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Advances in Nonlinear Analysis

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


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Noncoercive resonant (p,2)-equations with concave terms

Nikolaos S. Papageorgiou / Chao Zhang
  • Corresponding author
  • Department of Mathematics and Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China
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Published Online: 2018-10-11 | DOI: https://doi.org/10.1515/anona-2018-0175

Abstract

We consider a nonlinear Dirichlet problem driven by the sum of a p-Laplace and a Laplacian (a (p,2)-equation). The reaction exhibits the competing effects of a parametric concave term plus a Caratheodory perturbation which is resonant with respect to the principle eigenvalue of the Dirichlet p-Laplacian. Using variational methods together with truncation and comparison techniques and Morse theory (critical groups), we show that for all small values of the parameter, the problem has as least six nontrivial smooth solutions all with sign information (two positive, two negative and two nodal (sign changing)).

Keywords: Concave term; resonance; nonlinear regularity theory; nonlinear maximum principle,strong comparison principle; truncation; critical groups; constant sign and nodal solutions

MSC 2010: 35J20; 35J60; 58E05

1 Introduction

Let ΩN be a bounded domain with a C2-boundary Ω. In this paper, we study the following parametric (p,2)-equation:

{-Δpu(z)-Δu(z)=λ|u(z)|q-2u(z)+f(z,u(z))in Ω,u|Ω=0,λ>0, 1<q<2<p<.(1.1)

In this problem, for r(1,+), by Δr we denote the r-Laplace differential operator defined by

Δru=div(|Du|r-2Du)for all uW01,r(Ω).

When r=2, we have the usual Laplacian Δ2=Δ defined by

Δu=div(Du)for all uH01(Ω).

So, in problem (1.1) the differential operator is the sum of a p-Laplacian and a Laplacian (a (p,2)-equation). Such an operator is nonhomogeneous and this is a source of difficulties in the analysis of problem (1.1). In the reaction (right-hand side of (1.1)), we have the competing effects of two nonlinearities. One is a concave term uλ|u|q-2u (recall 1<q<2<p) and the other is a Caratheodory perturbation f(z,x) (that is, for all x, zf(z,x) is measurable and for a.a. zΩ, xf(z,x) is continuous). We assume that asymptotically as x±, f(z,) is resonant with respect to the principle eigenvalue λ^1(p)>0 of (-Δp,W01,p(Ω)). The resonance occurs from the right of the principal eigenvalue λ^1(p) in the sense that

λ^1(p)|x|p-pF(z,x)-uniformly for a.a. zΩ as x±,

where F(z,x)=0xf(z,x)𝑑s. This makes the energy functional of problem (1.1) indefinite. Our goal is to prove a multiplicity theorem for problem (1.1), providing sign information for all the solutions produced. We show that for all λ>0 small, problem (1.1) has at least six nontrivial smooth solutions, two positive, two negative and two nodal (that is, sign changing).

Our tools come from critical point theory, together with suitable truncation and comparison techniques and Morse theory (critical groups).

Boundary value problems driven by a combination of differential operators of different nature (such as (p,2)-equations) arise in the problems of mathematical physics. We mention the works of Benci, Fortunato and Pisani [3] concerning models of particle physics (existence of soliton-type solutions), Cherfils and Ilyason [5] on reaction-diffusion systems, and Wilhelmsson [29] on plasma physics. A survey of recent works on (p,q)-equations (1<q<p<) can be found in the paper of Marano and Mosconi [16]. The particular case of (p,2)-equations presents special interest and stronger results can be proved for such equations. In this direction we mention the works of Aizicovici, Papageorgiou and Staicu [2], Cingolani and Degiovanni [6], Papageorgiou and Radulescu [21, 22], Papageorgiou, Radulescu and Repovs [23, 24], Sun [27], and Sun, Zhang and Su [28]. From the aforementioned works, only [22] deals with equations which are resonant with respect to λ^1(p) from the right (noncoercive problems), where the existence of two nontrivial solutions with no sign information is proved . Resonant (p,2)-equations with parametric concave terms were considered by Papageorgiou and Winkert [25]. However, in their equation, the concave term enters with a negative sign and this leads to a coercive energy functional.

2 Mathematical background and hypotheses

Let X be a Banach space and X* its topological dual. By , we denote the duality brackets for the pair (X*,X). Given φC1(X,), we say that φ satisfies the “Cerami condition” (the “C-condition” for short), if the following property holds: Every sequence {un}n1X such that {φ(un)}n1 is bounded and

(1+unX)φ(un)0in X* as n

admits a strongly convergent subsequence.

This is a compactness-type condition on the functional φ, more general than the usual Palais–Smale condition. It leads to a deformation theorem from which one can deduce the minimax theory of the critical values of φ. One of the main results in that theory is the so-called “mountain pass theorem”, which we recall here.

Theorem 2.1.

If X is a Banach space, φC1(X,R) satisfies the C-condition, u0,u1X, u1-u0X>ρ>0,

max{φ(u0),φ(u1)}<inf[φ(u):u-u0X=ρ]=ηρ,

and

c=infγΓmax0t1φ(γ(t)),with Γ={γC([0,1],X):γ(0)=u0,γ(1)=u1},

then cηρ and c is a critical value of φ.

In the analysis of problem (1.1), we will use the Sobolev spaces W01,p(Ω) and H01(Ω). Since 2<p, we have that W01,p(Ω)H01(Ω) densely. By we denote the norm of W01,p(Ω). By the Poincaré inequality, we have

u=Dupfor all uW01,p(Ω).

Also we will use the Banach space C01(Ω¯)={uC1(Ω¯):u|Ω=0}. This is an ordered Banach space with positive (order) cone

C+={uC01(Ω¯):u(z)0 for all zΩ¯}.

This cone has a nonempty interior given by

intC+={uC+:u(z)>0 for all zΩ,un|Ω<0}.

Here un=(Du,n)N, with n() being the outward unit normal on Ω.

For every r(1,+), let Ar:W01,r(Ω)W-1,r(Ω)=W01,r(Ω)* (1r+1r=1) be the map defined by

Ar(u),h=Ω|Du|r-2(Du,Dh)N𝑑zfor all u,hW01,r(Ω).

For this map, we have the following result (see [18, p. 40]).

Proposition 2.2.

The map Ar() is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (hence maximal monotone, too) and of type (S)+, that is, if un𝑤u in W01,r(Ω) and

lim supnAr(un),un-u0,

then unu in W01,r(Ω). If r=2, then A2=AL(H01(Ω),H-1(Ω)).

Suppose that f0:Ω× is a Caratheodory function such that

|f0(z,x)|a0(z)[1+|x|r-1]for a.a. zΩ and all x,

with a0L(Ω)+ and

1<rp*={NpN-pif p<N,+if Np

(the critical Sobolev exponent for p). We set F0(z,x)=0xf0(z,s)𝑑s and we consider the C1-functional φ0:W01,p(Ω) defined by

φ0(u)=1pDupp+12Du22-ΩF0(z,u)𝑑zfor all uW01,p(Ω).

The next result can be found in [18, p. 409].

Proposition 2.3.

If u0W01,p(Ω) is a local C01(Ω¯)-minimizer of φ0, that is, there exists ρ0>0 such that

φ0(u0)φ0(u0+h)for all hC01(Ω¯), with hC01(Ω¯)ρ0,

then u0C01,α(Ω¯) for some α(0,1) and u0 is also a local W01,p(Ω)-minimizer of φ0, that is, there exists ρ1>0 such that

φ0(u0)φ0(u0+h)for all hW01,p(Ω), with hρ1.

To make an effective use of Proposition 2.3, we need a strong comparison principle, which is provided by the next proposition and is a particular case of [9, Proposition 3].

Given h1,h2L(Ω) we write h1h2 if for every KΩ compact, we can find ε=ε(K)>0 such that h1(z)+εh2(z) for a.a. zK. Evidently, if h1,h2C(Ω), and h1(z)<h2(z) for all zΩ, then h1h2.

Proposition 2.4.

If ξ,h1,h2L(Ω),ξ(z)0 for a.a. zΩ,h1h2,uC01(Ω¯){0}, vintC+ and

-Δpu(z)-Δu(z)+ξ(z)|u(z)|p-2u(z)=h1(z)for a.a. zΩ,-Δpv(z)-Δv(z)+ξ(z)v(z)p-1=h2(z)for a.a. zΩ,

then v-uintC+.

Next we recall some basic facts about the spectrum of the Dirichlet p-Laplacian. So, we consider the following nonlinear eigenvalue problem:

-Δpu(z)=λ^|u(z)|p-2u(z)in Ω,u|Ω=0.(2.1)

We say that λ^ is an eigenvalue of (-Δp,W01,p(Ω)) if problem (2.1) admits a nontrivial solution u^W01,p(Ω), known as an eigenfunction corresponding to the eigenvalue λ^. By σ^(p) we denote the spectrum of (-Δp,W01,p(Ω)) (that is, the set of eigenvalues). We know that there exists a smallest eigenvalue λ^1(p), which has the following properties:

  • λ^1(p)>0 and it is isolated in the spectrum σ^(p) (that is, there exists ε>0 such that (λ^1(p),λ^1(p)+ε)σ^(p)=),

  • λ^1(p) is simple (that is, if u^1,u^2 are eigenfunctions corresponding to λ^1(p), then u^1=ξu^2 with ξ0),

  • we have

    λ^1(p)=inf[Duppupp:uW01,p(Ω),u0].(2.2)

The infimum in (2.2) is realized on the corresponding one-dimensional eigenspace. Evidently, the elements of this eigenspace have constant sign. By u^1(p) we denote the positive Lp-normalized (that is, u^1(p)p=1) eigenfunction corresponding to the eigenvalue λ^1(p)>0. The nonlinear regularity theory and the nonlinear maximum principle (see [10, pp. 737–738]) imply that u^1intC+. It is easy to see that σ^(p)(0,+) is closed and using the Ljusternik–Schnirelmann minimax scheme, we produce a whole sequence of distinct eigenvalues {λ^k(p)}k of (-Δp,W01,p(Ω)) such that λ^k(p)+ as k+. These eigenvalues are known as “variational eigenvalues” and we do not know if they exhaust σ^(p). This is the case if N=1 (scalar eigenvalue problem) or if p=2 (linear eigenvalue problem). All eigenvalues λ^λ^1(p) have nodal eigenfunctions.

In what follows, for notational simplicity, we write

λ^1=λ^1(p)andu^1=u^1(p)intC+.

We will also encounter a weighted version of the eigenvalue problem (2.1). Namely, let ηL(Ω), with η(z)0 for a.a. zΩ, η0. We consider the following nonlinear eigenvalue problem:

-Δpu(z)=λ~η(z)|u(z)|p-2u(z)in Ω,u|Ω=0.

The same results are true for this problem. So, there exists a smallest eigenvalue λ~1(p,η)>0, which is isolated, simple and admits the following variational characterization:

λ~1(p,η)=inf[DuppΩη(z)|u|p𝑑z:uW01,p(Ω),u0].

These properties lead to the following monotonicity property for the map ηλ~1(p,η) (see [18, p. 250]).

Proposition 2.5.

If η,η^L(Ω), 0η(z)η^(z) for a.a. zΩ,η0,η^η, then λ~1(p,η^)<λ~1(p,η).

Next let us recall some basic definitions and facts from the theory of critical groups (Morse theory). So, let X be a Banach space, φC1(X,) and c. We introduce the following sets:

Kφ={uX:φ(u)=0}(the critical set of φ),Kφc={uKφ:φ(u)=c}(the critical set of φ at the level c),φc={uX:φ(u)c}(the sublevel set of φ at c).

Let (Y1,Y2) be a topological pair such that Y2Y1X. For any k0, by Hk(Y1,Y2) we denote the kth-relative singular homology group with integer coefficients for the pair (Y1,Y2). Recall that Hk(Y1,Y2)=0 for all k-. If uKφc is isolated, then the critical groups of φ at u are defined by

Ck(φ,u)=Hk(φcU,φcU{u})for all k0,

with U being a neighborhood of u such that KφφcU={u}. The excision property of singular homology implies that this definition of critical groups above is independent of the choice of the neighborhood U.

If uKφ is isolated and of mountain pass-type (see Theorem 2.1), then C1(φ,u)0. Moreover, if φC2(X,), then from [21] we know that

Ck(φ,u)=δk,1for all k0,

with δk,m being the Kronecker symbol defined by

δk,m={1if k=m,0if km,k,m0.

Next we fix our notation. Given x, we set x±=max{±x,0}. Then for uW01,p(Ω), we define

u±()=u()±.

We know that u±W01,p(Ω), u=u+-u-, |u|=u++u-. Also, given a measurable function g:Ω× (for example, a Caratheodory function), we define

Ng(u)()=g(,u())for all uW01,p(Ω),

which is the Nemytskii operator corresponding to g. By ||N we denote the Lebesgue measure on N, and if 1<r<+, then 1r+1r=1.

If u,u^W1,p(Ω) and uu^, then we define

[u,u^]={yW01,p(Ω):u(z)y(z)u^(z) for a.a. zΩ}.

By intC01(Ω¯)[u,u^], we denote the interior of [u,u^]C01(Ω¯) in the C01(Ω¯)-norm. Also, if k:(0,+)C01(Ω¯), then we say that k() is strictly increasing if λ<λ implies that k(λ)-k(λ)intC+. Finally, if uW1,p(Ω), then

[u)={yW01,p(Ω):u(z)y(z) for a.a. zΩ}.

Now we can introduce the hypotheses on the perturbation term f(z,x).

Hypothesis 2.6.

f:Ω× is a Caratheodory function such that f(z,0)=0 for a.a. zΩ, which satisfies the following properties:

  • (i)

    For every ρ>0, there exists aρL(Ω) such that

    |f(z,x)|aρ(z)for a.a. zΩ and all |x|ρ.

  • (ii)

    We have

    λ^1lim infx±f(z,x)|x|p-2xlim supx±f(z,x)|x|p-2xη^uniformly for a.a. zΩ.

  • (iii)

    If F(z,x)=0xf(z,s)𝑑s, then

    limx±f(z,x)x-pF(z,x)x2=-uniformly for a.a. zΩ.

  • (iv)

    There exist functions w±W1,p(Ω)L(Ω), with Δpw±,Δw±Lp(Ω), and θ±>0 such that, for a.a. zΩ,

    w-(z)-c-<0<c+w+(z),-Δpw-(z)-Δw-(z)0-Δpw+(z)-Δw+(z),θ+w+(z)q-1+f(z,w+(z))-c^+<0c^-θ-|w-(z)|q-2w-(z)+f(z,w-(z)).

  • (v)

    If c0=min{c+,c-}>0, then we can find δ0(0,c0) such that for all KΩ compact and all 0<m<δ0, we have

    0<cKmf(z,x)xqF(z,x)for a.a. zK and all m|x|δ0.

  • (vi)

    For every ρ>0, there exists ξ^ρ>0 such that for a.a. zΩ, the function

    xf(z,x)+ξ^ρ|x|p-2x

    is nondecreasing on [-ρ,ρ].

Remark 2.7.

Hypotheses 2.6 (i), (ii) imply that

|f(z,x)|c1[1+|x|p-1]for a.a. zΩ and all x, with c1>0.

Hypothesis 2.6 (ii) says that at ± we can have resonance with respect to the principal eigenvalue λ^1>0. In the process of the proof we shall see that Hypothesis 2.6 (iii) implies that the resonance occurs from the right of λ^1>0 in the sense that

λ^1|x|p-pF(z,x)-uniformly for a.a. zΩ as x±.

This makes the problem noncoercive and so the direct method of the calculus of variations is not directly applicable on (1.1). Hypothesis 2.6 (iv) is satisfied if we can find t-<0<t+ such that

f(z,t+)-c^+<0<c^-f(z,t-)for a.a. zΩ.

Therefore, Hypotheses 2.6 (iv), (v) dictate an oscillatory behavior for f(z,) near zero. Hypothesis 2.6 (vi) is satisfied if, for example, for a.a. zΩ, f(z,) is differentiable, and for every ρ>0, we can find ξ^ρ>0 such that

fx(z,x)x2-ξ^ρ|x|pfor a.a. zΩ,|x|ρ.

Example 2.8.

The following function satisfies Hypothesis 2.6 (for the sake of simplicity, we drop the z-dependence):

f(x)={|x|τ-2x-2|x|η-2xif |x|1,β|x|p-2x+|x|μ-2x+(β+2)|x|γ-2xif |x|>1,

with 1<τ<η<, βλ^1 and 1<γ2<μ<p.

3 Solutions of constant sign

In this section we produce solutions of constant sign (positive and negative solutions) of problem (1.1).

We start by considering the following auxiliary nonlinear parametric Dirichlet (p,2)-equation:

-Δpu(z)-Δu(z)=λ|u(z)|q-2u(z)in Ω,λ>0,u|Ω=0.(3.1)

Proposition 3.1.

For every λ>0, problem (3.1) has a unique positive solution u~λintC+, and since (3.1) is odd, v~λ=-u~λ-intC+ is the unique negative solution of (3.1); moreover, λu~λ is strictly increasing and

u~λC01(Ω¯)0+as λ0+.

Proof.

The existence of a positive solution for problem (3.1) can be established using the direct method of the calculus of variations. More precisely, let ψλ:W01,p(Ω) be the C1-functional defined by

ψλ(u)=1pDupp+12Du22-λqu+qqfor all uW01,p(Ω).

Since 1<q<2<p, we see that ψλ() is coercive. Also, using the Sobolev embedding theorem, we show that ψλ() is sequentially weakly lower semicontinuous. So, using the Weierstrass–Tonelli theorem, we can find u~λW01,p(Ω) such that

ψλ(u~λ)=inf[ψλ(u):uW01,p(Ω)].(3.2)

Since q<2<p, for t(0,1) small, we have ψλ(tu^1)<0, hence ψλ(uλ)<0=ψλ(0) (see (3.2)), and thus uλ0. From (3.2) we have ψλ(u~λ)=0, that is,

Ap(u~λ),h+A(u~λ),h=λΩ(u~λ+)q-1h𝑑zfor all hW01,p(Ω).(3.3)

In (3.3) we choose h=-u~λ-W01,p(Ω). Then Du~λ-pp+Du~λ-22=0, and thus u~λ0 and u~λ0. From (3.3) it follows that

-Δpu~λ(z)-Δu~λ(z)=λu~λ(z)q-1for a.a. zΩ,u~λ|Ω=0,

therefore u~λ is a positive solution of (3.1).

From [18, Corollary 8.6, p. 208], we have u~λL(Ω). Then, invoking [15, Theorem 1], we infer that u~λC+{0}. The nonlinear strong maximum principle of Pucci and Serrin [26, pp. 111, 120] implies that u~λintC+.

Next we show that this positive solution is unique. To this end, we introduce the integral functional j:L1(Ω)¯={+} defined by

j(u)={1pDu1/2pp+12Du1/222if u0,u1/2W01,p(Ω),+otherwise.

From [7, Lemma 1], we know that j() is convex.

Suppose that y~λW01,p(Ω) is another positive solution of (3.1). Again we have y~λintC+. Then for any hC01(Ω¯) and for |t|<1 small, we have

u~λ+thdomjandy~λ+thdomj,

where domj={uL1(Ω):j(u)<+} (the effective domain of j()). We can easily see that j() is Gateaux differentiable at u~λ2 and at y~λ2 in the direction of h. Moreover, the chain rule and Green’s identity (see [10, p. 211]) imply that

jλ(u~λ2)(h)=12Ω-Δpu~λ-Δu~λu~λh𝑑z,jλ(y~λ2)(h)=12Ω-Δpy~λ-Δy~λy~λh𝑑zfor all hC01(Ω¯).

The convexity of j() implies the monotonicity of j(). Therefore,

0Ω(-Δpu~λ-Δu~λu~λ-Δpy~λ-Δy~λy~λ)(u~λ-y~λ)𝑑z=Ωλ[1u~λ2-q-1y~λ2-q](u~λ-y~λ)𝑑z,

and hence u~λ=y~λ (since q<2). This proves the uniqueness of the positive solution of problem (3.1).

Let 0<η<λ and consider the Caratheodory function

kη(z,x)={η(x+)q-1if xu~λ(z),ηu~λ(z)q-1if u~λ(z)<x.(3.4)

We set Kη(z,x)=0xkη(z,s)𝑑s and consider the C1-functional ψ^η:W01,p(Ω) defined by

ψ^η(u)=1pDupp+12Du22-ΩKη(z,u)𝑑zfor all uW01,p(Ω).

Evidently ψ^η() is coercive (see (3.4)) and sequentially weakly lower semicontinuous. So, we can find u¯ηW01,p(Ω) such that

ψ^η(u¯η)=inf[ψ^η(u):uW01,p(Ω)].(3.5)

Since 1<q<2<p, we see that ψ^η(u¯η)<0=ψ^η(0), therefore u¯η0.

From (3.5) we have ψ^η(u¯η)=0, that is,

Ap(u¯λ),h+A(u¯λ),h=Ωkλ(z,u¯η),dzfor all hW01,p(Ω).(3.6)

In (3.6), first we choose h=-u¯η-W01,p(Ω). Then Du¯η-pp+Du¯η-22=0 (see (3.4)), hence u¯η0, u¯η0. Also, in (3.6), we choose h=(u¯η-u~λ)+W01,p(Ω). Then (from (3.4) and since η<λ)

Ap(u¯η),(u¯η-u~λ)++A(u¯η),(u¯η-u~λ)+=Ωηu~λq-1(u¯η-u~λ)+𝑑zΩλu~λq-1(u¯η-u~λ)+𝑑z=Ap(u~λ),(u¯η-u~λ)++A(u~λ),(u¯η-u~λ)+,

hence u¯ηu~λ (see Proposition 2.2).

So, we have proved that

u¯η[0,u~λ],u¯η0.(3.7)

From (3.4), (3.6) and (3.7), we infer that u¯η=u~ηintC+, thus

u~ηu~λ.(3.8)

We have (see (3.8) and recall that η<λ, u~ηintC+)

-Δpu~η-Δu~η=ηu~ηq-1=λu~ηq-1-(λ-η)u~ηq-1<λu~λq-1=-Δpu~λ-Δu~λfor a.a. zΩ.(3.9)

Let h1=ηu~ηq-1 and h2=λu~λq-1. Evidently h1,h2C01(Ω¯). For all KΩ compact, let cK=minKu~η>0 (recall u~ηintC+). Then (see (3.8))

h2(z)-h1(z)=λu~λ(z)q-1-ηu~η(z)q-1(λ-η)u~η(z)q-1(λ-η)cK>0for a.a. zK.

From (3.9) and Proposition 2.4 it follows that u~λ-u~ηintC+, therefore λu~λ is strictly increasing from (0,+) into C01(Ω¯).

Finally, let λ>0 and let u~λintC+ be the unique solution of (3.1). Then we have

Ap(u~λ),h+A(u~λ),h=λΩu~λq-1h𝑑zfor all hW01,p(Ω).

Choosing h=u~λW01,p(Ω), we obtain Du~λppλu~λqq, hence u~λp-qλc2 for some c2>0 (recall that q<p). Therefore, given μ>0, we see that

{u~λ}λ>0W01,p(Ω) is bounded,u~λ0as λ0+.(3.10)

Invoking [18, Corollary 8.6, p. 208], we can find c3>0 such that

u~λc3for all 0<λμ.

Then [15, Theorem 1] implies that there exist α(0,1) and c4>0 such that

u~λC01,α(Ω¯),u~λC01,α(Ω¯)c4for all 0<λμ.(3.11)

From (3.10), (3.11) and the compact embedding of C01,α(Ω¯) into C01(Ω¯), we conclude that

u~λ0in C01(Ω¯) as λ0+.

Since problem (3.1) is odd, v~λ=-u~λ-intC+ is the unique negative solution of (3.1) for all λ>0. Also, λv~λ is strictly decreasing from (0,+) into C01(Ω¯) and v~λC01(Ω¯)0 as λ0+. ∎

Let δ0>0 be as postulated by Hypothesis 2.6 (v). On account of Proposition 2.5, we can find λ±>0 such that

u~λ(z)[0,δ0]for all zΩ¯, 0<λλ+,v~λ(z)[-δ0,0]for all zΩ¯, 0<λλ-.(3.12)

With θ±>0 as in Hypothesis 2.6 (iv), we set

λ+*=min{λ+,θ+}andλ-*=min{λ-,θ-}.

Proposition 3.2.

If Hypothesis 2.6 holds, then

  • (a)

    for all 0<λλ+* , problem ( 1.1 ) admits a positive solution u0intC01(Ω¯)[u~λ,w+],

  • (b)

    for all 0<λλ-* , problem ( 1.1 ) admits a negative solution v0intC01(Ω¯)[w-,v~λ].

Proof.

(a) Recall that δ0<c0 (see Hypothesis 2.6 (v)). This fact and (3.12) permit the definition of the Caratheodory function

eλ+(z,x)={λu~λ(z)q-1+f(z,u~λ(z))if x<u~λ(z),λxq-1+f(z,x)if u~λ(z)xw+(z),λw+(z)q-1+f(z,w+(z))if w+(z)<x.(3.13)

We set Eλ+(z,x)=0xeλ+(z,x)𝑑s and consider the C1-functional φ^λ+:W01,p(Ω) defined by

φ^λ+(u)=1pDupp+12Du22-ΩEλ+(z,u)𝑑zfor all uW01,p(Ω).

From (3.13) it is clear that φ^λ+() is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find u0W01,p(Ω) such that φ^λ+(u0)=inf[φ^λ+(u):uW01,p(Ω)], hence (φ^λ+)(u0)=0, and therefore

Ap(u0),h+A(u0),h=Ωeλ+(z,u0)h𝑑zfor all hW01,p(Ω).(3.14)

In (3.14), first we choose h=(u~λ-u0)+W01,p(Ω). Then (see (3.12), (3.13), Hypothesis 2.6 (v) and Proposition 3.1)

Ap(u0),(u~λ-u0)++A(u0),(u~λ-u0)+=Ω[λu~λq-1+f(z,u~λ)](u~λ-u0)+𝑑zΩλu~λq-1(u~λ-u0)+𝑑z=Ap(u~λ),(u~λ-u0)++A(u~λ),(u~λ-u0)+,

hence u~λu0 (see Proposition 2.2).

Next in (3.14) we choose h=(u0-w+)+W01,p(Ω). Then (see (3.13), Hypothesis 2.6 (iv) and recall that 0<λλ+*θ+)

Ap(u0),(u0-w+)++A(u0),(u0-w+)+=Ω[λw+q-1+f(z,w+)](u0-w+)+𝑑zΩ[θ+w+q-1+f(z,w+)](u0-w+)+𝑑xAp(w+),(u0-w+)++A(w+),(u0-w+)+,

thus u0w+ (see Proposition 2.2).

We have proved that

u0[u~λ,w+].(3.15)

From (3.13), (3.14), (3.15), it follows that

-Δpu0(z)-Δu0(z)=λu0(z)q-1+f(z,u0(z))for a.a. zΩ,u0|Ω=0.(3.16)

From (3.15), (3.16) and [15, Theorem 1], we infer that

u0[u~λ,w+]intC+.(3.17)

Now let ρ=u0 and let ξ^ρ>0 be as postulated by Hypothesis 2.6 (vi). Then we have (see (3.12), (3.16), (3.17), Hypotheses 2.6 (v)–(vi) and Proposition 3.1)

-Δpu0(z)-Δu0(z)+ξ^ρu0(z)p-1=λu0(z)q-1+f(z,u0(z))+ξ^ρu0(z)p-1λu~λ(z)q-1+f(z,u~λ(z))+ξ^ρu~λ(z)p-1λu~λ(z)q-1+ξ^ρu~λ(z)p-1=-Δpu~λ(z)-Δu~λ(z)+ξ^ρu~λ(z)p-1 for a.a. zΩ.(3.18)

Set

h1(z)=λu~λ(z)q-1+ξ^ρu~λ(z)p-1,h2(z)=λu0(z)q-1+f(z,u0(z))+ξ^ρu0(z)p-1.

Evidently h1,h2L(Ω). Also, for KΩ compact, we have (recall that u0intC+)

0<mKu0(z)for all zK.

Therefore (see (3.17) and Hypothesis 2.6 (v)),

h2(z)-h1(z)f(z,u0(z))cK>0for a.a. zΩ.

Then (3.18) and Proposition 2.4 imply that u0-u~λintC+.

Also, we have (see (3.16), (3.17), Hypotheses 2.6 (iv), (vi), and recall that 0<λθ+)

-Δpu0(z)-Δu0(z)+ξ^ρu0(z)p-1=λu0(z)q-1+f(z,u0(z))+ξ^ρu0(z)p-1θ+w+(z)q-1+f(z,w+(z))+ξ^ρw+(z)p-1-c^1+ξ^ρw+(z)p-1-Δpw+(z)-Δw+(z)+ξ^ρw+(z)p-1for a.a. zΩ.(3.19)

From (3.19) and Proposition 2.4, we have w+-u0intC+. We conclude that u0intC01(Ω¯)[u~λ,w+].

(b) In the negative semiaxis, we consider the Caratheodory function

eλ-(z,x)={λ|w-(z)|q-2w-(z)+f(z,w-(z))if x<w-(z),λ|x|q-2x+f(z,x)if w-(z)xv~λ(z),λ|v~λ|q-2v~λ(z)+f(z,v~λ(z))if v~λ(z)<x.(3.20)

We set Eλ-(z,x)=0xeλ-(z,s)𝑑s and consider the C1-functional φ^λ-:W01,p(Ω) defined by

φ^λ-(u)=1pDupp+12Du22-ΩEλ-(z,u)𝑑zfor all uW01,p(Ω).

Arguing as in part (a), using this time the functional φ^λ- and (3.20), we produce a negative solution v0 for problem (1.1), with λ(0,λ-*], such that v0intC01(Ω¯)[w-,v~λ].

Next, using u0intC+, v0-intC+ together with suitable variational, truncation and comparison arguments, we will generate a second pair of constant sign smooth solutions.

Proposition 3.3.

If Hypothesis 2.6 holds, then

  • (a)

    for all 0<λλ+* , problem ( 1.1 ) has a second positive solution u^intC+ , with u^u0, u^-u~λintC+ ;

  • (b)

    for all 0<λλ-* , problem ( 1.1 ) has a second negative solution v^-intC+ , with v^v0, v~λ-v^intC+.

Proof.

(a) Let u~λintC+ be the unique positive solution of (3.1) (see Proposition 3.1). We introduce the following truncation of the reaction in problem (1.1):

rλ+(z,x)={λu~λ(z)q-1+f(z,u~λ(z))if xu~λ(z),λxq-1+f(z,x)if u~λ(z)<x.(3.21)

This is a Caratheodory function. Set Rλ+(z,x)=0xrλ+(z,s)𝑑s and consider the C1-functional φλ+:W01,p(Ω) defined by

φλ+(u)=1pDupp+12Du22-ΩRλ+(z,u)𝑑zfor all uW01,p(Ω).

From (3.13) and (3.21), it is clear that

φλ+|[u~λ,w+]=φ^λ+|[u~λ,w+].(3.22)

Let u0intC+ be the positive solution of problem (1.1) produced in Proposition 3.2. From the proof of that proposition, we know that u0intC+ is a minimizer of φ^λ+, and u0intC01(Ω¯)[u~λ,w+]. Hence, from (3.22), it follows that u0intC+ is a local C01(Ω¯)-minimizer of φλ+, and thus (see Proposition 2.3)

u0intC+ is a local W01,p(Ω)-minimizer of φλ+.(3.23)

Claim 1: Kφλ+[u~λ)intC+ and u-u~λintC+ for all uKφλ+.

Let uKφλ+. Then

Ap(u),h+A(u),h=Ωrλ+(z,u)h𝑑zfor all hW01,p(Ω).(3.24)

In (3.24), we choose h=(u~λ-u)+W01,p(Ω). Then (see (3.21), (3.12), Hypothesis 2.6 (v) and Proposition 3.1)

Ap(u),(u~λ-u)++A(u),(u~λ-u)+=Ω[λu~λq-1+f(z,u~λ)](u~λ-u)+𝑑zΩλu~λq-1(u~λ-u)+𝑑z=Ap(u~λ),(u~λ-u)++A(u~λ),(u~λ-u)+,

thus u~λu (see Proposition 2.2).

The nonlinear regularity theory of Lieberman [15] implies uintC+. In addition, as in the proof of Proposition 3.2, using Proposition 2.4, we show that u-u~λintC+. This proves claim 1.

On account of claim 1, we may assume that

Kφλ+ is finite.(3.25)

Otherwise, we already have an infinity of smooth solutions of problem (1.1) satisfying u-u~λintC+ for all the solutions u (see (3.21) and claim 1). Hence, we are done.

From (3.23) and (3.25) it follows that we can find ρ(0,1) small such that

φλ+(u0)<inf[φλ+(u):u-u0=ρ]=mλ+(3.26)

(see the proof of [1, Proposition 29]).

From Hypothesis 2.6 (iii), we see that given β>0, we can find M1=M1(β)>0 such that

f(z,x)x-pF(z,x)-βx2for a.a. zΩ,|x|M1.

Then we have

ddx[F(z,x)|x|p]=f(z,x)|x|p-p|x|p-2xF(z,x)|x|2p=f(z,x)x-pF(z,x)|x|px{-βxp-1if xM1,β|x|p-1if x-M1,

hence

F(z,x)|x|p-F(z,y)|y|pβp-2[1|x|p-2-1|y|p-2]for a.a. zΩ,|x||y|M1.(3.27)

From Hypothesis 2.6 (ii), we have

λ^1plim infx±F(z,x)|x|plim supx±F(z,x)|x|pη^puniformly for a.a. zΩ.(3.28)

So, if in (3.27) we let x± and use (3.28), then

λ^1p-F(z,y)|y|p-βp-21|y|p-2,

and thus

λ^1|y|p-pF(z,y)y2-βp-2-βfor a.a. zΩ,|y|M1.

Since β>0 is arbitrary, we conclude that

λ^1|x|p-pF(z,x)x2-uniformly for a.a. zΩ as x±.(3.29)

Claim 2: φλ+(tu^1)- as t+.

For t>0, we have

pφλ+(tu^1)=tpλ^1+pt22Du^122-ΩRλ+(z,tu^1)𝑑ztpλ^1+pt22Du^122-pλtqqDu^1qq-ΩpF(z,tu^1)𝑑z+c4=Ω[λ^1(tu^1)p-pF(z,tu^1)]𝑑z+pt22Du^122-pqDu^1qq+c4

for some c4>0 (see (3.21)), therefore

pφλ+(tu^1)t2=Ω[λ^1(tu^1)p-pF(z,tu^1)](tu^1)2u^12𝑑z+p2D(tu^1)22-pq1λ2-qDu^1qq+c4t2.

Passing to the limit as t+ and using (3.29), Fatou’s lemma and the fact that q<2, we obtain

pφλ+(tu^1)t2-as t+,

hence

φλ+(tu^1)-as t+.

This proves claim 2.

Claim 3: φλ+ satisfies the C-condition.

We consider a sequence {un}n1W01,p(Ω) such that

|φλ+(un)|M2for some M2>0 and all n(3.30)

and

(1+un)(φλ+)(un)0in W-1,p(Ω) as n.(3.31)

From (3.31) we have

|Ap(un),h+A(un),h-Ωrλ+(z,un)h𝑑z|εnh1+un(3.32)

for all hW01,p(Ω), with εn0+.

In (3.32), we choose h=-vn-W01,p(Ω). Using (3.21), we have

Dun-pp+Dun-22-Ω[λu~λq-1+f(z,u~λ)](-un-)𝑑zεnfor all n,

which implies un-pc5un- for some c5>0 and all n (see Hypothesis 2.6 (i)), hence

{un-}n1W01,p(Ω) is bounded.(3.33)

Next we show that {un+}W01,p(Ω) is bounded. Arguing by contradiction, we assume that at least for a subsequence, we have un++ as n. We let yn=un+un+,n. Then yn=1,yn0 for all n. We may assume that

yn𝑤yin W01,p(Ω),ynyin Lp(Ω),y0.

Multiplying (3.32) with 1un+p-1 and using (3.33), we obtain

|Ap(yn),h+A(yn),hun+p-2-Ωrλ+(z,un+)un+p-1h𝑑z|εnh

for all hW01,p(Ω), with εn0+. Using (3.21), we have

|Ap(yn),h+A(yn),hun+p-2-Ω[λ(un+)q-1+f(z,un+)]un+p-1h𝑑z|εn′′h(3.34)

for all hW01,p(Ω), with εn′′0+. On account of (3.2), we see that

{λ(un+)q-1+f(z,un+)un+p-1}n1Lp(Ω)

is bounded. Therefore, we may assume that

λ(un+)q-1+f(z,un+)un+p-1𝑤η0(z)yp-1in Lp(Ω),(3.35)

with λ^1η(z)η^ for a.a. zΩ (see Hypothesis 2.6 (ii) and recall that q<2<p).

In (3.34), we choose h=yn-yW01,p(Ω). Passing to the limit as n, we obtain (recall that 2<p)

limnAp(yn),yn-y=0,

therefore (see Proposition 2.2)

ynyin W01,p(Ω)y=1,y0.(3.36)

So, if in (3.34) we pass to the limit as n and use (3.35), (3.36), then

Ap(y),h=Ωη0(z)yp-1h𝑑z for all hW01,p(Ω),

and hence

-Δpy(z)=η0(z)y(z)p-1for a.a. zΩ,y|Ω=0.(3.37)

Suppose that η0λ^1 (see (3.35)). We have λ~1(p,η0)<λ~1(p,λ^1)=1 (see Proposition 2.5), and thus y must be nodal (see (3.37)), which contradicts (3.36).

Next assume that η0(z)=λ^1 for a.a. zΩ. From (3.37) it follows that y=ξu^1intC+ (ξ>0), hence y(z)>0 for all zΩ, and thus un+(z)+ for all zΩ. Therefore (see Hypothesis 2.6 (iii)),

f(z,un+(z))un+(z)-pF(z,un+(z))un+(z)2-for a.a. zΩ,

and thus (by Fatou’s lemma)

Ωf(z,un+)un+-pF(z,un+)un+2𝑑z-.(3.38)

From (3.30) and (3.33), we have

Dun+pp+p2Dun+22-Ω[λpq|un+|q+pF(z,un+)]𝑑z-M3(3.39)

for some M3>0 and all n (see (3.21)). Similarly, if we use (3.33) in (3.32) and also choose h=un+W01,p(Ω), then

-Dun+pp-Dun+22+Ω[λ(un+)q+f(z,un+)un+]𝑑z-M4(3.40)

for some M4>0 and all n (see (3.21)). We add (3.39) and (3.40) and obtain

(p2-1)Dun+22+Ω[f(z,un+)un+-pF(z,un+)]𝑑z-M5+λ(pq-1)un+qq,

with M5=M3+M4>0, for all n, hence

(p2-1)Dyn22+Ωf(z,un+)un+-pF(z,un+)un+2𝑑z-M5un+2+λ(pq-1)ynqqun+2-q.

Since q<2<p, passing to the limit as n, we have a contradiction (see (3.38)). Therefore, {un+}n1W01,p(Ω) is bounded, and hence {un}n1W01,p(Ω) is bounded (see (3.33)).

We may assume that

un𝑤uin W01,p(Ω),unuin Lp(Ω).(3.41)

In (3.32) we choose h=un-uW01,p(Ω), pass to the limit as n and use (3.41). Then

limn[Ap(un),un-u+A(un),un-u]=0,

which implies (recall that A() is monotone)

lim supn[Ap(un),un-u+A(u),un-u]0.

Hence (see (3.41)),

lim supnAp(un),un-u0,

and thus (see Proposition 2.2)

unuin W01,p(Ω).

Therefore, φλ+ satisfies the C-conditions and this proves claim 3.

Then (3.26) and claims 2 and 3, permit the use of Theorem 2.1 (the mountain pass theorem). So, we can find u^W01,p(Ω) such that u^Kφλ+[u~λ)intC+ (see claim 1) and mλ+φλ+(u^) (see (3.26)). From (3.21) and (3.26) we infer that u^ is a positive solution of (1.1) and u^u0. As in the proof of Proposition 3.2, using Proposition 2.4, we can show that u^-u~λintC+.

(b) Let v~λ=-u~λ-intC+ be the unique negative solution of (3.1) (see Proposition 3.1). We consider the following truncation of the reaction in problem (3.1):

rλ-(z,x)={λ|x|q-2x+f(z,x)if xv~λ(z),λ|v~λ(z)|q-2v~λ(z)+f(z,v~λ(z))if v~λ(z)<x.

This is a Caratheodory function. Set Rλ-(z,x)=0xrλ-(z,s)𝑑s and consider the C1-function φλ-:W01,p(Ω) defined by

φλ-(u)=1pDupp+12Du22-ΩRλ-(z,u)𝑑zfor all uW01,p(Ω).

Working with φλ- as in part (a), we see that for λ(0,λ-*], we can find v^-intC+, a second negative solution of (1.1), such that v^v0 and v~λ-vintC+. ∎

We will show that we have extremal constant sign solutions, that is, there is a smallest positive solution for problem (1.1), with λ(0,λ+*], and a biggest negative solution for (1.1), with λ(0,λ-*]. These extremal constant sign solutions will be used in the next section to generate nodal solutions.

We introduce the following solution sets:

  • Sλ+ is the set of positive solutions for problem (1.1),

  • Sλ- is the set of negative solutions for problem (1.1).

From Proposition 3.2 we know that

for all 0<λλ+*,Sλ+intC+,for all 0<λλ-*,Sλ--intC+.

Proposition 3.4.

If Hypothesis 2.6 holds, then

  • (a)

    for all λ(0,λ+*] , problem ( 1.1 ) has a smallest positive solution u¯λintC+ , that is, u¯λu for all uSλ+,

  • (b)

    for all λ(0,λ-*] , problem ( 1.1 ) has a biggest negative solution v¯λ-intC+ , that is, vv¯λ for all vSλ-.

Proof.

(a) First we show that

u~λufor all uSλ+(0<λλ+*).(3.42)

To this end, let uSλ+ and consider the Caratheodory function

βλ(z,x)={λ(x+)q-1if xu(z),λu(z)q-1if u(z)<x.(3.43)

We set Bλ(z,x)=0xβλ(z,s)𝑑s and consider the C1-functional ψ^λ:W01,p(Ω) defined by

ψ^λ(u)=1pDupp+12Du22-ΩBλ(z,u)𝑑zfor all uW01,p(Ω).

From (3.43) it is clear that ψ^λ() is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find u~λ*W01,p(Ω) such that

ψ^λ(u~λ*)=inf[ψ^λ(u):uW01,p(Ω)].(3.44)

Let yintC+. Since uSλ+intC+, using [17, Proposition 2.1], we can find t(0,1] small such that tyu. Then, from (3.43) and since q<2<p, we see that by choosing t(0,1] even smaller if necessary, we have ψ^λ(ty)<0, which implies ψ^λ(u~λ*)<0=ψ^λ(0) (see (3.44)), and thus u~λ*0.

From (3.44) we have ψ^λ(u~λ*)=0, hence

Ap(u~λ*),h+A(u~λ*),h=Ωβλ(z,u~λ*)h𝑑zfor all hW01,p(Ω).(3.45)

In (3.45), we choose h=-(u~λ*)-W01,p(Ω). Then D(u~λ*)-pp+D(u~λ*)-22=0 (see (3.43)), and therefore u~λ*0 and u~λ*0.

Also, in (3.45), we choose h=(u~λ*-u)+W01,p(Ω). Then (see (3.12), (3.43), Hypothesis 2.6 (v) and note that uSλ+)

Ap(u~λ*),(u~λ*-u)++A(u~λ*),(u~λ*-u)+=Ωλuq-1(u~λ*-u)+𝑑zΩ[λuq-1+f(z,u)](u~λ*-u)+𝑑z=Ap(u),(u~λ*-u)++A(u),(u~λ*-u)+,

hence u~λ*u.

We have proved that

u~λ*[0,u],u~λ*0.(3.46)

From (3.46), (3.45), (3.43) and Proposition 3.1, we conclude that u~λ*=u~λ. Therefore, (3.42) is true.

From [8], we know that Sλ+ is downward directed (that is, if u1,u2Sλ+, then we can find vSλ+ such that uu1,uu2). Invoking [13, Lemma 3.10, p. 178], we can find a decreasing sequence {un}n1Sλ+ such that infSλ+=infn1un. Evidently, {un}n1W01,p(Ω) is bounded (see Hypothesis 2.6 (i) and recall that 0unu1 for all n). So, we may assume that

un𝑤u¯λin W01,p(Ω),unu¯λin Lp(Ω).(3.47)

For every n, we have

Ap(un),h+A(un),h=Ω[λunq-1+f(z,un)]h𝑑zfor all hW01,p(Ω).(3.48)

In (3.48), we choose h=un-u¯λW01,p(Ω), pass to the limit as n, use (3.47), and reasoning as in the proof of Proposition 3.3, via Proposition 2.2, we conclude that

unu¯λin W01,p(Ω).(3.49)

So, in (3.48), we pass to the limit as n and use (3.49). Then

Ap(u¯λ),h+A(u¯λ),h=Ω[λu¯λq-1+f(z,u¯λ)]h𝑑zfor all hW01,p(Ω).

Also, from (3.42) and (3.49), we have u~λu¯λ. Therefore,

u¯λSλ+,u¯λufor all uSλ+.

(b) Similarly for the set Sλ-, we have vv~λ for all vSλ-. Reasoning as in part (a), we produce a v¯λSλ- such that vv¯λ for all vSλ-. ∎

Next we examine the maps λu¯λ and λv¯λ.

Proposition 3.5.

If Hypothesis 2.6 holds, then

  • (a)

    the map λu¯λ from (0,λ+*] into C+ is strictly increasing (that is, for 0<θ<λλ+*, u¯λ-u¯θintC+ ) and left continuous;

  • (b)

    the map λv¯λ from (0,λ-*] into -C+ is strictly decreasing (that is, for 0<θ<λλ-*, u¯θ-u¯λintC+ ) and left continuous.

Proof.

(a) First we show that λu¯λ is increasing. We consider the Caratheodory function

mθ(z,x)={θ(x+)q-1+f(z,x+)if xu¯λ(z),θu¯λ(z)q-1+f(z,u¯λ(z))if u¯λ(z)<x.(3.50)

We set Γθ(z,x)=0xmθ(z,s)𝑑s and consider the C1-functional σθ:W01,p(Ω) defined by

σθ(u)=1pDupp+12Du22-ΩΓθ(z,u)𝑑zfor all uW01,p(Ω).

This is coercive (see (3.50)) and sequentially weakly lower semicontinuous. Hence, we can find yθW01,p(Ω) such that

σθ(yθ)=inf[σθ(u):uW01,p(Ω)].(3.51)

Exploiting as before the fact that q<2<p, we have σθ(yθ)<0=σθ(0). Hence, yθ0. From (3.51) we have σθ(yθ)=0, therefore

Ap(yθ),h+A(yθ),h=Ωmθ(z,yθ)h𝑑zfor all hW01,p(Ω).(3.52)

In (3.52), first let h=-yθ-W01,p(Ω). Then, using (3.50), we have yθ0, yθ0. Also, in (3.52), we choose h=(yθ-u¯λ)+W01,p(Ω). Then (see (3.50) and note that θ<λ)

Ap(yθ),(yθ-u¯λ)++A(yθ),(yθ-u¯λ)+=Ω[θu¯λq-1+f(z,u¯λ)](yθ-u¯λ)+𝑑zΩ[λu¯λq-1+f(z,u¯λ)](yθ-u¯λ)+𝑑z=Ap(u¯λ),(yθ-u¯λ)++A(u¯λ),(yθ-u¯λ)+,

hence yθu¯λ.

We have proved that

yθ[0,u¯λ],yθ0,(3.53)

which implies yθSθ+intC+ (see (3.50), (3.52)), and thus u¯θyθu¯λ (see (3.53)). Hence, λu¯λ is increasing. Now let ρ=u¯λ and let ξ^ρ>0 be as postulated by Hypothesis 2.6 (vi). Then we have (see Hypothesis 2.6 (vi) and recall that θ<λ)

-Δpu¯θ-Δu¯θ+ξ^ρu¯θp-1=θu¯θq-1+f(z,u¯θ)+ξ^ρu¯θp-1=λu¯θq-1+f(z,u¯θ)+ξ^ρu¯θp-1-(λ-θ)u¯θq-1λu¯θq-1+f(z,u¯θ)+ξ^ρu¯θp-1=-Δpu¯λ-Δu¯λ+ξ^ρu¯λp-1for a.a. zΩ.

Let

h1(z)=θu¯θ(z)q-1+f(z,u¯θ(z))+ξ^ρu¯θ(z)p-1,h2(z)=λu¯λ(z)q-1+f(z,u¯λ(z))+ξ^ρu¯λ(z)p-1.

Evidently, h1,h2L(Ω) and (λ-θ)u¯θ(z)q-1h2(z)-h1(z) for a.a. zΩ. Since u¯θintC+, it follows that h1h2 and so, using Proposition 2.4, we conclude that u¯λ-u¯θintC+. We have proved that λu¯λ is strictly increasing.

Next we show that the map λu¯λ is left continuous. To this end, let λnλ-,λλ+*. For each n, let u¯n=u¯λnSλn+intC+ be the minimal positive solution of problem (1.1) (see Proposition 3.3). Then we have

Ap(u¯n),h+A(u¯n),h=Ω[λnu¯nq-1+f(z,u¯n)]h𝑑z(3.54)

for all hW01,p(Ω), all n, and

u¯nu¯λfor all n.(3.55)

In (3.54), we choose h=u¯nW01,p(Ω). Then from Hypothesis 2.6 (i) and (3.55) we infer that {u¯n}n1W01,p(Ω) is bounded. Using [18, Corollary 8.6, p. 208], we can find c6>0 such that

u¯nc6for all n.

Invoking [15, Theorem 1], we can find α(0,1) and c7>0 such that

u¯nC01,α(Ω¯),u¯C01,α(Ω¯)c7for all n.

Since C01,α(Ω¯)C01(Ω¯) compactly, we may assume that

u¯nu^λin C01(Ω¯).(3.56)

Passing to the limit as n in (3.54) and using (3.56), we obtain u^λSλ+. We claim that u¯λ=u^λ. If this is not true, then we can find z0Ω such that u¯λ(z0)<u^λ(z0), hence (see (3.56))

u¯λ(z0)<u¯n(z0)for all nn0.

This contradicts the fact that λu¯λ is increasing. Hence, u^λ=u¯λ and so we conclude the left continuity of the map λu¯λ. ∎

Summarizing the results obtained in this section, we can state the following theorem.

Theorem 3.6.

If Hypothesis 2.6 holds, then

  • (a)

    there exists λ+*>0 such that for λ(0,λ+*] , we have

    • (i)(a)

      problem ( 1.1 ) has at least two positive solutions

      u0,u^intC+,u0u^,u0-u~λ,u^-u~λintC+,

    • (i)(b)

      problem ( 1.1 ) has a smallest positive solution u¯λintC+ and λu¯λ is strictly increasing and left continuous;

  • (b)

    there exists λ-*>0 such that for λ(0,λ-*] , we have

    • (ii)(a)

      problem ( 1.1 ) has at least two negative solutions

      v0,v^-intC+,v0v^,v~λ-v0,v~λ-v^intC+,

    • (ii)(b)

      problem ( 1.1 ) has a biggest negative solution v¯λ-intC+ and λv¯λ is strictly decreasing and left continuous.

4 Nodal solutions

In this section we focus on nodal (sign changing) solutions for problem (1.1). So, let λ*=min{λ+*,λ-*} (see Theorem 3.6). Let λ(0,λ*] and consider the C1-functional τ^λ:H01(Ω) defined by

τ^λ(u)=12Du22-λquqq-ΩF(z,u)𝑑zfor all uH01(Ω).

Hypothesis 2.6 (v) and Proposition 2.1 of [14] imply the following result.

Proposition 4.1.

If Hypothesis 2.6 holds and λ>0, then Ck(τ^λ,0)=0 for all kN0.

Let τλ=τ^λ|W01,p(Ω). Since W01,p(Ω)H01(Ω) densely, we can apply [19, Theorem 16] (see also [4, p. 14]) and have Ck(τλ,0)=Ck(τ^λ,0) for all k0, thus (see [14])

Ck(τλ,0)=0for all k0.(4.1)

Let φλ:W01,p(Ω) be the energy functional for problem (1.1) defined by

φλ(u)=1pDupp+12Du22-λquqq-ΩF(z,u)𝑑zfor all uW01,p(Ω).

Then we have

|φλ(u)-τλ(u)|=1pDupp=1pup,|φλ(u)-τλ(u),h|=|Ap(u),h|Dupp-1hpfor all hW01,p(Ω).

Therefore,

φλ(u)-τλ(u)*Dupp-1=up-1.

Then the C1-continuity property of critical groups (see [11, Theorem 5.126, p. 836]) implies that

Ck(φλ,0)=Ck(τλ,0)for all k0,

hence (see (4.1))

Ck(φλ,0)=0for all k0.(4.2)

Let λ(0,λ*] and let u¯λintC+,v¯λ-intC+ be two extremal constant sign solutions for problem (1.1) (see Theorem 3.6). We introduce the following trunction of the reaction in problem (1.1):

gλ(z,x)={λ|v¯λ(z)|q-2v¯λ(z)+f(z,v¯λ(z))if xv¯λ(z),λ|x|q-1x+f(z,x)if v¯λ(z)xu¯λ(z)λu¯λ(z)q-1+f(z,u¯λ(z))if u¯λ(z)<x.(4.3)

This is a Caratheodory function. Set Gλ(z,x)=0xgλ(z,s)𝑑s and consider the C1-functional φ~λ:W01,p(Ω) defined by

φ~λ(u)=1pDupp+12Du22-ΩGλ(z,u)𝑑zfor all uW01,p(Ω).

We know that φ~λC1(W01,p(Ω)). Using (4.3), we can easily show that

Kφ~λ[v¯λ,u¯λ]C01(Ω¯).(4.4)

Therefore, the nontrivial critical points of φ~λ, distinct from v¯λ and u¯λ, are nodal solutions of (1.1). So, we assume that Kφ~λ is finite. Otherwise, we already have an infinity of nodal solutions for problem (1.1).

We also consider the positive and negative truncations of φ~λ, namely, we consider the C1-functionals φ~λ±:W01,p(Ω) defined by

φ~λ±(u)=1pDupp+12Du22-ΩGλ(z,±u±)𝑑zfor all uW01,p(Ω).

We can easily show that

Kφ~λ+[0,u¯λ]C01(Ω¯),Kφ~λ-[v¯λ,0]C01(Ω¯).

The extremality of u¯λ and v¯λ implies that

Kφ~λ+={0,u¯λ},Kφ~λ-={0,v¯λ}.(4.5)

We compute the critical groups of φ~λ.

Proposition 4.2.

If Hypothesis 2.6 holds and λ(0,λ*], then Ck(φλ,0)=Ck(φ~λ,0) for all kN0.

Proof.

We consider the homotopy h^(t,u) defined by

h^(t,u)=(1-t)φλ(u)+tφ~λ(u)for all t[0,1],uW01,p(Ω).

Suppose we can find {tn}n1[0,1] and {un}n1W01,p(Ω) such that

tnt,un0in W01,p(Ω),hu(tn,un)=0for all n.(4.6)

Then we have

Ap(un),h+A(un),h=(1-tn)Ω[λ|un|q-2un+f(z,un)]h𝑑z+tnΩgλ(z,un)h𝑑z

for all hW01,p(Ω) and all n, which implies

{-Δpun(z)-Δun(z)=(1-tn)[λ|un(z)|q-2un(z)+f(z,un(z))]+tngλ(z,un(z))for a.a. zΩ,un|Ω=0.(4.7)

Evidently, {un}n1W01,p(Ω) is bounded (see (4.6), (4.7)). Then as before using (4.7) and [15, Theorem 1], we can find α(0,1) and c8>0 such that

unC01,α(Ω¯),unC01,α(Ω¯)c8for all n.(4.8)

From (4.6), (4.8) and the compact embedding of C01,α(Ω¯) into C01(Ω¯), we have

un0in C01(Ω¯),

hence

un[v¯λ,u¯λ]for all nn0,

and therefore {un}nn0Kφ~λ (see (4.3)), a contradiction to our assumption that Kφ~λ is finite. Therefore, (4.6) can not occur, and from homotopy invariance of critical groups (see [11, Theorem 5.125, p. 836]), we have Ck(φλ,0)=Ck(φ~λ,0) for all k0. ∎

From Proposition 4.2 and (4.2), we infer that

Ck(φ~λ,0)=0for all k0.(4.9)

Proposition 4.3.

If Hypothesis 2.6 holds and 0<λλ*, then problem (1.1) admits two nodal solutions

y0,y^C01(Ω¯),y0infC01(Ω¯)[v¯λ,u¯λ].

Proof.

From (4.4) and (4.5) we know that

Kφ~λ[v¯λ,u¯λ]C01(Ω¯),Kφ~λ+={0,u¯λ},Kφ~λ-={0,v¯λ}.(4.10)

Claim: u¯λintC+ and v¯λ-intC+ are local minimizers of φ~λ.

From (4.3) it is clear that φ~λ+ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find u¯λ*W01,p(Ω) such that

φ~λ+(u¯λ*)=inf[φ~λ+(u):uW01,p(Ω)].(4.11)

As before, since q<2<p, we have φ~λ+(u¯λ*)<0=φ~λ+(0), hence u¯λ*0 and u¯λ*Kφ~λ+ (see (4.11)), and thus u¯λ*=u¯λ (see (4.10)). It is clear from (4.3) that φ~λ|C+=φ~λ+|C+. Since u¯λintC+, it follows that u¯λ is a local C01(Ω¯)-minimizer of φ~λ, hence u¯λ is a local W01,p(Ω)-minimizer of φ~λ (see Proposition 2.3). We can argue in a similar manner for v¯λ-intC+, using this time the functional φ~λ-. This proves the claim.

Without loss of generality, we may assume that

φ~λ(v¯λ)φ~λ(u¯λ).(4.12)

Recall that Kφ~λ is finite. So, on account of the claim and (4.12), we can find ρ(0,1) small such that

φ~λ(v¯λ)φ~λ(u¯λ)<inf[φ~λ(u):u-u¯λ=ρ]=m~λ,v¯λ-u¯λ>ρ.(4.13)

We know that φ~λ is coercive (see (4.3)). Therefore, we have that

φ~λ satisfies the C-condition(4.14)

(see [17, Proposition 2.2]). From (4.13), (4.14) we see that we can use Theorem 2.1 (the mountain pass theorem). Therefore, we can find y0W01,p(Ω) such that y0Kφ~λ[v¯λ,u¯λ]C01(Ω¯) (see (4.10)) and m~λφ~λ(y0) (see (4.13)), thus y0{v¯λ,u¯λ} (see (4.13)). Since y0 is a critical point of mountain pass type for φ~λ, we have

C1(φ~λ,y0)0(4.15)

(see [18, Proposition 6.100, p. 176]). From (4.9) and (4.15), we infer that y00, hence y0C01(Ω¯) is a nodal solution of (1.1).

Let a(y)=|y|p-2y+y for all yN. Evidently, aC(N,N) (recall that 2<p) and

diva(Du)=Δpu+Δufor all uW01,p(Ω).

We have

a(y)=|y|p-2[id+yy|y|2]+idfor all yN,

hence

(a(y)ξ,ξ)N|ξ|2for all y,ξN.(4.16)

We know that

v¯λy0u¯λ,with y0v¯λ,y0u¯λ.

Using (4.16) and the tangency principle of Pucci and Serrin [26, p. 35], we have

v¯λ(z)<y0(z)<u¯λ(z)for all zΩ.(4.17)

Let ρ=max{u¯λ,v¯λ} and let ξ^ρ>0 be as postulated by Hypothesis 2.6 (vi). Then (see (4.17) and Hypothesis 2.6 (vi))

-Δpy0-Δy0+ξ^ρ|y0|p-2y0=λ|y0|p-2y0+f(z,y0)+ξ^ρ|y0|p-2y0λu¯λq-1+f(z,u¯λ)+ξ^ρu¯λp-1=-Δpu¯λ-Δu¯λ+ξ^ρu¯λp-1for a.a. zΩ.(4.18)

Let

h1(z)=λ|y0(z)|q-2y0(z)+f(z,y0(z))+ξ^ρ|y0(z)|p-2y0(z),h2(z)=λu¯λ(z)q-1+f(z,u¯λ(z))+ξ^ρu¯λ(z)p-1.

Evidently, h1,h2L(Ω) and

h2(z)-h1(z)λ[uλ(z)q-1-|y0(z)|q-2y0(z)],

which implies h1h2 (see Hypothesis 2.6 (vi), (4.17) and recall u¯λintC+,y0C01(Ω¯)).

Then from (4.18) and Proposition 2.4, we infer that

u¯λ-y0intC+.

Similarly, we show that

y0-v¯λintC+.

Therefore,

y0intC01(Ω¯)[v¯λ,u¯λ].(4.19)

On the other hand, from [12, Proposition 17] (see also [20]), we have a nodal solution of (1.1) such that

y^intC01(Ω¯)[v¯λ,u¯λ].(4.20)

From (4.19) and (4.20) it follows that y0y^. ∎

We can state the following multiplicity result for problem (1.1).

Theorem 4.4.

If Hypothesis 2.6 holds, then there exists λ*>0 such that for all λ(0,λ*], problem (1.1) has at least six nontrivial smooth solutions

u0,u^intC+,u0u^,v0,v^-intC+,v0v^,y0,y^C01(Ω¯)both nodal.

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About the article

Received: 2018-08-04

Accepted: 2018-08-16

Published Online: 2018-10-11

Published in Print: 2019-03-01


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11671111

Funding Source: Natural Science Foundation of Heilongjiang Province

Award identifier / Grant number: LBHQ16082

This work was supported by the NSFC (No. 11671111) and Heilongjiang Province Postdoctoral Startup Foundation (Grant No. LBHQ16082).


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 228–249, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0175.

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