We prove the uniqueness property for a class of entire solutions to the equation
{−divA(x,∇u)=σ,u≥0in Rn,lim inf|x|→∞u=0,
where σ is a nonnegative locally finite measure in Rn, absolutely continuous with respect to the p-capacity, and divA(x,∇u) is the A-Laplace operator, under standard growth and monotonicity assumptions of order p (1<p<∞) on A(x,ξ) (x,ξ∈Rn); the model case A(x,ξ)=ξ|ξ|p−2 corresponds to the p-Laplace operator Δp on Rn. Our main results establish uniqueness of solutions to a similar problem,
{−divA(x,∇u)=σuq+μ,u≥0in Rn,lim inf|x|→∞u=0,
in the sub-natural growth case 0<q<p−1, where μ,σ are nonnegative locally finite measures in Rn, absolutely continuous with respect to the p-capacity, and A(x,ξ) satisfies an additional homogeneity condition, which holds in particular for the p-Laplace operator.
Citation: Nguyen Cong Phuc, Igor E. Verbitsky. Uniqueness of entire solutions to quasilinear equations of p-Laplace type[J]. Mathematics in Engineering, 2023, 5(3): 1-33. doi: 10.3934/mine.2023068
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We prove the uniqueness property for a class of entire solutions to the equation
{−divA(x,∇u)=σ,u≥0in Rn,lim inf|x|→∞u=0,
where σ is a nonnegative locally finite measure in Rn, absolutely continuous with respect to the p-capacity, and divA(x,∇u) is the A-Laplace operator, under standard growth and monotonicity assumptions of order p (1<p<∞) on A(x,ξ) (x,ξ∈Rn); the model case A(x,ξ)=ξ|ξ|p−2 corresponds to the p-Laplace operator Δp on Rn. Our main results establish uniqueness of solutions to a similar problem,
{−divA(x,∇u)=σuq+μ,u≥0in Rn,lim inf|x|→∞u=0,
in the sub-natural growth case 0<q<p−1, where μ,σ are nonnegative locally finite measures in Rn, absolutely continuous with respect to the p-capacity, and A(x,ξ) satisfies an additional homogeneity condition, which holds in particular for the p-Laplace operator.
We prove the uniqueness property for a class of reachable solutions to the equation
{−Δpu=σ,u≥0in Rn,lim inf|x|→∞u=0, | (1.1) |
where σ≥0 is a locally finite Borel measure in Rn absolutely continuous with respect to the p-capacity, and Δpu=div(∇u|∇u|p−2) (1<p<∞) is the p-Laplace operator.
More general A-Laplace operators divA(x,∇u) in place of Δp, under standard growth and monotonicity assumptions of order p on A(x,ξ) (x,ξ∈Rn), are treated as well (see Section 2). All solutions u of (1.1) are understood to be A-superharmonic (or, equivalently, locally renormalized) solutions in Rn (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) W1,pσ. Criteria of existence of solutions to (1.1), which ensure that W1,pσ≢∞, 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 A-Laplace operators similar to (1.1) in arbitrary domains Ω⊆Rn (not necessarily bounded), but with various additional restrictions on A(x,ξ) and data σ. We refer to [2] for σ∈L1(Ω), and [25] for measures σ with finite total variation in Ω. Notice that in general only local analogues of the Kilpeläinen and Malý pointwise estimates are known for solutions in domains Ω⫋Rn. For our purposes, we need global pointwise estimates, which at the moment are available only for Ω=Rn, or in the case p=2 for linear operators div(A(x)∇u) in terms of positive Green's functions in domains Ω⫋Rn.
In Section 4, we prove uniqueness of nontrivial reachable solutions to the problem
{−Δpu=σuq+μ,u≥0in Rn,lim inf|x|→∞u=0, | (1.2) |
in the sub-natural growth case 0<q<p−1, where μ,σ are nonnegative locally finite measures in Rn absolutely continuous with respect to the p-capacity. We observe that such a uniqueness property generally fails in the case q≥p−1.
When we treat the uniqueness problem for solutions of equations of type (1.2), but with more general A-Laplace operators divA(x,∇⋅) in place of Δp, we impose the additional homogeneity condition A(x,λξ)=λp−1A(x,ξ), for all ξ∈Rn and λ>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 A-Laplacian, as well as homogeneity of order q of the term σuq, 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 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 A-Laplace operator div(A(x)∇u), with bounded measurable coefficients A∈L∞(Rn)n×n, in place of Δ. In other words, all entire A-superharmonic solutions to such equations are unique. For similar problems in domains Ω⊆Rn 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∈C(¯Ω) in bounded smooth domains Ω⊂Rn and μ,σ∈C(¯Ω), 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≠2, all p-superharmonic solutions u to (1.1) or (1.2) are reachable, and hence unique, if, for instance, the condition lim inf|x|→∞u=0 in (1.1) or (1.2), respectively, is replaced with lim|x|→∞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 A-superharmonic solutions to (1.2) were obtained in [37]. (See also earlier results in [9] involving minimal solutions in the case μ=0.) In particular, it is known that the measure σ is necessarily absolutely continuous with respect to the p-capacity provided there exists a nontrivial u≥0 such that −Δpu≥σuq ([9], Lemma 3.6).
We remark that the proofs of the main existence results in [9,37] for (1.2) in the case μ=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).
Let Ω⊆Rn, n≥2, be an open set. By M+(Ω) we denote the cone of nonnegative locally finite Borel measures in Ω, and by M+b(Ω) the subcone of finite measures in M+(Ω). For μ∈M+(Ω), we set ‖μ‖M+(Ω)=μ(Ω) even if μ(Ω)=+∞. The space of finite signed Borel measures in Ω is denoted by Mb(Ω). By ‖μ‖Mb(Ω) we denote the total variation of μ∈Mb(Ω).
Let A:Rn×Rn→Rn be a Carathéodory function in the sense that the map x→A(x,ξ) is measurable for all ξ∈Rn, and the map ξ→A(x,ξ) is continuous for a.e. x∈Rn. Throughout the paper, we assume that there are constants 0<α≤β<∞ and 1<p<n such that for a.e. x in Rn,
A(x,ξ)⋅ξ≥α|ξ|p,|A(x,ξ)|≤β|ξ|p−1,∀ξ∈Rn,[A(x,ξ1)−A(x,ξ2)]⋅(ξ1−ξ2)>0,∀ξ1,ξ2∈Rn,ξ1≠ξ2. | (2.1) |
In the uniqueness results of Sec. 4, we assume additionally the homogeneity condition
A(x,λξ)=λp−1A(x,ξ),∀ξ∈Rn,λ>0. | (2.2) |
Such homogeneity conditions are often used in the literature (see [18,21]).
For an open set Ω⊂Rn, it is well known that every weak solution u∈W1,ploc(Ω) to the equation
−divA(x,∇u)=0inΩ | (2.3) |
has a continuous representative. Such continuous solutions are said to be A-harmonic in Ω. If u∈W1,ploc(Ω) and
∫ΩA(x,∇u)⋅∇φdx≥0, |
for all nonnegative φ∈C∞0(Ω), i.e., −divA(x,∇u)≥0 in the distributional sense, then u is called a supersolution to (2.3) in Ω.
A function u:Ω→(−∞,∞] is called A-superharmonic if u is not identically infinite in each connected component of Ω, u is lower semicontinuous, and for all open sets D such that ¯D⊂Ω, and all functions h∈C(¯D), A-harmonic in D, it follows that h≤u on ∂D implies h≤u in D.
A typical example of A(x,ξ) is given by A(x,ξ)=|ξ|p−2ξ, which gives rise to the p-Laplacian Δpu=div(|∇u|p−2∇u). In this case, A-superharmonic functions will be called p-superharmonic functions.
We recall here the fundamental connection between supersolutions of (2.3) and A-superharmonic functions discussed in [18].
Proposition 2.1 ([18]). (ⅰ) If u∈W1,ploc(Ω) is such that
−divA(x,∇u)≥0inΩ, |
then there is an A-superharmonic function v such that u=v a.e. Moreover,
v(x)=esslim infy→xv(y),x∈Ω. | (2.4) |
(ⅱ) If v is A-superharmonic, then (2.4) holds. Moreover, if v∈W1,ploc(Ω), then
−divA(x,∇v)≥0inΩ. |
(ⅲ) If v is A-superharmonic and locally bounded, then v∈W1,ploc(Ω), and
−divA(x,∇v)≥0inΩ. |
Note that if u is A-superharmonic, then the gradient of u may not exist in the sense of distributions in the case 1<p≤2−1/n. On the other hand, if u is an A-superharmonic function, then its truncation uk=min{u,k} is A-superharmonic as well, for any k>0. Moreover, by Proposition 2.1(ⅲ) we have uk∈W1,ploc(Ω). Using this we define the very weak gradient
Du:=limk→∞∇[min{u,k}]a.e.inΩ. |
If either u∈L∞(Ω) or u∈W1,1loc(Ω), 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 A-superharmonic in Ω and 1≤q<nn−1. Then both |Du|p−1 and A(⋅,Du) belong to Lqloc(Ω). Moreover, if p>2−1n, then Du coincides with the distributional gradient of u.
Note that by Proposition 2.2 and the dominated convergence theorem, we have
−divA(x,∇u)(φ):=∫ΩA(x,Du)⋅∇φdx=limk→∞∫ΩA(x,∇min{u,k})⋅∇φdx≥0, |
whenever φ∈C∞0(Ω), φ≥0, and u is A-superharmonic in Ω. It follows from Riesz's representation theorem (see [18,Theorem 21.2]) that there exists a unique measure μ[u]∈M+(Ω) called the Riesz measure of u such that
−divA(x,∇u)=μ[u]inΩ. |
In this section, we investigate the problems of existence and uniqueness of A-superharmonic solutions in the entire space Rn to the equation
{−divA(x,∇u)=σ,u≥0in Rn,lim inf|x|→∞u=0, | (3.1) |
with measures σ∈M+(Rn) (not necessarily finite).
There has been a lot of work addressing the existence and uniqueness problem for quasilinear equations of the form
{−divA(x,∇u)=σin Ω,u=0on ∂Ω, | (3.2) |
in a bounded domain Ω⊂Rn, where σ∈L1(Ω), or, more generally, σ∈Mb(Ω); see, e.g., [2,4,5,6,10,11,12,22,25]. For arbitrary domains, including Rn, we refer to the papers [2] (for L1 data) and [25] (for data in Mb(Ω)). In these papers one can find the notions of { entropy solutions (see [2,6,22]), SOLA (solutions obtained as limits of approximations) for L1 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 σ<<capp, i.e., σ is absolutely continuous with respect to the p-capacity in the sense that σ(K)=0 for any compact set K⊂Ω such that capp(K)=0. The p-capacity capp(⋅) is a natural capacity associated with the p-Laplacian defined by
capp(K):=inf{∫Ω|∇h|pdx:h∈C∞0(Ω),h≥1 on K}, |
for any compact set K⊂Ω.
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 σ∈M+b(Ω) and nonnegative solutions. Recall that we may use the decomposition σ=σ0+σs, where both σ0 and σs are nonnegative measures such that σ0<<cap0, and σs is concentrated on a set of zero p-capacity.
Definition 3.1. Let σ∈M+b(Ω), where Ω⊂Rn is a bounded open set. Then u≥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 Tk(u) belongs to W1,p0(Ω) for every k>0, where Tk(s):=min{k,s}, s≥0.
(b) The gradient Du of u satisfies |Du|p−1∈Lq(Ω) for all q<nn−1.
(c) For any h∈W1,∞(R) with compact support, and any φ∈W1,p(Ω)∩L∞(Ω) such that h(u)φ∈W1,p0(Ω),
∫ΩA(x,Du)⋅∇(h(u)φ)dx=∫Ωh(u)φdσ0, |
and for any φ∈C0b(Ω) (the space of bounded and continuous functions in Ω),
limm→∞1m∫{m≤u≤2m}A(x,Du)⋅Duφdx=∫Ωφdσs. |
Definition 3.2. Let σ∈M+b(Ω), where Ω⊂Rn is a bounded open set. Then u≥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∈W1,∞(R) with such that h′ has compact support, and any φ∈W1,r(Ω)∩L∞(Ω), r>n, such that h(u)φ∈W1,p0(Ω),
∫ΩA(x,Du)⋅∇(h(u)φ)dx=∫Ωh(u)φdσ0+h(+∞)∫Ωφdσs. |
Here h(+∞):=lims→+∞h(s).
Definition 3.3. Let σ∈M+b(Ω), where Ω⊂Rn is a bounded open set. Then u≥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 λk∈M+b(Ω) concentrated on the set {u=k} such that λk<<capp, and λk→σs in the narrow topology of measures in Ω as k→∞, i.e.,
limk→∞∫Ωφdλk=∫Ωφdσs,∀φ∈C0b(Ω). |
(d) For every k>0,
∫{u<k}A(x,Du)⋅∇φdx=∫{u<k}φdσ0+∫Ωφdλk |
for all φ∈W1,p0(Ω)∩L∞(Ω).
We shall also need the notion of a local renormalized (nonnegative) solution on a general open set Ω⊆Rn (not necessarily bounded) associated with a measure σ∈M+(Ω) (not necessarily finite). We recall the following equivalent definitions (see [3]), adapted to the case of nonnegative solutions.
Definition 3.4. Let σ∈M+(Ω), where Ω⊆Rn is an open set. Then a nonnegative function u is said to be a local renormalized solution of the equation −divA(x,∇u)=σ, if the following conditions hold:
(a) The function u is measurable and finite almost everywhere, and Tk(u) belongs to W1,ploc(Ω), for every k>0, where Tk(s):=min{k,s}, s≥0.
(b) The gradient Du of u satisfies |Du|p−1∈Lqloc(Ω) for all 0<q<nn−1, and up−1∈Lsloc(Ω) for all 0<s<nn−p.
(c) For any h∈W1,∞(R) with compact support, and any φ∈W1,p(Ω)∩L∞(Ω) with compact support in Ω such that h(u)φ∈W1,p(Ω),
∫ΩA(x,Du)⋅∇(h(u)φ)dx=∫Ωh(u)φdσ0, |
and for any φ∈C0b(Ω) with compact support in Ω,
limm→∞1m∫{m≤u≤2m}A(x,Du)⋅Duφdx=∫Ωφdσs. |
Definition 3.5. Let σ∈M+(Ω), where Ω⊆Rn is an open set. Then a nonnegative function u is said to be a local renormalized solution of the equation −divA(x,∇u)=σ, 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 λk<<capp, concentrated on the sets {u=k}, such that λk→σs weakly as measures in Ω as k→∞, i.e.,
limk→∞∫Ωφdλk=∫Ωφdσs, |
for all φ∈C0b(Ω) with compact support in Ω.
(d) For every k>0,
∫{u<k}A(x,Du)⋅∇φdx=∫{u<k}φdσ0+∫Ωφdλk |
for all φ in W1,p0(Ω)∩L∞(Ω) with compact support in Ω.
We now discuss solutions of (3.1) for general measures σ∈M+(Rn). It is known that a necessary and sufficient condition for (3.1) to admit an A-superharmonic solution is the finiteness condition
∫∞1(σ(B(0,ρ))ρn−p)1p−1dρρ<+∞; | (3.3) |
(see, e.g., [33,34]). Thus, it is possible to solve (3.1) for a wide and optimal class of measures σ satisfying (3.3) that are not necessarily finite.
We mention that (3.3) is equivalent to the condition W1,pσ(x)<+∞ for some x∈Rn (or equivalently quasi-everywhere in Rn with respect to the p-capacity), where
W1,pσ(x):=∫∞0(σ(B(x,ρ))ρn−p)1p−1dρρ |
is the Havin–Maz'ya–Wolff potential of σ (often called the Wolff potential); see [17,26].
By the fundamental result of Kilpeläinen and Malý [20,21], any A-superharmonic solution u to Eq (3.1) satisfies the following global pointwise estimates,
1KW1,pσ(x)≤u(x)≤KW1,pσ(x),∀x∈Rn, | (3.4) |
where K>0 is a constant depending only on n,p and the structural constants α and β 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 σ, and uniqueness is guaranteed as long as σ<<capp (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 σ is continuous with respect to the p-capacity.
Theorem 3.6. Let σ∈M+(Rn), where σ<<capp. Suppose that (3.3) holds. Then there exists a minimal A-superharmonic solution to Eq (3.1).
Proof. Condition (3.3) implies that
∫∞1(σ(B(x,ρ))ρn−p)1p−1dρρ<+∞ |
for all x∈Rn. Thus,
{W1,pσ=∞}={x∈Rn:W11,pσ:=∫10(σ(B(x,ρ))ρn−p)1p−1dρρ=∞}. |
This yields
capp({W1,pσ=∞})=limj→∞capp({W1,pσ=∞}∩Bj(0))=limj→∞capp({x∈Bj(0):W11,p(σ|Bj+1(0))=∞})=0. |
Here we used the fact that, if μ∈M+b(Rn), then capp({W1,pμ=∞})=0 (see [1,Proposition 6.3.12]). It follows that σ({W1,pσ=∞})=0, since σ<<capp.
Let σk (k=1,2,…) be the restriction of σ to the set Bk(0)∩{W1,pσ<k}. We then have that σk weakly converges to σ, and
∫RnW1,pσkdσk≤kσ(Bk(0))<+∞. |
Hence, σk∈W−1,p′(Bk(0)) (1/p+1/p′=1), and for each k>0, there exists a unique nonnegative solution uk∈W1,p0(Bk(0)) to the problem
{−divA(x,∇uk)=σkinBk(0),uk=0on∂Bk(0). | (3.5) |
If we set uk=0 in Rn∖Bk(0), then the sequence {uk} is non-decreasing, and by [33,Theorem 2.1{ }],
uk≤KW1,pσ<∞dσ−a.e. |
By [20,Theorem 1.17], it follows that the function u:=limk→∞uk is A-superharmonic in Rn. Moreover, u≤KW1,pσ, and consequently
lim inf|x|→∞u(x)≤Klim inf|x|→∞W1,pσ(x)=0. |
Thus, u is an A-superharmonic solution of (3.1).
To show the minimality of u, let v be another A-superharmonic solution of (3.1). From the construction of u, it is enough to show that uk≤v for any k≥1. To this end, let νj, j=1,2,…, be the Riesz measure of min{v,j}. Since v is A-superharmonic, it is also a local renormalized solution to −divA(x,∇v)=σ in Rn (see [19]). Hence, by a result of [3,12] and the fact that σ<<capp, we obtain
νj=σ|{v<j}+αj |
for αj∈M+(Rn) concentrated in the set {v=j}.
Using the estimate v≤KW1,pσ, we deduce
νj≥σ|{v<j}≥σ|{KW1,pσ<j}≥σ|{W1,pσ<k}≥σk, |
provided j/K>k. Since uk∈W1,p0(Bk(0)) and min{v,j}∈W1,p(Bk(0)), by the comparison principle (see [9,Lemma 5.1]), we estimate
uk≤min{v,j}≤v, |
provided j≥Kk. Thus, u=limk→∞uk≤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 σ,˜σ∈M+(Rn), where σ≤˜σ and σ<<capp, 1<p<n. Then u≤˜u, where u is the minimal A-superharmonic solution of (3.1) and ˜u is any A-superharmonic solution of (3.1) with datum ˜σ in place of σ.
Proof. Let σ′k, k=1,2,…, be the restriction of σ to the set Bk(0)∩{W1,p˜σ<k}. Since σ<<capp we have that σ′k weakly converges to σ. Moreover, as W1,pσ′k≤W1,pσ≤W1,p˜σ<k on the set {W1,p˜σ<k}, it follows that
∫RnW1,pσ′kdσ′k≤kσ(Bk(0))<+∞. |
Hence, σ′k∈W−1,pp−1(Bk(0)), and for each k>0 there exists a unique nonnegative solution u′k∈W1,p0(Bk(0)) to the problem
{−divA(x,∇u′k)=σ′kinBk(0),u′k=0on∂Bk(0). |
Letting u′k=0 in Rn∖Bk(0), we have that the sequence {u′k} is non-decreasing, and by [33,Theorem 2.1],
u′k≤KW1,pσ. |
Then u′k converges pointwise to an 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
u′k≤uk,∀k≥1, |
where uk is defined in (3.5). Hence, letting k→∞, we get u′≤u, which yields u′=u by the minimality of u.
We now let ˜σj (j=1,2,…) be the Riesz measure of min{˜u,j}. Recall that the Riesz measure ˜σ of ˜u can be decomposed as
˜σ=˜σ0+˜σs, |
where ˜σ0∈M+(Rn), ˜σ0<<capp, and ˜σs∈M+(Rn) is concentrated on a set of zero p-capacity. Then by a result of [3,12], we have
˜σj=˜σ0|{˜u<j}+˜αj, | (3.6) |
where ˜αj∈M+(Rn) is concentrated in the set {˜u=j}. On the other hand, since ˜σs({˜u<∞})=0 (see [19,Lemma 2.9]), we can rewrite (3.6) as
˜σj=˜σ|{˜u<j}+˜αj. | (3.7) |
Now using the estimate ˜u≤KW1,p˜σ and (3.7), we have
˜σj≥˜σ|{˜u<j}≥σ|{KW1,p˜σ<j}≥σ|{W1,p˜σ<k}≥σ′k, |
provided j/K>k.
Since u′k∈W1,p0(Bk(0)) and min{˜u,j}∈W1,p(Bk(0)), by the comparison principle of [9,Lemma 5.1] we find
u′k≤min{˜u,j}≤˜u, |
provided we choose a j such that j≥Kk. Letting k→∞, we obtain u≤˜u as desired.
Theorem 3.6 justifies the existence (and hence uniqueness) of the minimal A-superharmonic solution to (3.1) provided condition (3.3) holds and σ<<capp. 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 σ<<capp all 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 σ such that σ<<capp. Our definition is an adaptation of the notion of the reachable solution of [10,Definition 2.3].
Definition 3.8. Let σ∈M+(Rn). We say that a function u:Rn→[0,+∞] is an A-superharmonic reachable solution to Eq (3.1) if u is an A-superharmonic solution of (3.1), and there exist two sequences {ui} and {σi}, i=1,2,…, such that
(ⅰ) Each σi∈M+(Rn) is compactly supported in Rn, and σi≤σ;
(ⅱ) Each ui is an A-superharmonic solution of (3.1) with datum σi in place of σ;
(ⅲ) ui→u a.e. in Rn.
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 L1 data over bounded domains. By (ⅲ) and the weak continuity result of [36], we see that σi→σ weakly as measures in Rn. The extra requirement σi≤σ in our definition plays an important role in the proof of uniqueness in the case when the datum σ is absolutely continuous with respect to capp.
Theorem 3.10. Suppose σ∈M+(Rn), and suppose (3.3) holds. Then there exists an A-superharmonic reachable solution to (3.1). Moreover, if additionally σ<<capp, then any A-superharmonic reachable solution is unique and coincides with the minimal solution.
Proof. Existence: Suppose that (3.3) holds. Then W1,pσ<+∞ quasi-everywhere and hence almost everywhere. For each i=1,2,…, let uji be an A-superharmonic renormalized solution (see [12]) to
{−divA(x,∇uji)=σ|Bi(0)inBj(0),uji=0on∂Bj(0). |
Note that σ|Bi(0)≤σ and σ|Bi(0)→σ weakly as measures in Rn. Also, by [33], we have
uji≤KW1,p(σ|Bi(0)). |
Hence, by [20,Theorem 1.17], there exist an A-superharmonic function ui in Rn with
ui≤KW1,p(σ|Bi(0))≤KW1,pσ<+∞a.e., | (3.8) |
and a subsequence {ujki}k such that ujki→u and Dujki→Dui a.e. as k→∞. These estimates yield that the Riesz measure of ui is σ|Bi(0) and
lim inf|x|→∞ui=0. |
Using again [20,Theorem 1.17] and (3.8), we find a subsequence of {ui} that converges a.e. to an A-superharmonic reachable solution u of (3.1).
Uniqueness: We now assume further that σ<<capp. Let u be an A-superharmonic reachable solution in the sense of Definition 3.8 with approximating sequences {ui} and {σi}. Let us fix an i∈{1,2,…}. Then there exists a positive integer N=N(ⅰ) such that supp(σi)⊂BN(0). Let v be the minimal A-superharmonic solution to (3.1). Also, let vN be the minimal A-superharmonic solution to (3.1) with datum σ|BN(0) in place of σ. We have, by Theorem 3.7,
u≥v≥vN. |
Thus, as uj→u a.e., it is enough to show that
vN≥ui. | (3.9) |
Note that since σi≤σ and supp(σi)⊂BN(0) we have that
σi≤σ|BN(0). | (3.10) |
For R>0, let 0≤Θ=ΘR≤1 be a cutoff function such that
Θ∈C∞0(BR(0)),Θ≡1onBR/2(0),and|∇Θ|≤C/R. |
For any k>0, we set
T+k(t)={tif 0≤t≤k,kif t>k,0if t<0. |
Also, for any m>0, we define the following Lipschitz function with compact support on R:
hm(t)={1if 0≤|t|≤m,0if |t|≥2m,−tm+2if m<t<2m,tm+2if −2m<t<−m. |
As ui and vN are both local renormalized solutions (see [3,19]), we may use
hm(ui)hm(vn)T+k(ui−vN)Θ,m,k>0, |
as test functions and thus obtaining
∫RnA(x,Dui)⋅∇[hm(ui)hm(vN)T+k(ui−vN)Θ]dx=∫Rnhm(ui)hm(vn)T+k(ui−vN)Θdσi, |
and
∫RnA(x,DvN)⋅∇[hm(ui)hm(vN)T+k(ui−vN)Θ]dx=∫Rnhm(ui)hm(vn)T+k(ui−vN)Θdσ|BN(0). |
Let
Ⅰ=∫RnA(x,Dui)⋅∇[hm(ui)hm(vN)T+k(ui−vN)Θ]dx, |
and
Ⅱ=∫RnA(x,DvN)⋅∇[hm(ui)hm(vN)T+k(ui−vN)Θ]dx. |
Then by (3.10) we have
Ⅰ−Ⅱ≤0. | (3.11) |
On the other hand, we can write
Ⅰ−Ⅱ=∫Rn[A(x,Dui)−A(x,DvN)]⋅∇T+k(ui−vN)hm(ui)hm(vN)Θdx+∫Rn[A(x,Dui)−A(x,DvN)]⋅Duih′m(ui)T+k(ui−vN)hm(vN)Θdx+∫Rn[A(x,Dui)−A(x,DvN)]⋅DvNh′m(vN)T+k(ui−vN)hm(ui)Θdx+∫Rn[A(x,Dui)−A(x,DvN)]⋅∇ΘT+k(ui−vN)hm(ui)hm(vN)dx. |
Thus, in view of (3.11), it follows that
∫{0<ui−vN<k}[A(x,Dui)−A(x,DvN)]⋅(Dui−DvN)hm(ui)hm(vN)Θdx≤−∫Rn[A(x,Dui)−A(x,DvN)]⋅Duih′m(ui)T+k(ui−vN)hm(vN)Θdx−∫Rn[A(x,Dui)−A(x,DvN)]⋅DvNh′m(vN)T+k(ui−vN)hm(ui)Θdx−∫Rn[A(x,Dui)−A(x,DvN)]⋅∇ΘT+k(ui−vN)hm(ui)hm(vN)dx=:Am+Bm+Cm. |
To estimate |Am|, we observe that |hm(t)|≤1 and |h′m(t)|≤1/m. Hence,
|Am|≤βkm∫{m<ui<2m,0<vN<2m}[|Dui|p−1+|DvN|p−1]|Dui|Θdx≤Ckm∫{0<ui<2m,0<vN<2m}[|Dui|p+|DvN|p]Θdx. |
On the other hand, using T+2m(ui)Θ as a test function for the equation of ui and invoking condition (2.1), we estimate
α∫0<ui<2m|Dui|pΘdx≤∫RnT+2m(ui)Θdσi+β∫Rn|Dui|p−1T+2m(ui)|∇Θ|dx. |
Since T+2m(ui)/m≤2, and T+2m(ui)/m converges to zero quasi-everywhere, we deduce
limm→∞1m∫0<ui<2m|Dui|pΘdx=0. |
Similarly,
limm→∞1m∫0<vN<2m|DvN|pΘdx=0. |
Hence,
limm→∞|Am|=0. | (3.12) |
A similar argument gives
limm→∞|Bm|=0. |
To estimate |Cm|, we first use the pointwise bound (3.4) to obtain
|Cm|≤cR∫AR[|Dui|p−1+|DvN|p−1]min{W1,p(σ|BN(0)),k}dx, |
where AR is the annulus
AR={R/2<|x|<R}. |
Note that for R>4N we have
W1,p(σ|BN(0))(x)=∫∞R/4[σ(Bt(x)∩BN(0))tn−p]1p−1dtt≈Rp−np−1 |
for all x∈AR. Thus,
|Cm|≤cRp−np−1R−1∫AR[|Dui|p−1+|DvN|p−1]dx≤cRp−np−1R−1Rn−p+1[(infARui)p−1+(infARvN)p−1]≤cRp−np−1R−1Rn−p+1Rp−n=cRp−np−1, |
where we used the Caccioppoli inequality and the weak Harnack inequality in the second bound. This gives
limR→∞lim supm→∞|Cm|=0. | (3.13) |
Since hm(ui)hm(vN)→1 a.e. as m→∞, and Θ(x)→1 everywhere as R→∞, it follows from (3.12)–(3.13) and Fatou's lemma that
∫{0<ui−vN<k}[A(x,Dui)−A(x,DvN)]⋅(Dui−DvN)dx≤0. |
Letting k→∞, we deduce
∫{ui−vN>0}[A(x,Dui)−A(x,DvN)]⋅(Dui−DvN)dx≤0. |
Since the integrand is strictly positive whenever Dui≠DvN, we infer that Dui=DvN a.e. on the set {ui−vN>0}.
We next claim that the function T+k(ui−vN) belongs to W1,ploc(Rn) for any k>0. To see this, for any m>k, we compute
∇T+k(T+m(ui)−T+m(vN))=[∇T+m(ui)−∇T+m(vN)]χ{0<T+m(ui)−T+m(vN)<k}=[Duiχ{0<ui<m}−DvNχ{0<vN<m}]χ{0<T+m(ui)−T+m(vN)<k}=[Duiχ{0<ui<m}−DvNχ{0<vN<m}]χ{0<m−vN<k,ui≥m,vN<m}=−DvNχ{0<vN<m}χ{0<m−vN<k,u≥m,v<m}, | (3.14) |
where χA is the characteristic function of a set A. Thus,
∫Rn|∇T+k(T+m(ui)−T+m(vN))|pΘdx≤∫{m−k<vN<m}|DvN|pΘdx. |
On the other hand, using Hm,k(vN)Θ as a test function for the equation of vN, where
Hm,k(t)={1if 0≤|t|≤m−k,0if |t|≥m,−tk+mkif m−k<t<m,tk+mkif −m<t<−(m−k), |
we have
αk∫{m−k<vN<m}|DvN|pΘdx≤∫RnΘdσ|BN(0)+β∫Rn|DvN|p−1|∇Θ|dx. |
Thus, for each fixed k>0, the sequence {T+k(T+m(ui)−T+m(vN))}m is uniformly bounded in W1,ploc(Rn). Since T+k(T+m(ui)−T+m(vN))→T+k(ui−vN) a.e. as m→∞, we see that T+k(ui−vN)∈W1,ploc(Rn).
We are now ready to complete the proof of the theorem. Since T+k(ui−vN)=T+k(T+m(ui)−T+m(vN)) a.e. on the set {ui<m,vN<m} and the two functions belong to W1,ploc(Rn), by (3.14) we have
∇T+k(ui−vN)=∇T+k(T+m(ui)−T+m(vN))=0 |
a.e. on the set {ui<m,vN<m} for any m>0. Thus, ∇T+k(ui−vN)=0 a.e. in Rn, which implies the existence of a constant κ≥0 such that
max{ui−vN,0}=κ |
a.e. in the entire space Rn. Note that if κ≠0, then ui=vN+κ in Rn, which violates the condition at infinity, lim inf|x|→∞ui(x)=0. It follows that κ=0, which yields (3.9), as desired.
The following version of the comparison principle in Rn is an immediate consequence of Theorems 3.7 and 3.10.
Corollary 3.11. Let σ,˜σ∈M+(Rn), where σ≤˜σ and σ<<capp, 1<p<n. Let u be an A-superharmonic reachable solution of (3.1), and ˜u any A-superharmonic solution of (3.1) with datum ˜σ in place of σ. Then u≤˜u in Rn.
For σ∈M+(Rn) such that that σ<<capp, sometimes it is desirable to know when an A-superharmonic solution to (3.1) is also the A-superharmonic reachable solution to (3.1), and hence also the minimal 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 σ.
Theorem 3.12. Let σ∈M+(Rn), where σ<<capp. Suppose that any one of the following conditions holds:
(ⅰ) |Du|∈Lγ,∞(Rn) for some (p−1)n/(n−1)≤γ<p, where Lγ,∞(Rn) is the weak Lγ space in Rn;
(ⅱ) |Du|∈Lp(Rn);
(ⅲ) σ∈M+b(Rn).
Then any A-superharmonic solution u to the Eq (3.1) coincides with the minimal A-superharmonic solution.
Proof. Let v be the minimal A-superharmonic solution of (3.1). Our goal is to show that u≤v a.e. Let Θ(x)=ΘR(x),R>0, T+k(t),k>0, and hm(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 ui and v in place of vN, we have
∫{0<ui−vN<k}[A(x,Du)−A(x,Dv)]⋅(Du−Dv)hm(u)hm(v)Θdx≤Am+Bm+Cm, |
where now
Am=−∫Rn[A(x,Du)−A(x,Dv)]⋅Duh′m(u)T+k(u−v)hm(v)Θdx, |
Bm=−∫Rn[A(x,Du)−A(x,Dv)]⋅Dvh′m(v)T+k(u−v)hm(u)Θdx, |
and
Cm=−∫Rn[A(x,Du)−A(x,Dv)]⋅∇ΘT+k(u−v)hm(u)hm(v)dx. |
As in the proof of Theorem 3.10, we have
limm→∞(|Am|+|Bm|)=0. | (3.15) |
As for Cm, we have
|Cm|≤cR∫AR[|Du|p−1+|Dv|p−1]min{W1,pσ,k}dx, |
where, as above, AR is the annulus
AR={R/2<|x|<R}. |
Suppose now that condition (ⅰ) holds. Then |Du|p−1∈Lqq−1,∞(Rn) for some q∈(p,n]. Set
m=n(p−1)qn(q−1)−(p−1)q>0, |
and note that W1,pσ∈Lm,∞(Rn). A proof of this fact in the 'sublinear' case (p−1)q/(q−1)≤1 can be found in [32].
We have that either m≤q or m>q. In the case m≤q, for any ϵ>0 we find
1R∫AR|Du|p−1min{W1,pσ,k}dx≤1R‖|Du|p−1‖Lqq−1,∞(AR)‖min{W1,pσ,k}‖Lq,1(AR)≤1R‖|Du|p−1‖Lqq−1,∞(Rn)‖min{W1,pσ,k}‖Lq+ϵ,∞(AR)|AR|ϵq(q+ϵ)≤Ckq+ϵ−mq+ϵ‖|Du|p−1‖Lqq−1,∞(Rn)‖W1,pσ‖mq+ϵLm,∞(Rn)Rnϵq(q+ϵ)−1. |
Here we shall choose ϵ>0 such that
nϵq(q+ϵ)−1<0. |
In the case m>q, we have
1R∫AR|Du|p−1min{W1,pσ,k}dx≤1R‖|Du|p−1‖Lqq−1,∞(AR)‖min{W1,pσ,k}‖Lq,1(AR)≤1R‖|Du|p−1‖Lqq−1,∞(Rn)‖W1,pσ‖Lm,∞(Rn)|AR|m−qqm≤C‖|Du|p−1‖Lqq−1,∞(Rn)‖W1,pσ‖Lm,∞(Rn)R(m−q)nqm−1. | (3.16) |
Note that, since q<p,
(m−q)nmq−1=n(p−1)−n(q−1)+(p−1)q(p−1)q−1<0. |
Hence, in both cases we have, for any fixed k>0,
limR→∞1R∫AR|Du|p−1min{W1,pσ,k}dx=0, | (3.17) |
and likewise,
limR→∞1R∫AR|Dv|p−1min{W1,pσ,k}dx=0. | (3.18) |
On the other hand, suppose now that condition (ⅱ) holds, i.e., |Du|∈Lp(Rn). Then W1,pσ∈Lnpn−p(Rn), and as in (3.16) we have
1R∫AR|Du|p−1min{W1,pσ,k}dx≤C‖|Du|p−1‖Lpp−1(AR)‖W1,pσ‖Lnpn−p(AR), |
and likewise for v. Thus (3.17) and (3.18) also hold under condition (ⅱ).
Finally, suppose that (ⅲ) holds. For any 1<r<nn−1 and ϵ∈(0,1) such that ϵrr−1<n(p−1)n−p, we have
1R∫AR|Du|p−1min{W1,pσ,k}dx≤k1−ϵ1R(∫BR(0)|Du|(p−1)rdx)1r(∫BR(0)(W1,pσ)ϵrr−1dx)r−1r≤Ck1−ϵRnr−1(infBR(0)uR)p−1Rnr−1r(infBR(0)u)ϵ, |
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
1R∫AR|Du|p−1min{W1,pσ,k}dx≤Ck1−ϵRn−p(∫∞R(σ(Bt(0))tn−p)1p−1dtt)p−1(infBR(0)u)ϵ≤Ck1−ϵσ(Rn)(infBR(0)u)ϵ. | (3.19) |
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
limR→∞lim supm→∞|Cm|=0. | (3.20) |
Using (3.15) and (3.20), we deduce
∫{0<u−v<k}[A(x,Du)−A(x,Dv)]⋅(Du−Dv)dx≤0 |
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≤v a.e. as desired.
We now provide a criterion for reachability by requiring only the finiteness of the approximating measures σi.
Corollary 3.13. Let u be an A-superharmonic solution of (3.1), where σ∈M+(Rn), and σ<<capp. Suppose that there exist two sequences {ui} and {σi}, i=1,2,…, such that the following conditions hold:
(ⅰ) each σi∈M+b(Rn), and σi≤σ;
(ⅱ) each ui is an A-superharmonic solution of (3.1) with datum σi in place of σ;
(ⅲ) ui→u a.e. in Rn.
Then u is an A-superharmonic reachable solution of (3.1), and thus coincides with the minimal solution.
Proof. By Theorem 3.12, each ui 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 |Du|∈Lγ,∞(Rn) for 0<γ<(p−1)n/(n−1) as in this case σ=0. The proof of this fact, especially in the case 0<γ≤p−1, requires some results obtained recently in [31].
Theorem 3.14. If u is an A-superharmonic function in Rn such that |Du|∈Lγ,∞(Rn) for some 0<γ<(p−1)n/(n−1), then σ=0 where σ is the Riesz measure of u.
Proof. Let Qr(x), r>0, denote the open cube Qr(x):=x+(−r,r)n with center x∈Rn and side-length 2r. Using Φ∈C∞0(Qr(x)), Φ≥0, Φ=1 on Qr/2(x), and |∇Φ|≤C/r, as a test function we have
σ(Qr/2(x))≤Cr∫Qr(x)|Du|p−1dy. | (3.21) |
Thus if γ∈(p−1,(p−1)n/(n−1)), for any R>0 we use Hölder's inequality to get
σ(QR/2(0))≤CR‖Du‖p−1Lγ,∞(Rn)Rnγ−p+1γ. |
Note that nγ−p+1γ<1 and thus letting R→∞ we get σ=0.
We now consider the case 0<γ≤p−1. Let γ1 be a fixed number in (p−1),(p−1)n/(n−1)). By [31,Lemma 2.3], for any cube Qρ(x)⊂Rn, 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≤n after taking into account the comparison estimates of [14,27,28].
Hence, it follows that
![]() |
where we used (3.21) with r=4ρ in the last inequality. This allows us to employ a covering/iteration argument as in [16,Remark 6.12] to obtain that
![]() |
(3.22) |
for any ϵ>0.
Thus, if 0<γ<p−1, in view of (3.21), (3.22), and Hölder's inequality, we get
σ(QR/2(0))≤CR‖Du‖p−1γLγ,∞(Rn)Rn−np−1γ→0, |
as R→∞. Hence, σ=0. The case γ=p−1 is treated similarly, starting with the inequality
σ(QR/2(0))≤CR(∫QR(0)|Du|(p−1)(1+ϵ)dx)11+ϵRnϵ1+ϵ |
for a sufficiently small ϵ>0.
Due to the results of [13,14,24,30] (see also [15,29]), under some additional regularity conditions on the nonlinearity A(x,ξ), one has
|Du(x)|≤C[I1σ(x)]1p−1a.e.x∈Rn, |
provided u is an A-superharmonic solution to the Eq (3.1). This gradient estimate holds in particular for A(x,ξ)=|ξ|p−2ξ, i.e., the p-Laplacian Δp, which yields the following corollary.
Corollary 3.15. Let σ∈M+(Rn). Suppose that one of the following conditions holds:
(ⅰ) σ<<capp and I1σ∈Ls,∞(Rn) for some n/(n−1)<s<p/(p−1). This holds in particular if σ=f∈Lt,∞(Rn) for some 1<t<np/(np−n+p);
(ⅱ) I1σ∈Lp/(p−1)(Rn), i.e., σ is of finite energy.
Then any p-superharmonic solution u to the equation
{−Δpu=σ,u≥0inRn,lim inf|x|→∞u=0, |
coincides with the minimal p-superharmonic solution.
Finally, we show that if the condition at infinity, lim inf|x|→∞u=0 in (3.1), is replaced with the stronger one lim|x|→∞u=0, then all A-superharmonic solutions are indeed reachable.
Theorem 3.16. Suppose that u is an A-superharmonic solution of the equation
{−divA(x,∇u)=σ,u≥0inRn,lim|x|→∞u=0, | (3.23) |
where σ∈M+(Rn), and σ<<capp. Then u is the unique A-superharmonic solution of (3.23), which coincides with the minimal A-superharmonic reachable solution of (3.1).
Proof. First notice that the condition lim|x|→∞u=0 in (3.23) yields, in view of (3.4),
lim|x|→∞W1,pσ(x)=0. |
For any ϵ>0, let
Ωϵ:={x∈Rn:u(x)>ϵ}, |
and
uϵ:=max{u,ϵ}−ϵ. |
Clearly, Ωϵ is a bounded open set, uϵ=u−ϵ on Ωϵ, and uϵ=0 in Rn∖Ωϵ.
Let v be the minimal solution of (3.1), which is also the minimal solution of (3.23), since v≤KW1,pσ, and hence lim|x|→∞v=0. It is enough to show that
uϵ≤v | (3.24) |
in Ωϵ, as this will yield that u≤v in Rn after letting ϵ→0+.
Now by Lemma 4.1 below, to verify (3.24), it suffices to show that uϵ is a renormalized solution of
{−divA(x,∇uϵ)=σin Ωϵ,uϵ=0on ∂Ωϵ. |
Note that, for any k>0, Tk(uϵ)=Tk+ϵ(u)−ϵ. We have Tk(uϵ)∈W1,ploc(Rn), Tk(uϵ) is quasi-continuous in Rn, and Tk(uϵ)=0 everywhere in Rn∖Ωϵ. Thus Tk(uϵ)∈W1,p0(Ωϵ) (see [18,Theorem 4.5]).
As u is a local renormalized solution in Rn, for every k>0 there exists a nonnegative measure λk+ϵ<<capp, concentrated on the sets {u=k+ϵ}, such that λk+ϵ→0 weakly as measures in Rn as k→∞. Since Ωϵ is bounded, this implies that λk+ϵ→0 in the narrow topology of measures in Ωϵ.
Moreover, for k>0,
∫{u<k+ϵ}A(x,Du)⋅∇φdx=∫{u<k+ϵ}φdσ0+∫Rnφdλk+ϵ, |
for every φ∈W1,p0(Rn)∩L∞(Rn) with compact support in Rn. In particular, we have
∫{uϵ<k}∩ΩϵA(x,Du)⋅∇φdx=∫{uϵ<k}∩Ωϵφdσ0+∫{uϵ=k}∩Ωϵφdλk+ϵ |
for every φ∈W1,p0(Ωϵ)∩L∞(Ωϵ).
Thus, we conclude that uϵ is a renormalized solution in Ωϵ, as desired.
In this section, we study solutions to the equation
{−divA(x,∇u)=σuq+μ,u≥0in Rn,lim inf|x|→∞u=0, | (4.1) |
in the sub-natural growth case 0<q<p−1, with μ,σ∈M+(Rn).
We consider nontrivial A-superharmonic solutions to (4.1) such that 0<u<∞ dσ-a.e., which implies u∈Lqloc(Rn,σ), so that σuq+μ∈M+(Rn) (see [37]).
As was noted in the Introduction, σ<<capp whenever there exists a nontrivial solution u to (4.1), for any μ (in particular, μ=0).
The existence and uniqueness of nontrivial reachable A-superharmonic solutions to (4.1), under the additional assumption μ<<capp, are proved below. Without this restriction on μ, 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 μ=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 Ω be a bounded open set in Rn. Suppose that μ,ν∈M+b(Ω), where μ≤ν and μ<<capp. If u≥0 is a renormalized solution of
{−divA(x,∇u)=μinΩ,u=0on∂Ω, | (4.2) |
and if v≥0 is an A-superharmonic function in Ω with Riesz measure ν such that min{v,k}∈W1,p(Ω) for any k>0, then u≤v a.e.
Proof. Let νj, j>0, be the Riesz measure of min{v,j}. Since min{v,j}∈W1,p(Ω) we see that νj belongs to the dual of W1,p0(Ω) (see [18,Theorem 21.6]). As in (3.7), we have
νj=ν|{v<j}+αj |
for a measure αj∈M+b(Ω) concentrated in the set {v=j}. Thus the measure μj:=μ|{v<j}≤ν|{v<j}≤νj for any j>0. This implies that μj also belongs to the dual of W1,p0(Ω), and hence there exists a unique solution uj to the equation
−divA(x,∇uj)=μj,uj∈W1,p0(Ω). | (4.3) |
Then by the comparison principle (see [9,Lemma 5.1]) we find
0≤u1≤u2≤⋯≤uj≤min{v,j} |
for any integer j>0. Thus there is a function ˜u on Ω such that 0≤˜u≤v a.e. and uj→˜u as j→∞. We now claim that ˜u is also a renormalized solution to Eq (4.2). If this is verified then, as μ<<capp, we must have that ˜u=u a.e. (see [12,25]) and thus u≤v a.e. as desired.
To show that ˜u is the renormalized solution of (4.2), we first use T+k(uj), k>0, as a test function for (4.3) to obtain
α∫Ω|∇T+k(uj)|pdx≤kμj(Ω)≤kμ(Ω). | (4.4) |
Since T+k(uj)→T+k(˜u) a.e. as j→∞, we see that T+k(˜u)∈W1,p0(Ω) and
α∫Ω|∇T+k(˜u)|pdx≤kμ(Ω) |
for any k>0. By [2,Lemmas 4.1 and 4.2], this yields
˜u∈Ln(p−1)n−p,∞(Ω)andD˜u∈Ln(p−1)n−1,∞(Ω). |
Moreover, arguing as in Step 4 of the proof of Theorem 3.4 in [12], we see that {∇uj} is a Cauchy sequence in measure which converges to D˜u a.e. in Ω. There is no need to take a subsequence here as the limit is independent of any subsequence.
Moreover, for any Lipschitz function h:R→R such that h′ has compact support and for any function φ∈W1,r(Ω)∩L∞(Ω), r>n, such that h(˜u)φ∈W1,p0(Ω), we have
∫ΩA(x,∇uj)⋅D˜uh′(˜u)φdx+∫ΩA(x,∇uj)⋅∇φh(˜u)dx=∫Ωh(˜u)φdμj. |
Thus if the support of h′ is in [−M,M], M>0, then, using 0≤uj≤˜u, we can rewrite the above equality as
∫ΩA(x,∇T+M(uj))⋅∇T+M(˜u)h′(˜u)φdx+∫ΩA(x,∇uj)⋅∇φh(˜u)dx=∫Ωh(˜u)φdμj=∫{0≤v<j}h(˜u)φdμ. |
Note that by (4.4) and [2,Lemma 4.2], we have that ∇uj is uniformly bounded in Ln(p−1)n−1,∞(Ω) and ∇T+M(uj) is uniformly bounded in Lp(Ω). Thus by the Vitali Convergence Theorem, the left-hand side of the above equality converges to
∫ΩA(x,∇T+M(˜u))⋅∇T+M(˜u)h′(˜u)φdx+∫ΩA(x,D˜u)⋅∇φh(˜u)dx=∫ΩA(x,D˜u)⋅D˜uh′(˜u)φdx+∫ΩA(x,D˜u)⋅∇φh(˜u)dx. |
On the other hand, by the Lebesgue Dominated Convergence Theorem we have
limj→∞∫{0≤v<j}h(˜u)φdμ=∫Ωh(˜u)φdμ. |
Thus, we get
∫ΩA(x,D˜u)⋅D˜uh′(˜u)φdx+∫ΩA(x,D˜u)⋅∇φh(˜u)dx=∫Ωh(˜u)φdμ, |
which yields that ˜u is the renormalized solution of (4.2) (see Definition 3.2).
We recall that by ϰ=ϰ(Rn) we denote the least constant in the weighted norm inequality (see [9,37])
(∫Rn|φ|qdσ)1q≤ϰ‖−divA(x,∇φ)‖1p−1M+(Rn), | (4.5) |
for all A-superharmonic functions φ≥0 in Rn such that lim inf|x|→∞φ(x)=0. Notice that by estimates (3.4), K−1φ≤W1,pμ≤Kφ, where μ=−divA(x,∇φ)∈M+(Rn). Here we may assume without loss of generality that μ∈M+b(Rn), so that W1,pμ≢∞. Consequently, (4.5) is equivalent to the inequality
(∫Rn(W1,pμ)qdσ)1q≤κ‖μ‖1p−1M+(Rn)forallμ∈M+b(Rn), |
where K−1ϰ≤κ≤Kϰ. In particular, one can replace divA(x,∇φ) in (4.5) by Δp, up to a constant which depends only on K.
By ϰ(B), where B is a ball in Rn, we denote the least constant in a similar localized weighted norm inequality with the measure σB in place of σ, where σB=χBσ is the restriction of σ to B.
The so-called intrinsic nonlinear potential Kp,qσ, introduced in [9], is defined by
Kp,qσ(x)=∫∞0(ϰ(B(x,t))q(p−1)p−1−qtn−p)1p−1dtt,x∈Rn. |
Here B=B(x,t) is a ball in Rn of radius t>0 centered at x. As was noticed in [9], Kp,qσ≢+∞ if and only if
∫∞1(ϰ(B(0,t))q(p−1)p−1−qtn−p)1p−1dtt<∞. | (4.6) |
By [9,Theorem 1.1], there exists a nontrivial A-superharmonic solution to the homogeneous equation (4.1) in the case μ=0 if and only if W1,pσ≢+∞ and Kp,qσ≢+∞, 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 σ∈M+(Rn). Then the nontrivial minimal A-superharmonic solution u of
{−divA(x,∇u)=σuq,u≥0inRn,lim inf|x|→∞u=0, | (4.7) |
constructed in the proof of [9,Theorem 1.1] under the conditions (3.3) and (4.6), is an A-superharmonic reachable solution.
Proof. We start with the same construction as in the proof of [9,Theorem 1.1] for the minimal A-superharmonic solution u, but with datum σ|Bm(0) in place of σ (m=1,2,…).
For a fixed m, let vm be the minimal A-superharmonic solution to the equation
{−divA(x,∇vm)=σ|Bm(0)vqm,vm≥0,in Rn,lim inf|x|→∞vm=0, |
We recall from the construction in [9] that
vm=limj→∞(limk→∞vkj,m), |
where vk1,m (k=0,1,2,…) is the A-superharmonic renormalized solution of
{−divA(x,∇vk1,m)=σ|Bm(0)∩B2k(0)wq0,min B2k(0),vk1,m=0on ∂B2k(0), |
with w0,m=c0(W1,p(σ|Bm(0)))p−1p−1−q, and vkj,m (k=0,1,2,…, j=2,3,…) is the A-superharmonic renormalized solution of
{−divA(x,∇vkj,m)=σ|Bm(0)∩B2k(0)(limi→∞vij−1,m)qin B2k(0),vkj,m=0on ∂B2k(0). |
Here c0 is a fixed constant such that
0<c0≤min{(cqp−1−qK−1)p−1p−1−q,CK1−pp−1−q}, | (4.8) |
where C is the constant in (3.9) of [9], and c is the constant in (3.10) of [9] with α=1.
We also recall from [9] that
u=limj→∞(limk→∞ukj), |
where ukj are the A-superharmonic renormalized solutions of the corresponding problems in B2k(0) with σ in place of σ|Bm(0). In particular, min(ukj,l)∈W1,p0(B2k(0)) and min(vkj,m,l)∈W1,p0(B2k(0)) for all l>0.
Thus, by the above version of the comparison principle (Lemma 4.1) we see that
vkj,m1≤vkj,m2≤ukjinB2k(0), |
whenever m1≤m2.
This yields
0≤v1≤v2≤⋯≤vm≤uinRn. |
Letting now m→∞, we obtain an A-superharmonic reachable solution
v:=limm→∞vm |
to (4.7) such that v≤u in Rn. As u is the minimal A-superharmonic solution of (4.7), we see on the other hand that u≤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 c0. This choice of c0 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 A-superharmonic solution of (4.1) exists if and only if W1,pσ≢∞, Kp,qσ≢∞, and W1,pμ≢∞, i.e., the following three conditions hold:
∫∞1(σ(B(0,ρ))ρn−p)1p−1dρρ<+∞, | (4.9) |
∫∞1(ϰ(B(0,ρ))q(p−1)p−1−qtn−p)1p−1dρρ<∞, | (4.10) |
∫∞1(μ(B(0,ρ))ρn−p)1p−1dρρ<+∞. | (4.11) |
Theorem 4.3. Let 0<q<p−1, and let μ,σ∈M+(Rn), where μ<<capp. Then, under the conditions (4.9), (4.10), and (4.11), there exists a nontrivial minimal reachable A-superharmonic solution of (4.1).
Proof. Since the case μ=0 was treated in Theorem 4.2 above, without loss of generality we may assume that μ≠0. We recall that in the proof of [37,Theorem 1.1], a nontrivial A-superharmonic solution u of (4.1), was constructed using the following iteration process. We set u0=0, and for j=0,1,2,… construct the iterations
{−divA(x,∇uj+1)=σuqj+μin Rn,lim inf|x|→∞uj+1=0, | (4.12) |
where uj∈Lqloc(Rn,dσ). We observe that, for each j, the solution uj+1 was chosen in [37] so that uj≤uj+1 (j=0,1,2,…) by a version of the comparison principle (see [34,Lemma 3.7 and Lemma 3.9]). Then u:=limj→∞uj is a nontrivial A-superharmonic solution of (4.1).
We now modify this argument as follows to obtain a minimal nontrivial A-superharmonic solution of (4.1). Notice that μ<<capp by assumption, and, as mentioned above, σ<<capp, since a solution exists. Hence, clearly the measure σuqj+μ<<capp as well. By Theorem 3.6, uj+1 can be chosen as the minimal A-superharmonic solution to (4.12).
It follows by induction that uj≤uj+1 (j=0,1,2,…). Indeed, this is trivial when j=0, and then by the inductive step,
σuqj−1+μ≤σuqj+μ,j=1,2,…, |
which is obvious when j=1. From this, using Theorem 3.7 we deduce uj≤uj+1 for all j=1,2,….
Similarly, if ˜u is any A-superharmonic solution of (4.1), then again arguing by induction and using Theorem 3.7, we deduce that uj+1≤˜u (j=0,1,2,…), since
σuqj−1+μ≤σ˜uq+μ,j=1,2,…. |
Consequently, u≤˜u, i.e., u is the minimal A-superharmonic solution of (4.1).
We next show that u is a reachable solution. Using a similar iteration process with σ|Bm(0) in place of σ and μ|Bm(0) in place of μ (m=1,2,…), we set v0,m=0 and define vj,m to be the minimal A-superharmonic solution to the equation
{−divA(x,∇vj+1,m)=σ|Bm(0)vqj,m+μ|Bm(0),vj,m≥0,in Rn,lim inf|x|→∞vj,m=0, |
where vj,m≤vj+1,m for each m=1,2,….
As above, arguing by induction and using Theorem 3.7, we deduce
vj,m1≤vj,m2≤uj,j=1,2,…, |
whenever m1≤m2. It follows that vm:=limj→∞vj,m≤u (m=1,2,…) is an A-superharmonic solution of the equation
{−divA(x,∇vm)=σ|Bm(0)vqm+μ|Bm(0),vm≥0in Rn,lim inf|x|→∞vm=0, |
where vm1≤vm2≤u if m1≤m2.
Thus, letting m→∞, we obtain an A-superharmonic reachable solution v:=limm→∞vm to (4.7) such that v≤u. Since u is the minimal 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 μ,σ∈M+(Rn), where μ<<capp. Suppose A satisfies conditions (2.1) and (2.2). Then nontrivial A-superharmonic reachable solutions of (4.1) are unique.
Proof. Let u,v be two nontrivial A-superharmonic solutions of (4.1) in Rn. Then by [37,Theorem 1.1] and [37,Remark 4.3], there exists a constant C≥1, depending only on p, q and n, such that
C−1u≤v≤CuinRn. |
Hence, clearly,
−divA(x,∇v)=σvq+μ≤Cq(σuq+μ)=−divA(x,∇(Cqp−1u)). |
Notice that here by definition u,v∈Lqloc(Rn,σ). Suppose that v is a reachable solution of (4.1) in Rn. Then by Corollary 3.11 with σvq+μ in place of σ, and ˜σ=Cq(σuq+μ), it follows that v≤Cqp−1u.
By iterating this argument, we deduce
v≤C(qp−1)juinRn,j=1,2,…. |
Since 0<q<p−1, letting j→∞ in the preceding inequality, we obtain v≤u in Rn. Interchanging the roles of u and v, we see that actually u=v in Rn.
Corollary 4.5. Nontrivial A-superharmonic solutions u of (4.1) are unique under the assumptions of Theorem 4.4, provided any one of the following conditions holds:
(ⅰ) u∈Lq(Rn,dσ) and μ∈M+b(Rn), or equivalently ϰ(Rn)<∞ and μ∈M+b(Rn);
(ⅱ) lim|x|→∞u(x)=0;
(ⅲ) |Du|∈Lp(Rn), or |Du|∈Lγ,∞(Rn) for some (p−1)n/(n−1)≤γ<p.
Proof. Suppose first that (ⅰ) holds. By [9,Theorem 4.4], ϰ(Rn)<∞ if and only if there exists a nontrivial A-superharmonic solution u∈Lq(Rn,dσ) of (4.7). In particular, since by [37,Theorem 4.1],
u≥C[W1,pσ+(Kp,qσ)p−1p−1−q], |
it follows that
W1,pσ∈Lq(p−1)p−1−q(Rn,dσ)andKp,qσ∈Lq(Rn,dσ). |
Next, we denote by φ an A-superharmonic solution to the equation
{−divA(x,∇φ)=μ,φ≥0in Rn,lim inf|x|→∞φ=0, |
where μ is the Riesz measure of φ. Notice that φ≥K−1W1,pμ by the lower bound in inequality (3.4). Since μ∈Mb(Rn) and ϰ(Rn)<∞, using φ as a test function in inequality (4.5) yields W1,pμ∈Lq(Rn,dσ).
Hence, by [37,Theorem 1.1] and [37,Remark 4.3], we deduce that there exists a nontrivial A-superharmonic solution of (4.1) u∈Lq(Rn,dσ), and, for any such a solution, σuq+μ∈Mb(Rn). It follows that u is a reachable 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 A-superharmonic solution of (4.1) by Theorem 3.10 and Theorem 3.12 (ⅰ), (ⅱ).
In all these cases, reachable A-superharmonic solutions are unique by Theorem 4.4.
Remark 4.6. Uniqueness of finite energy solutions u to (4.1) such that |Du|∈Lp(Rn) 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 μ=0.) Solutions of finite energy to (4.1) exist if and only if W1,pσ∈L(1+q)(p−1)p−1−q(Rn,dσ) and W1,pμ∈L1(Rn,dμ) ([35,Theorem 1.1]).
Remark 4.7. Under the assumptions of Theorem 4.3, but without the restriction μ<<capp, it is still possible to prove the existence of an 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 {um}m≥1 of A-superharmonic solutions of
{−divA(x,∇um)=σ|Bm(0)uqm+μ|Bm(0),um≥0,in Rn,lim inf|x|→∞um=0. |
Then by [37,Theorem 1.1 and Remark 4.3],
um(x)≤C(W1,pμ(x)+Kp,qσ(x)+[W1,pσ(x)]p−1p−1−q),x∈Rn. |
It follows from [20,Theorem 1.17] that um→u pointwise everywhere, where u is an A-superharmonic reachable solution of (4.1).
The construction of {um}m≥1 can be done as follows. It suffices to demonstrate only how to construct u1 and u2 such that u2≥u1, since the construction of um for m≥3 is completely analogous. Let v1 be an A-superharmonic solution of
{−divA(x,∇v1)=μ|B1(0),v1≥0,in Rn,lim inf|x|→∞v1=0. |
Here as above v1 is an a.e. pointwise limit of a subsequence of {v(k)1}k≥1, where each v(k)1 is a nonnegative A-superharmonic renormalized solution of
{−divA(x,∇v(k)1)=μ|B1(0)in Bk(0),v(k)1=0on ∂Bk(0). |
Next, for any j≥1, let vj+1 be an A-superharmonic solution of
{−divA(x,∇vj+1)=σ|B1(0)vqj+μ|B1(0),vj+1≥0,in Rn,lim inf|x|→∞vj+1=0. |
Notice that vj+1 is an a.e. pointwise limit of a subsequence of {v(k)j+1}k≥1, where each v(k)j+1 is a nonnegative A-superharmonic renormalized solution of
{−divA(x,∇v(k)j+1)=σ|B1(0)vqj+μ|B1(0)in Bk(0),v(k)j+1=0on ∂Bk(0). |
By [33,Lemma 6.9] we may assume that v(k)2≥v(k)1 for all k≥1, and hence v2≥v1. In the same way, by induction we deduce that v(k)j+1≥v(k)j for all j,k≥1. It follows that vj+1≥vj, and
vj+1≤CW1,p(σvqj+1)+CW1,p(μ). |
Then by [37,Theorem 4.1], for any j≥1, we obtain the bound
vj+1(x)≤C(W1,pμ(x)+Kp,qσ(x)+[W1,pσ(x)]p−1p−1−q),x∈Rn. | (4.13) |
Thus, the nondecreasing sequence {vj}j≥1 converges to an A-superharmonic solution u1 of
{−divA(x,∇u1)=σ|B1(0)uq1+μ|B1(0),u1≥0,in Rn,lim inf|x|→∞u1=0. |
To construct u2 such that u2≥u1, let w1 be an A-superharmonic solution of
{−divA(x,∇w1)=μ|B2(0),w1≥0,in Rn,lim inf|x|→∞w1=0. |
Notice that w1 is an a.e. pointwise limit of a subsequence of {w(k)1}k≥1, where each w(k)1 is a nonnegative A-superharmonic renormalized solution of
{−divA(x,∇w(k)1)=μ|B2(0)in Bk(0),w(k)1=0on ∂Bk(0). |
Again, by [33,Lemma 6.9] we may assume that w(k)1≥v(k)1 for all k≥1, and hence w1≥v1.
Next, for any j≥1, let wj+1 be an A-superharmonic solution of
{−divA(x,∇wj+1)=σ|B2(0)wqj+μ|B2(0),wj+1≥0,in Rn,lim inf|x|→∞wj+1=0. |
Notice that wj+1 is an a.e. pointwise limit of a subsequence of {w(k)j+1}k≥1, where each w(k)j+1 is a nonnegative A-superharmonic renormalized solution of
{−divA(x,∇w(k)j+1)=σ|B2(0)wqj+μ|B2(0)in Bk(0),w(k)j+1=0on ∂Bk(0). |
We can ensure here that w(k)j+1≥max{v(k)j+1,w(k)j} for all j,k≥1. Indeed, since w1≥v1 and w1≥0, by Lemma 4.8 below we may assume that w(k)2≥max{v(k)2,w(k)1} for all k≥1, and hence w2≥max{v2,w1}. Repeating this argument by induction we obtain w(k)j+1≥max{v(k)j+1,w(k)j} for all j,k≥1.
It follows that wj+1≥max{vj+1,wj} for all j≥1. As in (4.13) we have
wj+1(x)≤C(W1,pμ(x)+Kp,qσ(x)+[W1,pσ(x)]p−1p−1−q),x∈Rn, |
and hence the nondecreasing sequence {wj} converges to an A-superharmonic solution u2 of
{−divA(x,∇u2)=σ|B2(0)uq2+μ|B2(0),u2≥0,in Rn,lim inf|x|→∞u2=0 |
such that u2≥u1, as desired.
The following lemma invoked in the argument presented above is an extension of [33,Lemma 6.9].
Lemma 4.8. Let Ω be a bounded open set in Rn and let μ1,μ2∈M+b(Ω). Suppose that ui (i=1,2) is a renormalized solution of
{−divA(x,∇ui)=μiinΩ,ui=0on∂Ω. |
Then for any measure ν∈M+b(Ω) such that ν≥μ1 and ν≥μ2, there is a renormalized solution v of
{−divA(x,∇v)=νinΩ,v=0on∂Ω, |
such that v≥u1 and v≥u2 a.e.
Proof. For i=1,2, let ui,k=min{ui,k} (k=1,2,…). Then ui,k is the bounded renormalized solution of
{−divA(x,∇ui,k)=μi0|{ui<k}+λi,kin Ω,ui,k=0on ∂Ω. |
Here μi=μi0+μis (i=1,2) is the decomposition of μi used in Section 3 above, where μi0,μis∈M+b(Ω), μi0<<capp, and μis is concentrated on a set of zero p-capacity. Moreover, λi,k∈M+b(Ω) and λi,k→μis in the narrow topology of measures as k→∞ (see Definition 3.3).
Now let vk (k=1,2,…) be a renormalized solution of
{−divA(x,∇vk)=2∑i=1(μi0+λi,k)+(ν−μ1)+(ν−μ2)in Ω,vk=0on ∂Ω. |
Then by [33,Lemma 6.8] we deduce vk≥max{u1,k,u2,k} for all k≥1. Finally, we use the stability results of [12] to find a subsequence of {vk} that converges a.e. to a desired function v.
Nguyen Cong Phuc is supported in part by the Simons Foundation, award number 426071.
The authors declare no conflict of interest.
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