Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Nonlinear Analysis

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


IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

SCImago Journal Rank (SJR) 2018: 3.215
Source Normalized Impact per Paper (SNIP) 2018: 3.225

Mathematical Citation Quotient (MCQ) 2018: 3.18

Open Access
Online
ISSN
2191-950X
See all formats and pricing
More options …

On fractional p-Laplacian problems with local conditions

Anran Li / Chongqing Wei
Published Online: 2016-11-23 | DOI: https://doi.org/10.1515/anona-2016-0105

Abstract

In this paper, we deal with fractional p-Laplacian equations of the form

{(-Δ)psu=λf(x,u),xΩ,u(x)=0,xNΩ,

where λ(0,+), 0<s<1<p<+ and ΩN, N2, is a bounded domain with smooth boundary. With assumptions on f(x,t) just in Ω×(-δ,δ), where δ>0 is small, existence and multiplicity of nontrivial solutions are obtained via variational methods.

Keywords: local conditions; variational methods.

MSC 2010: 35R11; 35A15; 35B38

1 Introduction and main results

In this paper we are concerned with the existence and multiplicity of nontrivial solutions of the following fractional p-Laplacian equation:

{(-Δ)psu=λf(x,u),xΩ,u(x)=0,xNΩ,(Pλ)

where λ(0,+), 0<s<1<p<+, ΩN is a bounded domain with smooth boundary, N2, and f(x,t) is a Carathéodory function defined on Ω×(-δ,δ), with δ>0 being small.

When p=2, much attention has been paid to the semi-linear problem

{(-Δ)su=g(x,u),xΩ,u(x)=0,xNΩ,

from the point of view of existence, non-existence and regularity, where g:Ω× is a Carathéodory function satisfying suitable growth conditions. Several existence results via variational methods are proved in a series of papers [18, 20, 19, 21]. The issues of regularity and non-existence of solutions are studied in [4, 15, 16, 14, 3]. The corresponding equations in N have also been widely studied, see, for example [17, 5, 9, 1, 10] and references therein.

Very recently, a new nonlocal and nonlinear operator was considered, namely, for p(1,+), s(0,1) and u smooth enough,

(-Δ)psu(x)=2limε0+NBε(x)|u(x)-u(y)|p-2(u(x)-u(y))|x-y|N+sp𝑑y,xN,

which is consistent, up to some normalization constant depending upon N and s, with the linear fractional Laplacian (-Δ)s in the case p=2. This operator, known as the fractional p-Laplacian, leads naturally to the quasilinear problem (P1). One typical feature of this operator is the nonlocality, in the sense that the value of (-Δ)psu at any point xΩ depends not only on the values of u in Ω, but actually on the whole N, since u(x) represents the expected value of a random variable ties to a process randomly jumping arbitrarily far from the point x. While in the classical case, by the continuity properties of the Brownian motion, at the exit time from Ω one necessarily is on Ω, due to the jumping nature of the process, at the exit time one could end up anywhere outside Ω. In this sense, the natural nonhomogeneous Dirichlet boundary condition consists in assigning the values of u in NΩ rather than merely on Ω. Then, it is reasonable to search for solution in the space of functions uWs,p(N) vanishing outside Ω. It should be pointed out that in a bounded domain, this is not the only possible way to provide a formulation for the problem. In the works of [11, 13], the eigenvalue problem associated with (-Δ)ps is studied, and particularly some properties of the first eigenvalue and of the higher order (variational) eigenvalues are obtained. From the point of view of regularity theory, some results can be found in [13]. This work is most focused on the case where p is large and the solutions inherit some regularity directly from the functional embedding theorems. In [7, 2], relevant results about the local boundedness and Hölder continuity for the solutions to the problem of finding (s,p)-harmonic functions u were obtained. Very recently, in [12], Iannizzotto et al. established a priori L-bounds for the solutions of problem (P1) under suitable growth conditions on the nonlinearity.

In all the works mentioned above, the nonlinearity is assumed on the whole Ω×. Motivated by [12, 22, 6], we can consider problem () with local conditions on the nonlinearity f. Firstly, we assume that f satisfies a p-sublinear condition at the origin, without any growth condition at infinity. In particular, we assume the following:

  • (f0)

    f(x,t) is a Carathéodory function defined on Ω×(-δ,δ).

  • (f1)

    There exists positive constant p1(p2ps*,p) such that

    lim|t|0F(x,t)|t|p1=+uniformly for a.e. xΩ,

    where F(x,t)=0tf(x,ξ)𝑑ξ.

  • (f2)

    There exists positive constant p2(p2ps*,p) such that

    lim|t|0F(x,t)|t|p2=0uniformly for a.e. xΩ.

  • (f3)

    There exists α(p1,p) such that

    αF(x,t)-f(x,t)t0for a.e. xΩ, with |t|small.

The first two results of our paper read as follows.

Theorem 1.1.

Let (f0)(f3) hold. Then there exists λ0(0,1] such that problem () has at least a nontrivial weak solution uλ for all 0<λ<λ0.

Furthermore, if f(x,t) is also odd in t, then for every λ(0,+), problem () has infinitely many nontrivial weak solutions.

Theorem 1.2.

Let (f0)(f3) hold, and assume that f(x,t) is odd in t. Then, for every λ(0,+), problem () has a sequence of weak solutions {unλ} satisfying unλ0 as n.

Remark 1.3.

Assumptions similar to (f1)(f3) were used in [22], where similar results were obtained for elliptic equation problems on a bounded domain. Condition (f3) is a local version of the subquadratic condition.

Remark 1.4.

In our assumptions p1,p2>p2ps* will be just used to give an L-estimate for the solutions. Based on the proof of [12, Theorem 3.1], we can get a more suitable L-estimate to our problem (). As far as we know (see [12]), similar bounds were obtained before only in some special cases, namely, for a semilinear fractional Laplacian equation with the reaction term independent of u, and for the eigenvalue problem of some fractional elliptic operators.

Secondly, we consider () with p-superlinear nonlinearity. In particular, we make the following assumptions on f just near origin:

  • (f1’)

    There exists q1(p,2pps*p+ps*) such that

    lim|t|0f(x,t)t|t|q1=0uniformly for a.e. xΩ.

  • (f2’)

    There exists q2(p,p+γ) such that

    lim|t|0F(x,t)|t|q2=+uniformly for a.e. xΩ, where γ=q1(2-q1ps*)-q12p.

  • (f3’)

    There exists ϖ(p,ps*) such that

    0<ϖF(x,t)f(x,t)tfor a.e. xΩ,with |t| small.

Theorem 1.5.

Let (f0), (f1’)(f3’) hold. Then there exists Λ0>1 such that problem () has at least a nontrivial weak solution uλ for all λ>Λ0.

Remark 1.6.

In our assumptions q1<2pps*p+ps*, q2<p+γ will also just be used to give an L-estimate for the solutions.

We will prove our results via a variational approach following the methods of [22, 6]. The strategy is to modify and extend f to an appropriate f~, and to show for the associated modified functional the existence of solutions with bounded L norm, therefore to obtain solutions for the original problem (). So the L-estimate of the solution is very important. However, there are no Lp-estimates for fractional Laplace problems as the classic Laplace problem. Recently, in [12], Iannizzotto et al. proved a priori L bounds on the weak solutions of problem (P1). By this estimate and the Sobolev embedding, we are able to get a more suitable estimate of L norm of solutions and avoid further restriction on the behavior of f at infinity.

Throughout the paper, we denote by C various positive constants, whose values are not essential to the problem, and may be different from line to line. We denote the usual norm of Lq(Ω) by q for 1q. Moreover, let 0<s<1<p< be real numbers, and the fractional critical exponent be defined as

ps*={NpN-spif sp<N,if spN.

The paper is organized as follows. In Section 2, we introduce some preliminary notions and notations, and set the functional framework of our problem. In Section 3, we will prove Theorem 1.1. Section 4 is devoted to the proof Theorem 1.2. The proof of Theorem 1.5 will be given in Section 5.

2 Preliminary

In this preliminary section, for the reader’s convenience, we collect some basic results that will be used in the forthcoming sections.

Firstly, we introduce a variational setting for problem (). The Gagliardo seminorm is defined, for all measurable function u:N, by

[u]s,p=(2N|u(x)-u(y)|p|x-y|N+sp𝑑x𝑑y)1p.

The fractional Sobolev space

Ws,p(N):={uLp(N):u is measurable,[u]s,p<}

is endowed with the norm

us,p=(N|u|p𝑑x+[u]s,p)1p.

In this paper, we will work in the closed linear subspace

X(Ω)={uWs,p(N):u(x)=0 for a.e. xNΩ},

which can be equivalently renormed by setting =[]s,p (see [8, Theorem 7.1]). It is readily seen that (X(Ω),) is a uniformly convex Banach space and the following Sobolev embedding theorem holds.

Lemma 2.1 ([8]).

The embedding X(Ω)Lq(Ω) is continuous for all q[1,ps*], and compact for q(1,ps*).

In [12], the fractional p-Laplacian is redefined variationally as the nonlinear operator A:X(Ω)X(Ω)* defined, for all u,vX(Ω), by

A(u),v=2N|u(x)-u(y)|p-2(u(x)-u(y))(v(x)-v(y))|x-y|N+sp𝑑x𝑑y.

A weak solution of problem () is a function of uX(Ω) such that

A(u),v=λΩf(x,u)v𝑑xfor all vX(Ω).

Since X(Ω) is uniformly convex, A satisfies the following compactness condition:

  • (S)

    If {un} is a sequence in X(Ω) such that unuinX(Ω) and A(un),un-u0, then unuinX(Ω).

Definition 2.2.

Let X be a Banach space. We say that the functional IC1(X,) satisfies the Palais–Smale condition at the level c ((PS)c in short) if any sequence {un}X, satisfying I(un)c, I(un)0 as n, has a convergent subsequence. The functional I satisfies the (PS) condition if it satisfies the (PS)c condition for any c.

When f is odd in t, we need the following critical point theorem, see [22, Lemma 2.4].

Lemma 2.3.

Let X be a Banach space. Let also IC1(X,R), and assume that I satisfies the (PS) condition, and that is bounded from below and even, with I(0)=0. If for any kN, there exist a k-dimensional subspace Xk and ρk>0 such that

supXkSρkI<0,

where Sρk={uX:u=ρk}, then I has a sequence of critical values ck<0 satisfying ck0 as k.

3 Proof of Theorem 1.1

In this section we will prove Theorem 1.1. Since (f1)(f3) give the behavior of f just in Ω×(-δ,δ), the functional ΩF(x,u)𝑑x is not well-defined in X(Ω). To overcome this difficulty, we need to modify and extend f to an appropriate f~, in the spirit of the arguments developed in [22]. For this purpose, we first observe that (f1) and (f2) imply, for small |t|, that

|F(x,t)|>|t|p1,|F(x,t)|<|t|p2for a.e. xΩ.(3.1)

Let ρ(t)C1(,[0,1]) be an even cut-off function verifying tρ(t)0 and

ρ(t)={1if |t|τ,0if |t|2τ,(3.2)

where 0<τ<δ2 is chosen such that (3.1), (3.2) and (f3) hold for |s|2τ. Set

F~(x,t)=ρ(t)F(x,t)+(1-ρ(t))|t|p1,f~(x,t)=tF~(x,t).

It is easy to check that the following properties on f~ hold.

Lemma 3.1.

Let (f0), (f1)(f3) be satisfied. Then the following hold:

  • (i)

    There exists a constant C>0 such that

    |f~(x,t)|C(|t|p1-1+|t|p2-1)for all (x,t)Ω×.

  • (ii)

    We have

    αF~(x,t)-f~(x,t)t0for all (x,t)Ω×.

  • (iii)

    f~(x,t) is odd in t for all t , if f(x,t) is odd in t for t(-δ,δ).

Proof.

By the chosen of ρ, (i) and (iii) are simple. We will just show (ii) for p1α<p.

On the one hand, for 0|t|τ and |t|2τ, we have

f~(x,t)t=ρ(t)tf(x,t)+p1(1-ρ(t))|t|p1-ρ(t)t(|t|p1-F(x,t)),αF~(x,t)-f~(x,t)t=ρ(t)(αF(x,t)-tf(x,t))+(α-p1)(1-ρ(t))|t|p1+ρ(t)t(|t|p1-F(x,t)),

and the conclusion follows. On the other hand, by (f1), F(x,t)|t|p1 for τ|t|2τ. Using the fact that ρ(t)t0 and αp1, by (f3), we get the conclusion. ∎

By Lemma 3.1, we can modify and extend f to get f~C(Ω×,) satisfying all properties listed in Lemma 3.1. Now we define

I~λ(u)=1pup-λΩF~(x,u)𝑑xfor uX(Ω).(3.3)

Then I~λ(u)C1(X(Ω),). We remark that f~(x,t)=f(x,t) for (x,t)Ω×[-τ,τ], and that a critical point u of I~λ is a solution of () if and only if uτ.

Now we investigate the properties of the functional I~λ.

Lemma 3.2.

I~λ(u)=I~λ(u),u=0 if and only if u=0.

Proof.

By (3.3), it is easy to see that

0=I~λ(u)-1αI~λ(u),u=(1p-1α)up-λΩ(F~(x,u)-1αf~(x,u)u)𝑑x.

Since p1α<p and F~(x,u)-1αf~(x,u)u0 for u, by Lemma 3.1, we get u=0. ∎

Lemma 3.2 implies that the trivial solution 0 of () is the unique critical point of I~λ at the level 0.

Next we check that I~λ(u) is coercive, i.e., I~λ(u) as u, and I~λ satisfies the (PS) condition.

Lemma 3.3.

The functional I~λ is bounded from below and satisfies the (PS) condition.

Proof.

From Lemma 3.1, we have

F~(x,t)C(|t|p1+|t|p2)for (x,t)Ω×,

where C is a positive constant. Therefore,

I~λ(u)=1pup-λΩF~(x,u)𝑑x1pup-λCΩ(|u|p1+|u|p2)𝑑x1pup-λC(up1+up2).

Since p2<p1<p, it follows that

I~λ(u)+as u,(3.4)

that is, I~λ(u) is coercive, and then is bounded from below.

Now we prove that I~λ satisfies the (PS) condition. Let {un}X(Ω) be a (PS) sequence. Then there exists M>0, such that

|I~λ(un)|M,I~λ(un)0as n.(3.5)

By (3.4), it follows that {un} is bounded in X(Ω). By Lemma 2.1, we can assume that, up to a subsequence, for some uX(Ω)

unuin X(Ω),unuin Lq(Ω),q(1,ps*),un(x)u(x)for a.e. xΩ.

It follows, from (3.5) and the fact that unu in X(Ω), that

I~λ(un),un-u0as n.

As p1,p2(1,ps*), by Lemma 3.1 (i) and Hölder’s inequality, we have

|Ωf~(x,un)(un-u)𝑑x|Ω|f~(x,un)||un-u|𝑑xCΩ(|un|p1-1+|un|p2-1)|un-u|𝑑xC(unp1p1-1un-up1+unp2p2-1un-up2)0as n.

Then, for any fixed λ>0,

A(un),un-u=I~λ(un),un-u+λΩf~(x,un)(un-u)𝑑x0as n.

Therefore, by condition (S), un-u0 as n, and the functional I~λ satisfies the (PS) condition. ∎

Now we are in the position to give the proof of Theorem 1.1.

Proof of Theorem 1.1.

By Lemma 3.3, for each λ>0, there exists some uλX(Ω) such that

I~λ(uλ)=infvX(Ω)I~λ(v).

We have uλ0, since the trivial solution 0 is not a local minimizer. By Lemmas 2.1 and 3.1, and the fact that I~λ(uλ),uλ=0, it follows that for some C>0,

uλp=λΩf~(x,uλ)uλ𝑑xλC(uλp1+uλp2).(3.6)

We claim that uλ1 uniformly for λ>0 small. Otherwise, we have a sequence of λn0 such that uλn>1. Thus, uλnp2uλnp1 since p2<p1<p. By (3.6), we obtain

uλnp-p12Cλn0as n,

a contradiction with the assumption uλn>1. Now, it follows from (3.6) that

uλp-p2λC0as λ0.

Next we show that there exists λ0>0 small enough such that uλτ for λ(0,λ0). We modify the proof of [12, Theorem 3.1]. By Lemma 3.1 (i) and (f2), we can easily get that

|f~(x,t)|C(|t|p2-1+|t|p-1),(x,t)Ω×, 1+p2p>1+pps*.

Then all the conditions of [12, Theorem 3.1] hold. Fix a weak solution uλX(Ω), with uλ+0, of the following problem:

{(-Δ)psu=λf~(x,u),xΩ,u(x)=0,xNΩ.

We choose ρmax{1,uλp-1}, set vλ=(ρuλp)-1uλ, so vλX(Ω),vλp=ρ-1, and vλ is a weak solution of the auxiliary problem

{(-Δ)psvλ=λ(ρuλp)1-pf~(x,ρuλpvλ),xΩ,vλ(x)=0,xNΩ.(3.7)

For all n, we set vn,λ=(vλ-τ+τ2-n)+. Then vn,λX(Ω),v0,λ=vλ+, and for all n, we have that 0vn+1,λ(x)vn,λ(x) and vn,λ(x)(vλ(x)-τ)+ for a.e. xΩ as n. Moreover, the following inclusion holds (up to a Lebesgue null set):

{vn+1,λ>0}{0<vλ<(2n+1-1)vn,λ}{vn,λ>τ2-n-1}.(3.8)

For all n, we set Rn,λ=vn,λpp, so R0,λ=vλ+ppρ-p, and {Rn,λ} is a nonincreasing sequence in [0,1]. We shall prove that Rn,λ0 as n. By Hölder’s inequality, the fractional Sobolev inequality (see [8, Theorem 6.5]), (3.8), and Chebyshev’s inequality, for all n, we have

Rn+1,λ|{vn+1,λ>0}|1-pps*vn+1,λps*pC|{vn,λ>τ2-n-1}|1-pps*vn+1,λpCτp(pps*-1)2(p-p2ps*)(n+1)Rn,λ1-pps*vn+1,λpC2(p-p2ps*)(n+1)Rn,λ1-pps*vn+1,λp.(3.9)

So, what we need now is an estimate of vn+1,λ. Using the elementary inequality

|ζ+-η+|p|ζ-η|p-2(ζ-η)(ζ+-η+),ζ,η,

testing (3.7) with vn+1,λ, and applying (3.8), for any fixed λ(0,1), we obtain

vn+1,λpA(vλ),vn+1,λλΩ(ρuλp)1-pf~(x,ρuλpvλ)vn+1,λ𝑑xC{vn+1,λ>0}((ρuλp)p2-p|vλ|p2-1+|vλ|p-1)vn+1,λ𝑑xC{vn+1,λ>0}((2n+1-1)p2-1|vn,λ|p2+(2n+1-1)p-1|vn,λ|p)𝑑xC2(n+1)(p-1)Rn,λp2p.(3.10)

Combining (3.9) with (3.10), we have

Rn+1,λC2(2p-1-p2ps*)(n+1)Rn,λ1+p2p-pps*=C2(2p-1-p2ps*)HnRn,λ1+βHn(C0Rn,λ)1+β,(3.11)

where

H=22p-1-p2ps*,β=p2p-pps*andC0>1 is large enough.

Similar to [12], provided that

ρ=max{(C01+βν)1pβ,1uλp}

is large enough, we can prove, for all n, that

Rn,λνnρp,where ν=H-1β(0,1).(3.12)

We argue by induction. We already know that R01ρp. Assuming that (3.12) holds for some n, by (3.11), we have

Rn+1,λHn(C0Rn,λ)1+βHnC01+β(νnρp)1+βC01+βρpβνnρpνn+1ρp.

By (3.12), we have Rn,λ0. This implies that vn,λ(x)0 as n for a.e. xΩ. So, vλ(x)τ for a.e. xΩ. An analogous argument applies to -vλ, so we have vλτ, hence uλL(Ω), and by the fractional Sobolev embedding,

uλτρuλp=τfor λ small enough such that ρ=1uλp.

That is to say, we can find λ0>0 such that uλτ for 0<λ<λ0. Hence, uλ is a nontrivial solution of the original problem (). The proof is complete. ∎

4 Proof of Theorem 1.2

In this section, we deal with the case where f is odd near the origin. First of all, for any fixed λ>0, we will get the existence of the infinitely many critical points of the functional I~λ, by Lemma 2.3. By Lemmas 3.1 and 3.3, and (f2), we have in hand the facts that I~λ satisfies the (PS) condition, and that is even and bounded from below, with I~λ(0)=0. Hence, it suffices to find, for any k, a subspace Xk and ρk>0 such that

supuXkSρkI~λ(u)<0.(4.1)

For any k, we find k independent functions φiX(Ω), i=1,,k, and set Xk:=span{φ1,φ2,,φk}. By (f1) and the definition of F~(x,t), we have F~(x,t)C|t|p1 for t. Then

I~λ(u)=1pup-λΩF~(x,u)dx1pup-λCΩ|u|p1dx=1pup-λCup1p1.

Since all norms on Xk are equivalent, p1<p, by choosing ρk>0 small enough, (4.1) holds. With all conditions of Lemma 2.3 being verified, we get a sequence of critical points unλX(Ω) with I~λ(unλ)=cnλ0 and cnλ<0 as n. In particular, {unλ} is a (PS)0 sequence of I~λ and has a convergent subsequence, still denoted by {unλ}. By Lemma 3.2,

unλ0as n.

Next, following similar arguments as in the proof of Theorem 1.1, we can choose n large enough such that ρ=1/unλp. Then we also get unλτ for n large enough. That is to say, for n large enough, unλ are solutions of the original problem (). The proof is complete.∎

5 Proof of Theorem 1.5

In this section, we deal with the superlinear case and give the proof of Theorem 1.5. We need to modify the nonlinearity as before. Similarly, we first observe that (f1’) and (f2’) imply that for |t| small enough,

F(x,t)|t|q2for a.e. xΩ(5.1)

and

F(x,t)|t|q1for a.e. xΩ.(5.2)

Let ρ(s) be the cut-off function defined in (3.1), where 0<τ<δ2 is chosen such that (5.1), (5.2) and (f3’) hold for |t|2τ. Set

F¯(x,t)=ρ(t)F(x,t)+(1-ρ(t))|t|q1,f¯(x,t)=tF¯(x,t).

As in Lemma 3.1, it is easy to check that f¯ has the following properties.

Lemma 5.1.

  • (i)

    There exists C>0 such that

    |f¯(x,t)|C|t|q1-1for t and a.e. xΩ.

  • (ii)

    We have that

    0<θF¯(x,t)f¯(x,t)tfor all t0 and a.e. xΩ,(5.3)

    where θ=min(q1,ϖ).

We introduce the functional

I¯λ(u)=1pup-λΩF¯(x,u)𝑑x,(5.4)

which is well defined on X(Ω), by Lemma 5.1.

Lemma 5.2.

The functional I¯λ satisfies the (PS) condition.

Proof.

Let {un} be a (PS) sequence for I¯λ, that is,

|I¯λ(un)|MandI¯λ(un)0as n.

From (5.3), we observe that since θ>p, for n large enough,

M+1+unI¯λ(un)-1θI¯λ(un),un(1p-1θ)unp.(5.5)

Hence, {un} is bounded, and the rest of proof is standard as in Lemma 3.3. ∎

The functional I¯λ enjoys the mountain pass geometry.

Lemma 5.3.

  • (i)

    There exist ρλ=ρ(λ)>0 and βλ=β(ρλ,λ)>0 such that

    I¯λ(u)βλfor u=ρλ.

  • (ii)

    Let eX(Ω) with 0<e<2τ . Then there exists Λ0>0 such that e>ρλ and I¯λ(e)<0 for λ>Λ0.

Proof.

By Lemma 5.1 and (5.4), we see that

I¯λ(u)1pup-Cλ|u|q1q1up(1p-Cλuq1-p),

where C is independent of λ. Taking ρλ=(2pCλ)1p-q1, we have

I¯λ(u)ρλp2p>0for uX(Ω),u=ρλ.

We also see that e>ρλ for λ large enough, since ρλ0 as λ. Moreover, we have

I¯λ(e)=1pep-λΩF¯(x,e)𝑑x.

Hence, there exists Λ0>0 such that I¯λ(e)<0 for λ>Λ0. ∎

From Lemmas 5.2 and 5.3, and the mountain pass theorem, we see that I¯λ has a critical value cλ and

cλ=infγΓmaxt[0,1]I¯λ(γ(t)),

where Γ={γC([0,1],E):γ(0)=0,γ(1)=e}.

In order to get the estimate of the critical level cλ, we introduce the following functional:

Jλ(u)=1pup-λΩ|u|q2𝑑x.(5.6)

Lemma 5.4.

There exists C>0 independent of λ>1 such that

cλCλ-pq2-p.

Proof.

By direct calculations, from (5.6), we obtain that

maxt0Jλ(te)=q2-ppq2(1λq2)pq2-p(epeq2q2)pq2-p.

Since te<2τ, t[0,1], and (5.1) holds, there exists C>0 independent of λ>1 such that

cλmaxt[0,1]I¯λ(te)maxt[0,1]Jλ(te)Cλ-pq2-p.

Now we give the proof of Theorem 1.5.

Proof of Theorem 1.5.

Let uλ be a critical point of I¯λ with critical value cλ. Similar to (5.5), by Lemma 5.4, we have

uλpθθ-pI¯λ(uλ)=pθθ-pcλ0as λ+.

Next, similar to Theorem 1.1, we will prove that for λ large enough, the critical point uλ is also the solution of the original problem (). By Lemma 5.1 (i) and (f1’), we can easily get that

|f¯(x,t)|C(|t|q1-1+|t|p-1),(x,t)Ω×, 2>q1p+q1ps*.

Then all the conditions of [12, Theorem 3.1] hold. Fix a weak solution uλX(Ω), with uλ+0, of the following problem:

{(-Δ)psu=λf¯(x,u),xΩ,u(x)=0,xNΩ.

We choose ρmax{1,uλq1-1} and set vλ=(ρuλq1)-1uλ. Then vλX(Ω),vλq1=ρ-1 and vλ is a weak solution of the auxiliary problem

{(-Δ)psvλ=λ(ρuλq1)1-pf¯(x,ρuλq1vλ),xΩ,vλ(x)=0,xNΩ.(5.7)

For all n, we set vn,λ=(vλ-τ+τ2-n)+. Then vn,λX(Ω), v0,λ=vλ+ and for all n, we have that 0vn+1,λ(x)vn,λ(x) and vn,λ(x)(vλ(x)-τ)+ for a.e. xΩ as n. Moreover, the following inclusion holds (up to a Lebesgue null set):

{vn+1,λ>0}{0<vλ<(2n+1-1)vn,λ}{vn,λ>τ2-n-1}.(5.8)

For all n, we set Rn,λ=vn,λq1q1. Then R0,λ=vλ+q1q1ρ-q1 and {Rn,λ} is a nonincreasing sequence in [0,1]. We shall prove that Rn,λ0 as n. By Hölder’s inequality, the fractional Sobolev inequality (see [8, Theorem 6.5]), (5.8) and Chebyshev’s inequality, for all n, we have

Rn+1,λ|{vn+1,λ>0}|1-q1ps*vn+1,λps*q1C|{vn,λ>τ2-n-1}|1-q1ps*vn+1,λq1Cτq1(q1ps*-1)2(q1-q12ps*)(n+1)Rn,λ1-q1ps*vn+1,λq1C2(q1-q12ps*)(n+1)Rn,λ1-q1ps*vn+1,λq1.(5.9)

So, what we need now is an estimate of vn+1,λ. Using the elementary inequality

|ζ+-η+|p|ζ-η|p-2(ζ-η)(ζ+-η+),ζ,η,

testing (5.7) with vn+1,λ, and applying (5.8), for any fixed large λ>1, we obtain

vn+1,λpA(vλ),vn+1,λλΩ(ρuλq1)1-pf¯(x,ρuλq1vλ)vn+1,λ𝑑xλC{vn+1,λ>0}((ρuλq1)q1-p|vλ|q1-1+|vλ|p-1)vn+1,λ𝑑xλC(ρuλq1)q1-p{vn+1,λ>0}((2n+1-1)q1-1|vn,λ|q1+(2n+1-1)p-1|vn,λ|p)𝑑xλC2(n+1)(q1-1)(ρuλq1)q1-pRn,λpq1.(5.10)

Combining (5.9) with (5.10), we have

Rn+1,λλC2(q1+q12p-q1p-q12ps*)(n+1)(ρuλq1)q12p-q1Rn,λ2-q1ps*=λC2(q1+q12p-q1p-q12ps*)(n+1)(ρuλq1)q12p-q1HnRn,λ1+βHn(ρuλq1)q12p-q1λ(C0Rn,λ)1+β,(5.11)

where

H=2(q1+q12p-q1p-q12ps*),β=1-q1ps*andC0>1 is large enough.

Similar to [12], provided that

ρ=max{1uλq1,ν-1γ(λC01+βuλq1(q12p-q1))1γ}

is big enough, we can prove that for all n,

Rn,λνnρq1,where ν=H-1β(0,1),γ=q1β+q1-q12p.(5.12)

We argue by induction. We already know that R01ρp. Assuming that (5.12) holds for some n, by (5.11), we have

Rn+1,λHn(ρuλq1)q12p-q1λ(C0Rn,λ)1+βHn(ρuλq1)q12p-q1λC01+β(νnρq1)1+βλ(C0)1+β(uλq1)q12p-q1ργνnρq1νn+1ρq1.

By (5.12), we have Rn,λ0. This implies that vn,λ(x)0 as n for a.e. xΩ. So vλ(x)τ for a.e. xΩ. An analogous argument applies to -vλ, so we have vλ<τ. Hence, uλL(), and by the fractional Sobolev embedding

uλτρuλq1=τfor λ large enough such that ρ=1uλq1,

since q2<γ+p. That is to say, we can find Λ0>0 such that uλτ for λ>Λ0. Hence, uλ is a nontrivial solution of the original problem (). The proof is complete. ∎

References

  • [1]

    G. Autuoria and P. Pucci, Elliptic problems involving the fractional Laplacian in N, J. Differential Equations 255 (2013), 2304–2362.  Google Scholar

  • [2]

    L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J. 37 (2014), 769–799.  CrossrefWeb of ScienceGoogle Scholar

  • [3]

    X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), 23–53.  CrossrefGoogle Scholar

  • [4]

    L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260.  CrossrefGoogle Scholar

  • [5]

    X. Chang and Z.-Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity 26 (2013), 479–494.  CrossrefWeb of ScienceGoogle Scholar

  • [6]

    D. Costa and Z.-Q. Wang, Multiplicity results for a class of superlinear elliptic problems, Proc. Amer. Math. Soc. 133 (2005), 787–794.  CrossrefGoogle Scholar

  • [7]

    A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 5, 1279–1299.  CrossrefGoogle Scholar

  • [8]

    E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.  Web of ScienceCrossrefGoogle Scholar

  • [9]

    S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche (Catania) 1 (2013), 201–216.  Google Scholar

  • [10]

    P. Felmer, A. Quaas and J. G. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), 1237–1262.  CrossrefGoogle Scholar

  • [11]

    G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S.) 5 (2014), no. 2, 373–386.  Google Scholar

  • [12]

    A. Iannizzotto, S. B. Liu, K. Perera and M. Squassina, Existence results for fractional p-Laplacian problem via morse theory, Adv. Calc. Var. 9 (2016), no. 2, 101–125.  Google Scholar

  • [13]

    E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014), 795–826.  Web of ScienceCrossrefGoogle Scholar

  • [14]

    X. Ros-Oton and J. Serra, Fractional Laplacian: Pohožaev identity and nonexistence results, C. R. Math. Acad. Sci. Paris. 350 (2012), 505–508.  CrossrefGoogle Scholar

  • [15]

    X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9) 101 (2014), 275–302.  CrossrefWeb of ScienceGoogle Scholar

  • [16]

    X. Ros-Oton and J. Serra, The Pohožev identity for the fractional Laplacian, Arch. Rat. Mech. Anal. 213 (2014), 587–628.  CrossrefGoogle Scholar

  • [17]

    S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in N, J. Math. Phys. 54 (2013), Article ID 031501.  Google Scholar

  • [18]

    R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), 887–898.  CrossrefWeb of ScienceGoogle Scholar

  • [19]

    R. Servadei and E. Valdinoci, Lewy–Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam. 29 (2013), 1091–1126.  Web of ScienceCrossrefGoogle Scholar

  • [20]

    R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (2013), 2105–2137.  Web of ScienceGoogle Scholar

  • [21]

    R. Servadei and E. Valdinoci, The Brezis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015), 67–102.  Google Scholar

  • [22]

    Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, NoDEA Nonlinear Differential Equations Appl. 8 (2001), 15–33.  CrossrefGoogle Scholar

About the article

Received: 2016-05-04

Accepted: 2016-09-11

Published Online: 2016-11-23

Published in Print: 2018-11-01


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11526126

Award identifier / Grant number: 11271264

Award identifier / Grant number: 11571209

This work was supported by NSFC grant no. 11526126, and was partly supported by NSFC grants nos. 11271264 and 11571209.


Citation Information: Advances in Nonlinear Analysis, Volume 7, Issue 4, Pages 485–496, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0105.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Jianfang Qin, Guotao Wang, Lihong Zhang, and Bashir Ahmad
Boundary Value Problems, 2019, Volume 2019, Number 1
[3]
Dimitri Mugnai and Edoardo Proietti Lippi
Nonlinear Analysis, 2019, Volume 188, Page 455

Comments (0)

Please log in or register to comment.
Log in