb T}(v_n))\to |\psi|^{p-1} \mathrm{sign}(\psi)$ in $L^1(\Omega).$

Now set

Then ${{\mathcal O}}$ is a closed, convex subset of $L^1({\Omega})$ and by (4.8), ${\mathbb S}({{\mathcal O}}) \subset {{\mathcal O}}$. Thus we can apply Schauder fixed point theorem to obtain the existence of a function $v \in {{\mathcal O}}$ such that ${\mathbb S}(v) = v$. This means that $u = v^{\frac{1}{p-1}} \mathrm{sign}(v)$ is a solution of (4.1) satisfying (4.3). □

Proof of Theorem 1.4. Let $\{ \rho_n\}_{n = 1}^\infty$ be a sequence of mollifiers. Set $\tau_n = \rho_n*\tau$ and $g_n = \max(-n, \min(g, n)).$ Then $g_n$ satisfies (1.9) with the same constant $\Lambda_g.$ Thus, there exists a weak solution $u_n\in W^{s, p}_0(\Omega)$ of

In addition, it satisfies

where $t_0 > 0$ depends on $N, {\Omega}, {\Lambda}_g, \Lambda_K, a, s, p, d.$

By (4.24), we have that

Hence by Proposition 3.5,

where $C$ depends only on $N, {\Omega}, {\Lambda}_g, \Lambda_K, a, s, p, d$ and $t_0.$ This, together with Proposition 2.7, implies that for any $q\in(p-1, \frac{N(p-1)}{N-s})$ and $h\in(0, s),$ there exists a positive constant $c = c(N, s, p, \Lambda_K, s, h, q, | \Omega|)$ such that

Therefore, in view of the proof of Proposition 2.8, we may show that there exists a subsequence, still denoted by the same notation, such that $u_n\rightarrow u$ in $W^{h, q}({\mathbb R}^N)$ and a.e. in ${\mathbb R}^N.$ Now, we will show that $g_n(u_n)\to g(u)$ in $L^1(\Omega).$ We will prove it by using Vitali's convergence theorem. Let $E\subset \Omega$ be a Borel set. Then, by Lemma 3.5 and (4.24), we have

Let $\varepsilon > 0,$ then there exists $s_0$ such that

Set $\delta = \frac{ \varepsilon}{2(1+g(s_0)-g(-s_0))} > 0.$ Then for any Borel set $E$ with $|E|\leq \delta,$ we have

Hence, by the last three inequalities, we may invoke Vitali's convergence theorem in order to prove that $g_n(u_n)\to g(u)$ in $L^1(\Omega).$

In view of the proof of Proposition 2.8, we may deduce that $u$ is a very weak solution of (1.16). Furthermore, by Fatou's lemma, we can easily show that $u$ satisfies (1.17) and (1.18). □

4.2.   Power nonlinearities: proof of Proposition 1.5 and Theorem 1.6

Proof of Proposition 1.5. Let $w = ACW_{s, p}^{2 \mathrm{diam}\, (\Omega)}[ \rho\tau],$ where $C$ is the constant in (2.29) and $A > 1$ is a constant that will be determined later. Set ${{\mathrm{d}}} \nu = w^ \kappa {{\mathrm{d}}} x+ \rho {\mathrm{d}}\tau,$ then by (1.20), we obtain

If we choose $A = 2^{\frac{1}{p-1}+1}$ and $\rho$ small enough such that $(AC)^\frac{ \kappa}{p-1}M \rho^{\frac{ \kappa-p+1}{(p-1)^2}}+1 < 2,$ we deduce that

Now, let $x_0\in \Omega$ be such that $W_{s, p}^{2 \mathrm{diam}\, (\Omega)}[\tau](x_0) < \infty.$ If $0\leq v\leq c_0W_{s, p}^{2 \mathrm{diam}\, (\Omega)}[\tau]$ a.e. in ${\mathbb R}^N,$ for some constant $c_0 > 0,$ then we have

Thus $v\in L^ \kappa(\Omega).$

Let $u_0\geq0$ be a very weak solution of

satisfying $C^{-1} W_{s, p}^{\frac{d(x)}{8}}[ \mu_{n-1}](x)\leq u_0(x)\leq CW_{s, p}^{2 \mathrm{diam}\, (\Omega)}[ \rho\tau](x)$ a.e. in $\Omega.$ We may construct a nondecreasing sequence $\{u_n\}_{n\geq0},$ such that $u_n$ is a very weak solution to problem

and satisfies

for any $n\in{\mathbb N},$ where ${{\mathrm{d}}} \mu_{n-1} = u_{n-1}^ \kappa {{\mathrm{d}}} x+ \rho{{\mathrm{d}}}\tau.$ In addition, by (4.26) and the above inequality, there holds

where the positive constant $C^{-1}$ depends only on $N, p, s, q.$ Finally, $u_n$ satisfies (2.15)–(2.17) with ${{\mathrm{d}}} \mu = w^ \kappa {{\mathrm{d}}} x+ \rho {\mathrm{d}}\tau.$

Proceeding as in the proof of Proposition 2.8, we may show that there exists a subsequence, still denoted by $\{u_n\},$ such that $u_n\to u$ a.e. in ${\mathbb R}^N$ and $u$ is a very weak solution of problem (1.19). By (4.27) and Fatou's Lemma, we obtain estimate (1.21). The proof is complete. □

Proof of Theorem 1.6. We will first prove that $(i)$ implies $(ii)$ by using some ideas from ^[30]. Without loss of generality we assume that $\rho = 1.$ Extend $\mu$ to whole ${\mathbb R}^N$ by setting $\mu({\mathbb R}^N\setminus \Omega) = 0.$

Let $0\leq g\in L^\frac{ \kappa}{p-1}({\mathbb R}^N; \mu).$ We set

It is well known that there exists a positive constant $c_1$ depending only on $N, p, \kappa$ such that

(see, e.g., ^[14]). Also,

Let $K = \mathrm{supp}\, \tau.$ By the assumption, we have that $r_0: = \mathrm{dist}\, (K, \partial \Omega) > 0.$ Set $g = fanxiexian_myfh_K \tilde g,$ for any nonnegative $\tilde g\in L^\frac{ \kappa}{p-1}({\mathbb R}^N; \mu_{\lfloor K}).$ We first note that $B_{\frac{r_0}{8}}(x)\cap K = \emptyset$ if $x\in \Omega$ with $d(x) < \frac{r_0}{8}$ or if $x\in{\mathbb R}^N\setminus \Omega,$ which implies

if $x\in \Omega$ with $d(x) < \frac{r_0}{24}$ or if $x\in{\mathbb R}^N\setminus \Omega.$ Therefore, by the above equality and (4.29), we have

Also, by [5, Theorem 2.3] (see also [1, Corollary 3.6.3]), we have

where the implicit constant depends only on $s, p, N, \kappa$ and $r_0.$

Hence, combining the last two displays, we may show that there exists a positive constant $c_3 = c_3(N, p, s, \kappa, r_0)$ such that

Let $f\in L^{\frac{ \kappa}{ \kappa-p+1}}({\mathbb R}^N).$ Then, for any $\tilde g\in L^\frac{ \kappa}{p-1}({\mathbb R}^N; \mu_{\lfloor K}),$ there holds

The last inequality implies,

By [1, Theorem 7.2.1], the above inequality is equivalent to

for any compact $F\subset{\mathbb R}^N.$ (1.23) follows by the above inequality and the fact that $\tau\leq \mu_{\lfloor K}.$

Next, we prove that (ⅱ) implies (ⅲ). We note that proceeding as above, in the opposite direction, we may prove that (1.23) implies

By (4.30) and taking $\tilde g = fanxiexian_myfh_B,$ we can easily show that there exists a positive constant $C$ depending only on $N, s, p, \Omega$ such that

We will show that (ⅲ) implies (ⅳ). Let $R = 2 \mathrm{diam}\, (\Omega)$ and $C_3$ be the constant in (1.24). In the spirit of the proof of [31, Theorem 2.10], we need to prove that there exists a positive constant $c_0 = c_0(N, p, \kappa, s, C_3, R, \tau(\Omega)) > 0$ such that

for any $t\leq R$ and $\forall x\in \Omega.$

Concerning the proof of the above inequality, we first note that for any $y\in B_t(x)$ and $t\leq \frac{R}{4},$ there holds

By the above inequality, we deduce

where in the last inequality we used (1.24). This implies (4.33).

For any $x\in \Omega$ and $t < R,$ we set

and

Then we can easily prove that

Now, we treat the first term on the right hand in $(4.35).$ By (1.24), we have

which implies

Next, we treat the second term on the right hand in $(4.35).$ First we note that

which implies

where we have used integration by parts in the last equality. By (4.33), we have

Combining the last two displays, we obtain

The desired result follows by (4.35), (4.37), (4.38) and the fact that

□

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

The author wishes to thank Professor L. Véron for useful discussions. The author would like to thank the anonymous referee for a careful reading of the manuscript and helpful comments. The research project was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the "2nd Call for H.F.R.I. Research Projects to support Post-Doctoral Researchers" (Project Number: 59).

Conflict of interest

The author declares no conflict of interest in this paper.