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Advances in Nonlinear Analysis

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


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A fractional Kirchhoff problem involving a singular term and a critical nonlinearity

Alessio FiscellaORCID iD: https://orcid.org/0000-0001-6281-4040
Published Online: 2017-08-03 | DOI: https://doi.org/10.1515/anona-2017-0075

Abstract

In this paper, we consider the following critical nonlocal problem:

{M(2N|u(x)-u(y)|2|x-y|N+2s𝑑x𝑑y)(-Δ)su=λuγ+u2s*-1in Ω,u>0in Ω,u=0in NΩ,

where Ω is an open bounded subset of N with continuous boundary, dimension N>2s with parameter s(0,1), 2s*=2N/(N-2s) is the fractional critical Sobolev exponent, λ>0 is a real parameter, γ(0,1) and M models a Kirchhoff-type coefficient, while (-Δ)s is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is, when the Kirchhoff function M is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.

Keywords: Kirchhoff-type problems; fractional Laplacian; singularities; critical nonlinearities; perturbation methods

MSC 2010: 35J75; 35R11; 49J35; 35A15; 45G05; 35S15

1 Introduction

This paper is devoted to the study of a class of Kirchhoff-type problems driven by a nonlocal fractional operator and involving a singular term and a critical nonlinearity. More precisely, we consider

{(2N|u(x)-u(y)|2|x-y|N+2s𝑑x𝑑y)θ-1(-Δ)su=λuγ+u2s*-1in Ω,u>0in Ω,u=0in NΩ,(1.1)

where Ω is an open bounded subset of N with continuous boundary, dimension N>2s with parameter s(0,1), 2s*=2N/(N-2s) is the fractional critical Sobolev exponent, λ>0 is a real parameter, θ(1,2s*/2), while γ(0,1). Here (-Δ)s is the fractional Laplace operator defined, up to normalization factors, by the Riesz potential as

(-Δ)sφ(x)=N2φ(x)-φ(x+y)-φ(x-y)|y|N+2s𝑑y,xN,

along any φC0(Ω); we refer to [11] and the recent monograph [22] for further details on the fractional Laplacian and the fractional Sobolev spaces Hs(N) and H0s(Ω).

As is well explained in [11, 22], problem (1.1) is the fractional version of the following nonlinear problem:

{-M(Ω|u(x)|2𝑑x)Δu=λuγ+u2*-1in Ω,u>0in Ω,u=0in Ω,(1.2)

where Δ denotes the classical Laplace operator while, just for a general discussion, M(t)=tθ-1 for any t0+. In literature, problems like (1.1) and (1.2) are called of Kirchhoff type whenever the function M:0+0+ models the Kirchhoff prototype, given by

M(t)=a+btθ-1,a,b0,a+b>0,θ1.(1.3)

In particular, when M(t)constant>0 for any t0+, Kirchhoff problems are said to be non-degenerate and this happens for example if a>0 in the model case (1.3). While if M(0)=0 but M(t)>0 for any t+, Kirchhoff problems are called degenerate. Of course, for (1.3) this occurs when a=0.

This kind of nonlocal problems has been widely studied in recent years. We refer to [17, 18, 19, 21, 20] for different Kirchhoff problems with M like in (1.3), driven by the Laplace operator and involving a singular term of type u-γ. In [21], Liu and Sun study a Kirchhoff problem with a singular term and a Hardy potential by using the Nehari method. The same approach is used in [19] for a singular Kirchhoff problem with also a subcritical term. In [17], strongly assuming a>0 in (1.3), Lei, Liao and Tang prove the existence of two solutions for a Kirchhoff problem like (1.2) by combining perturbation and variational methods. While in [18], Liao, Ke, Lei and Tang provide a uniqueness result for a singular Kirchhoff problem involving a negative critical nonlinearity by a minimization argument. By arguing similarly to [17], Liu, Tang, Liao and Wu [20] give the existence of two solutions for a critical Kirchhoff problem with a singular term of type |x|-βu-γ.

Problem (1.1) has been studied by Barrios, De Bonis, Medina and Peral [4] when θ=1, namely without a Kirchhoff coefficient. They prove the existence of two solutions by applying the sub/supersolutions and Sattinger methods. In [8], Canino, Montoro, Sciunzi and Squassina generalize the results of [4, Section 3] to the delicate case of the p-fractional Laplace operator (-Δ)ps. While in the last section of [1], Abdellaoui, Medina, Peral and Primo provide the existence of a solution for nonlinear fractional problems with a singularity like u-γ and a fractional Hardy term by perturbation methods. Concerning fractional Kirchhoff problems involving critical nonlinearities, we refer to [2, 9, 13, 14, 16, 23] for existence results and to [5, 12, 24, 25, 29] for multiplicity results. In particular, in [9, 13, 14, 23] different singular terms appear, but are given by the fractional Hardy potential.

Inspired by the above works, we study a multiplicity result for problem (1.1). As far as we know, a fractional Kirchhoff problem involving a singular term of type u-γ has not been studied yet. We can state our result as follows.

Theorem 1.1.

Let s(0,1), N>2s, θ(1,2s*/2), γ(0,1) and let Ω be an open bounded subset of RN with Ω continuous. Then there exists λ¯>0 such that for any λ(0,λ¯) problem (1.1) has at least two different solutions.

The first solution of problem (1.1) is obtained by a suitable minimization argument, where we must pay attention to the nonlocal nature of the fractional Laplacian. Concerning the second solution, because of the presence of u-γ, we can not apply the usual critical point theory to problem (1.1). For this, we first study a perturbed problem obtained by truncating the singular term u-γ. Then by approximation we get our second solution of (1.1).

Finally, we observe that Theorem 1.1 generalizes in several directions the first part of [4, Theorem 4.1] and [17, Theorem 1.1].

The paper is organized as follows: In Section 2, we discuss the variational formulation of problem (1.1), and we introduce the perturbed problem. In Section 3, we prove the existence of the first solution of (1.1), and we give a possible generalization of this existence result at the end of the section. In Section 4, we prove the existence of a mountain pass solution for the perturbed problem. In Section 5, we prove Theorem 1.1.

2 Variational setting

Throughout this paper, we assume without further mentioning that s(0,1), N>2s, θ(1,2s*/2), γ(0,1) and Ω is an open bounded subset of N with Ω continuous. As a matter of notations, we denote with φ+=max{φ,0} and φ-=max{-φ,0} respectively the positive and negative part of a function φ.

Problem (1.1) has a variational structure, and the natural space where to find solutions is the homogeneous fractional Sobolev space H0s(Ω). In order to study (1.1) it is important to encode the “boundary condition” u=0 in NΩ in the weak formulation, by considering also that the interaction between Ω and its complementary in N gives a positive contribution in the so-called Gagliardo norm, given as

uHs(N)=uL2(N)+(2N|u(x)-u(y)|2|x-y|N+2s𝑑x𝑑y)1/2.(2.1)

The functional space that takes into account this boundary condition will be denoted by X0 and it is defined as

X0={uHs(N):u=0 a.e. in NΩ}.

We refer to [26] for a general definition of X0 and its properties. We also would like to point out that when Ω is continuous, by [15, Theorem 6] the space C0(Ω) is dense in X0, with respect to the norm (2.1). This last point will be used to overcome the singularity in problem (1.1).

In X0 we can consider the norm

uX0=(2N|u(x)-u(y)|2|x-y|N+2s𝑑x𝑑y)1/2,

which is equivalent to the usual one defined in (2.1) (see [26, Lemma 6]). We also recall that (X0,X0) is a Hilbert space, with the scalar product defined as

u,vX0=2N(u(x)-u(y))(v(x)-v(y))|x-y|N+2s𝑑x𝑑y.

From now on, in order to simplify the notation, we will denote X0 and ,X0 by and ,, respectively, and Lq(Ω) by q for any q[1,].

In order to present the weak formulation of (1.1) and taking into account that we are looking for positive solutions, we will consider the following Kirchhoff problem:

{(2N|u(x)-u(y)|2|x-y|N+2s𝑑x𝑑y)θ-1(-Δ)su=λ(u+)γ+(u+)2s*-1in Ω,u=0in NΩ.(2.2)

We say that uX0 is a (weak) solution of problem (2.2) if u satisfies

u2(θ-1)u,φ=λΩφ(u+)γ𝑑x+Ω(u+)2s*-1φ(2.3)

for any φX0. Problem (2.2) has a variational structure and Jλ:X0, defined by

Jλ(u)=12θu2θ-λ1-γΩ(u+)1-γ𝑑x-12s*u+2s*2s*,

is the underlying functional associated to (2.2). Because of the presence of a singular term in (2.2), the functional Jλ is not differentiable on X0. Therefore, we can not apply directly the usual critical point theory to Jλ in order to solve problem (2.2). However, it is possible to find a first solution of (2.2) by using a local minimization argument. In order to get the second solution of (2.2) we have to study an associated approximating problem. That is, for any n, we consider the following perturbed problem:

{(2N|u(x)-u(y)|2|x-y|N+2s𝑑x𝑑y)θ-1(-Δ)su=λ(u++1n)γ+(u+)2s*-1in Ω,u=0in NΩ.(2.4)

For this, we say that uX0 is a (weak) solution of problem (2.4) if u satisfies

u2(θ-1)u,φ=λΩφ(u++1n)γ𝑑x+Ω(u+)2s*-1φ(2.5)

for any φX0. In this case, solutions of (2.4) correspond to the critical points of the functional Jn,λ:X0, set as

Jn,λ(u)=12θu2θ-λ1-γΩ[(u++1n)1-γ-(1n)1-γ]𝑑x-12s*u+2s*2s*.(2.6)

It is immediate to see that Jn,λ is of class C1(X0).

We conclude this section by recalling the best constant of the fractional Sobolev embedding, which will be very useful to study the compactness property of the functional Jn,λ. That is, we consider

S=infvHs(N)v02N|v(x)-v(y)|2|x-y|N+2s𝑑x𝑑y(N|v(x)|2s*𝑑x)2/2s*,(2.7)

which is well defined and strictly positive, as shown in [10, Theorem 1.1].

3 A first solution for problem (1.1)

In this section, we prove the existence of a solution for problem (1.1) by a local minimization argument. For this, we first study the geometry of the functional Jλ.

Lemma 3.1.

There exist numbers ρ(0,1], λ0=λ0(ρ)>0 and α=α(ρ)>0 such that Jλ(u)α for any uX0, with u=ρ, and for any λ(0,λ0]. Furthermore, set

mλ=inf{Jλ(u):uB¯ρ},

where B¯ρ={uX0:uρ}. Then mλ<0 for any λ(0,λ0].

Proof.

Let λ>0. From the Hölder inequality and (2.7) for any uX0 we have

Ω(u+)1-γ𝑑x|Ω|(2s*-1+γ)/2s*u2s*1-γ|Ω|(2s*-1+γ)/2s*S-(1-γ)/2u1-γ.(3.1)

Hence, using again (2.7) and (3.1), we get

Jλ(u)12θu2θ-S-2s*/22s*u2s*-λ1-γ|Ω|(2s*-1+γ)/2s*S-(1-γ)/2u1-γ.

Since 1-γ<1<2θ<2s*, the function

η(t)=12θt2θ-1+γ-S-2s*/22s*t2s*-1+γ,t[0,1],

admits a maximum at some ρ(0,1] small enough, that is, maxt[0,1]η(t)=η(ρ)>0. Thus, let

λ0=(1-γ)S(1-γ)/22|Ω|(2s*-1+γ)/2s*η(ρ).

Then for any uX0 with u=ρ1 and for any λλ0, we get Jλ(u)ρ1-γη(ρ)/2=α>0.

Furthermore, fixed vX0 with v+0, for t(0,1) sufficiently small we have

Jλ(tv)=t2θ2θv2θ-t1-γλ1-γΩ(v+)1-γ𝑑x-t2s*2s*v+2s*2s*<0

since 1-γ<1<2θ<2s*. ∎

We are now ready to prove the existence of the first solution of (1.1).

Theorem 3.2.

Let λ0 be given as in Lemma 3.1. Then for any λ(0,λ0] problem (1.1) has a solution u0X0 with Jλ(u0)<0.

Proof.

Fix λ(0,λ0] and let ρ be as given in Lemma 3.1. We first prove that there exists u0B¯ρ such that Jλ(u0)=mλ<0. Let {uk}kB¯ρ be a minimizing sequence for mλ, that is, such that

limkJλ(uk)=mλ.(3.2)

Since {uk}k is bounded in X0, by applying [26, Lemma 8] and [6, Theorem 4.9], there exist a subsequence, still denoted by {uk}k, and a function u0B¯ρ such that, as k, we have

{uku0 in X0,uku0 in L2s*(Ω),uku0 in Lp(Ω) for any p[1,2s*),uku0 a.e. in Ω.(3.3)

Since γ(0,1), by the Hölder inequality, for any k we have

|Ω(uk+)1-γ𝑑x-Ω(u0+)1-γ𝑑x|Ω|uk+-u0+|1-γ𝑑xuk+-u0+21-γ|Ω|(1+γ)/2,

which yields, by (3.3),

limkΩ(uk+)1-γ𝑑x=Ω(u0+)1-γ𝑑x.(3.4)

Let wk=uk-u0; by [7, Theorem 2] it holds true that

uk2=wk2+u02+o(1),uk2s*2s*=wk2s*2s*+u02s*2s*+o(1)(3.5)

as k. Since {uk}kB¯ρ, by (3.5) for k sufficiently large, we have wkB¯ρ. Lemma 3.1 implies that for any uX0, with u=ρ, we get

12θu2θ-12s*u+2s*2s*α>0,

and from this, since ρ1, for k sufficiently large we have

12θwk2θ-12s*wk+2s*2s*>0.(3.6)

Thus, by (3.2), (3.4)–(3.6) and considering θ1, it follows that, as k,

mλ=Jλ(uk)+o(1)=12θ(wk2+u02)θ-λ1-γΩ(u0+)1-γ𝑑x-12s*(wk+2s*2s*+u0+2s*2s*)+o(1)Jλ(u0)+12θwk2θ-12s*wk+2s*2s*+o(1)Jλ(u0)+o(1)mλ

since u0B¯ρ. Hence, u0 is a local minimizer for Jλ, with Jλ(u0)=mλ<0, which implies that u0 is nontrivial.

Now, we prove that u0 is a positive solution of (2.2). For any ψX0, with ψ0 a.e. in N, let us consider a t>0 sufficiently small so that u0+tψB¯ρ. Since u0 is a local minimizer for Jλ, we have

0Jλ(u0+tψ)-Jλ(u0)=12θ(u0+tψ2θ-u02θ)-λ1-γΩ[((u0+tψ)+)1-γ-(u0+)1-γ]𝑑x-12s*(u0+tψ2s*2s*-u0+2s*2s*).

From this, by dividing by t>0 and passing to the limit as t0+, it follows that

lim inft0+λ1-γΩ((u0+tψ)+)1-γ-(u0+)1-γt𝑑xu02(θ-1)u0,ψ-Ω(u0+)2s*-1ψ𝑑x.(3.7)

We observe that

11-γ((u0+tψ)+)1-γ-(u0+)1-γt=((u0+ξtψ)+)-γψa.e. in Ω,

with ξ(0,1) and ((u0+ξtψ)+)-γ(u0+)-γψ a.e. in Ω as t0+. Thus, by the Fatou lemma, we obtain

λΩ(u0+)-γψ𝑑xlim inft0+λ1-γΩ((u0+tψ)+)1-γ-(u0+)1-γt𝑑x.(3.8)

Therefore, combining (3.7) and (3.8), we get

u02(θ-1)u0,ψ-λΩ(u0+)-γψ𝑑x-Ω(u0+)2s*-1ψ𝑑x0(3.9)

for any ψX0 with ψ0 a.e. in N.

Since Jλ(u0)<0 and by Lemma 3.1, we have u0Bρ. Hence, there exists δ(0,1) such that (1+t)u0B¯ρ for any t[-δ,δ]. Let us define Iλ(t)=Jλ((1+t)u0). Since u0 is a local minimizer for Jλ in B¯ρ, the functional Iλ has a minimum at t=0, that is,

Iλ(0)=u02θ-λΩ(u0+)1-γ𝑑x-Ω(u0+)2s*𝑑x=0.(3.10)

For any φX0 and any ε>0, let us define ψε=u0++εφ. Then by (3.9) we have

0u02(θ-1)u0,ψε+-λΩ(u0+)-γψε+𝑑x-Ω(u0+)2s*-1ψε+𝑑x=u02(θ-1)u0,ψε+ψε--λΩ(u0+)-γ(ψε+ψε-)𝑑x-Ω(u0+)2s*-1(ψε+ψε-)𝑑x.(3.11)

We observe that, for a.e. x,yN, we obtain

(u0(x)-u0(y))(u0-(x)-u0-(y))=-u0+(x)u0-(y)-u0-(x)u0+(y)-[u0-(x)-u0-(y)]2-|u0-(x)-u0-(y)|2,(3.12)

from which we immediately get

(u0(x)-u0(y))(u0+(x)-u0+(y))|u0(x)-u0(y)|2.

From the last inequality it follows that

u0,ψε+ψε-=2N(u0(x)-u0(y))(ψε(x)+ψε-(x)-ψε(y)-ψε-(y))|x-y|N+2s𝑑x𝑑y2N|u0(x)-u0(y)|2|x-y|N+2s𝑑x𝑑y+ε2N(u0(x)-u0(y))(φ(x)-φ(y))|x-y|N+2s𝑑x𝑑y+2N(u0(x)-u0(y))(ψε-(x)-ψε-(y))|x-y|N+2s𝑑x𝑑y.(3.13)

Hence, denoting Ωε={xN:u0+(x)+εφ(x)0} and combining (3.11) with (3.13), we get

0u02θ+εu02(θ-1)u0,φ+u02(θ-1)u0,ψε--λΩ(u0+)-γ(u0++εφ)𝑑x-Ω(u0+)2s*-1(u0++εφ)𝑑x+λΩε(u0+)-γ(u0++εφ)𝑑x+Ωε(u0+)2s*-1(u0++εφ)𝑑xu02θ-λΩ(u0+)1-γ𝑑x-Ω(u0+)2s*𝑑x+u02(θ-1)u0,ψε-+ε[u02(θ-1)u0,φ-λΩ(u0+)-γφ𝑑x-Ω(u0+)2s*-1φ𝑑x]=u02(θ-1)u0,ψε-+ε[u02(θ-1)u0,φ-λΩ(u0+)-γφ𝑑x-Ω(u0+)2s*-1φ𝑑x],(3.14)

where the last equality follows from (3.10). Arguing similarly to (3.12), for a.e. x,yN we have

(u0(x)-u0(y))(u0+(x)-u0+(y))|u0+(x)-u0+(y)|2.(3.15)

Thus, denoting

𝒰ε(x,y)=(u0(x)-u0(y))(ψε-(x)-ψε-(y))|x-y|N+2s,

by the symmetry of the fractional kernel and (3.15), we get

u0,ψε-=Ωε×Ωε𝒰ε(x,y)𝑑x𝑑y+2Ωε×(NΩε)𝒰ε(x,y)𝑑x𝑑y-ε(Ωε×Ωε𝒰(x,y)𝑑x𝑑y+2Ωε×(NΩε)𝒰(x,y)𝑑x𝑑y)2εΩε×N|𝒰(x,y)|𝑑x𝑑y,(3.16)

where we set

𝒰(x,y)=(u0(x)-u0(y))(φ(x)-φ(y))|x-y|N+2s.

Clearly 𝒰L1(2N), so that for any σ>0 there exists Rσ sufficiently large such that

(suppφ)×(NBRσ)|𝒰(x,y)|𝑑x𝑑y<σ2.

Also, by the definition of Ωε, we have Ωεsuppφ and |Ωε×BRσ|0 as ε0+. Thus, since 𝒰L1(2N), there exist δσ>0 and εσ>0 such that for any ε(0,εσ],

|Ωε×BRσ|<δσandΩε×BRσ|𝒰(x,y)|𝑑x𝑑y<σ2.

Therefore, for any ε(0,εσ],

Ωε×N|𝒰(x,y)|𝑑x𝑑y<σ,

from which we get

limε0+Ωε×N|𝒰(x,y)|𝑑x𝑑y=0.(3.17)

Combining (3.14) with (3.16), dividing by ε, letting ε0+ and using (3.17), we obtain

u02(θ-1)u0,φ-λΩ(u0+)-γφ𝑑x-Ω(u0+)2s*-1φ𝑑x0

for any φX0. By the arbitrariness of φ, we prove that u0 verifies (2.3), that is, u0 is a nontrivial solution of (2.2).

Finally, considering φ=u0- in (2.3) and using (3.12), we see that u0-=0, which implies that u0 is nonnegative. Moreover, by the maximum principle in [28, Proposition 2.17], we can deduce that u0 is a positive solution of (2.2), and so also solves problem (1.1). This concludes the proof. ∎

We end this section by observing that the result in Theorem 3.2 can be extended to more general Kirchhoff problems. That is, we can consider the problem

{M(2N|u(x)-u(y)|p|x-y|N+ps𝑑x𝑑y)(-Δ)psu=λuγ+ups*-1in Ω,u>0in Ω,u=0in NΩ,(3.18)

where ps*=pN/(N-ps), with N>ps and p>1, while the Kirchhoff coefficient M satisfies the following condition:

  • ${(\mathcal{M})}$

    M:0+0+ is continuous and nondecreasing. There exist numbers a>0 and ϑ such that for any t0+,

    (t):=0tM(τ)dτatϑ,with {ϑ(1,ps*/p)if M(0)=0,ϑ=1if M(0)>0.

The main operator (-Δ)ps is the fractional p-Laplacian which may be defined, for any function φC0(Ω), as

(-Δ)psφ(x)=2limε0+NBε(x)|φ(x)-φ(y)|p-2(φ(x)-φ(y))|x-y|N+ps𝑑y,xN,

where Bε(x)={yN:|x-y|<ε}. Then, arguing as in the proof of Theorem 3.2 and observing that we have not used yet the assumption that Ω is continuous, we can prove the following result.

Theorem 3.3.

Let s(0,1), p>1, N>ps, γ(0,1) and let Ω be an open bounded subset of RN. Let M satisfy (M). Then there exists λ0>0 such that for any λ(0,λ0] problem (3.18) admits a solution.

4 A mountain pass solution for problem (2.4)

In this section, we prove the existence of a positive solution for the perturbed problem (2.4) by the mountain pass theorem. For this, throughout this section we assume n without further mentioning. Now, we first prove that the related functional Jn,λ satisfies all the geometric features required by the mountain pass theorem.

Lemma 4.1.

Let ρ(0,1], λ0=λ0(ρ)>0 and α=α(ρ)>0 be given as in Lemma 3.1. Then, for any λ(0,λ0] and any uX0 with uρ, one has Jn,λ(u)α. Furthermore, there exists eX0, with e>ρ, such that Jn,λ(e)<0.

Proof.

Since γ(0,1), by the subadditivity of tt1-γ, we have

(u++1n)1-γ-(1n)1-γ(u+)1-γa.e. in Ω(4.1)

for any uX0 and any n. Thus, we have Jn,λ(u)Jλ(u) for any uX0 and the first part of the lemma directly follows by Lemma 3.1.

For any vX0, with v+0, and t>0, we have

Jn,λ(tv)=t2θ2θv2θ-λ1-γΩ[(tv++1n)1-γ-(1n)1-γ]𝑑x-t2s*2s*v+2s*2s*-as t

since 1-γ<1<2θ<2s*. Hence, we can find eX0, with e>ρ sufficiently large, such that Jn,λ(e)<0. ∎

We discuss now the compactness property for the functional Jn,λ, given by the Palais–Smale condition. We recall that {uk}kX0 is a Palais–Smale sequence for Jn,λ at level c if

Jn,λ(uk)candJn,λ(uk)0  in (X0) as k.(4.2)

We say that Jn,λ satisfies the Palais–Smale condition at level c if any Palais–Smale sequence {uk}k at level c admits a convergent subsequence in X0.

Before proving this compactness condition, we introduce the following positive constants, which will help us for a better explanation:

D1=(12θ-12s*)S2s*θ/(2s*-2θ),D2=[(11-γ+12s*)|Ω|(2s*-1+γ)/2s*S-(1-γ)/2]2θ/(2θ-1+γ)(12θ-12s*)(1-γ)/(2θ-1+γ).(4.3)

Lemma 4.2.

Let λ>0. Then the functional Jn,λ satisfies the Palais–Smale condition at any level cR verifying

c<D1-D2λ2θ/(2θ-1+γ),(4.4)

with D1, D2>0 given as in (4.3).

Proof.

Let λ>0 and let {uk}k be a Palais–Smale sequence in X0 at level c, with c satisfying (4.4). We first prove the boundedness of {uk}k. By (4.2), there exists σ>0 such that, as k,

c+σuk+o(1)Jn,λ(uk)-12s*Jn,λ(uk),uk=(12θ-12s*)uk2θ-λ1-γΩ[(uk++1n)1-γ-(1n)1-γ]𝑑x+λ2s*Ω(uk++1n)-γuk𝑑x(12θ-12s*)uk2θ-λ(11-γ+12s*)Ω|uk|1-γ𝑑x(12θ-12s*)uk2θ-λ(11-γ+12s*)|Ω|(2s*-1+γ)/2s*S-(1-γ)/2uk1-γ,

where the last two inequalities follow by (2.7), (4.1) and the Hölder inequality. Therefore, {uk}k is bounded in X0 since 1-γ<1<2θ. Also, {uk-}k is bounded in X0, and by (4.2) we have

limkJn,λ(uk),-uk-=limkuk2(θ-1)uk,-uk-=0.

Thus, by inequality (3.12) we deduce that uk-0 as k, which yields

Jn,λ(uk)=Jn,λ(uk+)+o(1)andJn,λ(uk)=Jn,λ(uk+)+o(1)  as k.

Hence, we can suppose that {uk}k is a sequence of nonnegative functions.

By the boundedness of {uk}k and by using [26, Lemma 8] and [6, Theorem 4.9], there exist a subsequence, still denoted by {uk}k, and a function uX0 such that

{uku in X0,ukμ,uku in L2s*(Ω),uk-u2s*,uku in Lp(Ω) for any p[1,2s*),uku a.e. in Ω,ukh a.e. in Ω,(4.5)

as k, with hLp(Ω) for a fixed p[1,2s*). If μ=0, then immediately uk0 in X0 as k. Hence, let us assume that μ>0.

Since n, by (4.5) it follows that

|uk-u(uk+1n)γ|nγ(h+|u|)a.e. in Ω,

so by the dominated convergence theorem and (4.5) we have

limkΩuk-u(uk+1n)γ𝑑x=0.(4.6)

By (4.5) and [7, Theorem 2], we have

uk2=uk-u2+u2+o(1),uk2s*2s*=uk-u2s*2s*+u2s*2s*+o(1)(4.7)

as k. Consequently, we deduce from (4.2), (4.5), (4.6) and (4.7) that, as k,

o(1)=Jn,λ(uk),uk-u=uk2(θ-1)uk,uk-u-λΩuk-u(uk+1n)γ𝑑x-Ωuk2s*-1(uk-u)𝑑x=μ2(θ-1)(μ2-u2)-uk2s*2s*+u2s*2s*+o(1)=μ2(θ-1)uk-u2-uk-u2s*2s*+o(1).

Therefore, we have proved the crucial formula

μ2(θ-1)limkuk-u2=limkuk-u2s*2s*.(4.8)

If =0, since μ>0, by (4.5) and (4.8) we have uku in X0 as k, concluding the proof.

Thus, let us assume by contradiction that >0. By (2.7), the notation in (4.5) and (4.8), we get

2s*Sμ2(θ-1)2.(4.9)

Noting that (4.8) implies in particular that

μ2(θ-1)(μ2-u2)=2s*,

by using (4.9), it follows that

(2s*)2s/N=(μ2(θ-1))2s/N(μ2-u2)2s/NSμ2(θ-1).

From this we obtain

μ4s/N(μ2-u2)2s/NS(μ2(θ-1))(N-2s)/N.

Considering N<2sθ/(θ-1)=2sθ, we have

μ2SN/(2sθ-N(θ-1)).(4.10)

Indeed, the restriction N/(2θ)<s follows directly from the fact that 1<θ<2s*/2=N/(N-2s). By (4.1), considering that n, for any k we have

Jn,λ(uk)-12s*Jn,λ(uk),uk(12θ-12s*)uk2θ-λ(11-γ+12s*)Ωuk1-γ𝑑x.

From this, as k, since θ1, by (4.2), (4.5), (4.7), (4.10), the Hölder inequality and the Young inequality, we obtain

c(12θ-12s*)(μ2θ+u2θ)-λ(11-γ+12s*)|Ω|(2s*-1+γ)/2s*S-(1-γ)/2u1-γ(12θ-12s*)(μ2θ+u2θ)-(12θ-12s*)u2θ-(12θ-12s*)-(1-γ)/(2θ-1+γ)[λ(11-γ+12s*)|Ω|(2s*-1+γ)/2s*S-(1-γ)/2]2θ/(2θ-1+γ)(12θ-12s*)S2s*θ/(2s*-2θ)-(12θ-12s*)-(1-γ)/(2θ-1+γ)[λ(11-γ+12s*)|Ω|(2s*-1+γ)/2s*S-(1-γ)/2]2θ/(2θ-1+γ),

which contradicts (4.4) since (4.3). This concludes the proof. ∎

We now give a control from above for the functional Jn,λ under a suitable situation. For this, we assume, without loss of generality, that 0Ω. By [10], we know that the infimum in (2.7) is attained at the function

uε(x)=ε(N-2s)/2(ε2+|x|2)(N-2s)/2with ε>0,(4.11)

that is, it holds true that

2N|uε(x)-uε(y)|2|x-y|N+2s𝑑x𝑑y=SuεL2s*(N)2.

Let us fix r>0 such that B4rΩ, where B4r={xN:|x|<4r}, and let us introduce a cut-off function ϕC(N,[0,1]) such that

ϕ={1 in Br,0 in NB2r.(4.12)

For any ε>0, we set

ψε=ϕuεϕuε2s*2X0.(4.13)

Then we can prove the following result.

Lemma 4.3.

There exist ψX0 and λ1>0 such that for any λ(0,λ1),

supt0Jn,λ(tψ)<D1-D2λ2θ/(2θ-1+γ),

with D1, D2>0 given as in (4.3).

Proof.

Let λ,ε>0. Let uε and ψε be as in (4.11) and (4.13), respectively. By (2.6), we have Jn,λ(tψε)- as t, so that there exists tε>0 such that Jn,λ(tεψε)=maxt0Jn,λ(tψε). By Lemma 4.1, we know that Jn,λ(tεψε)α>0. Hence, by the continuity of Jn,λ there exist two numbers t0, t*>0 such that t0tεt*.

Now, since uεL2s*(N) is independent of ε, by [27, Proposition 21] we have

ψε22N|uε(x)-uε(y)|2|x-y|N+2s𝑑x𝑑yϕuε2s*2=S+O(εN-2s),

from which, by the elementary inequality

(a+b)pap+p(a+1)p-1b,for any a>0,b[0,1],p1,

with p=2θ, it follows that, as ε0+,

ψε2θSθ+O(εN-2s).

Hence, by the last inequality, (2.6) and since t0tεt*, for any ε>0 sufficiently small, we have

Jn,λ(tεψε)tε2θ2θSθ+C1εN-2s-λ1-γΩ[(t0ψε+1n)1-γ-(1n)1-γ]𝑑x-tε2s*2s*,(4.14)

with a suitable positive constant C1. We observe that

maxt0(t2θ2θSθ-t2s*2s*)=(12θ-12s*)S2s*θ/(2s*-2θ).

Thus, by (4.14) it follows that

Jn,λ(tεψε)(12θ-12s*)S2s*θ/(2s*-2θ)+C1εN-2s-λ1-γΩ[(t0ψε+1n)1-γ-(1n)1-γ]𝑑x.(4.15)

Now, let us consider a positive number q, less than 1, satisfying

(N-2s)(1-γ)-2q(N-2s)(1-γ)+2s*qN2s*2θ2θ-1+γ1N-2s+1-2θ2θ-1+γ<0,(4.16)

that is, since 2<2θ<2s*, N>2s and γ(0,1), such that

0<q<min{(N-2s)(2s*-2θ)(1-γ)2θN(2s*-2)+4θNγ+8θs(1-γ),1}.

By the elementary inequality

a1-γ-(a+b)1-γ-(1-γ)b(1-γ)/pa(p-1)(1-γ)/pfor any a>0,b>0 large enough,p>1,

with p=2s*/2, considering ε<r1/q sufficiently small, with r given by (4.12), and since q<1, we have

-11-γ{xΩ:|x|εq}[(t0ψε+1n)1-γ-(1n)1-γ]𝑑x-C~ε(N-2s)(1-γ)/2s*{xΩ:|x|εq}[1(|x|2+ε2)(N-2s)/2]2(1-γ)/2s*𝑑x-C2ε((N-2s)(1-γ)-2q(N-2s)(1-γ)+2s*qN)/2s*,(4.17)

with two positive constants C~ and C2 independent of ε. By combining (4.15) with (4.17), we get

Jn,λ(tεψε)(12θ-12s*)S2s*θ/(2s*-2θ)+C1εN-2s-C2λε((N-2s)(1-γ)-2q(N-2s)(1-γ)+2s*qN)/2s*.(4.18)

Thus, let us consider λ*>0 such that

D1-D2λ2θ/(2θ-1+γ)>0for any λ(0,λ*),

and let us set

ν1=2qθ(2θ-1+γ)(N-2s),ν2=2θ[(N-2s)(1-γ)-2q(N-2s)(1-γ)+2s*qN]2s*(2θ-1+γ)(N-2s)+1,ν3=ν2-2θ2θ-1+γ,λ1=min{λ*,r1/ν1,(C2C1+D2)-1/ν3},

where r and q are given in (4.12) and (4.16), respectively, while we still consider D1 and D2 as defined in (4.3). Then, by considering ε=λν1/q in (4.18), since (4.16) implies that ν3<0, for any λ(0,λ1) we have

Jn,λ(tεψε)D1+C1λ2θ/(2θ-1+γ)-C2λν2=D1+λ2θ/(2θ-1+γ)(C1-C2λν3)<D1-D2λ2θ/(2θ-1+γ),

which concludes the proof. ∎

We can now prove the existence result for (2.4) by applying the mountain pass theorem.

Theorem 4.4.

There exists λ¯>0 such that, for any λ(0,λ¯), problem (2.4) has a positive solution vnX0 with

α<Jn,λ(vn)<D1-D2λ2θ/(2θ-1+γ),(4.19)

where α, D1 and D2 are given in Lemma 3.1 and (4.3), respectively.

Proof.

Let λ¯=min{λ0,λ1}, with λ0 and λ1 given in Lemmas 3.1 and 4.3, respectively. Let us consider λ(0,λ¯). By Lemma 4.1, the functional Jn,λ verifies the mountain pass geometry. For this, we can set the critical mountain pass level as

cn,λ=infgΓmaxt[0,1]Jn,λ(g(t)),

where

Γ={gC([0,1],X0):g(0)=0,Jn,λ(g(1))<0}.

By Lemmas 4.1 and 4.3, we get

0<α<cn,λsupt0Jn,λ(tψ)<D1-D2λ2θ/(2θ-1+γ).

Hence, by Lemma 4.2 the functional Jn,λ satisfies the Palais–Smale condition at level cn,λ. Thus, the mountain pass theorem gives the existence of a critical point vnX0 for Jn,λ at level cn,λ. Since

Jn,λ(vn)=cn,λ>α>0=Jn,λ(0),

we obtain that vn is a nontrivial solution of (2.4). Furthermore, by (2.5) with test function φ=vn- and inequality (3.12), we can see that vn-=0, which implies that vn is nonnegative. By the maximum principle in [28, Proposition 2.17], we have that vn is a positive solution of (2.4), concluding the proof. ∎

5 A second solution for problem (1.1)

In this last section, we prove the existence of a second solution for problem (1.1), as a limit of solutions of the perturbed problem (2.4). For this, here we need the assumption that Ω is continuous in order to apply a density argument for the space X0.

Proof of Theorem 1.1.

Let us consider λ¯ as given in Theorem 4.4, and let λ(0,λ¯). Since λ¯λ0, by Theorem 3.2 we know that problem (1.1) admits a solution u0 with Jλ(u0)<0.

In order to find a second solution for (1.1) let {vn}n be a family of positive solutions of (2.4). By (2.7), (4.1), (4.19) and the Hölder inequality, we have

D1-D2λ2θ/(2θ-1+γ)>Jn,λ(vn)-12s*Jn,λ(vn),vn=(12θ-12s*)vn2θ-λ1-γΩ[(vn+1n)1-γ-(1n)1-γ]𝑑x+λ2s*Ω(vn+1n)-γvn𝑑x(12θ-12s*)vn2θ-λ1-γΩvn1-γ𝑑x(12θ-12s*)vn2θ-λ1-γ|Ω|(2s*-1+γ)/2s*S-(1-γ)/2vn1-γ,

which yields that {vn}n is bounded in X0 since 1-γ<1<2θ. Hence, by using [26, Lemma 8] and [6, Theorem 4.9], there exist a subsequence, still denoted by {vn}n, and a function v0X0 such that

{vnv0 in X0,vnμ,vnv0 in L2s*(Ω),vn-v02s*,vnv0 in Lp(Ω) for any p[1,2s*),vnv0 a.e. in Ω.(5.1)

We want to prove that vnv0 in X0 as n. When μ=0, by (5.1) we have vn0 in X0 as n. For this, we suppose μ>0. We observe that

0vn(vn+1n)γvn1-γa.e. in Ω,

so by the Vitali convergence theorem and (5.1) it follows that

limnΩvn(vn+1n)γ𝑑x=Ωv01-γ𝑑x.(5.2)

Using (2.5) for vn and test function φ=vn, by (5.1) and (5.2), as n, we have

μ2θ-λΩv01-γ𝑑x+vn2s*2s*=o(1).(5.3)

For any n, by an immediate calculation in (2.4) we see that

vn2(θ-1)(-Δ)svnmin{1,λ2γ}in Ω.

Thus, since {vn}n is bounded in X0 and by using a standard comparison argument (see [3, Lemma 2.1]) and the maximum principle in [28, Proposition 2.17], for any Ω~Ω there exists a constant cΩ~>0 such that

vncΩ~>0,a.e. in Ω~ and for any n.(5.4)

Now, let φC0(Ω) with suppφ=Ω~Ω. By (5.4), we have

0|φ(vn+1n)γ||φ|cΩ~γa.e. in Ω,

so that by (5.1) and the dominated convergence theorem we obtain

limnΩφ(vn+1n)γ𝑑x=Ωv0-γφ𝑑x.(5.5)

Thus, by considering (2.5) for vn, sending n and using (5.1) and (5.5), for any φC0(Ω) it follows that

μ2(θ-1)v0,φ-λΩv0-γφ𝑑x+Ωv02s*-1φ𝑑x=0.(5.6)

However, since Ω is continuous, by [15, Theorem 6] the space C0(Ω) is dense in X0. Thus, by a standard density argument, (5.6) holds true for any φX0. By combining (5.3) and (5.6) with test function φ=v0, as n we get

μ2(θ-1)(μ2-v02)=vn2s*2s*-v02s*2s*+o(1),

and by (5.1) and [7, Theorem 2] we have

μ2(θ-1)limnvn-v02=2s*.(5.7)

If =0, then vnv0 in X0 as n since μ>0.

Let us suppose >0 by contradiction. Arguing as in Lemma 4.2, by (5.1) and (5.7) we get (4.10). Therefore, since θ1, by (4.1), (4.10), (4.19), (5.1), the Hölder inequality and the Young inequality we have

D1-D2λ2θ/(2θ-1+γ)>Jn,λ(vn)-12s*Jn,λ(vn),vn(12θ-12s*)(μ2θ+v02θ)-λ(11-γ+12s*)|Ω|(2s*-1+γ)/2s*S-(1-γ)/2v01-γ(12θ-12s*)S2s*θ/(2s*-2θ)-(12θ-12s*)-(1-γ)/(2θ-1+γ)[λ(11-γ+12s*)|Ω|(2s*-1+γ)/2s*S-(1-γ)/2]2θ/(2θ-1+γ),

which is the desired contradiction, thanks to (4.3).

Therefore, vnv0 in X0 as n, and by (2.3) and (2.5) we immediately see that v0 is a solution of problem (2.2). Furthermore, by (4.19) we have Jλ(v0)α>0, which also implies that v0 is nontrivial. Reasoning as at the end of the proof of Theorem 4.4, we conclude that v0 is a positive solution of (2.2), and so v0 also solves problem (1.1). Finally, v0 is different from u0 since Jλ(v0)>0>Jλ(u0). ∎

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About the article

Received: 2017-03-24

Revised: 2017-05-18

Accepted: 2017-06-05

Published Online: 2017-08-03


Funding Source: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Award identifier / Grant number: PNPD–CAPES 33003017003P5

The author is supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior through the fellowship PNPD–CAPES 33003017003P5. The author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica “G. Severi” (INdAM).


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 645–660, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0075.

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