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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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The Gelfand problem for the 1-homogeneous p-Laplacian

José Carmona TapiaORCID iD: https://orcid.org/0000-0001-9319-4382 / Alexis Molino SalasORCID iD: https://orcid.org/0000-0003-2819-7282 / Julio D. RossiORCID iD: https://orcid.org/0000-0001-7622-2759
Published Online: 2017-07-21 | DOI: https://doi.org/10.1515/anona-2016-0233

Abstract

In this paper, we study the existence of viscosity solutions to the Gelfand problem for the 1-homogeneous p-Laplacian in a bounded domain ΩN, that is, we deal with

-1p-1|u|2-pdiv(|u|p-2u)=λeu

in Ω with u=0 on Ω. For this problem we show that, for p[2,], there exists a positive critical value λ*=λ*(Ω,N,p) such that the following holds:

  • If λ<λ*, the problem admits a minimal positive solution wλ.

  • If λ>λ*, the problem admits no solution.

Moreover, the branch of minimal solutions {wλ} is increasing with λ. In addition, using degree theory, for fixed p we show that there exists an unbounded continuum of solutions that emanates from the trivial solution u=0 with λ=0, and for a small fixed λ we also obtain a continuum of solutions with p[2,].

Keywords: Gelfand problem; elliptic equations; viscosity solutions

MSC 2010: 35J15; 35J60; 35J70

1 Introduction

This paper deals with the Gelfand problem corresponding to the 1-homogeneous p-Laplacian,

{-ΔpNu=λeuin Ω,u=0on Ω,(${P_{\lambda,p}}$)

where ΩN is a regular bounded domain, p[2,] and the operator ΔpN is the 1-homogeneous p-Laplacian (it is also called the normalized p-Laplacian) defined, for p<, by

ΔpNu:=1p-1|u|2-pdiv(|u|p-2u)=αΔu+βΔu,

with α=1/(p-1) and β=(p-2)/(p-1), and for p=,

ΔuΔNu=u|u|(D2uu|u|)

is the 1-homogeneous infinity Laplacian. These kinds of elliptic operators for 2p< have 1 and 1/(p-1) as ellipticity constants, hence there is a lack of uniform ellipticity when we let p. Therefore, the theory of uniformly elliptic operators can not be applied. Moreover, we remark the lack of variational structure and differentiability of this operator, in contrast to what happens with the classical p-Laplacian. This fact implies that the theory concerning “stable solutions” can not be applied to our problem.

Note that the 1-homogeneous p-Laplacian is a convex combination of Laplacian and infinity Laplacian operators since α+β=1. Moreover, α=1, β=0 if p=2, and α0, β1 as p. This operator appears when one considers Tug-of-War games with noise; see [18, 23, 24], where the Poisson problem is studied. Moreover, the sublinear problem and the eigenvalue problem associated to the 1-homogeneous p-Laplacian, namely, the problem with right-hand side λuq for 0<q1 , has been studied in [20] and [19]. In view of these two references it seems natural to deal with the superlinear case (which for this operator is challenging due to the fact that there is no variational structure and no Sobolev spaces framework).

Concerning the Gelfand problem, since it is a classical problem, there is a large number of references. We refer to [2, 4, 5, 10, 21] and the references therein for the Laplacian, and to [25] for the fractional Laplacian.

Our first result for this problem reads as follows.

Theorem 1.1.

For every fixed p[2,+], there exists a positive extremal parameter λ*=λ*(Ω,N,p) such that

Moreover, the branch of minimal solutions {wλ} is increasing with λ. Even more, in the case of a ball, Ω=Br, the minimal solution is radial.

One of our main tools for the proof of this result is a comparison principle (that we prove here) adapted to the particular structure of the 1-homogeneous p-Laplacian (see Theorem 3.3). This result generalizes previous ones in [20, 3]. We believe that this comparison principle is of independent interest.

Using arguments from degree theory, we can obtain the following result concerning solutions that are not necessarily the minimal one. Remark that we even obtain a continuum of solutions for a fixed p using λ as parameter, or for fixed λ small taking p as parameter. More precisely, for fixed p we denote by 𝒮p the solution set

𝒮p={(λ,u)[0,λ*(Ω,N,p)]×𝒞(Ω¯):u solves (Pλ,p)}.

Analogously, for fixed λ we denote by 𝒮λ the solution set

𝒮λ={(p,u)[2,]×𝒞(Ω¯):u solves (Pλ,p)}.

Theorem 1.2.

For every fixed p[2,], there exists an unbounded continuum of solutions CSp that emanates from λ=0, u=0, i.e., (0,0)C. Moreover, for every fixed λ<λ0=min{λ*(Ω,N,2),(2d2e)-1}, where d is the diameter of Ω, there exists a continuum of solutions DSλ with Proj[2,+]D=[2,+] and u1 for all (p,u)D.

We remark that, as a consequence of the previous theorem, there is a lower bound for the extremal parameter found in Theorem 1.1: 0<λ0λ*(Ω,N,p) for every p[2,+].

The use of degree theory is new for this kind of operators. Here we perform homotopies both in the parameters λ and p. The deformation in p is needed in order to start the argument with the trivial solution u=0 for the problem with p=2 and λ=0, Δu=0, which is known to have degree 1. Note that, due to the nonsmoothness of the operator, there is a nontrivial difficulty in the computation of the degree of the trivial solution to ΔpNu=0. Also note that the necessary compactness is nontrivial; we rely here on results from [7].

Remark 1.3.

Our results can be generalized to handle the equation

-ΔpNu=λf(u),

with a general continuous nonlinearity f that verifies

f(0)>0,f(s) is increasing,f(s)sk>0.

To simplify the exposition we just write the details for f(s)=es and we make a comment at the end of the paper on how to deal with this general case.

The rest of this paper is organized as follows: In Section 2, we collect some preliminaries and state the definition of a viscosity solution to our equation. In Section 3, we prove our comparison result. Finally, in Sections 4 and 5 we prove our main results concerning the Gelfand problem.

2 Preliminaries

In this section, we introduce the notion of a viscosity solution for problem ((${P_{\lambda,p}}$)). Actually, we give the definition for a more general family of nonlinearities and we consider the following boundary value problem:

{-ΔpNu=λf(x,u)in Ω,u=0on Ω,(2.1)

where f:Ω× is a continuous function.

Since the normalized infinity Laplacian

Δu=u|u|(D2uu|u|)

is not well defined at the points where |u(x)|=0, we have to use the semicontinuous envelopes of the operator

(ξ,X)ξ|ξ|(Xξ|ξ|),ξN,X𝕊N,

in order to define viscosity solutions for problem (2.1) (see [8, 9]). To this end, we denote the largest and the smallest eigenvalue for A𝕊N by M(A) and m(A), respectively. That is,

M(A)=max|η|=1η(Aη),m(A)=min|η|=1η(Aη).

Let us denote by USC(ω) the set of upper semicontinuous functions u:ωN, and we denote by LSC(ω) the set of lower semicontinuous functions.

Definition 2.1.

  • (i)

    u¯USC(Ω) is a viscosity subsolution of the equation -ΔpNu=λf(x,u) if whenever x0Ω and φ𝒞2(Ω) such that φ(x0)=u¯(x0) and φ-u¯>0 in Ω{x0}, then

    {-ΔpNφ(x0)λf(x0,φ(x0))if φ(x0)0,-αΔφ(x0)-βM(D2φ(x0))λf(x0,φ(x0))if φ(x0)=0.

    If, in addition, u¯USC(Ω¯) and u¯0 on Ω, we say that u¯ is a subsolution of (2.1).

  • (ii)

    u¯LSC(Ω) is a viscosity supersolution of the equation -ΔpNu=λf(x,u) if whenever x0Ω and ψ𝒞2(Ω) such that ψ(x0)=u¯(x0) and u¯-ψ>0 in Ω{x0}, then

    {-ΔpNψ(x0)λf(x0,ψ(x0))if ψ(x0)0,-αΔψ(x0)-βm(D2ψ(x0))λf(x0,ψ(x0))if ψ(x0)=0.

    If, in addition, u¯LSC(Ω¯) and u¯0 on Ω, we say that u¯ is a supersolution of (2.1).

  • (iii)

    A continuous function u:Ω¯ is a viscosity solution of (2.1) if it is both a viscosity supersolution and a viscosity subsolution.

In what follows, φ stands for test functions whose graph touches the graph of u from above, and ψ denotes test functions whose graph touches the graph of u from below. Notice that the inequalities φ-u¯>0 and u¯-ψ>0 have to be satisfied in a neighborhood of {x0} and not necessarily in the whole Ω{x0}.

Remark 2.2.

Let u be a classical subsolution of (2.1), that is, u𝒞2(Ω¯), u0 on Ω and for every xΩ it satisfies

{-ΔpNu(x)λf(x,u(x))if u(x)0,-αΔu(x)-βM(D2u(x))λf(x,u(x))if u(x)=0.

Then u is a viscosity subsolution. Indeed, let φ𝒞2(Ω) be such that φ(x0)=u(x0) and φ-u>0 in Ω{x0}; then (φ-u)(x0)=0 and D2(φ-u)(x0) is a positive definite N×N matrix. Therefore,

η(D2φ(x0)η)η(D2u(x0))η,ηN,

and tr(D2φ(x0))tr(D2u(x0)) (i.e., Δφ(x0)Δu(x0)). Hence, if u(x0)0, we obtain

-αΔφ(x0)-βΔNφ(x0)-αΔu(x0)-βΔNu(x0)λf(x0,φ(x0)).

Finally, using that M(D2φ(x0))M(D2u(x0)) for u(x0)=0, it follows that u is a viscosity subsolution. We can proceed analogously with the supersolution case. Thus, classical solutions of (2.1) are solutions in the viscosity sense.

Let us observe that u¯USC(Ω) is a viscosity subsolution of -ΔpNu=λf(x,u) if

{-αtr(X)-βη|η|(Xη|η|)λf(x0,φ(x0))if η0,-αtr(X)-βM(X)λf(x0,φ(x0))if η=0,(2.2)

whenever x0Ω and (η,X)=(φ(x0),D2φ(x0))N×𝕊N for some φ𝒞2(Ω) such that φ(x0)=u¯(x0) and φ-u¯>0 in Ω{x0}. Thus, as in [9], we can characterize viscosity sub- and supersolutions by using the concept of upper and lower semijets in the sense of the following definition.

Definition 2.3.

For uUSC(𝒪) and x0𝒪, we define the upper semijet

J𝒪2,+u(x0)={(φ(x0),D2φ(x0)):φ𝒞2(𝒪),φ(x0)=u(x0) and φ-u has a local minimum at x0}.

Analogously, for uLSC(𝒪) and x0𝒪 we define the lower semijet

J𝒪2,-u(x0)={(ψ(x0),D2ψ(x0)):ψ𝒞2(𝒪),ψ(x0)=u(x0) and ψ-u has a local maximum at x0}.

Finally, we introduce the sets J¯𝒪2,+u(x0), J¯𝒪2,-u(x0) as follows: (p,X)J¯𝒪2,+u(x0) if there exist xnBr(x0) and (pn,Xn)J𝒪2,+u(xn), such that u(xn)u(x0) and (xn,pn,Xn)(x0,p,X) as n. An analogous statement holds for J¯𝒪2,-u(x0).

Remark 2.4.

It is clear that u¯USC(Ω) is a viscosity subsolution of -ΔpNu=λf(x,u) if (2.2) is verified for every (η,X)JΩ2,+u¯(x0). Moreover, if u¯ is a subsolution, then (2.2) is verified for every (η,X)J¯Ω2,+u¯(x0). The analogous statement holds for supersolutions.

Remark 2.5.

In [12], a parabolic equation of the form

ut=|u|γ(Δu+(p-2)ΔNu)

was studied using viscosity solutions. The definition of viscosity solutions given there (inspired by [22]) differs from ours. In fact, in [12] Imbert, Lin and Silvestre restrict the class of test functions in order to give sense to the equation when the gradient vanishes (note that this parabolic problem can be singular or degenerate according to the value of γ). In our definition we do not restrict the test functions but we give a meaning to ΔNu in terms of the largest and the smallest eigenvalue of D2u at points where the gradient vanishes. With our definition we can prove a comparison principle in the following section.

3 Comparison principle and uniqueness

In this section, we start giving sufficient conditions on f to prove a comparison principle and hence obtain uniqueness for (2.1).

Definition 3.1.

Given a positive function hC1(0,+) such that hL1(0,1) and h(s)/h2(s) is nondecreasing, we say that f:Ω× satisfies the h-decreasing condition if for every xΩ,

h(s)f(x,s) is decreasing with respect to s.(3.1)

Remark 3.2.

Observe that if f(x,s)=f0(x)>0, that is, f does not depend on s, then f satisfies the h-decreasing condition for h(s)=1/sq for any 0<q<1. In addition, when f(x,s)=f0(x)sq>0 for some 0q<1, then f satisfies the h-decreasing condition for h(s)=1/sq+ε for any 0<ε<1-q. Moreover, taking a decreasing function h, we obtain that any function 0<f𝒞1(Ω×) nonincreasing with respect to s also satisfies the h-decreasing condition (since h(s)f(x,s)+h(s)fs(x,s)<0 in this case).

Theorem 3.3.

Assume that 0<fC(Ω×R) satisfies the h-decreasing condition. Let u¯,u¯C(Ω¯) be a sub- and a supersolution, respectively, of -ΔpNu=f(x,u) such that u¯>0 in Ω and u¯u¯ on Ω. Then u¯u¯ in Ω¯.

Proof.

We argue by contradiction following closely the ideas in [9]. Suppose that Ω+={xΩ¯:u¯(x)>u¯(x)} is nonempty. Let

H(s)=0sh(t)𝑑t

for s0. By hypothesis, u¯u¯ on Ω. Using that u¯,u¯C(Ω¯), we have that there exists x^Ω+ with

H(u¯(x^))-H(u¯(x^))=supxΩ+H(u¯(x))-H(u¯(x))>0.

Since Ω+ is an open set, we can take Ω^, an open neighborhood of x^, such that Ω^¯Ω+. Now, let w¯ and w¯ be the positive functions defined for xΩ^ by

w¯(x)=H(u¯(x))andw¯(x)=H(u¯(x)).

Clearly w¯,w¯C(Ω^¯) and

w¯(x)>w¯(x)>0,xΩ^.(3.2)

Now, we claim that w¯ and w¯ are a sub- and a supersolution (in the viscosity sense) of the equation

-ΔpNw+h(H-1(w))h2(H-1(w))|w|2=h(H-1(w))f(x,H-1(w))in Ω^.($Q$)

Indeed, we proceed to show that w¯ is a subsolution (the fact that w¯ is a supersolution can be proved in the same way). For every x0Ω^, we take φ𝒞2(Ω^) with φ(x0)=w¯(x0) and φ(x)>w¯(x) for every xΩ^{x0}. If φ(x0)0 and we take φ~=H-1(φ), then it is easy to check that

-ΔpNφ(x0)+h(H-1(φ(x0)))h2(H-1(φ(x0)))|φ(x0)|2=-αΔφ(x0)-βΔφ(x0)+h(φ~(x0))|φ~(x0)|2=-αΔφ~(x0)h(φ~(x0))-αh(φ~(x0))|φ~(x0)|2-βΔφ~(x0)h(φ~(x0))-βh(φ~(x0))|φ~(x0)|2+h(φ~(x0))|φ~(x0)|2=-ΔpNφ~(x0)h(φ~(x0)).

Now, taking into account that φ~(x0)=u¯(x0) and (φ~-u¯)(x)>0 in Ω^{x0}, it follows that φ~ is a test function touching from above u at x0. Thus, since u¯ is a subsolution of -ΔpNu=f(x,u), we get

-ΔpNφ~(x0)f(x0,H-1(φ~(x0))).

Consequently,

-ΔpNφ(x0)+h(H-1(φ(x0)))h2(H-1(φ(x0)))|φ(x0)|2h(H-1(φ(x0)))f(x0,H-1(φ(x0))).

In the case φ(x0)=0, since φ~(x0)=0 and D2φ(x0)=h(φ~(x0))D2φ~(x0), we have

-αΔφ(x0)-βM(D2φ(x0))=-αΔφ~(x0)h(φ~(x0))-βM(D2φ~(x0))h(φ~(x0))h(H-1(φ(x0)))f(x0,H-1(φ(x0))).

Therefore, we conclude that w¯ is a subsolution of problem ($Q$), which was our claim.

Now, consider the sequence of functions

Ψn(x,y)=w¯(x)-w¯(y)-n4|x-y|4,(x,y)Ω^¯×Ω^¯,n.

For every n, let (xn,yn)Ω^¯×Ω^¯ be such that

Ψn(xn,yn)=supΩ^¯×Ω^¯Ψn(x,y).

We note that Ψn(xn,yn) is finite since w¯-w¯ is continuous and Ω^¯ is compact. Moreover,

Ψn(xn,yn)Ψ(x,x)=w¯(x)-w¯(x)>0

Furthermore, we can assume that xn,ynx^,y^, w¯(xn)w¯(x^) and w¯(yn)w¯(y^) as n, and that x^=y^ (see [9, Lemma 3.1 and Proposition 3.7]). Next, by [9, Theorem 3.2], there exist Xn,Yn𝕊N satisfying

  • (i)

    XnYn,

  • (ii)

    (ηn,Xn)J¯Ω^2,+(w¯(xn)), (ηn,Yn)J¯Ω^2,-(w¯(yn)),

  • (iii)

    Xn0Yn for xn=yn,

where ηn=n|xn-yn|2(xn-yn).

Hence, if xnyn, having in mind that w¯ and w¯ are sub- and supersolution of ($Q$) and using Remark 2.4, we obtain that

h(H-1(w¯(yn)))f(yn,H-1(w¯(yn)))-αtr(Yn)-βηn|ηn|(Ynηn|ηn|)+h(H-1(w¯(yn)))h2(H-1(w¯(yn)))|ηn|2-αtr(Xn)-βηn|ηn|(Xnηn|ηn|)+h(H-1(w¯(xn)))h2(H-1(w¯(xn)))|ηn|2+(h(H-1(w¯(yn)))h2(H-1(w¯(yn)))-h(H-1(w¯(xn)))h2(H-1(w¯(xn))))|ηn|2h(H-1(w¯(xn)))f(xn,H-1(w¯(xn)))+(h(H-1(w¯(yn)))h2(H-1(w¯(yn)))-h(H-1(w¯(xn)))h2(H-1(w¯(xn))))|ηn|2.

Letting n, by the continuity of w¯, w¯, f, h, h, and using that h/h2 is nondecreasing, we get

h(H-1(w¯(x^)))f(x^,H-1(w¯(x^)))h(H-1(w¯(x^)))f(x^,H-1(w¯(x^))).

This is a contradiction to (3.2) since it implies, by using (3.1), that

h(H-1(w¯(x^)))f(x^,H-1(w¯(x^)))>h(H-1(w¯(x^)))f(x^,H-1(w¯(x^))).

If xn=yn for nn0, then ηn=0 and by (iii) we have

h(H-1(w¯(yn)))f(yn,H-1(w¯(yn)))-αtr(Yn)-βm(Yn)-αtr(Xn)-βM(Xn)h(H-1(w¯(xn)))f(xn,H-1(w¯(xn)));

arguing as above, this leads to a contradiction. ∎

Let us extract easy consequences of this comparison principle.

Proposition 3.4 (Uniqueness).

Assume that 0<fC(Ω×R) satisfies the h-decreasing condition. Then there exists at most one positive viscosity solution of

{-ΔpNu(x)=f(x,u)in Ω,u=0on Ω.(${P}$)

Proof.

Suppose that there exist two solutions u1,u20 of (${P}$). Using Theorem 3.3 twice, we obtain that u1u2 and u2u1, and we conclude that u1=u2. ∎

The next result improves [20], where a starshaped condition on the domain Ω was required.

Corollary 3.5.

As a particular case, we can assert that there exists a unique positive solution of

{-ΔpNu(x)=λuqin Ω,u=0on Ω

for every λ>0 and 0<q<1. Moreover, for λ=0, the problem admits as unique solution u=0.

Proof.

For λ>0, the uniqueness is due to Proposition 3.4 and the existence due to [20, Theorem 3.1] (which can be extended to the case p= by using the same iterative procedure as in [20, Theorem 3.1]). For λ=0, we observe that u is a solution of -ΔpNu=0 if and only if -Δpu=0 in the viscosity sense, (this holds since it is enough to test the equation -Δpu=0 with test functions with φ0; see [15]). Thus, the trivial solution u=0 is the unique solution when λ=0. ∎

4 Existence of minimal solutions for the Gelfand problem

The first result of this section shows how one can pass to the limit in a sequence of viscosity solutions of a sequence of problems to obtain a viscosity solution of the limit problem.

Lemma 4.1.

Let un, fnC(Ω) and pn[2,] be three sequences satisfying

-ΔpnNun=fn,(4.1)

in the viscosity sense, such that fnf, unu uniformly for every ωΩ and pnp[2,]. Then u is a viscosity solution to the problem

-ΔpNu=f.

Proof.

First, we prove that u is a subsolution. For every x0Ω, we take φ𝒞2(Ω) such that φ(x0)=u(x0) and φ-u>0 in Ω{x0}. Fix δ>0 such that Bδ(x0)¯Ω, and for every n we consider xn as the strict minimum point (not necessarily unique) of φ-un in Bδ(x0)¯, i.e.,

(φ-un)(xn)(φ-un)(x)for all xBδ(x0)¯.

Up to a subsequence, we can assume that xnx*Bδ(x0)¯. Using that un is continuous and that the sequence un uniformly converges to u, we deduce that un(xn)u(x*). We obtain, taking limits in the above inequality, that

(φ-u)(x*)(φ-u)(x)for all xBδ(x0)¯,

and we can assert that x*=x0. We set

φn(x)=φ(x)+un(xn)-φ(xn)+x-xn4,xBδ(x0)¯.

It is easy to check that φn satisfies

φn(xn)=un(xn),φn(xn)=φ(xn),D2φn(xn)=D2φ(xn),(φn-un)(x)>0

in a neighborhood of xn. Thus, using that un is a subsolution of (4.1) and taking φn as test function, we obtain the following:

  • (i)

    If φn(xn)0, then -αnΔφn(xn)-βnΔφn(xn)fn(xn), and thus

    -αnΔφ(xn)-βnΔφ(xn)fn(xn).(4.2)

  • (ii)

    If φn(xn)=0, then -αnΔφn(xn)-βnM(D2φn(xn))fn(xn), and thus

    -αnΔφ(xn)-βnM(D2φ(xn))fn(xn),(4.3)

where αn=1/(pn-1), βn=(pn-2)/(pn-1) if pn<+, and αn=0, βn=1 if pn=.

Now, denoting α=1/(p-1), β=(p-2)/(p-1) if p<+, and α=0, β=1 in the other case, we distinguish three different cases. Case 1: φ(x0)0. In this case, we can suppose that, up to a subsequence, φn(xn)0 for nn0 and, taking into account that φ𝒞2 and the continuity and uniform convergence of fn, we can pass to the limit in (4.2) as n to obtain

-αΔφ(x0)-βΔφ(x0)f(x0).

Case 2: φ(x0)=0 and, up to a subsequence, φn(xn)0 for nn0. In this case, since

Δφ(xn)M(D2φ(xn)),

replacing in (4.2), we get (4.3). Taking limits, we obtain the desired inequality

-αΔφ(x0)-βM(D2φ(x0))f(x0).(4.4)

Case 3: φ(x0)=φn(xn)=0 for nn0. We obtain (4.4) directly from (4.3).

On the other hand, to prove that u is a supersolution, we argue in a similar way. To be more specific, for every x0Ω we take the test function ψ𝒞2(Ω) satisfying that u-ψ has a strict minimum at x0 with ψ(x0)=u(x0). Now, taking xn, the strict minimum of un-ψ in Bδ(x0)¯Ω, we set

ψn(x)=ψ(x)+un(xn)-ψ(xn)-x-xn4

as the test function in (4.1) touching the graph of un from below in xn. The rest of the proof runs as before. ∎

Now we can prove the existence of minimal solutions of ((${P_{\lambda,p}}$)) for λ small and the nonexistence of solutions for λ large, that is, we prove Theorem 1.1.

Proof of Theorem 1.1.

Let z𝒞2([0,1]) be a classical solution to the problem

{-z′′(r)-α(N-1)z(r)r=λez(r),r in (0,1),z(1)=0,z(0)=0,(4.5)

with

α=1p-1if p<+ andα=0 in the other case.

Then u(x):=z(|x|) is a solution to the problem

{-ΔpNu=λeuin B1,u>0in B1,u=0on B1,(4.6)

in the sense of Definition 2.1 (iii) (see also Remark 2.2). Due to [14], it is well known that there exists a positive number λ~(B1), depending only on p, N, such that problem (4.5) has no solution for λ>λ~(B1). Moreover, for every 0λ<λ~(B1) there exists a classical solution z𝒞2([0,1]) (see also [13] for a complete description of the multiplicity of solutions). Observe that for any classical solution z𝒞2([0,1]), λ0, of (4.5) it holds that λλ~(B1) (we refer again to [13] for a complete description of the multiplicity of solutions).

Note also that the relationship between classical solutions of (4.5) and viscosity radial solutions of (4.6) is bidirectional. Given a solution u𝒞(B1¯) of (4.6) radially symmetric and decreasing, then z(r)=u(|x|) for some xΩ with |x|=r satisfies (4.5) in the weak sense (which is equivalent to be a classical solution in this case).

By taking into account Remark 2.2, u is also a solution to our problem in the viscosity sense.

Now, for any fixed R>0, we can rescale the problem and consider

v(r):=z(rR).

It is easy to check that we arrive at the ODE

{-v′′(r)-α(N-1)v(r)r=λR2ev(r)in (0,R),v(R)=0,v(0)=0.(4.7)

Summarizing, we have that there exists a positive value

λ~(BR)=λ~(B1)R2>0,

which is decreasing with respect to R, such that problem ((${P_{\lambda,p}}$)) admits at least a solution for every λ<λ~(BR) in the ball of radius R, Ω=BR.

Let now Ω be a bounded domain and R1>0 given by

R1=min{R>0:ΩBR}.

Notice that if uR1 is a solution in BR1 for some Λ<λ~(BR1), then it is a supersolution in Ω for λΛ<λ~(BR1). We claim that there exists a solution of problem ((${P_{\lambda,p}}$)) with λ=Λ. Indeed, to prove this fact we use a standard monotone iteration argument: let w0=0, and for every n1 we define the recurrent sequence {wn} by

{-ΔpNwn=λewn-1in Ω,wn>0in Ω,wn=0on Ω.(${Q_{n}}$)

The sequence {wn}𝒞(Ω¯) is well defined by [18, 24]; see also [17]. Note that we are solving a problem of the form -ΔpNwn=f in Ω, with f>0 and wn=0 on Ω as boundary condition. Then the existence is a consequence of a limit procedure involving game theory (in this problem the right-hand side, f, enters into the problem as a running payoff and the boundary condition wn=0 as the final payoff). The existence of such a solution can also be proved directly by using Perron’s method thanks to our general comparison principle.

Moreover, the sequence {wn} is increasing with n. Indeed, taking into account that 0<w1, we obtain λew0λew1, and by using the comparison principle in Theorem 3.3, it follows that w1w2. By an inductive argument, we get 0<w1w2wn for all n1. From the fact that uR1 is a supersolution of problem ((${P_{\lambda,p}}$)), with a similar inductive argument, we prove that wnuR1 for every n.

Since uR1L(Ω), the sequence {wn(x)} is increasing and bounded by uR1(x); therefore, there exists

wλ(x):=limnwn(x).

In addition, thanks to the subtle Krylov–Safonov 𝒞0,α-estimates of wn for every p[2,] (here we refer to [6, 7]), we obtain that wnwλ uniformly. Taking fn=λewn-1 and pn=p in Lemma 4.1, we get that wλ is a solution of problem ((${P_{\lambda,p}}$)).

To prove that the obtained solution wλ is minimal let vλ be a solution of problem ((${P_{\lambda,p}}$)). By a similar argument, using the comparison principle and induction in n, we have wnvλ for all n. As wλ(x)=limnwn(x) (we use again comparison here), we obtain wλvλ.

We have thus proved that for every λ<λ~(BR1) there exists a minimal solution wλ of problem ((${P_{\lambda,p}}$)). In particular,

0<λ~(BR1)λ*(Ω,N,p)=sup{λ>0: a minimal solution of (Pλ,p)}.

Now to ensure that λ*(Ω,N,p)< let

R2=max{R>0:BRΩ};

we remark that without loss of generality we can assume that 0Ω. In that way, taking wλ, the minimal solution in Ω, as a supersolution in BR2 and applying the above argument again, with Ω replaced by BR2, we obtain that λ*(Ω,N,p)λ*(BR2,N,p).

Note that in the case Ω=Br we can perform the previous argument starting with w0=0 and obtain that the minimal solution is radial. In fact, by uniqueness, in this case wn is radial for every n. Remark that in this case the unique minimal solution leads to a solution of the ODE (4.7), and thus λ*(BR2,N,p)λ~(BR2). ∎

Remark 4.2.

The arguments used in the previous proof show that the extremal parameter verifies

λ*(Ω,N,p)=sup{λ>0:there exists a minimal solution of (Pλ,p)}=sup{λ>0:there exists a solution of (Pλ,p)}=sup{λ>0:there exists a nonnegative supersolution of (Pλ,p)}.

Also note that

λ*(Ω1,N,p)λ*(Ω2,N,p) when Ω2Ω1,

and that the extremal value for a ball, Ω=BR, is the one that corresponds to the existence of a radial solution; we refer to [13, 14] for the analysis of the resulting ODE.

In addition, we note that, if we have a solution to our problem, then the inequality

-ΔpNu=λeuλu

holds. Therefore, we must have λλ1,p(Ω), where λ1,p(Ω) is the first eigenvalue of the operator -ΔpN with Dirichlet boundary conditions. We conclude that

λ*(Ω,N,p)λ1,p(Ω).

5 Unbounded continua of solutions

For the reader’s convenience, we recall the following general results from the theory of global continua of solutions using degree theory, which will be essential for our analysis. For the proofs we refer to [1, 26, 16].

Theorem 5.1 (Continuation theorem of Leray–Schauder).

Let X be a real Banach space, O an open bounded subset of X and assume that T:R×XX is completely continuous (i.e., relatively compact and continuous). Furthermore, assume that for λ=λ0 we have that uT(λ0,u) for every uO and deg(I-T(λ0,),O,0)0. Let

Σ={(λ,u)[λ0,)×X:u=T(λ,u)}.

Then there exists a maximal connected and closed CΣ. Moreover, the following statements are valid:

  • (i)

    𝒞{λ0}×𝒪.

  • (ii)

    Either 𝒞 is unbounded or 𝒞{λ0}×X𝒪¯.

Theorem 5.2 (Homotopy property).

Let X be a real Banach space, let O be an open subset of X and let TC([0,1]×O¯,X) be completely continuous in [0,1]×O¯. If b:[0,1]X is continuous and b(t)u-T(t,u) in [0,1]×O, then deg(I-T,O,b(t)) remains constant for all t[0,1].

Theorem 5.3 (Classical Leray–Schauder’s theorem).

Let X be a real Banach space, let OX be an open and bounded subset of X and let Φ:[a,b]×O¯X be given by Φ(t,u)=u-T(t,u), with T being completely continuous. We also assume that

Φ(t,u)ufor all (t,u)[a,b]×𝒪.

Then, if deg(Φ(a,),O,0)0, the following assertions hold:

  • (i)

    The equation Φ(t,u)=0 with uX has a solution in 𝒪 for every atb.

  • (ii)

    There exists a closed and connected set Σa,b{(t,u)[a,b]×X:u=T(t,u)} that intersects t=a and t=b.

Let us consider the operator

K:[0,1]××𝒞(Ω¯)𝒞(Ω¯)

by defining u:=K(t,λ,w), for every t[0,1], λ and w𝒞(Ω¯), as the unique solution in 𝒞(Ω¯) of the problem

{-Δp(t)Nu=λ+ew+in Ω,u=0on Ω,

where

p(t)=t-2t-1.

That is, -Δp(t)Nu=-(1-t)Δu-tΔu. Notice that every p(t)[2,] is labeled by a unique t[0,1] (and conversely), thus K is well defined.

Now, we prove that K is completely continuous, which allows us to apply the Leray–Schauder degree techniques (see [16]), in order to study the existence of “continua of solutions” of ((${P_{\lambda,p}}$)), i.e., connected and closed subsets in the solution set

𝒮p={(λ,u)[0,)×𝒞(Ω¯):K(p-2p-1,λ,u)=u}

for every fixed p[2,+], or, if we fixed λ instead, in

𝒮λ={(p,u)[2,]×𝒞(Ω¯):K(p-2p-1,λ,u)=u}.

Lemma 5.4.

Let us assume that unC(Ω¯) satisfies

{-Δp(tn)Nun=λnewnin Ω,un=0on Ω,

with tn[0,1] and 0λn, wn bounded in R×C(Ω¯). Then, up to a subsequence, un is strongly convergent to uC(Ω¯). If, in addition, λnλ, tnt and wn converges in C(Ω¯) to w, then u is a solution of the problem

{-Δp(t)Nu=λewin Ω,u=0on Ω.

Proof.

If λn=0, then un=0 is the unique solution (Corollary 3.5) and the proof is immediate. In the other case, since 0<λnewnC for some positive constant, un is a subsolution of the problem

{-Δp(tn)Nv=Cin Ω,v=0on Ω.

It is well known, by the theory of uniformly elliptic fully nonlinear equations, that, for every fixed n, un𝒞0,ν(n)(Ω¯) whenever 2p(tn)M for some M sufficiently large (for instance, greater than the dimension N), with 0<ν(n)<1 (see [6, 11]). We stress that this Hölder estimates depend on the ratio between the ellipticity constants, which in this case is p(tn)-1 and, consequently, it blows-up as p(tn). However, for p(tn)[M,] it is shown in [7, Theorem 7] that

un𝒞0,ρ(n)(Ω¯)for ρ(n)=p(tn)-Np(tn)-1.

Thus, we can assert that the sequence un𝒞0,γ(Ω¯), where γ=min{ν(n),ρ(n):n}. Hence, the Ascolí–Arzelá theorem gives that un possesses a subsequence converging in 𝒞(Ω¯), which concludes the first part of the lemma. Finally, the second part is a direct consequence of the uniqueness of solutions by Proposition 3.4 and Lemma 4.1. ∎

Proof of Theorem 1.2.

For fixed R>0, let 𝒪R be the open ball of radius R of 𝒞(Ω¯), and we fix some λR with

0<λR<R2d2eR,

where d is the diameter of Ω.

By Lemma 5.4, we obtain that K𝒞([0,1]×[0,λR]×𝒪R¯,𝒞(Ω¯)) and K(t,λ,) is completely continuous for every (t,λ)[0,1]×[0,λR]. Now, in order to apply Theorem 5.2 twice for the parameters (t,λ) with b(t,λ)0𝒞(Ω¯), we must check an a priori bound of the solutions of the equation u=K(t,λ,u). That is, uK(t,λ,u) in [0,1]×[0,λR]×𝒪R. In fact, we argue by contradiction: Suppose that u=R and there exist t[0,1] and λ[0,λR] such that u satisfies the equation

-Δp(t)Nu=λeuin Ω,

hence u is a subsolution of problem

-Δp(t)Nv=λeRin Ω.

On the other hand, a simple computation of [7, Theorem 1 and Theorem 3] shows that if v𝒞(Ω¯) is a nonnegative subsolution of the Poisson problem

-ΔpNv=f(x)in Ω,

with 0f𝒞(Ω¯) and p[2,], then v2d2f. Applying this last result, we get the following contradiction:

R=u2d2λeR2d2λReR<R.

In this way, due to the homotopy property, we obtain

deg(I-K(t,λ,),𝒪R,0)=constfor all (t,λ)[0,1]×[0,λR].

Moreover, since

K(0,λ,w)=(-Δ)-1(λew+)

is the inverse of the Laplacian operator and it is well known that

deg(I-K(0,0,),𝒪R,0)=1,

we get

1=deg(I-K(0,0,),𝒪R,0)=deg(I-K(t,λ,),𝒪R,0).

In order to conclude this proof, we apply the continuation theorem of Leray–Schauder (Theorem 5.1) with T(λ,u)=K(t,λ,u) for every fixed t[0,1], which is completely continuous (Lemma 5.4). Therefore, using that deg(I-T(0,),𝒪R,0)=10, we can assert that there exists a maximal connected subset 𝒞 of 𝒮p that contains (0,0). Furthermore, 𝒞 is not bounded since 0 is the unique solution for λ=0. Finally, since for every λ such that there is a solution of ((${P_{\lambda,p}}$)) we can construct a minimal solution, we can state that 𝒞[0,λ*]×𝒞(Ω¯).

With the same arguments, using Theorem 5.3 with T(t,u)=K(t,λ,u) and [a,b]=[0,1], for every fixed

λ(0,λ0=min{λ*(Ω,N,2),12d2e})

we can obtain the existence of a continuum of solutions moving p[2,]. More precisely, since

deg(I-K(0,λ,),𝒪1,0)=1,

we can apply Theorem 5.3 obtaining the existence of a continuum Σ0,1{(t,u)[0,1]×𝒪1:u=T(t,u)} such that Proj[0,1]Σ0,1=[0,1]. Note that the upper bound for λ is used to ensure an a priori bound. Thus, we finish the proof by taking

𝒟={(t-2t-1,u)[2,+]×𝒪1:(t,u)Σ0,1}.

Remark 5.5.

Now we briefly comment on possible extensions for more general nonlinearities. Note that we can also deal with the equation

-ΔpNu=λf(u),

with a general continuous nonlinearity f that verifies f(0)>0, f(s)/sk>0 and is increasing. In fact, we only need to show the existence and nonexistence of radial solutions (the rest of the arguments can be extended without much difficulties). Hence we arrive at the problem

{-z′′(r)-θz(r)r=λf(z(r)),r(0,1),z(r)>0,r(0,1),z(1)=z(0)=0,

where θ=(N-1)/(p-1)[0,) due to the fact that p[2,]. Multiplying by rθ and integrating twice, we obtain

z(r)=λr11τθ0τsθf(z(s))𝑑s𝑑τλr11τθ0rsθf(z(s))𝑑s𝑑τλr11τθ0rsθf(z(r))𝑑s𝑑τ.

Therefore, for every r(0,1) it must hold that

1kz(r)f(z(r))λr10r(sτ)θ𝑑s𝑑τ:=λFθ(r).

As Fθ(r) is positive in (0,1) and is bounded above, we conclude that λ1/(c(θ)k). Hence there is no solution for λ greater than a constant that depends only on p and N.

To look for the existence of solutions for small λ we can use degree theory for the operator

T:[0,)×C([0,1])C([0,1])

given by

T(λ,u)=λr11τθ0τsθf(u(s))𝑑s𝑑τ.

Since f is assumed to be continuous, it is easy to check that T is completely continuous. Now, as T(0,u)=0 for every uC([0,1]), using Leray–Schauder’s theorem, we obtain the existence of a continuum of solutions 𝒞[0,)×C([0,1]) that is unbounded with (0,0)𝒞. In particular, there exist solutions for values of λ close to 0.

Acknowledgements

Part of this work was done during a visit of the second author at Universidad de Buenos Aires (UBA). He thanks for the nice atmosphere and hospitality.

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About the article

Received: 2016-11-01

Revised: 2017-03-18

Accepted: 2017-04-17

Published Online: 2017-07-21


The first author was partially supported by MINECO–FEDER Grant MTM2015-68210-P (Spain) and Junta de Andalucía FQM-194 (Spain). The second author was partially supported by MINECO–FEDER Grant MTM2015-68210-P (Spain), Junta de Andalucía FQM-116 (Spain) and MINECO Grant BES-2013-066595 (Spain). The third author was partially supported by CONICET (Argentina) and MINECO–FEDER Grant MTM2015-70227-P (Spain).


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 545–558, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0233.

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