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

Advanced Nonlinear Studies

Editor-in-Chief: Ahmad, Shair


IMPACT FACTOR 2018: 1.650

CiteScore 2018: 1.49

SCImago Journal Rank (SJR) 2018: 1.422
Source Normalized Impact per Paper (SNIP) 2018: 0.865

Mathematical Citation Quotient (MCQ) 2018: 1.19

Online
ISSN
2169-0375
See all formats and pricing
More options …
Volume 16, Issue 1

Issues

Multiple Solutions to (p,q)-Laplacian Problems with Resonant Concave Nonlinearity

Salvatore A. Marano
  • Corresponding author
  • Department of Mathematics and Computer Science, University of Catania, Viale A. Doria 6, 95125 Catania, Italy
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Sunra J. N. Mosconi / Nikolaos S. Papageorgiou
Published Online: 2015-12-04 | DOI: https://doi.org/10.1515/ans-2015-5011

Abstract

The existence of multiple solutions to a Dirichlet problem involving the (p,q)-Laplacian is investigated via variational methods, truncation-comparison techniques, and Morse theory. The involved reaction term is resonant at infinity with respect to the first eigenvalue of -Δp in W01,p(Ω) and exhibits a concave behavior near zero.

Keywords: -Laplacian; Resonance from the Left (Right); Concave Nonlinearity; Multiple Solutions

MSC 2010: 35J20; 35J92; 58E05

1 Introduction

Let Ω be a bounded domain in N with a smooth boundary Ω, let 1<qp<+, and let μ0. Consider the Dirichlet problem

{-Δpu-μΔqu=f(x,u) in Ω,u=0 on Ω,(1.1)

where Δr, r>1, denotes the r-Laplacian, namely,

Δru:=div(ur-2u) for all uW01,r(Ω),

the reaction term f:Ω× satisfies Carathéodory’s conditions, while, as usual, p=q if and only if μ=0. Elliptic equations involving differential operators of the form

Au:=-Δpu-Δqu,

often called (p,q)-Laplacian, occur in many important concrete situations. For instance, this happens when one seeks stationary solutions to the reaction-diffusion system

ut+Au=c(x,u),

which exhibits a wide range of applications in physics and related sciences such as biophysics, quantum and plasma physics, and chemical reaction design; see [3, 6]. Consequently, they have been the subject of numerous investigations, both in bounded domains and on the whole space, mainly concerning the multiplicity of solutions or bifurcation-type results.

This paper falls within the first framework. We show that if, roughly speaking, f has a subcritical growth and, moreover,

  • (i)

    lim|t|+p|t|p0tf(x,ξ)𝑑ξ=λ1,p  uniformly in xΩ, where λ1,p denotes the first eigenvalue of (-Δp,W01,p(Ω)),

  • (ii)

    lim|t|+[f(x,t)t-p0tf(x,ξ)𝑑ξ]=+  uniformly in xΩ,

  • (iii)

    c|t|θf(x,t)tθ0tf(x,ξ)𝑑ξ  for all (x,t)Ω×[-δ,δ], where c>0, θ(1,q), while δ>0,

then (1.1) possesses at least three nontrivial solutions in C01(Ω¯), one greatest negative v-, another smallest positive u+, and a third nodal u0, such that v-u0u+; see Theorem 3.9 below.

Assumptions (i)(ii) directly give

lim|t|+[λ1,p|t|p-p0tf(x,ξ)𝑑ξ]=+uniformly in xΩ;(1.2)

see the proof of Lemma 3.1. Hence, resonance with respect to λ1,p from the left occurs and, a fortiori, the energy functional φ associated with (1.1) is coercive.

Now, the question of investigating what happens if there is resonance from the right of λ1,p, i.e., the limit in (1.2) equals -, naturally arises. Accordingly, φ turns out to be indefinite and direct methods no longer work. However, via linking arguments and, in place of (ii), via the hypothesis that

  • (iv)

    either μ>0 and

    lim inf|t|+1|t|η[p0tf(x,ξ)𝑑ξ-f(x,t)t]C>0uniformly in xΩ,

    where η(q,p], or μ=0 and

    lim|t|+[p0tf(x,ξ)𝑑ξ-f(x,t)t]=+uniformly in xΩ,

we still obtain a nontrivial solution u0C01(Ω¯) of (1.1); cf. Theorem 4.5 below.

It should be also noted that, in both settings, due to (ii), the nonlinearity f(x,) exhibits a concave behavior at the origin. Such a type of growth rate has been widely studied, also combined with further conditions, provided p=2 and μ=0, i.e., the equation is semilinear. As an example, besides the seminal paper [2], let us mention [21, 22, 16, 8]. A similar comment holds true also when p2 but μ=0, in which case the literature looks to be daily increasing; see for instance the very recent papers [12, 19, 14, 18] and, concerning the nonsmooth framework, [17, 13].

Another meaningful feature of (1.1) is the following. If μ>0, then the differential operator u-Δpu -μΔqu turns out to be nonhomogeneous. Hence, standard results for the p-Laplacian not always extend in a simple way to it.

Our approach is variational, based on critical point theory, together with appropriate truncation-comparison arguments and results from Morse theory.

2 Mathematical Background

Let (X,) be a real Banach space. Given a set VX, write V¯ for the closure of V and V for the boundary of V. If xX, δ>0, then Bδ(x):={zX:z-x<δ}, while Bδ:=Bδ(0). The symbol X* denotes the dual space of X, , indicates the duality brackets for the pair (X*,X), and xnx (respectively, xnx) in X means that ‘the sequence {xn} converges strongly (respectively, weakly) in X’. An operator A:XX* is called of type (S)+ provided

xnxin X,lim supn+A(xn),xn-x0implyxnxin X.

Let φC1(X) and let c. Put

φc:={xX:φ(x)c},φc:={xX:φ(x)c},Kφ:={xX:φ(x)=0},Kφc:={xKφ:φ(x)=c}.

We say that φ satisfies the Cerami condition when

  • (C)

    every sequence {xn}X such that {φ(xn)} is bounded and

    limn+(1+xn)φ(xn)=0in X*

    admits a strongly convergent subsequence.

This compactness-type assumption turns out to be weaker than the usual Palais–Smale condition. Nevertheless, it suffices to prove a deformation theorem, from which the minimax theory for the critical values of φ follows. In such a framework, the topological notion of linking sets plays a key role.

Suppose Q0,Q,E are three nonempty closed subsets of a Hausdorff topological space Y with Q0Q. The pair (Q0,Q) links E in Y if Q0E= and, for every γC0(Q,Y) such that γ|Q0=id|Q0, one has γ(Q)E.

The following general minimax principle is well known; see, e.g., [10, Theorem 5.2.5].

Let X be a Banach space, let Q0,Q, and E be such that the pair (Q0,Q) links E in X, and let φC1(X) satisfy condition (C). If, moreover, supQ0φ<infEφ and

c:=infγΓsupxQφ(γ(x)),where Γ:={γC0(Q,X):γ|Q0=id|Q0},

then cinfEφ and Kφc.

Appropriate choices of linking sets in Theorem 2.2 produce meaningful critical point results. For later use, we state here the famous Ambrosetti–Rabinowitz mountain pass theorem.

If (X,) is a Banach space, φC1(X) fulfills (C), x0,x1X, 0<ρ<x1-x0,

max{φ(x0),φ(x1)}<mρ:=infBρ(x0)φ,

and

c:=infγΓmaxt[-1,1]φ(γ(t)),where Γ:={γC0([0,1],X):γ(0)=x0,γ(1)=x1},

then cmρ and Kφc.

Let (Y1,Y2) be a topological pair such that Y2Y1X and let k be any nonnegative integer. We denote by Hk(Y1,Y2) the k-th relative singular homology group for the pair (Y1,Y2) with integer coefficients. Given an isolated critical point x0Kφc,

Ck(φ,x0):=Hk(φcU,φcU{x0}),k0,

is the k-th critical group of φ at x0. Here, U indicates any neighborhood of x0 fulfilling KφφcU={x0}. The excision property of singular homology ensures that this definition does not depend on the choice of U. The monographs [5, 11] are general references on this subject.

Hereafter, stands for the N-norm, while |A| denotes the N-dimensional Lebesgue measure of AN. If p[1,+), then p indicates the conjugate exponent of p and p is the usual norm of the Sobolev space W01,p(Ω), namely, thanks to the Poincaré inequality,

up:=uLp(Ω)for all uW01,p(Ω).

Let u,v:Ω and let t. The symbol uv means u(x)v(x) for almost every xΩ, t±:=max{±t,0}, as well as u±():=u()±. It is known that u±W01,p(Ω) provided uW01,p(Ω). Next, define

C01(Ω¯):={uC1(Ω¯):u|Ω=0}.

With the standard norm of C1(Ω¯), this set is an ordered Banach space whose positive cone

C+:={uC01(Ω¯):u(x)0 in Ω¯}

has nonempty interior given by

int(C+)={uC+:u(x)>0 for all xΩ,un(x)<0 for all xΩ},

where n() denotes the outward unit normal on Ω; see [10, Remark 6.2.10]. If

pr<p*:={NpN-pfor p<N,+otherwise,

then, due to the continuous embedding W01,p(Ω)Lr(Ω) and the Poincaré inequality, one has

uLr(Ω)cr,pupfor all uW01,p(Ω).(2.1)

Let W-1,p(Ω) be the dual space of W01,p(Ω) and let Ap:W01,p(Ω)W-1,p(Ω) be the nonlinear operator stemming from the negative p-Laplacian, i.e.,

Ap(u),v:=Ωu(x)p-2u(x)v(x)𝑑xfor all u,vW01,p(Ω).

Denote by λ1,p (respectively, λ2,p) the first (respectively, second) eigenvalue of the operator -Δp in W01,p(Ω). The following properties of λ1,p, λ2,p, and Ap can be found in [7, 15]; see also [10, Section 6.2].

  • (p1)

    0<λ1,p<λ2,p.

  • (p2)

    uLp(Ω)p1λ1,pupp for all uW01,p(Ω).

  • (p3)

    There is a unique eigenfunction u1,p corresponding to λ1,p such that

    u1,pint(C01(Ω¯)+),u1,pLp(Ω)=1.

    Any other eigenfunction is a scalar multiple of u1,p.

  • (p4)

    If U:={uW01,p(Ω):uLp(Ω)=1} and

    Γ0:={γC0([-1,1],U):γ(-1)=-u1,r=-γ(1)},

    then

    λ2,p=infγΓ0maxuγ([-1,1])upp.

  • (p5)

    The operator Ap is bounded, continuous, strictly monotone, and of type (S)+.

Now, with p,q,μ, and f as in Section 1, suppose that

|f(x,t)|c(1+|t|p-1),(x,t)Ω×,(2.2)

for appropriate c>0, put

F(x,t):=0tf(x,ξ)𝑑ξ,(2.3)

and consider the C1-functional φ:W01,p(Ω) given by

φ(u):=1pupp+μquqq-ΩF(x,u(x))𝑑x for all uW01,p(Ω).

The next result establishes a relation between local C01(Ω¯)-minimizers and local W01,p(Ω)-minimizers of φ. Its proof is the same as that of [1, Proposition 2], with the (p,q)-Laplacian instead of the differential operator considered therein. This idea goes back to the pioneering works of Brézis and Nirenberg [4] for p=2 and García Azorero, Manfredi, and Peral Alonso [9] when p2.

If u0W01,p(Ω) is a local C01(Ω¯)-minimizer of φ, then u0 lies in C01,α(Ω¯) for some α(0,1) and u0 turns out to be a local W01,p(Ω)-minimizer of φ.

Finally, we shall write Nf(u)():=f(,u()) for every uLp(Ω). The function

Nf:Lp(Ω)Lp(Ω)

is often called the Nemytskii operator associated with f. Moreover, given u:Ω and c,

Ω(uc):={xΩ:u(x)c}.

The meaning of Ω(u>c) etc. is analogous.

3 Resonance from the Left

To avoid unnecessary technicalities, ‘for every xΩ’ will take the place of ‘for almost every xΩ’ and the variable x will be omitted when no confusion may arise. Moreover, p=q if and only if μ=0 and f(x,0)0. We will posit the following assumptions, where F is given by (2.3).

  • (h1)

    For appropriate c>0, one has

    |f(x,t)|c(1+|t|p-1)for all (x,t)Ω×.

  • (h2)

    lim|t|+pF(x,t)|t|p=λ1,p uniformly in xΩ.

  • (h3)

    lim|t|+[f(x,t)t-pF(x,t)]=+ uniformly in xΩ.

  • (h4)

    There exist θ(1,q) and δ0, c0>0 such that

    c0|t|θf(x,t)tθF(x,t),(x,t)Ω×[-δ0,δ0].

The energy functional φ:W01,p(Ω) stemming from (1.1) is defined by

φ(u):=1pupp+μquqq-ΩF(x,u(x))𝑑xfor all uW01,p(Ω).

Clearly, φC1(W01,p(Ω)). Moreover, once

f+(x,t):=f(x,t+),f-(x,t):=f(x,-t-),F±(x,t):=0tf±(x,ξ)𝑑ξ,

one has F+(x,t)=F(x,t+), F-(x,t)=F(x,-t-), while the associated truncated functionals

φ±(u):=1pupp+μquqq-ΩF±(x,u(x))𝑑x,uW01,p(Ω),

turn out to be C1 as well.

If (h1)(h3) hold true, then φ, φ+, and φ- are coercive and weakly sequentially lower semicontinuous.

Proof.

We will verify the conclusion for φ+, the other cases being similar. The space W01,p(Ω) compactly embeds in Lp(Ω) while the Nemytskii operator Nf+ turns out to be continuous on Lp(Ω). Thus, a standard argument ensures that φ+ is weakly sequentially lower semicontinuous. In view of (h3), given any K>0, there exists δ>0 such that

f+(x,t)t-pF+(x,t)Kfor all (x,t)Ω×[δ,+),

which clearly means that

ddtF+(x,t)tpKtp+1.

After integration, we obtain

F+(x,s)sp-F+(x,t)tp-Kp(1sp-1tp)provided stδ.(3.1)

Thanks to (h2), letting s+ in (3.1) yields

λ1,pptp-F+(x,t)Kp,(x,t)Ω×[δ,+).

Therefore,

limt+[λ1,pptp-F+(x,t)]=+uniformly with respect to xΩ.(3.2)

Now, suppose by contradiction that there exists a sequence {un}W01,p(Ω) such that

limn+unp=+butφ+(un)C<+for all n.(3.3)

Write vn:=un+/un+p. Since vnp1, passing to a subsequence when necessary, one has

vnvin W01,p(Ω),vnvin Lp(Ω),vnv0a.e. in Ω.

Fix any ε>0 and, through (h2), choose δ>0 fulfilling

F+(x,t)λ1,p+εptp,(x,t)Ω×[δ,+).

Moreover, set M:=supΩ×[0,δ]F+. From (3.3) it evidently follows that

φ+(un+)Cfor all n,(3.4)

because F+(x,un(x))=0 as soon as un(x)0, while un+runr.

We claim that {un+} is bounded in W01,p(Ω). In fact, if the assertion were false, then, up to subsequences, un+p+. Dividing (3.4) by un+pp gives

1pvnpp+μqun+p-qvnqqCun+pp+ΩF(x,un+(x))un+pp𝑑xCun+pp+Ω(un+δ)F(x,un+(x))un+pp𝑑x+Ω(un+>δ)F(x,un+(x))un+pp𝑑xCun+pp+M|Ω|un+pp+λ1,p+εpΩ|un+(x)|pun+pp𝑑x.(3.5)

Recall next that pq, but p=q only when μ=0. As n+ and ε0+, we get

vppλ1,pvLp(Ω)p.

On account of (p3), this implies that v=ξu1,p for some ξ0. If ξ=0, then vn0 in Lp(Ω). Thus, by (3.5), vn0 in W01,p(Ω), which contradicts vnp1. So, suppose ξ>0, whence un+(x)+ for every xΩ. Through (p2), Fatou’s lemma, and (3.2), one gets

1pun+pp-ΩF+(x,un+(x))𝑑xΩ(λ1,pp|un+(x)|p-F(x,un+(x)))𝑑x+,

against (3.4). Consequently, the claim holds true.

Finally, also the sequence {un} is bounded in W01,p(Ω), because F+(x,-un-(x))0 and φ+(un)C for all n. This completes the proof. ∎

Let (h1)(h4) be satisfied. Then, (1.1) has at least two nontrivial constant-sign solutions u0int(C+), v0-int(C+), both local minimizers of φ.

Proof.

By Lemma 3.1, the functional φ+ possesses a global minimizer u0W01,p(Ω). If θ,δ0,c0 come from (h4), wint(C+), and wL(Ω)1, then

φ+(tw)tppwpp+μtqqwqq-c0θtθwLθ(Ω)θfor all t(0,δ0].

Since θ<qp but q=p if and only if μ=0, for sufficiently small t>0, the right-hand side in the above inequality turns out to be negative, which evidently forces φ+(u0)<0, namely, u00. Proceeding as in [20, Theorem 4.1] then gives u0int(C+). Moreover, u0 is a local C01(Ω¯)-minimizer of φ, because φ|C+=φ+|C+. Now, the conclusion follows from Proposition 2.4. A similar argument yields a function v0 with the asserted properties. ∎

To establish the existence of a third nodal solution, we will first show that there exist two extremal constant-sign solutions, i.e., a smallest positive one and a biggest negative one. In fact, through (h1) and (h4) one has

f(x,t)tc0|t|θ-c1|t|pin Ω×,(3.6)

where c1>0. Thus, it is quite natural to compare solutions of (1.1) with those of the auxiliary problem

-Δpu-μΔqu=c0|u|θ-2u-c1|u|p-2u,(3.7)

which, by [20, Lemma 2.2], possesses a unique positive solution u¯int(C+) and a unique negative solution v¯=-u¯. Reasoning as in the proof of [20, Lemma 2.2] yields the next result.

Under (h1)(h4) , any positive (respectively, negative) solution u to (1.1) fulfills uu¯ (respectively, u-u¯).

Proof.

Let u be a positive solution of (1.1). For every (x,t)Ω×, define the functions

j(x,t):={0if t0,c0tθ-1-c1tp-1if 0<tu(x),c0u(x)θ-1-c1u(x)p-1otherwise,(3.8)

J(x,t):=0tj(x,ξ)𝑑ξ, as well as

η(w):=1pwpp+μqwqq-ΩJ(x,w(x))𝑑x,wW01,p(Ω).

Obviously, the functional η belongs to C1(W01,p(Ω)), is coercive, and weakly sequentially lower semicontinuous. So, there exists u~W01,p(Ω) such that

η(u~)=minwW01,p(Ω)η(w).(3.9)

As in the above proof, for sufficiently small t>0, we have η(tu)<0, whence η(u~)<0 and, a fortiori, u~0. Now, from (3.9) it follows that

Ap(u~),w+μAq(u~),w=Ωj(x,u~(x))w(x)𝑑xfor all wW01,p(Ω).(3.10)

Setting w:=-u~- in (3.10), one obtains u~-=0, i.e., u~0. Likewise, if w:=(u~-u)+, then, on account of (3.10), (3.8), (3.6), and the properties of u, one gets

Ap(u~),(u~-u)++μAq(u~),(u~-u)+=Ω(c0uθ-1-c1up-1)(u~-u)+𝑑xΩf(x,u)(u~-u)+𝑑x=Ap(u),(u~-u)++μAq(u),(u~-u)+.

Therefore,

Ap(u~)-Ap(u),(u~-u)++μAq(u~)-Aq(u),(u~-u)+0.

By (p5), this evidently forces uu~. Through (3.10) and (3.8) we thus see that the function u~ is a nonnegative nontrivial solution of (3.7). Since, due to [23, Theorem 5.4.1 and Theorem 5.5.1], u~int(C+), while (3.7) possesses a unique positive solution, we get u~=u¯, and the desired inequality follows. A similar argument works for the other conclusion. ∎

Weaker versions of (h4) allow to achieve the last two lemmas, namely,

there exists θ(1,q) such that lim inft0F(x,t)|t|θ>0 uniformly in xΩ

for Lemma 3.2 and (3.6) for Lemma 3.3. So, instead of any comparison between F(x,t) and f(x,t)t, only the behavior of tf(x,t) and tF(x,t) for t close to zero needs to be prescribed.

From now on, Σ will denote the set of all solutions to (1.1), while

Σ+:=Σint(C+),Σ-:=Σ(-int(C+)).

Proceeding exactly as in the proof of [20, Lemma 4.2], one obtains the next result.

If (h1)(h4) hold true, then (1.1) has a smallest positive solution u+int(C+) and a greatest negative solution v--int(C+).

A mountain pass procedure can now provide a third solution, but in order to exclude that it is the trivial one, we need further information on the critical groups of φ at zero, which will be achieved as in [21]. This is the point where (h4) plays a crucial role.

Let (h1), (h4) be satisfied, let φ(u)0 for some uW01,p(Ω){0}, and let zero be an isolated critical point of φ. Then, Ck(φ,0)=0 for all k0.

Proof.

Observe that

ddτφ(τu)|τ=1=φ(u),u=upp+μuqq-Ωf(x,u(x))u(x)𝑑x(1-θp)upp+μ(1-θp)uqq+Ω[θF(x,u(x))-f(x,u(x))u(x)]𝑑x.

By (h1), (h4), and (2.1), one has

Ω|θF(x,u)-f(x,u)u|𝑑xc2Ω(|u|δ0)|u(x)|p𝑑xc3Ω|u(x)|r𝑑xc4upr,

where ci>0, i=2,3,4, are suitable constants, while p<r<p*. Consequently,

ddτφ(τu)|τ=1(1-θp)upp-c4upr>0

whenever up is sufficiently small, say uB¯2ρ{0} for some ρ>0. Thus, in particular, if τ0>0, τ0uB¯2ρ{0}, and φ(τ0u)0, then

ddτφ(τu)|τ=τ0=1τ0ddτφ(ττ0u)|τ=1>0.

This means that the C1-function τφ(τu), τ(0,+), turns out to be increasing at the point τ provided τu(B¯2ρ{0})φ0. So, it vanishes at most once in the open interval (0,2ρ/up). On the other hand, (h1) and (h4) force

F(x,t)c0θ|t|θ-c5|t|p,(x,t)Ω×,

with appropriate c5>0. Hence,

ΩF(x,τu(x))𝑑xc0θτθuLθ(Ω)θ-c5τpuLp(Ω)pfor all τ>0.

Since θ<qp, we get

limτ0+φ(τu)τθ-c0θuLθ(Ω)θ<0,(3.11)

i.e., φ(τu)<0 for τ>0 small enough. Summing up, given any uB¯2ρ{0}, either φ(τu)<0 as soon as τuB¯2ρ or

there exists a unique τ¯(u)>0 such that τ¯(u)uB¯2ρ{0},φ(τ¯(u)u)=0.(3.12)

Moreover, if u(B¯2ρ{0})φ0, then 0<τ¯(u)1 and

φ(τu)<0 for all τ(0,τ¯(u)),φ(τu)>0 for all τ>τ¯(u)with τuB¯2ρ.(3.13)

Let τ:B¯ρ{0}(0,+) be defined by

τ(u):={1when u(B¯ρ{0})φ0,τ¯(u)when u(B¯ρ{0})φ0.

We claim that the function τ(u) is continuous. This immediately follows once one knows that τ¯(u) turns out to be continuous on (B¯ρ{0})φ0, because, by uniqueness, uB¯ρ{0} and φ(u)=0 evidently imply τ¯(u)=1; cf. (3.12). Pick u^(B¯ρ{0})φ0. The function ϕ(t,u):=φ(tu) belongs to C1(×W01,p(Ω)) and, on account of (3.13), we have

ϕ(τ¯(u^),u^)=0,ϕu(τ¯(u^),u^)=τ¯(u^)φ(τ¯(u^)u^).

Since zero turns out to be an isolated critical point for φ, there is no loss of generality in assuming that KφB¯ρ={0}. So, the implicit function theorem furnishes σC1(Bε(u^)), ε>0, such that

ϕ(σ(u),u)=0 for all uBε(u^),σ(u^)=τ¯(u^).

Through 0<τ¯(u^)1, we thus get 0<σ(u)<2 for all uU, where UBε(u^) denotes a convenient neighborhood of u^. Consequently,

σ(u)uB¯2ρ{0}andφ(σ(u)u)=0provided u(B¯ρ{0})φ0U.

By (3.12), this results in σ(u)=τ¯(u), from which the continuity of τ¯(u) at u^ follows. As u^ was arbitrary, the function τ¯(u) turns out to be continuous on (B¯ρ{0})φ0.

Next, observe that τuB¯ρφ0 for all τ[0,1], uB¯ρφ0. Hence, if

h(t,u):=(1-t)u,(t,u)[0,1]×(B¯ρφ0),

then h([0,1]×(B¯ρφ0))B¯ρφ0, namely, B¯ρφ0 is contractible in itself. Moreover, the function

g(u):=τ(u)ufor all uB¯ρ{0}

is continuous and one has g(B¯ρ{0})(B¯ρφ0){0}. Since

g|(B¯ρφ0){0}=id|(B¯ρφ0){0},

the set (B¯ρφ0){0} turns out to be a retract of B¯ρ{0}. Being B¯ρ{0} contractible in itself, because W01,p(Ω) is infinite dimensional, we get (see, e.g., [11, p. 389])

Ck(φ,0):=Hk(B¯ρφ0,(B¯ρφ0){0})=0,k0.

This completes the proof. ∎

A careful inspection of the above argument shows that the second inequality in (h4) can be weakened to achieve the same conclusion, requiring instead

f(x,t)t-θF(x,t)c6|t|rin Ω×[-δ0,δ0]

for suitable θ<p<r and c6, δ0>0.

We are now ready to find a nodal solution of (1.1). Write, provided u,v lie in W01,p(Ω) and vu,

[v,u]={wW01,p(Ω):vwu}.

If (h1)(h4) hold true, then (1.1) admits a sign-changing solution u0C01(Ω¯)[v-,u+].

Proof.

For every (x,t)Ω×, define the function

f^(x,t):={f(x,v-(x))if t<v-(x),f(x,t)if v-(x)tu+(x),f(x,u+(x))if u+(x)<t,(3.14)

as well as

f^+(x,t):=f^(x,t+),f^-(x,t):=f^(x,-t-).

Moreover, provided uW01,p(Ω), set

φ^(u):=1pupp+μquqq-ΩF^(x,u(x))𝑑x,φ^±(u):=1pup+μquqq-ΩF^±(x,u(x))𝑑x,

where

F^(x,t):=0tf^(x,ξ)𝑑ξ,F^±(x,t):=0tf^±(x,ξ)𝑑ξ.

The same reasoning as in the proof of [20, Theorem 4.3] guarantees here that

Kφ^[v-,u+],Kφ^-{v-,0},Kφ^+{0,u+},(3.15)

besides

u+int(C+)andv--int(C+)are local W01,p(Ω)-minimizers for φ^.(3.16)

Since, by (3.15), one has Kφ^=Kφ[v-,u+], it suffices to find a nontrivial critical point of φ^. Suppose that φ^(v-)φ^(u+) (the opposite case is analogous). Due to (3.16) there exists ρ(0,1) such that

v--u+p>ρ,φ^(u+)<mρ:=infBρ(u+)φ^.(3.17)

Furthermore, the functional φ^ fulfills condition (C), because it is coercive by construction; cf. (3.14). Hence, Theorem 2.3 applies and we obtain a point u0W01,p(Ω) such that

u0Kφ^,mρφ^(u0).

The strict inequality in (3.17) and (3.15) forces u0[v-,u+]{v-,u+}. Now, if Kφ^ possesses infinitely many elements, then the conclusion follows at once. Otherwise, C1(φ^,u0)0, because u0 is a critical point of mountain pass type; see [5, p. 89]. Through u+int(C+), v--int(C+), and φ^|[v-,u+]=φ|[v-,u+], we infer that

Ck(φ^|C01(Ω¯),0)=Ck(φ|C01(Ω¯),0),k0.

Moreover, recalling that C01(Ω¯) turns out to be dense in W01,p(Ω),

Ck(φ^|C01(Ω¯),0)=Ck(φ^,0),Ck(φ|C01(Ω¯),0)=Ck(φ,0).

So, thanks to Theorem 3.6, Ck(φ^,0)=0 for all k0, whence u00. The solution u0 is nodal by the extremality of v- and u+, while standard nonlinear regularity results yield u0C01(Ω¯). ∎

Combining Lemma 3.5 with Theorem 3.8 directly produces the next result.

Let (h1)(h4) be satisfied. Then, (1.1) admits a smallest positive solution u+int(C+), a greatest negative solution v--int(C+), and a nodal solution u0C01(Ω¯) such that v-u0u+.

4 Resonance from the Right

The notation in this section is the same as in Section 3. Conditions (h2) and (h3) furnish that

lim|t|+[λ1,p|t|p-pF(x,t)]=+uniformly in xΩ;(4.1)

cf. (3.2). So, under these hypotheses, resonance with respect to λ1,p from the left occurs and, a fortiori, the energy functional φ turns out to be coercive (Lemma 3.1). Now, the question of investigating what happens when there is resonance from the right of λ1,p, i.e., the limit in (4.1) equals -, naturally arises. In this case, φ turns out to be indefinite and direct methods no longer work. However, the linking structure of suitably defined sets still fits our purpose.

The following assumption will take the place of (h3).

  • (h3’)

    If μ>0, then there exist η(q,p] and α0>0 such that

    lim inf|t|+pF(x,t)-f(x,t)t|t|ηα0>0uniformly in xΩ.

    If μ=0 then,

    lim inf|t|+[pF(x,t)-f(x,t)t]=+uniformly in xΩ.

Suppose (h1)(h3’) hold true. Then, φ satisfies condition (C).

Proof.

Since W01,p(Ω) compactly embeds in Lp(Ω), the Nemytskii operator Nf is continuous on Lp(Ω), and Ap enjoys property (p5), it suffices to show that every sequence {un}W01,p(Ω) fulfilling

|φ(un)|Cfor all n(4.2)

and

limn+(1+unp)φ(un)=0(4.3)

turns out to be bounded. If the assertion were false, then, along a subsequence when necessary, unp+. Let vn:=un/unp. We may evidently assume

vnvin W01,p(Ω),vnvin Lp(Ω),vn(x)v(x)for every xΩ,

because vnp1. Inequality (4.2) gives

1pvnpp+μq1unp-qvnqqCunpp+ΩF(x,un(x))unpp𝑑x.

Proceeding exactly as in the proof of Lemma 3.1, one obtains vppλ1,pvLp(Ω)p, which forces v=ξu1,p for appropriate ξ{0}. Therefore, |v|>0 and thus

|un|+a.e. in Ω.(4.4)

Through (4.3), we easily have φ(un),un0, whence

unpp+μunqq-Ωf(x,un(x))un(x)𝑑xεn,(4.5)

where εn0+. From (4.2) it follows that

-unpp-μpqunqq+ΩpF(x,un(x))𝑑xC.(4.6)

Combining (4.5) and (4.6) leads to

Ω[pF(x,un(x))-f(x,un(x))un(x)]𝑑xC+εn+μ(pq-1)unqq,(4.7)

i.e., after an elementary calculation,

Ω[pF(x,un(x))-f(x,un(x))un(x)]𝑑xc7(1+unpq)(4.8)

for all n. If μ>0, then, because of (h3’), Fatou’s lemma, and (4.4),

lim infn+1unpηΩ[pF(x,un)-f(x,un)un]𝑑x=lim infn+ΩpF(x,un)-f(x,un)un|un|η|vn|η𝑑xα0vLη(Ω)η>0.(4.9)

However, since η>q, dividing (4.8) by unpη and letting n+ produces

lim supn+1unpηΩ[pF(x,un(x))-f(x,un(x))un(x)]𝑑x0,

against (4.9). So, suppose μ=0. Thanks to (h3’), one has

limn+Ω[pF(x,un(x))-f(x,un(x))un(x)]𝑑x=+,

which contradicts (4.7). Therefore, the sequence {un} turns out to be bounded in W01,p(Ω), as required.∎

Let (h1)(h3’) be satisfied. Then, limt±φ(tu1,p)=-.

Proof.

Consider first the case μ>0. Without loss of generality, we may suppose η<p in (h3’). Thus, there exist α1,δ1>0 such that

α1|t|ηpF(x,t)-f(x,t)tfor every xΩ,|t|δ1.

Consequently,

ddtF(x,t)tp=f(x,t)t-pF(x,t)tp+1-α1tη-p-1,(x,t)Ω×[δ1,+).

After integration, this results in

F(x,t)tp-F(x,s)spα1p-η(tη-p-sη-p)provided tsδ1.

Letting t+, on account of (h2) we have

λ1,ppsp-F(x,s)-α1p-ηsηin Ω×[δ1,+),

which clearly implies that

λ1,ppsp-F(x,s)-α1p-ηsη+c8for all (x,s)Ω×[0,+).

Hence, for any t>0,

φ(tu1,p)=tppλ1,pu1,pLp(Ω)p+μtqqu1,pqq-ΩF(x,tu1,p(x))𝑑x-α1p-ηtηu1,pLη(Ω)η+μtqqu1,pqq+c8|Ω|,

namely, φ(tu1,p)- as t+. The proof for t- is analogous.

Now, let μ=0. By (h3’) again, to every K>0 corresponds δ>0 such that

pF(x,t)-f(x,t)t>K,(x,t)Ω×[δ,+).

The same argument as before yields here

λ1,ppsp-F(x,s)-Kpin Ω×[δ,+).

Define

M:=supΩ×[0,δ]|λ1,ppsp-F(x,s)|

and observe that

φ(tu1,p)=tppλ1,pu1,pLp(Ω)p-ΩF(x,tu1,p(x))𝑑xΩ(u1,pδt)(λ1,pptpu1,pp-F(x,tu1,p))𝑑x+Ω(u1,p>δt)(λ1,pptpu1,pp-F(x,tu1,p))𝑑xM|Ω(u1,pδt)|-Kp|Ω|

provided t>0. Since u1,p>0, letting t+ leads to

lim supt+φ(tu1,p)-Kp|Ω|.

As K>0 was arbitrary, we actually have limt+φ(tu1,p)=-. The case t- is quite similar. ∎

Next, write

E:={uW01,p(Ω):upp=λ2,puLp(Ω)p}.

Obviously, E turns out to be nonempty and closed.

If (h1)(h2) hold true, then φ|E is coercive.

Proof.

Pick ξ(λ1,p,λ2,p). The hypotheses give K>0 such that

F(x,t)ξp|t|p+Kfor all (x,t)Ω×.

Consequently, for any uE,

φ(u)1pupp+μquqq-ξpuLp(Ω)p-K|Ω|1p(1-ξλ2,p)upp-K|Ω|.

Since ξ<λ2,p, the assertion follows. ∎

Lemma 4.3 basically ensures that infEφ>-. Thanks to Lemma 4.2, we can find τ>0 fulfilling

φ(±τu1,p)<mE:=infEφ.(4.10)

Define

Q0:={±τu1,p},Q:={tu1,p:t[-τ,τ]}.

The pair (Q0,Q) links E in W01,p(Ω).

Proof.

One evidently has Q0E=. Moreover, if

U:={uW01,p(Ω):upp<λ2,puLp(Ω)p},

then Q0U, because λ1,p<λ2,p. Let us verify that -τu1,p and τu1,p lie in different pathwise connected components of U. Arguing by contradiction, there exists σC0([-1,1],U) such that σ(-1)=-τu1,p=-σ(1). On the other hand, (p4) forces

λ2,pmaxt[-1,1]σ(t)ppσ(t)Lp(Ω)p,

which leads to σ(t0)U for some t0(0,1). However, this is impossible. Hence, any γC0(Q,W01,p(Ω)) such that γ|Q0=id|Q0 must satisfy the condition γ(Q)U. Since UE, the proof is complete. ∎

We are now in a position to treat the existence of solutions to (1.1) when resonance from the right of λ1,p occurs. To the best of our knowledge, multiplicity is still an open question.

Under assumptions (h1)(h3’) and (h4), the problem (1.1) has at least one nontrivial solution u0C01(Ω¯).

Proof.

By Lemma 4.1, Lemma 4.4, and (4.10), one can apply Theorem 2.2. Thus, we get a point u0W01,p(Ω) such that

φ(u0)=0,infEφφ(u0)=infγΓsupuQφ(γ(u)),

where

Γ:={γC0(Q,W01,p(Ω)):γ|Q0=id|Q0}.

Moreover, C1(φ,u0)0, because u0 is a critical point of mountain pass type; see [5, p. 89]. On the other hand, due to Lemma 4.3 and Theorem 3.6, one has C1(φ,0)=0. Therefore, u00. Standard results from nonlinear regularity theory then ensure that u0C01(Ω¯). ∎

References

  • [1]

    Aizicovici S., Papageorgiou N. S. and Staicu V., On p-superlinear equations with a nonhomogeneous differential operator, NoDEA Nonlinear Differential Equations Appl. 20 (2013), no. 2, 151–175.  Google Scholar

  • [2]

    Ambrosetti A., Brézis H. and Cerami G., Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), no. 2, 519–543.  Google Scholar

  • [3]

    Benci V., Fortunato D. and Pisani L., Soliton like solutions of a Lorentz invariant equation in dimension 3, Rev. Math. Phys. 10 (1998), no. 3, 315–344.  Google Scholar

  • [4]

    Brézis H. and Nirenberg L., H1 versus C1 local minimizers, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 5, 465–472.  Google Scholar

  • [5]

    Chang K.-C., Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progr. Nonlinear Differential Equations Appl. 6, Birkhäuser, Boston, 1993.  Google Scholar

  • [6]

    Cherfils L. and Il’yasov Y., On the stationary solutions of generalized reaction diffusion equations with p&q-Laplacian, Comm. Pure Appl. Anal. 3 (2005), no. 1, 9–22.  Google Scholar

  • [7]

    Cuesta M., de Figueiredo D. and Gossez J.-P., The beginning of the Fučik spectrum for the p-Laplacian, J. Differential Equations 159 (1999), no. 1, 212–238.  Google Scholar

  • [8]

    de Paiva F. O. and Massa E., Multiple solutions for some elliptic equations with a nonlinearity concave at the origin, Nonlinear Anal. 66 (2007), no. 12, 2940–2946.  Google Scholar

  • [9]

    García Azorero J. P., Manfredi J. J. and Peral Alonso I., Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math. 2 (2000), no. 3, 385–404.  Google Scholar

  • [10]

    Gasiński L. and Papageorgiou N. S., Nonlinear Analysis, Ser. Math. Anal. Appl. 9, Chapman and Hall/CRC Press, Boca Raton, 2006.  Google Scholar

  • [11]

    Granas A. and Dugundji J., Fixed Point Theory, Springer Monogr. Math., Springer, New York, 2003.  Google Scholar

  • [12]

    Hu S. and Papageorgiou N. S., Multiplicity of solutions for parametric p-Laplacian equations with nonlinearity concave near the origin, Tohoku Math. J. (2) 62 (2010), no. 1, 137–162.  Google Scholar

  • [13]

    Iannizzotto A., Marano S. A. and Motreanu D., Positive, negative, and nodal solutions to elliptic differential inclusions depending on a parameter, Adv. Nonlinear Stud. 13 (2013), no. 2, 431–445.  Google Scholar

  • [14]

    Iannizzotto A. and Papageorgiou N. S., Existence, nonexistence and multiplicity of positive solutions for parametric nonlinear elliptic equations, Osaka J. Math. 51 (2014), no. 1, 179–203.  Google Scholar

  • [15]

    Lê A., Eigenvalue problems for the p-Laplacian, Nonlinear Anal. 64 (2006), no. 5, 1057–1099.  Google Scholar

  • [16]

    Li S., Wu S. and Zhou H.-S., Solutions to semilinear elliptic problems with combined nonlinearities, J. Differential Equations 185 (2002), no. 1, 200–224.  Google Scholar

  • [17]

    Marano S. A., On a Dirichlet problem with p-Laplacian and set-valued nonlinearity, Bull. Aust. Math. Soc. 86 (2012), no. 1, 83–89.  Google Scholar

  • [18]

    Marano S. A., Motreanu D. and Puglisi D., Multiple solutions to a Dirichlet eigenvalue problem with p-Laplacian, Topol. Methods Nonlinear Anal. 42 (2013), no. 2, 277–291.  Google Scholar

  • [19]

    Marano S. A. and Papageorgiou N. S., Multiple solutions to a Dirichlet problem with p-Laplacian and nonlinearity depending on a parameter, Adv. Nonlinear Anal. 1 (2012), no. 3, 257–275.  Google Scholar

  • [20]

    Marano S. A. and Papageorgiou N. S., Constant sign and nodal solutions of coercive (p,q)-Laplacian problems, Nonlinear Anal. 77 (2013), 118–129.  Google Scholar

  • [21]

    Moroz V., Solutions of superlinear at zero elliptic equations via Morse theory, Topol. Methods Nonlinear Anal. 10 (1997), no. 2, 387–397.  Google Scholar

  • [22]

    Perera K., Multiplicity results for some elliptic problems with concave nonlinearities, J. Differential Equations 140 (1997), no. 1, 133–141.  Google Scholar

  • [23]

    Pucci P. and Serrin J., The Maximum Principle, Progr. Nonlinear Differential Equations Appl. 73, Birkhäuser, Basel, 2007.  Google Scholar

About the article

Received: 2014-11-07

Revised: 2015-01-13

Accepted: 2015-01-14

Published Online: 2015-12-04

Published in Print: 2016-02-01


The first two authors acknowledge the support of GNAMPA of INdAM, Italy.


Citation Information: Advanced Nonlinear Studies, Volume 16, Issue 1, Pages 51–65, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2015-5011.

Export Citation

© 2016 by De Gruyter.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]
N. S. Papageorgiou, C. Vetro, and F. Vetro
Journal of Mathematical Physics, 2019, Volume 60, Number 10, Page 101506
[2]
Dimitrios A. Kandilakis and Manolis Magiropoulos
Complex Variables and Elliptic Equations, 2019, Page 1
[3]
Antonella Nastasi
Journal of Mathematical Analysis and Applications, 2019, Volume 479, Number 1, Page 45
[4]
Antonella Nastasi and Calogero Vetro
Journal of Difference Equations and Applications, 2019, Volume 25, Number 3, Page 331
[5]
Fuyi Li, Ting Rong, and Zhanping Liang
Journal of Mathematical Physics, 2018, Volume 59, Number 12, Page 121506
[7]
Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, and Dušan D. Repovš
Applied Mathematics & Optimization, 2017
[8]
N. S. Papageorgiou and C. Vetro
Complex Variables and Elliptic Equations, 2017, Page 1
[9]
Salvatore A. Marano and Sunra J. N. Mosconi
Discrete and Continuous Dynamical Systems - Series S, 2017, Volume 11, Number 2, Page 279
[10]
Dumitru Motreanu, Calogero Vetro, and Francesca Vetro
Numerical Functional Analysis and Optimization, 2016, Volume 37, Number 12, Page 1551
[11]
Diego Averna, Dumitru Motreanu, and Elisabetta Tornatore
Applied Mathematics Letters, 2016, Volume 61, Page 102

Comments (0)

Please log in or register to comment.
Log in