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 …

Quasilinear equations with indefinite nonlinearity

Junfang Zhao
  • School of Science, China University of Geosciences, Beijing 100083, P. R. China; and Department of Mathematics, University of Texas-Rio Grande Valley, Edinburg, TX 78539, USA
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Xiangqing Liu
  • Department of Mathematics, Yunnan Normal University, Kunming, Yunnan 650500, P. R. China; and Department of Mathematics, University of Texas-Rio Grande Valley, Edinburg, TX 78539, USA
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Zhaosheng Feng
Published Online: 2018-06-20 | DOI: https://doi.org/10.1515/anona-2018-0010

Abstract

In this paper, we are concerned with quasilinear equations with indefinite nonlinearity and explore the existence of infinitely many solutions.

Keywords: Quasilinear equations; indefinite nonlinearity; Sobolev inequality; approximate solution

MSC 2010: 35J62; 35A15; 35B20; 35B38

1 Introduction

Consider the quasilinear equation with the indefinite nonlinearity

{Δu+12uΔu2-a-|u|r-2u+a+|u|s-2u=0in Ω,u=0on Ω,(1.1)

where ΩN, N3, is a bounded smooth domain, r>4, 4<s<4NN-2, and a± are nonnegative continuous functions in Ω¯. A great number of theoretical issues concerning nonlinear elliptic equations with indefinite nonlinearity have received considerable attention in the past few decades. In particular, the existence of solutions has been studied extensively. For example, the existence of positive solutions and their multiplicity was studied by variational techniques [2], and the existence of nontrivial solutions was investigated by two different approaches (one involving the Morse theory and the other using the min-max method) [1]. It was shown that the existence of positive solutions, negative solutions and sign-changing solutions could be established by means of the Morse theory [7, 8]. For the results on a priori estimates and more comparable relations among various solutions etc., we refer the reader to [3, 5, 6, 9, 10, 12] and the references therein. However, as far as one can see from the literature, not much has been known about the existence of solutions to quasilinear equations with indefinite nonlinearity. From the variational point of view, there are two main difficulties that arise in the study. One lies in the fact that there is no suitable space in which the corresponding functional enjoys both smoothness and compactness. Compared with quasilinear equations with the definite nonlinearity, the other one is to prove the boundedness of the associated Palais–Smale sequences. In this work, we will get over these obstacles by means of the variational techniques and the perturbation method to study the existence of infinitely many solutions of system (1.1).

Set

Ω±={xΩ¯a±(x)>0}andΩ0=Ω(Ω¯+Ω¯-).

Assume that

  • (a)

    Ω+ and Ω¯+Ω¯-=.

We are looking for uH01(Ω)L(Ω) satisfying equation (1.1) in the weak form

Ω(1+u2)uφdx+Ωu|u|2φdx+Ωa-|u|r-2uφdx=Ωa+|u|s-2uφdx(1.2)

for φH01(Ω)L(Ω), which is formally the variational formulation of the following functional:

I(u)=12Ω(1+u2)|u|2dx+1rΩa-|u|rdx-1sΩa+|u|sdx.

In view of the perturbation (regularization) approach [11], due to the lack of a suitable working space, we introduce the corresponding perturbed functionals, which are smooth functionals in the given space and satisfy the necessary compactness property. For μ(0,1], we define the perturbed functional Iμ on the Sobolev space W01,p(Ω) with p>N by

Iμ(u)=μ2(Ω(1+u2)p2|u|pdx)2p+I(u)=μ2(Ω(1+u2)p2|u|pdx)2p+12Ω(1+u2)|u|2dx+1rΩa-|u|rdx-1sΩa+|u|sdx.

Note that Iμ is a C1- functional. We shall show that Iμ satisfies the Palais–Smale condition. The critical points of Iμ will be used as the approximate solution of problem (1.1).

Now let us briefly summarize our main results of this paper.

Theorem 1.1.

Assume that condition (a) holds, r>4, 4<s<4NN-2, μn>0, μn0, unW01,p(Ω), DIμn(un)=0 and Iμn(un)C. Then the following assertions hold:

  • (i)

    There exists a function uH01(Ω)L(Ω) satisfying equation ( 1.1 ).

  • (ii)

    Up to a subsequence, there holds unLC, unu for a.e. xΩ, unu in H01(Ω) , and

    μn[Ω(1+un2)p2|un|pdx]2p0𝑎𝑛𝑑Iμn(un)I(u)  as n.

Theorem 1.2.

Assume that condition (a) holds, r>4 and 4<s<4NN-2. Then problem (1.1) has infinitely many solutions.

Consider the more general quasilinear equation

{i,j=1NDi(aij(x,u)Dj(u))-12i,j=1NDsaij(x,u)DiuDju,-a-(x)|u|r-2u+a+(x)|u|s-2u=0in Ω,u=0on Ω.(1.3)

In the weak form, we look for uH01(Ω)L(Ω) such that

Ωi,j=1Naij(x,u)DiuDjφdx+12Ωi,j=1NDsaij(x,u)DiuDjuφdx+Ωa-(x)|u|r-2uφdx=Ωa+(x)|u|s-2uφdx(1.4)

for φH01(Ω)L(Ω), where Di=xi and Dsaij(x,s)=saij(x,s). For the coefficients aij, i,j=1,,N, we make the following assumptions:

  • (a0)

    aij, DsaijC1(Ω¯×), aij=aji, i,j=1,,N.

  • (a1)

    There exist constants c1,c2>0 such that

    c1(1+s2)|ξ|2i,j=1Naij(x,s)ξiξjc2(1+s2)|ξ|2

    for xΩ¯, s and ξ=(ξi)N.

  • (a2)

    There exist δ>0 and 0<q<s such that

    δi,j=1Naij(x,s)ξiξji,j=1N[aij(x,s)+12sDsaij(x,s)]ξiξjq(12-δ)i,j=1Naij(x,s)ξiξj

    for xΩ¯, s and ξN.

  • (a3)

    There exists an M>0 such that

    i,j=1N[aij(x,s)+12Dsaij(x,s)]ξiξj2s2M+s2i,j=1Naij(x,s)ξiξj

    for xΩ¯, s and ξN.

  • (a4)

    There holds

    lim|s|aij(x,s)s2=Aij(x)

    uniformly in xΩ¯.

Theorem 1.3.

Assume that r>4, 4<s<4NN-2 and conditions (a0)(a4) hold. Then equation (1.3) has infinitely many solutions.

The rest of the paper is organized as follows: The proof of Theorem 1.1 is presented in Section 2, and the proof of Theorem 1.2 is shown in Section 3. Section 4 is dedicated to the existence of infinitely many solutions to the more general quasilinear equation (1.3).

2 Convergence theorem

To prove Theorem 1.1, we need the following two technical lemmas.

Lemma 2.1.

There holds that

μ(Ω(1+u2)p2|u|pdx)2p+Ω(1+u2)|u|2dx+Ωa-|u|rdx+Ωa+|u|sdxC{Iμ(u)+DIμ(u)u+μ[Ω(1+u2)p2|u|pdx]2p+Ω(1+u2)u2dx}.

Proof.

For φW01.p(Ω), we know that

DIμ(u),φ=μ[Ω(1+u2)p2|u|pdx]2p-1Ω[(1+u2)p2|u|p-2uφ+(1+u2)p2-1u|u|pφ]dx+Ω[(1+u2)uφ+u|u|2φ]dx+Ωa-|u|r-2uφdx-Ωa+|u|s-2uφdx.(2.1)

Since Ω¯-Ω¯+=, we can choose ψC0(N) such that ψ0, ψ(x)=1 for xΩ¯-, and ψ(x)=0 for xΩ¯+. Then

Ωa-|u|rψpdx=Ωa-|u|rdxandΩa+|u|sψpdx=0.

Taking ψ=uψp as the test function in (2.1), we get

μ(Ω(1+u2)p2|u|pdx)2p-1Ω(1+u2)p2-1(1+2u2)|u|pψpdx+Ω(1+2u2)|u|2ψpdx+Ωa-|u|rdx=DIμ(u),uψp-pΩ(1+u2)uuψp-1ψdx   -pμ(Ω(1+u2)p2|u|pdx)2p-1Ω(1+u2)p2|u|p-2uuψp-1ψdxCDIλ(u)u+εΩ(1+u2)|u|2ψpdx+CΩ(1+u2)u2|ψ|2dx   +μ(Ω(1+u2)p2|u|pdx)2p-1{εΩ(1+u2)p2|u|pψpdx+CΩ(1+u2)p2|u|p|ψ|pdx}.(2.2)

In view of the Sobolev inequality

Ω(1+u2)p2|u|pdxCΩ(1+u2)p2|u|pdx,(2.3)

it follows from (2.2) and (2.3) that

Ωa-|u|rdxC{DIμ(u)u+μ[Ω(1+u2)p2|u|pdx]2p+Ω(1+u2)u2dx}.

By choosing q(4,s), we deduce that

Iμ(u)-1qDIμ(u),u=12μ(Ω(1+u2)p2|u|pdx)2p+12Ω(1+u2)|u|2dx-1qμ(Ω(1+u2)p2|u|pdx)2p-1Ω(1+u2)p2-1(1+2u2)|u|pdx-1qΩ(1+2u2)|u|2dx-(1q-1r)Ωa-|u|rdx+(1q-1s)Ωa+|u|sdx(12-2q)μ(Ω(1+u2)p2|u|pdx)2p+(12-2q)(Ω(1+u2)p2|u|2dx)2p-(1q-1r)Ωa-|u|rdx+(1q-1s)Ωa+|u|sdx,

and thus

μ(Ω(1+u2)p2|u|pdx)2p+Ω(1+u2)|u|2dx+Ωa+|u|sdxC{Iμ(u)+DIμ(u)u+Ωa-|u|rdx}C{Iμ(u)+DIμ(u)u+μ[Ω(1+u2)p2|u|pdx]2p+Ω(1+u2)|u|2dx}.

Lemma 2.2.

Assume that μn>0, μn0, unW01,p(Ω), DIμn(un)=0 and Iμn(un)C0. Then there exists a constant C>0 independent of n such that

μn[Ω(1+un2)p2|un|pdx]2p+Ω(1+un2)un2dxC.

Moreover, there holds

μn[Ω(1+un2)p2|un|pdx]2p+Ω(1+un2)|un|2dx+Ωa-|un|rdx+Ωa+|un|sdxC.

Proof.

Assume that

ρn4=μn(Ω(1+un2)2p|un|pdx)2p+Ω(1+un2)un2dxas n.(2.4)

By Lemma 2.1, we have

μn[Ω(1+un2)p2|un|pdx]2p+Ω(1+un2)|un|2dx+Ωa-|un|rdx+Ωa+|un|sdxCρn4.

Let vn=ρn-1un. Then

Ωvn2|vn|2dx=ρn-4Ωun2|un|2dxC,Ωa-|vn|rdx=ρn-rΩa-|un|rdxCρn4-r0as n,Ωa+|vn|sdx=ρn-sΩa+|un|rdxCρn4-s0as n.

We have vnv in Lq(Ω), 1q<4NN-2, vn2v2 in H01(Ω), Ωa-|v|rdx=0 and Ωa+|v|sdx=0. Thus, v(x)=0 for xΩ¯+Ω¯- and v2H01(Ω0)H01(Ω).

If p>N, then W01,p(Ω)Cα(Ω) for some α(0,1). Given ψ0 and ψC0(Ω0), we take

φn=ψun1+un2W01,p(Ω0)W01,p(Ω),

and have

0=DIμn(un),φn=μn(Ω(1+un2)p2|un|pdx)2p-1Ω(1+un2)p2|un|p-2unψun1+un2dx+μn(Ω(1+un2)p2|un|pdx)2p-1Ω(1+un2)p2-2|un|pψdx+Ω(1+un2)unψun1+un2dx+Ω(1+un2)-1|un|2ψdxμn(Ω(1+un2)p2|un|pdx)2pΩ(1+un2)p-12|un|p-2unψun(1+un2)12dx+Ωununψdx.

We further obtain the estimates as

ρn-2μn(Ω(1+un2)p2|un|pdx)2p-1|Ω(1+un2)p-12|un|p-2unψun(1+un2)12dx|ρn-2μn(Ω(1+un2)p2|un|pdx)1p(Ω|ψ|pdx)1pCμn120as n,

and

ρn-2Ωununψdx=ΩvnvnψdxΩvvψdxas n.

Hence, it gives

Ωv2ψdx0for ψ0 and ψC0(Ω0).

Since v2H01(Ω0), we have v20 in Ω. It follows from (2.4) and Lemma 2.1 that

ρn-4[μn(Ω(1+un2)p2|un|pdx)2p+Ω(1+un2)|un|2dx]C,

and

ρn-4[μn(Ω(1+u2)p2|un|pdx)2p+Ω(1+un2)un2dx]=1.(2.5)

In view of 1q<4NN-2, by taking vnv=0 in Lq(Ω), we get

ρn-4Ωun4dx=Ωvn4dxΩv4dx=0,

and

ρn-4Ωun2dx12ρn-4Ω(1+un4)dx0.

Let znW01,p(Ω) and

Dzn=ρn-2μn12(1+un2)12Dun.

It is easy to see that

(Ω|Dzn|pdx)2pρn-4μn(Ω(1+un2)p2|un|pdx)2pC.(2.6)

So we further deduce that

C1ρn-2μn12(1+un2)12|un||zn|C2ρn-2μn12(1+un2)12|un|,|zn|C2μn12(ρn-2+vn2)12|vn|0for a.e. xΩ,ρn-4μn(Ω(1+un2)p2|un|pdx)2pC(Ω|zn|pdx)2p0as n.

Consequently, the left-hand side of (2.5) converges to zero, which yields a contradiction. ∎

Proof of Theorem 1.1.

By Lemmas 2.1 and 2.2, we have

μn[Ω(1+un2)p2|un|pdx]2p+Ω(1+un2)|un|2dxC.

Assume that unu in Lq(Ω), 1q<4NN-2 and un2u2 in H01(Ω). We separate the proof into three steps.

Step 1. Moser’s iteration shows that the sequence {un} is uniformly bounded.

Note that for p>N, the convergence W01,p(Ω)Cα(Ω¯) holds for some α(0,1). The function un belongs to L(Ω). Here we prove that unL(Ω) is uniformly bounded. The term a+|u|s-2u in equation (1.2) is subcritical (s<4NN-2), while the term of a-|u|r-2u in the equation does not cause any trouble to us at this step. The sequence {un} is bounded in L4N/(N-2)(Ω). Starting from this L4N/(N-2)(Ω)-bound, by Moser’s iteration we see the L(Ω)-bound.

Step 2. Choose a suitable test function and show that the limit function u satisfies equation (1.2).

Let ψ0, ψC0(Ω) and φ=ψe-unW01,p(Ω). Take φ as the test function. Then we have

0=DIμn(un),φ=μn(Ω(1+un2)p2|un|pdx)2p-1Ω(1+un2)p2|un|p-2unψe-undx-μn(Ω(1+un2)p2|un|pdx)2p-1Ω(1+un2)p2-1(1+un2-un)|un|pψe-undx+Ω(1+un2)unψe-undx-Ω(1+un2-un)|un|2ψdx+Ωa-|un|r-2unψe-undx-Ωa+|un|s-2unψe-undx.(2.7)

We estimate each term in (2.7). From (2.6) we get

|μn(Ω(1+un2)p2|un|pdx)2p-1Ω(1+un2)p2|un|p-2unψe-undx|μn(Ω(1+un2)p2|un|pdx)1p(Ω|ψ|1pdx)Cμn120as n.

By the weak convergence, it gives

Ω(1+un2)unψe-undxΩ(1+u2)uψe-udx.

By the lower semi-continuity, we have

limnΩ(1+un2-un)|un|2ψe-undxΩ(1+u2-u)|u|2ψe-udx.

Using Lebesgue’s dominated convergence theorem leads to

Ωa-|un|r-2unψe-undxΩa-|u|r-2uψe-udx

and

Ωa+|un|s-2unψe-undxΩa+|u|s-2uψe-udx.

It follows from (2.7) and the above estimates that

Ω(1+u2)uψe-udx-Ω(1+u2-u)|u|2ψe-udx+Ωa-|u|r-2uψe-udx-Ωa+|u|s-2uψe-udx0

for ψ0 and ψC0(Ω). Moreover, we get

Ω(1+u2)u(ψe-u)dx+Ωu|u|2ψe-udx3+Ωa-|u|r-2uψe-udx-Ωa+|u|s-2uψe-udx0.(2.8)

Given φ0 and φC0(Ω), we choose ψnC0(Ω) such that ψnφeu in H01(Ω), ψnC and ψnφe-u for a.e. xΩ. Taking ψn as the test function in (2.8) and letting n tend to infinity, we obtain

Ω(1+u2)uφdx+Ωu|u|2φdx+Ωa-|u|r-2uφdx-Ωa+|u|s-2uφdx0

for φ0 and φC0(Ω).

Processing in a similar manner, one can also obtain an inequality with an opposite direction. Equation (1.2) holds for all φ0 and φC0(Ω). By the density argument, equation (1.2) also holds for all functions φH01(Ω)L(Ω).

Step 3. Since uH01(Ω)L(Ω) satisfies equation (1.2), we have

Ω(1+2u2)|u|2dx+Ωa-|u|rdx=Ωa+|u|sdx.

By DIμn(un)=0, we have DIμn(un),un=0. That is,

μn(Ω(1+un2)p2|un|pdx)2p-1Ω(1+un2)p2-1(1+un2)|un|pdx+Ω(1+2un2)|un|2dx+Ωa-|un|rdx=Ωa+|un|sdx.

In view of s<4NN-2 and Ωa+|un|sdxΩa+|u|sdx, as n tends to infinity, by the lower semi-continuity we deduce that

μn(Ω(1+un2)p2|un|pdx)2p0,Ω|un|2dxΩ|u|2dx,Ωun2|un|2dxΩu2|u|2dx,Ωa-|un|rdxΩa-|u|rdx,

and

Iμn(un)I(u).

Hence, we obtain unu in H01(Ω) and ununuu in L2(Ω) as n. ∎

In the following context, we call c a critical value of the functional I, provided there exists a function uH01(Ω)L(Ω) satisfying equation (1.2) and I(u)=c. Theorem 1.1 implies that if μn0, cn is a critical value of Iμn and c=limncn, then c is a critical value of I.

3 Proof of Theorem 1.2

In this section, we prove Theorem 1.2. We construct a sequence of critical values of the functional Iμ with μ>0. The corresponding critical points will be used as the approximate solutions of equation (1.2).

Lemma 3.1.

Suppose that {un} is a Palais–Smale sequence of the functional Iμ with μ>0. Then un is bounded in W01,p(Ω).

Proof.

The proof is similar to the one of Lemma 2.2. Since μ>0 is fixed and DIμ(un)0, by Lemma 2.1 we have

μ(Ω(1+un2)p2|un|pdx)2p+Ω(1+un2)|un|2dx+Ωa-|un|rdx+Ωa+|un|sdxC[1+Iμ(un)+μ(Ω(1+un2)p2|un|pdx)2p+Ω(1+un2)un2dx].

As in the proof of Lemma 2.2, we prove it by way of contradiction. Assume that

ρn4=μ(Ω(1+un2)p2|un|pdx)2p+Ω(1+un2)un2dxas n.(3.1)

Then it is easy to see that

μ(Ω(1+un2)p2|un|pdx)2p+Ω(1+un2)|un|2dx+Ωa-|un|rdx+Ωa+|un|sdxCρn4.

Let vn=ρn-1un. As in the proof of Lemma 2.1, we have vnv in Lq(Ω), 1q<4NN-2, vn2v2 in H01(Ω), and v=0 in Ω¯+Ω¯-. Since p>N and μ is fixed, we find

μ(Ω|vn|p|vn|pdx)2pρn-4μ(Ω|un|p|un|pdx)2pC.

So we see vn2v2 in Cα(Ω¯) for some α(0,1). Given ψ0 and ψC0(Ω0), we take φn=ψun/(1+un2) as the test function, and thus have

φn=ψun1+un2-ψ1-un2(1+un2)2un,φn=(Ω|φn|pdx)1pC(Ω|ψ|pdx)1p+CψL(Ω)(Ω|un|pdx)1pCρn2ψ,

and

o(1)ψ=ρn-2DIμ(un),φnρn-2[μ(Ω(1+un2)p2|un|pdx)2p-1Ω(1+un2)p2-1|un|p-2ununψdx+Ωununψdx]=μ(Ω(ρn-2+vn2)p2|vn|pdx)2p-1Ω(ρn-2+vn2)p2-1|vn|p-2vnvnψdx+Ωvnvnψdx(3.2)

for ψ0 and ψC0(Ω0).

If

limnΩ(ρn-2+vn2)p2|vn|pdx=C0=0,

then

Ω|v|p|v|pdxlim¯nΩ(ρn-2+vn2)p2|vn|pdx=0.

So one can see v20. Otherwise, we assume that

lim¯nΩ(ρn-2+vn2)p2|vn|pdx=C0>0.

By a density argument, inequality (3.2) also holds for ψ0 and ψW01,p(Ω0).

Let

Cn=max{vn2(x)xΩ0}.

Since vn2v2 in Cα(Ω¯) and v2=0 in Ω¯+Ω¯-, we have v20 on Ω0 and Cn0 as n.

Define

ψnW01,p(Ω0)byψn(x)=(vn2(x)-Cn)+,

for xΩ¯0, |ψn|2|vnvn| and ψnC. Taking ψn as the test function in inequality (3.2), we have

o(1)μC02p-1Ω|vn|p-2vn|vn|p-2vn(vn2-Cn)+dx+Ωvnvn(vn2-Cn)+dx=μC02p-121-pΩ|(vn2-Cn)+|pdx+12Ω|(vn2-Cn)+|2dx.

Taking n, we obtain

0μC02p-121-pΩ|v2|pdx+12Ω|v2|2dx.

Hence, v2=0 in Ω¯, which leads to a contradiction by virtue of Lemma 2.2. In fact, it follows from (3.1) that

μ(Ω|v|pdx)2p+Ωv4dx=1.

Lemma 3.2.

The functional Iμ with μ>0 satisfies the Palais–Smale condition.

Proof.

Let {un} be a Palais–Smale sequence of the functional Iμ with μ>0. By Lemma 3.1, {un} is bounded in W01,p(Ω). When p>N, we have unu in Cα(Ω¯) for some α(0,1).

Let

limnΩ(1+un2)p2|un|pdx=C0.

If C0=0, then un0 in W01,p(Ω). Otherwise, there holds

o(1)=DIμ(un)-DIμ(um),un-um=μC02p-1Ω(1+u2)p2(|un|p-2un-|um|p-2um,un-um)dx+Ω(1+u2)|un-um|2dx+o(1)CΩ|un-um|pdx+o(1).

So, {un} is a Cauchy sequence of W01,p(Ω) . ∎

Proof of Theorem 1.2.

We define a sequence of critical values of the functional Iμ with μ(0,1] by

ck(μ)=infφΓksuptBkIμ(φ(t)),k=1,2,,

where

Γk={φφC(Bk,W01,p(Ω)),φ is odd and I1(φ(t))<0 for tBk},

and Bk is the unit ball of k. Then we have

Iμ(u)=12μ[Ω(1+u2)p2|u|pdx]2p+I(u)12Ω(1+u2)|u|2dx+1rΩa-|u|rdx-1sΩa+|u|sdx12Ω|w|2dx-CΩ|w|s2dx:=J(w),

where w is defined by Dw=(1+u2)1/2Du and wH01(Ω), and J is a C1-functional defined on H01(Ω).

Define the critical values of J by

αk=infφGksuptBkJ(φ(t)),

where

Gk={φφC(Bk,W01,p(Ω)),φ is odd and J(φ(t))<0 for tBk}.

By Lemma 3.2 and the symmetric mountain pass lemma [4], αk, k=1,2,, are critical values of J and αk as k. We have the estimate ck(μ)αk. On the other hand, let βk=ck(1). Then we get

αkck(μ)βk,k=1,2,, for μ(0,1].

Let

ck=limμ0ck(μ).

By virtue of Theorem 1.1, ck, k=1,2,, are the critical values of I and ck+ as k. ∎

4 More general cases

In this section, we consider the more general quasilinear equation (1.3) and prove Theorem 1.3.

Equation (1.3) has a variational structure, given by the functional

H(u)=12Ωi,j=1Nai,j(x,u)DiuDjudx+1rΩa-|u|rdx-1sΩa+|u|sdx.

Again we apply the perturbation method and introduce the perturbed functional Hμ with μ(0,1]:

Hμ(u)=12μ(Ω(1+u2)p2|u|pdx)2p+H(u)=12μ(Ω(1+u2)p2|u|pdx)2p+12Ωi,j=1Naij(x,u)DiuDjudx+1rΩa-|u|rdx-1sΩa+|u|sdx.

Note that Hμ is defined on the Sobolev space W01,p(Ω) with p>N. It is a C1-functional on W01,p(Ω), and satisfies

DHμ(u),φ=μ[Ω(1+u2)p2|u|pdx]2p-1Ω[(1+u2)p2|u|p-2uφ+(1+u2)p2-1u|u|pφ]dx+Ωi,j=1Naij(x,u)DiuDjφdx+12Ωi,j=1NDsaij(x,u)DiuDjφdx+Ωa-|u|r-2uφdx-Ωa+|u|s-2uφdx

for φW01,p(Ω).

As we have seen in the preceding section, for the quasilinear equations with indefinite nonlinearity, compared with ones with definite nonlinearity, the difficulty is to prove the boundedness of some associated sequences, either the sequence of approximate solutions (Lemmas 2.1 and 2.2) or the Palais–Smale sequence of the perturbed functional (Lemma 3.1). When we have proved the boundedness of these sequences, we can deal with the quasilinear equations as before to obtain the convergence and the existence results.

In the following, we will prove the boundedness of sequences of the approximate solutions, and the boundedness of the Palais–Smale sequences of the functional Hμ.

Lemma 4.1.

For Hμ(u), there holds

μ(Ω(1+u2)p2|u|pdx)2p+Ω(1+u2)|u|2dx+Ωa-|u|rdx+Ωa+|u|sdxC[Hμ(u)+DHμ(u)u+μ(Ω(1+u2)p2|u|pdx)2p+Ω(1+u2)u2dx].(4.1)

Proof.

The proof is closely analogous to the one of Lemma 2.1. Choose ψC0(N) such that ψ0, ψ(x)=1 for xΩ-, and ψ(x)=0 for xΩ+. Taking φ=uψp as the test function, we have

μ(Ω(1+u2)p2|u|pdx)2p-1Ω(1+u2)p2-1(1+2u2)|u|pψpdx   +Ωi,j=1N(aij(x,u)+12uDsaij(x,u))DiuDjuψpdx+Ωa-|u|rdx=DHμ(u),uψp-pμ(Ω(1+u2)p2|u|pdx)2p-1Ω(1+u2)p2|u|p-2uuψp-1ψdx   -pΩi,j=1Naij(x,u)Diuuψp-1Djψdxμ(Ω(1+u2)p2|u|pdx)2p-1[εΩ(1+u2)p2|u|pψpdx+CΩ(1+u2)p2|u|p|ψ|pdx]+CCHμ(u)u   +εΩi,j=1Naij(x,u)DiuDjuψpdx+CΩi,j=1Naij(x,u)DiψDjψu2ψp-2dx.

In view of assumptions (a2) and (a3), it follows from the Sobolev inequality (2.3) that

Ωa-|u|rdxC[DHμ(u)u+μ(Ω(1+u2)p2|u|pdx)2p+Ω(1+u2)u2dx].(4.2)

Hence, we further have

Hμ(u)-1qDHμ(u),u=-1qμ(Ω(1+u2)p2|u|pdx)2p-1Ω(1+2u2)2p-1(1+2u2)|u|pdx+Ωi,j=1N(12aij(x,u)-1q(aij(x,u)+12uDsaij(x,u)))DiuDjudx+12μ(Ω(1+u2)p2|u|pdx)2p-(1q-1r)Ωa-|u|rdx+(1q-1s)Ωa+|u|sdx(12-2q)(Ω(1+u2)p2|u|pdx)2p+δΩi,j=1Naij(x,u)DiuDjudx-(1q-1r)Ωa-|u|rdx+(1q-1s)Ωa+|u|sdx.(4.3)

By virtue of (4.2) and (4.3), we arrive at (4.1). ∎

Lemma 4.2.

Assume that μn>0, μn0, unW01,p(Ω), DHμn(un)=0 and Hμn(un)C. Then there exists a constant C independent of n such that

μn(Ω(1+un2)p2|un|pdx)2p+Ω(1+un2)un2dxC.

Moreover, we have

μn(Ω(1+un2)p2|un|pdx)2p+Ω(1+un2)|un|2dx+Ωa-|un|rdx+Ωa+|un|sdxC.

Proof.

As in the proof of Lemma 2.2, we apply the indirect argument by assuming that

ρn4=μn(Ω(1+un2)p2|un|pdx)2p+Ω(1+un2)|un|2dxas n.

By Lemma 4.1, we get

μn(Ω(1+un2)p2|un|pdx)2p+Ω(1+un2)|un|2dx+Ωa-|un|rdx+Ωa+|un|sdxCρn4.(4.4)

Let vn=ρn-1un. Then vnu in Lq(Ω), 1q<4NN-2 and vnDvnvv in L2(Ω), where v2H01(Ω0)H01(Ω). Given ψ0 and ψC0(Ω0), we set φn=ψun/(M+un2) with M being the constant given in condition (a3). From condition (a3) we have

0=DHμn(un),φn=μn(Ω(1+un2)p2|un|pdx)2p-1Ω(1+un2)p2|un|p-2unψunM+un2dx+μn(Ω(1+un2)p2|un|pdx)2p-1Ω(1+un2)p2|un|pψ(M(M+u2)2+(M-1)un2(1+un2)(M+un2)2)dx+Ωi,j=1Naij(x,un)M+un2unDiunDjψdx+Ωi,j=1N(aij(x,un)M-un2(M+un2)2+12unDsaij(x,un)1M+un2)DiunDjunψdxμn(Ω(1+un2)p2|un|pdx)2p-1Ω(1+un2)p2|un|p-2unψunM+un2dx+Ωi,j=1Naij(x,un)M+un2unDiunDjunψdx.

By (4.4), we find

ρn-2μn(Ω(1+un2)p2|un|pdx)2p-1|Ω(1+un2)p2|un|p-2unψunM+un2dx|ρn-2μn(Ω(1+un2)p2|un|pdx)1p(Ω|ψ|pdx)1pCμn120as n.

It follows from Lemma 4.3 that

ρn-2Ωi,j=1Naij(x,un)M+un2uniunjψdx=Ωi,j=1Naij(x,un)M+un2vnivnjψdxΩi,j=1NAij(x)vivjψdxas n.

That is,

Ωi,j=1NAij(x)vivjψdx0

for ψ0 and ψC0(Ω0).

Since v2H01(Ω0)H01(Ω), by the density argument we have

Ωi,j=1NAij(x)vivjv2dx=0

and v20 in Ω. The remainder is the same as that shown in Lemma 2.2, so we omit it here. ∎

Lemma 4.3.

For ψH01(Ω), there holds

limnρn-2Ωi,j=1Naij(x,un)M+un2uniunjψdx=Ωi,j=1NAij(x)vivjψdx.

Proof.

For T>0, let unT be the truncated function of un, that is, unT=un if |un|T, and unT=±T if ±unT. Taking unT as the test function, we have

0=DHμn(un),unT=μn(Ω(1+un2)p2|un|pdx)2p-1Ω[(1+un2)p2|unT|p+(1+un2)p2-1|un|2ununT]dx+Ωi,j=1N(aij(x,un)+12unDsaij(x,un))iunTjunTdx+12|un|Ti,j=1NunTDsaij(x,un)DiunDjundx+Ωa-|un|r-2ununTdx-Ωa+|un|s-2ununTdx.(4.5)

In view of assumption (a3), there exists a T0>0 such that

i,j=1NsDsaij(x,s)ξiξj0

for |s|T>T0 and xΩ¯, where ξN. By condition (a2) and (4.5), we get

Ω|unT|2dxCΩi,j=1N(aij(x,un)+12unDsaij(x,un))iunTjunTdxCΩa+|un|s-2ununTdxCTΩa+|un|s-1dxCT(Ωa+|un|sdx)s-1sCTρn4-4s.

Hence, we obtain the estimate

ρn-2Ωi,j=1Naij(x,un)M+un2uniunjψdx=ρn-2|un|Ti,j=1Naij(x,un)M+un2uniunjψdx+ρn-2|un|<Ti,j=1Naij(x,un)M+un2uniunjψdx=ρn-2(|un|TAij(x)uniunjψdx+oT(1)ununL2(Ω))+ρn-2CTunTL2(Ω)=ΩAij(x)vnivnjψdx+oT(1)+ρn-2CT32ρn2-2s=ΩAij(x)vnivnjψdx+on(1)ΩAij(x)vivjψdxfor ψH01(Ω).

The following proposition can be regarded as a counterpart of Theorem 1.1.

Proposition 4.4.

Assume that μn>0, μn0, unC01(Ω), DHμn(un)=0 and Hμn(un)C. Then the following assertions hold:

  • (i)

    There exists a function uH01(Ω)L(Ω) satisfying equation ( 1.4 ).

  • (ii)

    Up to a subsequence, there holds

    unLC,unu in H01(Ω),μn(Ω(1+un)p2|un|pdx)2p0,

    and

    Hμn(un)H(un).

By Lemma 4.2, there holds

μn(Ω(1+un2)p2|un|pdx)2p)+Ω(1+un2)|un|2dxC.(4.6)

Proposition 4.4 can be proved similarly to Theorem 1.1, so we omit it and refer to [11]. As we have seen in the proof of Theorem 1.1, with the help of estimate (4.6), the proof of Theorem 1.1 can also be done in the same way as that for quasilinear equations with definite nonlinearity.

Next, we consider the perturbed functional Hμ with μ(0,1].

Lemma 4.5.

Let {un} be a Palais–Smale sequence of the functional Hμ with μ>0. Then un is bounded in W01,p(Ω).

Proof.

By Lemma 4.1, we only need to prove

μ(Ω(1+un2)p2|un|pdx)2p+Ω(1+un2)|un|2dxC.

Otherwise, we have

ρn4=μ(Ω(1+un2)p2|un|pdx)2p+Ω(1+un2)|un|2dx+.

Let

vn=ρn-1un and (Ω|vn2|p|vn|pdx)2p=ρn-4(Ω|un|p|un|pdx)2pC.

Then vnv in Lq(Ω), 1q<4NN-2, vn2v2 in Cα(Ω¯) for some α(0,1), and vn2v2 in W01,p(Ω) (and H01(Ω)). As in the proof of Lemma 2.2, we know that v(x)=0 for xΩ¯+Ω¯-, and v2W01,p(Ω0)W01,p(Ω). For ψ0 and ψW01,p(Ω0), we let φn=ψun/(M+un2), φnW01,p(Ω) and

φn=ψunM+un2+ψM-un2(M+un2)2un,φnCρn2ψ.

Then we deduce that

o(1)ψ=ρn-2DHμ(un),φn=ρn-2μ(Ω(1+un2)p2|un|pdx)2p-1Ω(1+un2)p2|un|p-2unψunM+un2dx+ρn-2μ(Ω(1+un2)p2|un|pdx)2p-1Ω(1+un2)p|un|pψ(M(M+un2)2+(M-1)un2(1+un2)(M+un2)2)dx+ρn-2Ωi,j=1Naij(x,un)M+un2uniunjψdx+ρn-2Ωi,j=1N(aij(x,un)M-un2(M+un2)2+12Dsaij(x,un)unM+un2)DiunDjunψdxμ(Ω(ρn-2+vn2)p2|vn|p)2p-1Ω(ρn-2+vn2)p2|vn|p-2vnψvnM+un2dx+Ωi,j=1Naij(x,un)M+vn2vnDivnDjψdx.

Choose ψ=12(vn2-Cn)+W01,p(Ω0) and Cn=maxΩ0vn20. By the lower semi-continuity, we obtain

0μ(Ω|v|p|v|pdx)2p+cΩv2|v|2dx.

Hence, v2=0. This yields a contradiction according to Lemma 3.1. ∎

Proposition 4.6.

The functional Hμwith μ>0 satisfies the Palais–Smale condition.

Proof.

Let {un} be a Palais–Smale sequence of Hμ. By Lemma 4.5, un is bounded in W01,p(Ω). Assume that unu in Cα(Ω¯) for some α(0,1). Set

C0=limnΩ(1+un2)p2|un|pdx.

If C0=0, we are done. Otherwise, there holds

o(1)=DHμ(un)-DHμ(um),un-um=μC02p-1Ω(1+u2)p2(|un|p-2un-|um|p-2um,un-um)dx+Ωi,j=1Naij(x,u)Di(un-um)Dj(un-um)dx+o(1)cΩ|un-um|pdx+o(1).

So, {un} is a Cauchy sequence of W01,p(Ω). ∎

Proof of Theorem 1.3.

Define

ck(μ)=infφΓksuptBkHμ(φ(t)),k=1,2,,

where

Γk={φφC(Bk,W01,p(Ω)),φ is odd,H1(φ(t))<0 for tBk}.

A straightforward estimate on Hk(u) gives

Hk(u)=12μ(Ω(1+u2)p2|u|pdx)2p+H(u)12Ωi,j=1Naij(x,u)DiuDjudx+1rΩa-|u|rdx-1sa+|u|sdxc12Ω(1+u2)|u|2dx-1sΩa+|u|sdxc12Ω|w|2dx-cΩ|w|s2dx:=J(w).

One can find αkβk such that αk as k, and αkck(μ)βk.

Let

ck=limμ0ck(μ).

According to Proposition 4.4, ck, k=1,2,, are critical values of H and ck+ as k. Consequently, the proof is completed. ∎

References

  • [1]

    S. Alama and M. Del Pino, Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 1, 95–115.  CrossrefGoogle Scholar

  • [2]

    S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations 1 (1993), no. 4, 439–475.  CrossrefGoogle Scholar

  • [3]

    S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal. 141 (1996), no. 1, 159–215.  CrossrefGoogle Scholar

  • [4]

    A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.  CrossrefGoogle Scholar

  • [5]

    H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal. 4 (1994), no. 1, 59–78.  CrossrefGoogle Scholar

  • [6]

    H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl. 2 (1995), no. 4, 553–572.  CrossrefGoogle Scholar

  • [7]

    K.-C. Chang and M.-Y. Jiang, Dirichlet problem with indefinite nonlinearities, Calc. Var. Partial Differential Equations 20 (2004), no. 3, 257–282.  CrossrefGoogle Scholar

  • [8]

    K.-C. Chang and M.-Y. Jiang, Morse theory for indefinite nonlinear elliptic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 1, 139–158.  CrossrefGoogle Scholar

  • [9]

    Y. Du and S. Li, Nonlinear Liouville theorems and a priori estimates for indefinite superlinear elliptic equations, Adv. Differential Equations 10 (2005), no. 8, 841–860.  Google Scholar

  • [10]

    S. Li and Z.-Q. Wang, Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, J. Anal. Math. 81 (2000), 373–396.  CrossrefGoogle Scholar

  • [11]

    X.-Q. Liu, J.-Q. Liu and Z.-Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc. 141 (2013), no. 1, 253–263.  Google Scholar

  • [12]

    M. Ramos, S. Terracini and C. Troestler, Superlinear indefinite elliptic problems and Pohožaev type identities, J. Funct. Anal. 159 (1998), no. 2, 596–628.  CrossrefGoogle Scholar

About the article

Received: 2018-01-14

Accepted: 2018-04-07

Published Online: 2018-06-20


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11601493

Award identifier / Grant number: 11761082

Junfang Zhao is supported by National Science Foundation of China under No. 11601493 and Beijing Higher Education Young Elite Faculty Project. Xiangqing Liu is supported by National Science Foundation of China under No. 11761082 and Yunnan Young Academic and Technical Leaders’ Program (2015HB028). This work is also partially supported by UTRGV Faculty Research Council Award 110000327.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 1235–1251, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0010.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

Comments (0)

Please log in or register to comment.
Log in