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Advances in Nonlinear Analysis

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On viscosity and weak solutions for non-homogeneous p-Laplace equations

Maria Medina
  • Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile
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/ Pablo Ochoa
Published Online: 2017-06-04 | DOI: https://doi.org/10.1515/anona-2017-0005

Abstract

In this manuscript, we study the relation between viscosity and weak solutions for non-homogeneous p-Laplace equations with lower-order term depending on x, u and u. More precisely, we prove that any locally bounded viscosity solution constitutes a weak solution, extending results presented in Juutinen, Lindqvist and Manfredi [9], and Julin and Juutinen [6]. Moreover, we provide a converse statement in the full case under extra assumptions on the data.

Keywords: weak solutions; viscosity solutions; non-homogeneous equation

MSC 2010: 35J92; 35J70; 35D40; 35D30

1 Introduction and main results

In this work, we consider the following degenerate (or singular) elliptic equations of p-Laplacian type:

-div(|u|p-2u)=f(x,u,u),(1.1)

defined in an open and bounded set Ωn and for 1<p<. The modulus of ellipticity of the p-Laplace operator is |u|p-2. When p>2, the modulus vanishes whenever u=0, and the equation is called degenerate at those points where that occurs. On the other hand, for p<2, the modulus becomes infinite when u=0, and the equation is called singular at those points. Observe that the case p=2 is just the linear case and corresponds to the Laplace operator.

Different notions of solutions have been formulated for equation (1.1). We are interested in the relation between Sobolev weak solutions and viscosity solutions. For the homogeneous p-Laplace equation, this relation has already been studied by Juutinen, Lindqvist and Manfredi in [9], via the notion of p-harmonic, p-subharmonic and p-superharmonic functions. Roughly speaking, a p-harmonic function is a continuous function which solves, weakly, the homogeneous p-Laplace equation, and a p-superharmonic (p-subharmonic) function is a lower (upper) semicontinuous function that admits comparison with p-harmonic functions from below (above).

In [9], Juutinen, Lindqvist and Manfredi showed that the notion of p-harmonic solution is equivalent to the notion of viscosity solution. Moreover, it was shown in [13] that locally bounded p-harmonic functions are weak solutions. Conversely, every weak solution to the homogeneous p-Laplace equation has a representative which is lower semicontinuous and it is p-harmonic. We refer the interested reader to [5] for further details. In this way, there is an equivalence between the notion of weak and viscosity solutions for the homogeneous framework. It is worth to mention that a different and simpler proof of this equivalence was stated by Julin and Juutinen in [6] by using inf and sup convolutions. In turn, this reasoning was extended in [10] to more general second-order differential equations.

For the non-homogeneous case, the notion of p-harmonic functions is lost and we need to study directly the link between viscosity and Sobolev weak solutions. In [6], the authors showed that viscosity solutions of (1.1) are weak solutions in the case where f is continuous and depends only on x.

Our main goal in the present manuscript is to prove the equivalence of these two notions of solutions for the general structure (1.1). The implication that viscosity solutions are weak solutions is partially based on the work [6], but the non-homogeneous nature of the equation under consideration requires some extra effort to deal with the lower-order term.

On the other hand, the converse statement relies on comparison principles for weak solutions. To the best of our knowledge, the available comparison results for the full case f=f(x,s,η) require additional limitations in the degenerate case which do not appear in the singular context (compare Theorem A.1 and Theorem A.2). Moreover, we believe that the assumption that weak subsolutions and weak supersolutions belong to 𝒞1 or to the Sobolev space Wloc1, in order to have comparison is not a strong limitation since we are interested in the equivalence of weak and viscosity solutions, and for weak solutions the 𝒞1,α-regularity holds (see [4, 16]). Finally, in the quasi-linear case f=f(x,u) there is no need to impose higher regularity than Wloc1,p𝒞 on the solutions. We refer the reader to [15] for a survey of maximum principles and comparison results for general structures in divergence form.

Finally, we stress that the equivalence between weak and viscosity solutions may be used to prove relevant properties on the solutions. As an example, in [7], Juutinen and Lindqvist prove a Radó’s-type theorem for p-harmonic functions. Roughly speaking, they state that if a function u solves, weakly, the homogeneous p-Laplace equation in the complement of the set where u vanishes, then it is a solution in the whole set. It is an open problem to obtain a similar result for equations like (1.1). We shall return to this issue in a subsequent paper.

We recall that the p-Laplace operator is defined as

Δpu:=div(|u|p-2u).

Let us state the different type of solutions to (1.1) we will manage.

Definition 1.1 (Sobolev weak solution).

A function uWloc1,p(Ω) is a weak supersolution to (1.1) if

Ω|u|p-2uψΩf(x,u,u)ψ

for all non-negative ψC0(Ω). On the other hand, u is a weak subsolution if -u is a weak supersolution of the equation -Δpu=-f(x,-u,-u). We call u a weak solution if it is both a weak subsolution and a weak supersolution to (1.1).

Due to the non-homogeneous nature of (1.1), viscosity solutions are stated as in [6], considering semicontinuous envelopes of the p-Laplace operator. More precisely, we have the following definition.

Definition 1.2.

A lower semicontinuous function u:Ω(-,+] is a viscosity supersolution to (1.1) if u+ and for every ϕC2(Ω) such that ϕ(x0)=u(x0), u(x)ϕ(x) and ϕ(x)0 for all xx0, there holds

limr0supxBr(x0){x0}(-Δpϕ(x))f(x0,u(x0),ϕ(x0)).(1.2)

A function u is a viscosity subsolution if -u is a viscosity supersolution to the equation -Δpu=-f(x,-u,-u), and it is a viscosity solution if it is both a viscosity sub- and supersolution.

Remark 1.3.

Notice that condition (1.2) is established this way to avoid the problems derived from having ϕ(x0)=0 in the case 1<p<2. If p2, this condition can be simply replaced by

-Δpϕ(x0)f(x0,u(x0),ϕ(x0)).

We now list the main contributions of our work. The results are stated for supersolutions, but they hold for subsolutions as well.

Theorem 1.4.

Let 1<p<. Assume that f=f(x,s,η) is uniformly continuous in Ω×R×Rn, non-increasing in s, and satisfies the growth condition

|f(x,s,η)|γ(|s|)|η|p-1+ϕ(x),(1.3)

where γ0 is continuous, and ϕLloc(Ω). Hence, if uLloc(Ω) is a viscosity supersolution to (1.1), then it is a weak supersolution to (1.1).

A converse of Theorem 1.4 is given below.

Theorem 1.5.

Assume that f=f(x,s,η) is continuous in Ω×R×Rn, non-increasing in s, and locally Lipschitz continuous with respect to η. Hence we have the following:

  • (i)

    If 1<p2 and if uWloc1,(Ω) is a weak supersolution to ( 1.1 ), then it is a viscosity supersolution to ( 1.1 ) in Ω.

  • (ii)

    If p>2, f(x,s,0)=0 for xΩ and s , and if u𝒞1(Ω) is a weak supersolution to ( 1.1 ), then it is a viscosity supersolution to ( 1.1 ) in Ω.

  • (iii)

    Finally, if p>2 and if uWloc1,(Ω) is a weak supersolution to ( 1.1 ), with u0 in Ω , then it is a viscosity supersolution to ( 1.1 ).

Remark 1.6.

According to recent results (see [11]), it is possible to weak the locally Lipschitz assumption in Theorem 1.5 when f takes some particular forms or it satisfies extra convexity and coercivity assumptions. For instance, as a consequence of the results in [11], if

f(x,η)=c|η|q+ϕ(x),ϕ>0,c,ηn, and q[1,p],

then Theorem 1.5 holds for bounded supersolutions u in W1,p(Ω)𝒞(Ω). It is also a consequence of [11, Theorem 1.3] that the same conclusion is obtained when

f(x,s,η)=h(s)|η|p-1+ϕ(x),

where h0 is decreasing and ϕ0.

In view of the available regularity theory for weak solutions of (1.1), we have the following equivalence.

Corollary 1.7.

Let 1<p<. Assume that f=f(x,s,η) is uniformly continuous, locally Lipschitz in η, non-increasing in s, and satisfies the growth condition (1.3). Additionally, assume that f(x,s,0)=0 for xΩ and sR when p>2. Then u is a weak solution to (1.1) if and only if it is a viscosity solution to (1.1).

We point out that, in the degenerate case, it is possible to remove the assumption f(x,s,0)=0 by imposing the non-vanishing of the gradient of the weak solution in the whole Ω. This is a straightforward consequence of Theorem 1.5 (iii).

In the particular case where f does not depend on η, we have the following converse to Theorem 1.4 which does not require the locally Lipschitz regularity of the solutions.

Theorem 1.8.

Let 1<p<. Suppose that f=f(x,s) is continuous in Ω×R and non-increasing in s. If uWloc1,p(Ω)C(Ω) is a weak supersolution to (1.1), then it is a viscosity supersolution to (1.1).

Let us briefly discuss the above hypotheses on f. Firstly, assuming that f is non-increasing and introducing the operator

F(x,s,η,𝒳):=-|η|p-2(tr(𝒳)+p-2|η|2𝒳ηη)-f(x,s,η),p2,

we derive that F is proper, that is, F is non-increasing in 𝒳 and non-decreasing in s, which is a standard and useful assumption in the theory of viscosity solutions [1]. For instance, it allows to get the equivalence between classical solutions (𝒞2 functions which satisfy the equations pointwise) and 𝒞2 viscosity solutions. On the other hand, the growth property (1.3) implies the 𝒞1,α-regularity of weak solutions to (1.1) (see [4, 17, 16]). Moreover, under a regular Dirichlet boundary condition φ𝒞1,α, the 𝒞1,α-regularity up to the boundary of weak solutions follows. For further details, see the reference [12]. Finally, the extra assumption f(x,s,0)=0 appearing in Theorem 1.5 in the degenerate case is used to remove critical sets of points of the weak solution (see reference [8]). Hence, it allows the application of comparison results without assuming the non-vanishing of the gradients. We point out that other properties of f=f(x,s,η), as more regularity on s and η and convexity-like conditions, may be employed to ensure comparison for weak solutions. We refer the reader to [11] and the references therein for more details.

It is worth mentioning that many equations appearing in the literature have the structure of (1.1) with the lower-order term satisfying the above assumptions on f. We refer the reader to [14, 15, 3, 2] and the references therein for examples of such f.

The paper is organized as follows: In Section 2, we provide some preliminary results concerning properties of infimal convolutions (which will be the main tool in the proof of Theorem 1.4) and a convergence result. In addition, we prove a Caccioppoli-type estimate that will provide important uniform bounds, fundamental when using approximation arguments. This result is interesting in itself.

Section 3 contains the proof of the main result of the paper, Theorem 1.4, that states under which conditions on the non-homogeneous function f in (1.1) viscosity solutions are actually weak solutions. This proof is divided into two major cases: the singular and the degenerate scenario, thus, although both cases rely on the same idea, different approximations and estimates are needed depending on the range of p.

In Section 4, we prove the reverse statement, that is, weak solutions of (1.1) are viscosity solutions. This result is based on comparison arguments, and this will determine the conditions we will need to impose on f. Finally, in Appendix A we give, for the sake of completeness, precise references and state the comparison results that we use in Section 4.

2 Preliminary results

2.1 Infimal convolution

Let us define the infimal convolution of a function u as

uε(x):=infyΩ(u(y)+|x-y|qqεq-1),(2.1)

where q2 and ε>0.

We recall some useful properties of uε. Let u:Ω be bounded and lower semicontinuous in Ω. It is well known that uε is an increasing sequence of semiconcave functions in Ω, which converges pointwise to u. Hence, uε is locally Lipschitz and twice differentiable a.e. in Ω. Moreover, it is possible to write

uε(x)=infyBr(ε)(x)Ω(u(y)+|x-y|qqεq-1)

for r(ε)0 as ε0. For these and further properties, see [6, Lemma A.1.] and [1].

The next lemma is the counterpart of [6, Lemma A.1 (iii)] for our setting.

Lemma 2.1.

Suppose that u:ΩR is bounded and lower semicontinuous in Ω. Let f=f(x,s,η) be continuous in Ω×R×Rn and non-increasing in s. If u is a viscosity supersolution to

-Δpu=f(x,u,u)(2.2)

in Ω for 1<p<, then uε is a viscosity supersolution to

-Δpuε=fε(x,uε,uε)

in Ωε:={xΩ:dist(x,Ω)>r(ε)}, where

fε(x,s,η):=infyBr(ε)(x)f(y,s,η).

Proof.

We start by noticing that

uε(x)=infzBr(ε)(0)(u(z+x)+|z|qqεq-1),xΩε.

Let us see first that for every zBr(ε)(0), the function

ϕz(x):=u(z+x)+|z|qqεq-1

is a viscosity supersolution to -Δpϕz=fε in Ωε. Indeed, let x0Ωε and φ𝒞2(Ωε) so that

minΩε(ϕz-φ)=(ϕz-φ)(x0)=0.

We assume that φ(x)0 for all xx0 if 1<p<2. Making y:=z+x, y0:=z+x0 and

φ~(y):=φ(y-z)-|z|qqεq-1,

we derive that u-φ~ has a local minimum at y0, and indeed (u-φ~)(y0)=0. Since u is a viscosity supersolution to (2.2), there follows

limρ0supxBρ(y0){y0}(-Δpφ~(x))f(y0,φ~(y0),φ~(y0)).

Therefore,

limρ0supxBρ(x0){x0}(-Δpφ(x))=limρ0supxBρ(y0){y0}(-Δpφ~(x))f(y0,φ~(y0),φ~(y0))=f(z+x0,φ~(z+x0),φ(x0))=f(z+x0,φ(x0)-|z|qqεq-1,φ(x0))f(z+x0,φ(x0),φ(x0))fε(x0,φ(x0),φ(x0)),(2.3)

where we have used that f is non-increasing in the second variable. Let us see now that, since uε is an infimum of supersolutions, it is itself a supersolution (observe that uε is continuous, since it is locally Lipschitz). Let x0Ωε and ϕ𝒞2(Ωε) so that

minΩε(uε-ϕ)=(uε-ϕ)(x0)=0.(2.4)

Again, ϕ(x)0 for all xx0 in the singular scenario. Moreover, we may assume that the minimum is strict. For each n, there exists znBr(ε)(0) such that

u(zn+x0)+|zn|qqεq-1<uε(x0)+1n.(2.5)

Let xn be a sequence of points in B¯r(x0)Ωε so that

u(zn+xn)+|zn|qqεq-1-ϕ(xn)u(zn+x)+|zn|qqεq-1-ϕ(x)

for all xB¯r(x0), i.e., (ϕzn-ϕ) has a minimum in B¯r(x0) at xn. Up to a subsequence, xny0 as n. Furthermore, by (2.5),

uε(xn)-ϕ(xn)u(zn+xn)+|zn|qqεq-1-ϕ(xn)u(zn+x0)+|zn|qqεq-1-ϕ(x0)uε(x0)+1n-ϕ(x0).(2.6)

Taking liminf and using the lower semicontinuity of uε, we derive

uε(y0)-ϕ(y0)uε(x0)-ϕ(x0).

Since the minimum in (2.4) is strict, we must have y0=x0. Moreover, taking

φ(x):=ϕ(x)+(ϕzn-ϕ)(xn)

in (2.3), we have

limρ0supxBρ(xn){xn}(-Δpϕ(x))f(zn+xn,u(zn+xn)+|zn|qqεq-1,ϕ(xn)).

Since f is non-increasing with respect to the second variable, by (2.6) we obtain

limρ0supxBρ(xn){xn}(-Δpϕ(x))f(zn+xn,uε(x0)+1n-ϕ(x0)+ϕ(xn),ϕ(xn)).

As n, there holds

limρ0supxBρ(x0){x0}(-Δpϕ(x))f(z+x0,uε(x0),ϕ(x0))

for some zBr(0)¯. Therefore,

limρ0supxBρ(x0){x0}(-Δpϕ(x))fε(x0,ϕ(x0),ϕ(x0)),

and we conclude that uε is a viscosity supersolution of

-Δpuε=fε(x,uε,uε)in Ωε,

as desired. ∎

The next lemma states the weak convergence of the lower-order terms in the particular situation of infimal convolutions.

Lemma 2.2.

Let f=f(x,s,η) be a uniformly continuous function, which satisfies the growth condition (1.3). Assume that uWloc1,p(Ω) is locally bounded and lower semicontinuous in Ω. For each ε>0 define uε as in (2.1) and fε as in Lemma 2.1. Then, if uε converges to u in Llocp(Ω), the following holds:

limε0Ωfε(x,uε,uε)ψ𝑑x=Ωf(x,u,u)ψ𝑑x

for every non-negative ψC0(Ω).

Proof.

Let ψ𝒞0(Ω) and denote K:=spt(ψ). Consider ε>0 small enough so that

KKΩ,

where K:=xKBr(ε)(x)¯. Since f is uniformly continuous in K××n, for every ρ>0 there exists δ>0 such that

|f(x,uε(x),uε(x))-f(y,uε(x),uε(x))|<ρif |x-y|<δ,x,yK.

Choose ε0>0 so that r(ε)<δ for every ε<ε0. Thus, from the previous inequality we get

f(x,uε(x),uε(x))<ρ+f(y,uε(x),uε(x))

for every xK and yBr(ε)(x). In particular,

f(x,uε(x),uε(x))<ρ+fε(x,uε(x),uε(x)),

and therefore

0|f(x,uε(x),uε(x))-fε(x,uε(x),uε(x))|<ρ.

Hence we arrive at the estimate

Ω|f(x,uε,uε)-fε(x,uε,uε)|ψ𝑑xρψL(K)|K|.(2.7)

On the other hand, due to the continuity of f and the convergences of uε and uε,

f(x,uε(x),uε(x))f(x,u(x),u(x))a.e. in Ω.

Observe that

uε0uεu for all εε0.

Since uε0, u belong to Lloc(Ω), there exists a uniform constant C>0 so that

uεL(K)C,εε0.

Thus, in view of the growth estimate on f and the continuity of γ, we have, for an appropriate positive constant C,

|f(x,uε(x),uε(x))|C|uε(x)|p-1+ϕ(x).(2.8)

Since |uε|p-1Llocp/(p-1)(Ω), Hölder’s inequality and the strong convergence of uε imply

K|uε|p-1CuεLp(K)p-1Cfor all ε.

By (2.7), (2.8) and the Lebesgue dominated convergence theorem, we conclude

limε0Kf(x,uε,uε)ψ𝑑x=Kf(x,u,u)ψ𝑑x.

2.2 A Caccioppoli’s estimate

In the next lemma we provide a Caccioppoli’s estimate for the Llocp-norm of the gradients of weak solutions.

Lemma 2.3.

Let uW1,p(Ω) be a locally bounded weak supersolution to (1.1). Assume that f is continuous in Ω×R×Rn and satisfies the growth bound (1.3). Then there exists a constant C=C(p,Ω,ϕ,γ)>0 such that for all test function ξC0(Ω), 0ξ1, we have

Ω|u|pξp𝑑xC[(oscKu)pΩ(|ξ|p+1)𝑑x+oscKu],

where oscKu:=supKu-infKu, and K:=spt(ξ).

Proof.

Let ξ𝒞0(Ω) and KΩ be as in the lemma. Consider the test function

ψ(x):=(supKu-u(x))ξp(x),xΩ.

Then

Ωf(x,u,u)ψ𝑑xΩ|u|p-2uψdx=-Ω|u|p-2u[ξpu-pξp-1ξ(supKu-u)]𝑑x.

Therefore,

Ω|u|pξp𝑑xpΩξp-1|u|p-2uξ(supKu-u)𝑑x-Ωf(x,u,u)ψ𝑑x.(2.9)

Observe that

Ω||u|p-2u|𝑑xΩ|u|p-1𝑑x,

which shows that |u|p-2uLp/(p-1)(Ω). Hence, Young’s inequality

abδaq+δ-1/(q-1)bq,

where q and q are conjugate exponents, implies that the first integral on the right-hand side of (2.9) may be bounded by

δΩ|u|pξp+δ1-pΩpp|ξ|p(oscKu)p𝑑x.

Moreover, by (1.3) we have

f(x,u(x),u(x))-γ|u(x)|p-1-ϕL(K)(2.10)

for all x in the support of ξ, where γ:=supxK|γ(u(x))|. Therefore, the second integral in (2.9) is estimated from above by

γΩ|u|p-1(supKu-u)ξp𝑑x+C(Ω,ϕ)oscKu,

where C(Ω,ϕ) is a positive constant. The assumption ξ1 and Young’s inequality yield

γΩ|u|p-1(supKu-u)ξp𝑑xγΩ|u|p-1(supKu-u)ξp-1𝑑xδΩ|u|pξp𝑑x+δ1-pC(p,Ω,γ)(oscKu)p.

Therefore,

Ω|u|pξp𝑑x2δΩ|u|pξp𝑑x+δ1-pC(p,Ω,γ,ϕ)[(oscKu)pΩ(|ξ|p+1)𝑑x+oscKu].

Taking δ<1/2, we derive Caccioppoli’s estimate. ∎

3 Proof of Theorem 1.4

3.1 Degenerate case

We begin with the range p2.

Proof of Theorem 1.4.

Let uε be the infimal convolution defined in (2.1) with q=2. Then

ϕ(x):=uε(x)-C|x|2

is concave in Ωr(ε) (see [6, Lemma A.2.]). By Aleksandrov’s Theorem, ϕ is twice differentiable almost everywhere in Ωr(ε), and so is uε. Therefore by Lemma 2.1,

-Δpuε(x)fε(x,uε(x),uε(x))

a.e. in Ωr(ε). Furthermore,

Ω|uε|p-2uεψΩ(-Δpuε)ψ

for all non-negative test functions ψ (see the proof of [6, Theorem 3.1]). Hence, we derive

Ωfε(x,uε,uε)ψ𝑑xΩ|uε|p-2uεψdx

for all non-negative test functions ψ and all ε>0. We claim that, as ε0, there holds

Ωf(x,u,u)ψ𝑑xΩ|u|p-2uψdx.

To prove the claim, observe first that Caccioppoli’s estimate allows us to conclude that

|uε|p-2uε

converges weakly in Llocp/(p-1)(Ω). Indeed, for any compact set KΩ, choose an open set UΩ containing K and a non-negative test function 0ξ1 so that

KK:=sptξU

and ξ=1 in K. Then

K||uε|p-2uε|p/(p-1)𝑑xK|uε|p𝑑xΩ|uε|pξp𝑑x.(3.1)

Observe that since f satisfies (1.3), the lower term fε verifies the bound (2.10). Therefore, Lemma 2.3 applies and the right-hand side of (3.1) is bounded from above by

C[(oscKuε)pΩ(|ξ|p+1)𝑑x+oscKuε].(3.2)

Moreover, since uε is an increasing sequence and converges pointwise to u in Ω, we have

oscKuεsupKu-infKuε0

for all ε<ε0. Then in view of (3.1), (3.2) and the above comments, we can find a uniform bound for the integrals

K||uε|p-2uε|p/(p-1)𝑑x,Ω|uε|pξp𝑑x.

Hence |uε|p-2uε converges weakly in Llocp/(p-1)(Ω), and uε converges weakly in Llocp(Ω). Since uε converges pointwise to u, we derive that uWloc1,p(Ω) and uε converges weakly in Wloc1,p(Ω) to u.

More can be said: uε converges strongly in Llocp(Ω) to u. Indeed, take

ϕ(x):=(u(x)-uε(x))θ(x),xΩ,

where θ is a non-negative smooth test function compactly supported in Ω. From

Ω|uε|p-2uεϕdxΩfε(x,uε,uε)ϕ𝑑x

we get

Ω[|u|p-2u-|uε|p-2uε](u-uε)θdx-Ωfε(x,uε,uε)ϕ𝑑x+Ω|u|p-2u(u-uε)θdx.(3.3)

By the weak convergence of uε to u in Wloc1,p(Ω), the last integral in (3.3) tends to 0 as ε0. The left-hand side is given by

Ωθ[|u|p-2u-|uε|p-2uε](u-uε)dx+Ω(u-uε)[|u|p-2u-|uε|p-2uε]θdx.(3.4)

The second integral in (3.4) is estimated in absolute value by

θL(Ω)(sptθ|u-uε|p𝑑x)1/p[(sptθ|u|p𝑑x)(p-1)/p+(sptθ|uε|p𝑑x)(p-1)/p],

which tends to 0 as ε0. Moreover, since

-sptθfε(x,uε,uε)ϕ𝑑xγΩ|uε|p-1(u-uε)θ𝑑x+ϕL(spt(θ))Ω(u-uε)θ𝑑x,

with γ:=supxspt(θ)|γ(uε(x))|, does not depend on ε, it also holds that

lim supε0[-sptθfε(x,uε,uε)ϕ𝑑x]=0.

Hence

limε0Ωθ[|u|p-2u-|uε|p-2uε](u-uε)dx=0,(3.5)

where we have used the fact that the integrand is always non-negative. Therefore, using the inequality

2p-2|u(x)-uε(x)|p[|u(x)|p-2u(x)-|uε(x)|p-2uε(x)](u(x)-uε(x))

valid for all p2, and (3.5), we conclude the strong convergence of uε in Llocp(Ω). Finally, (3.5) together with [5, Lemma 3.73]), implies

|uε|p-2uε|u|p-2uin Llocp/(p-1)(Ω),

and, in turn, the strong convergence of the gradients uε and Lemma 2.2 gives

limε0Ωfε(x,uε,uε)ψ𝑑x=Ωf(x,u,u)ψ𝑑x.

This ends the proof of the claim and we deduce that u is a weak supersolution. ∎

3.2 The singular case: 1<p<2

Consider now the infimal convolution given in (2.1) choosing q>p/(p-1), i.e.,

uε(x):=infyΩ(u(y)+|x-y|qqεq-1).(3.6)

Notice that q>2 for 1<p<2.

We need the following auxiliary result, which is an adaptation of [6, Lemma 4.3].

Lemma 3.1.

Suppose that u is a bounded viscosity supersolution to (1.1). If there is x^Ωr(ε) such that uε is differentiable at x^ and uε(x^)=0, then fε(x^,uε(x^),uε(x^))0.

Proof.

From [6, Lemma 4.3] we know that uε(x^)=u(x^), and hence

u(y)+|x^-y|qqεq-1u(x^)for every yΩ.

Define

ψ(y):=u(x^)-|x^-y|qqεq-1,yΩ,

which satisfies ψ𝒞2(Ω), ψ(x^)=0 and

limr0supyBr(x^){x^}(-Δpψ(y))=0(3.7)

in view of q>p/(p-1). Since ψ(x^)=u(x^), ψ(y)u(y) for yΩ, ψ(x)0 for all xx^, and u is a viscosity supersolution to (1.1),

limr0supyBr(x^){x^}(-Δpψ(y))f(x^,ψ(x^),ψ(x^)).

Noticing that ψ(x^)=uε(x^) and ψ(x^)=uε(x^)=0, by (3.7) we conclude

0f(x^,ψ(x^),ψ(x^))=f(x^,uε(x^),uε(x^))fε(x^,uε(x^),uε(x^)),

as desired. ∎

We can prove now Theorem 1.4 in the case 1<p<2.

Proof of Theorem 1.4.

Let uε be defined in (3.6). Proceeding as in the degenerate case, by Aleksandrov’s theorem and Lemma 2.1 we obtain

-Δpuεfε(x,uε,uε)

a.e. in Ωr(ε){uε=0}. Performing the same approximation argument as in the proof of [6, Theorem 4.1], we reach that

Ω|uε|p-2uεψdxΩ{uε=0}fε(x,uε,uε)ψ𝑑x

for every ψ𝒞0(Ω), ψ0. Therefore and since by Lemma 3.1 we know fε0 in the set {xΩ:uε(x)=0}, we get

Ω|uε|p-2uεψdxΩfε(x,uε,uε)ψ𝑑x.(3.8)

Repeating the proof for the case p2 (and noticing that Lemma 2.3 works for every 1<p<), we obtain the uniform boundedness of uε in Llocp(Ω) and

limε0K[|u|p-2u-|uε|p-2uε](u-uε)dx=0(3.9)

for any compact set KΩ, and from here the convergence

|uε|p-2uε|u|p-2uin Llocp/(p-1)(Ω).(3.10)

Using Hölder’s inequality and the vector inequality (see [4, Chapter I])

|a-b|2(|a|+|b|)2-pC(|a|p-2a-|b|p-2b)(a-b),1<p<2,

with C=C(n,p) and a,bn, we obtain

K|u-uε|p𝑑x(K|u-uε|2(|u|+|uε|)2-p𝑑x)p/2(K(|u|+|uε|)p𝑑x)(2-p)/2C(K|u-uε|2(|u|+|uε|)2-p𝑑x)p/2C(K[|u|p-2u-|uε|p-2uε](u-uε)dx)p/2.

Thus, from (3.9) we deduce that uε converges to u in Llocp(Ω). By (3.10) and Lemma 2.2 we can pass to the limit in (3.8) to conclude

Ω|u|p-1uψdxΩf(x,u,u)ψ𝑑x.

4 Proofs of Theorem 1.5 and Theorem 1.8

Proof of Theorem 1.5 (i).

Let uWloc1,(Ω) be a weak supersolution to (1.1). To reach a contradiction, assume that u is not a viscosity supersolution. By assumption, there exist x0Ω and φ𝒞2(Ω) so that φ(x)0 for all xx0,

u(x0)=φ(x0),u(x)>φ(x)for all xx0,(4.1)

and

limr0supxBr(x0){x0}(-Δpφ(x))<f(x0,u(x0),φ(x0)).

Moreover, by the Wloc1,-regularity of u, we may assume that u is continuous in Ω. Thus, the map

xf(x,u(x),φ(x))

is continuous in Ω, and (4.1) yields

limr0supxBr(x0){x0}[-Δpφ(x)-f(x,u(x),φ(x))]<0.

Hence, there exists some r0>0 so that

-Δpφf(x,u(x),φ(x)),xBr0(x0){x0}.(4.2)

Let

m:=infBr0(x0)(u-φ).

Then by (4.1) we have m>0. Consider

φ~(x):=φ(x)+m,xΩ.

By (4.2), φ~ is a weak subsolution to

-Δpv=f~(x,v)(4.3)

in Br0(x0), where f~(x,η):=f(x,u(x),η). Observe that f~ is continuous in Ω×n and locally Lipschitz in η. Moreover, in the weak sense, we have

-Δpuf(x,u,u)=f~(x,u),

which shows that u is a weak supersolution to (4.3). In addition, uφ~ on Br0(x0). By the comparison Theorem A.2, we conclude that uφ~ in Br0(x0). This contradicts (4.1). ∎

Proof of Theorem 1.5 (ii).

For a given weak supersolution u𝒞1(Ω), by following the lines above and appealing to the comparison Theorem A.1, we can show that u is a viscosity supersolution in the non-critical set

{xΩ:u(x)0}.

By [8, Corollary 4.4], which holds true for sub- and supersolutions, u is a viscosity supersolution in the whole set Ω. ∎

Proof of Theorem 1.5 (iii).

The proof follows similarly just by using the assumption u0 in Ω together with the comparison Theorem A.1. ∎

Remark 4.1.

Observe that the Wloc1,-regularity of u is only needed to apply the comparison principles. In the rest of the proof, the continuity of u would be enough.

Proof of Theorem 1.8.

The result follows by reproducing the proof of Theorem 1.5 using the comparison Theorem A.3 instead of Theorems A.2 and A.1. ∎

A Appendix

A.1 Comparison principles for weak solutions

In this section, we provide the comparison principles for weak solutions of (1.1) that we use in the proof of Theorem 1.5. As we pointed out in Section 1, other comparison results may be employed (see [11]).

The first one is contained in [15, Corollary 3.6.3].

Theorem A.1.

Assume that f=f(x,s,η) is continuous in Ω×R×Rn, non-increasing in s, and locally Lipschitz continuous with respect to η in Ω×R×Rn. Let uWloc1,(Ω) be a weak supersolution and let vWloc1,(Ω) be a weak subsolution to (1.1) in Ω for 1<p<. Assume that |u|+|v|>0 in Ω. If uv on Ω, then uv in Ω.

In the singular case, the assumptions on the gradients may be removed. See [15, Corollary 3.5.2].

Theorem A.2.

Let 1<p2. Assume that f=f(x,s,η) is continuous in Ω×R×Rn, non-increasing in s, and that it is locally Lipschitz continuous in η on compact subsets of its variables. Then if uWloc1,(Ω) is a weak supersolution and if vWloc1,(Ω) is a weak subsolution to (1.1) in Ω so that uv on Ω, then uv in Ω.

Finally, in the case where f does not depend on η, we have the following result (see [15, Corollary 3.4.2]).

Theorem A.3.

Suppose that f=f(x,s) is continuous in Ω×R and non-increasing in s. Let uWloc1,p(Ω)C(Ω) be a supersolution and vWloc1,p(Ω)C(Ω) a subsolution so that uv on Ω. Then uv in Ω.

Remark A.4.

In [15], Theorem A.1 is stated in a more general framework of equations in divergence form as

div(A(x,u,u))=f(x,u,u),

where the operator A=A(x,s,η) is assumed to be continuous in Ω××n, continuously differentiable with respect to s and η for all s and all η0, and elliptic in the sense that A(x,s,η) is positive definite in Ω××(n{0}). In the particular case of the p-Laplace operators

Ap(η)=|η|p-2η,p2,Ap(η)={|η|p-2ηif η0,0,for η=0,1<p2,

all of the assumptions above are satisfied. The positive definiteness of Ap is a consequence of

i,j=1nAiηj(η)ξiξjc|η|p-2|ξ|2

for a positive constant c. Finally, observe that A is uniformly elliptic for 0<|η|C if p2. This allows the improved comparison result in Theorem A.2.

Acknowledgements

The authors would like to thank the anonymous referee for her/his comments.

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About the article

Received: 2017-01-13

Revised: 2017-02-28

Accepted: 2017-03-02

Published Online: 2017-06-04


The first author was supported by the grant FONDECYT Postdoctorado 2016, No. 3160077. The second author was partially supported by CONICET and grant PICT 2015-1701 AGENCIA.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 468–481, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0005.

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