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Advances in Nonlinear Analysis

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


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Constant sign and nodal solutions for parametric (p, 2)-equations

Nikolaos S. Papageorgiou / Andrea Scapellato
  • Corresponding author
  • Università degli Studi di Catania, Dipartimento di Matematica e Informatica, Viale Andrea Doria 6, 95125, Catania, Italy
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Published Online: 2019-06-06 | DOI: https://doi.org/10.1515/anona-2020-0009

1 Introduction

Let Ω ⊆ ℕR be a bounded domain with a C2-boundary ∂ Ω.

We study the following parametric (p, 2)-equation:

Δpu(z)Δu(z)=λ|u(z)|p2u(z)+f(z,u(z))in Ωu|Ω=0,p>2,λ>0.(Pλ)

For 1 < q < ∞, Δq denotes the q-Laplace differential operator defined by

Δqu=div|Du|q2Dufor all uW01,q(Ω).

When q = 2, we have the Laplace differential operator denoted by Δ.

In the right hand side (reaction) of the problem, we have a parametric term xλ |x|p−2x with λ > 0 being a parameter and also a perturbation f(z, x) which is a Caratheodory function (that is, for all x ∈ ℝ, zf(z, x) is measurable and for a.a. zΩ, xf(z, x) is continuous).

We do not impose any sign condition on f(z, ⋅) and we assume that for a.a. zΩ, f(z, ⋅) is (p − 1)-superlinear near ± ∞. However, we do not assume that it satisfies the usual in such cases Ambrosetti-Rabinowitz condition (the AR-condition for short).

Our aim is to prove multiplicity theorems providing sign information for all the solutions produced. To this end, first we look for constant sign solutions and we prove bifurcation-type results describing in a precise way the changes in the sets of positive and negative solutions respectively as the parameter λ moves in the positive semiaxis (0, +∞). We also show that there exist extremal constant sign solutions (that is, a smallest positive solution and a biggest negative solution). Then these extremal constant sign solutions are used to generate nodal (that is, sign changing) solutions. By strengthening the conditions on the perturbation f(z, ⋅) and using also tools from the theory of critical groups (Morse theory), we prove a multiplicity theorem for small values of the parameter λ > 0. So, we show that when the parameter λ > 0 is small, problem (Pλ) has at least seven nontrivial solutions all with sign information: two positive, two negative and three nodal.

We mention that (p, 2)-equations (that is, equations driven by a p-Laplacian and a Laplacian), arise in problems of mathematical physics (see, for example, Benci-D’Avenia-Fortunato-Pisani [1]). We also mention the work of Zhikov [2] who used (p, 2)-equations to describe phenomena in nonlinear elasticity. More precisely, Zhikov introduced models for strongly anisotropic materials in the context of homogenization. For this purpose Zhikov introduces the so-called double phase functional

Jp,q(u)=Ω|Du|p+a(z)|Du|qdz

with 0 ≤ a(z)≤ M for a.a. zΩ, 1 < q < p, uW01,p(Ω). Here the modulating coefficient a(z) dictates the geometry of the composite made of two different materials with hardening exponents p and q respectively.

Recently there have been some existence and multiplicity results for such equations. We mention the works of Aizicovici-Papageorgiou-Staicu [3, 4], Cingolani-Degiovanni [5], Gasiński-Papageorgiou [6, 7], He-Guo-Huang-Lei [8], Papageorgiou-Rădulescu [9, 10], Papageorgiou-Rădulescu-Repovš [11], Sun [12], Sun-Zhang-Su [13]. The multiplicity theorem here is the first one producing seven solutions of nonlinear nonhomogeneous equations.

Our approach combines variational methods based on the critical point theory, together with truncation and comparison techniques and Morse theory (critical groups).

2 Mathematical Background

The variational methods which we will use, involve the direct method of the calculus of variations and the mountain pass theorem, which for the convenience of the reader we recall below.

Suppose that X is a Banach space and X* its topological dual. By 〈⋅, ⋅〉, we denote the duality brackets for the pair (X*, X). Given φC1(X, ℝ), we say that φ(⋅) satisfies the Cerami condition (the C- condition for short), if the following property holds:

Every sequence {un}n≥1X such that

{φ(un)}n1Risbounded,

(1+un)φ(un)0inXasn,

admits a strongly convergent subsequence.

This compactness-type condition on the functional φ(⋅), leads to a deformation theorem from which one derives the minimax theory of the critical values of φ. One of the first and most important results in this theory, is the so-called mountain pass theorem.

Theorem 2.1

If X is a Banach space, φC1(X, ℝ), it satisfies the C-condition, u0, u1X, ∥u1u0X > ρ,

max{φ(u0),φ(u1)}<infφ(u):uu0X=ρ=mρ

and

c=infyΓmax0t1φ(y(t))withΓ={yC([0,1],X):y(0)=u0,y(1)=u1},

then, cmρ and c is a critical value of φ (that is, there exists ûX such that φ(û) = c and φ′(û) = 0).

In what follows for a given φC1(X, ℝ), by Kφ we denote the critical set of φ, that is,

Kφ={uX:φ(u)=0}.

The main spaces in the analysis of problem (Pλ), are the Sobolev spaces W01,p(Ω)andH01(Ω) and the Banach space C01(Ω¯)={uC1(Ω¯):u|Ω=0}.

We have

C01(Ω¯)W01,p(Ω)H01(Ω)(recall thatp>2)

and the space C01(Ω¯) is dense in both W01,p(Ω)andH01(Ω). By ∥⋅∥ we denote the norm of the Sobolev space W01,p(Ω). On account of the Poincaré inequality, we have

u=Dupfor alluW01,p(Ω).

The space C01(Ω¯) 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 and un|Ω<0.

Here un=(Du,n)RN is the normal derivative of u(⋅), with n(⋅) being the outward unit normal on ∂ Ω.

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

|f0(z,x)|a0(z)1+|x|r1for a.a. zΩ, all xR

with a0L(Ω) and

1<rp=NpNpif p<N+if pN(the critical Sobolev exponent corresponding top).

We set F0(z,x)=0xf0(z,s)ds and consider the C1-functional φ0 : W01,p(Ω) → ℝ defined by

φ0(u)=1pDupp+12Du22ΩF0(z,u)dzfor all uW01,p(Ω).

The next result is an outgrowth of the nonlinear regularity theory (see Lieberman [14], Theorem 1). It is a special case of a more general result of Papageorgiou-Rădulescu [15].

Proposition 2.1

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)forallhC01(Ω¯)ρ0,

then u0C01,α(Ω¯)=C1,α(Ω¯)C01(Ω¯) and it is also a local W01,p(Ω)-minimizer of φ0, that is, there exists ρ1 > 0 such that

φ0(u0)φ0(u0+h)forallhρ1.

This result is more effective when it is combined with the following strong comparison principle, which is a special case of a result of Gasiński- Papageorgiou [16] (Proposition 3.2).

If h1, h2L(Ω), then we write that h1h2 if for all KΩ compact, we have 0 < cKh2(z) − h1(z) for a.a. zK.

Proposition 2.2

If ξ, h1, h2L(Ω), ξ(z) ≥ 0 for a.a. zΩ, h1h2, and uC01(Ω¯) ∖ {0}, v ∈ int C+, uv satisfy

Δpu(z)Δu(z)+ξ(z)|u(z)|p2u(z)=h1(z),Δpv(z)Δv(z)+ξ(z)v(z)p1=h2(z)

for a.a. zΩ, then vu ∈ int C+.

For q ∈ (1, +∞), let Aq:W01,q(Ω)W1,q(Ω)=W01,q(Ω)(1q+1q=1) be the nonlinear map defined by

Aq(u),h=Ω|Du|q2(Du,Dh)RNdzfor all u,hW01,q(Ω).

The following proposition recalls the main properties of this map (see, for example, Motreanu-Motreanu-Papageorgiou [17], p. 40).

Proposition 2.3

The map Aq(⋅) is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (hence maximal monotone too) and of type (S)+ (that is, if un w u in W01,p(Ω) and lim supn+A(un),unu0, then unu in W01,p(Ω)).

If q = 2, then A2 = AL(H01(Ω),H1(Ω)).

We will need some basic facts about the spectrum of (Δ,H01(Ω)). So, we consider the following linear eigenvalue problem

Δu(z)=λ^u(z)in Ω,u|Ω=0.(2.1)

We say that λ̂ ∈ ℝ is an eigenvalue of (−Δ, H01(Ω)), if problem (2.1) admits a nontrivial solution ûH01(Ω) known as an eigenfunction corresponding to λ̂. Via the spectral theorem for compact self-adjoint operators, we show that the spectrum consists of a strictly increasing sequence {λ̂k(2)}k∈ℕ of eigenvalues and λ̂k(2) → ∞. The corresponding sequence {ûn(2)}n∈ℕH01(Ω) of eigenfunctions of (2.1), forms an orthonormal basis of H01(Ω) and an orthogonal basis of L2(Ω). Standard regularity theory implies that {ûn(2)}n∈ℕC01(Ω¯). By E(λ̂k(2)) we denote the eigenspace corresponding to the eigenvalue λ̂k(2), k ∈ ℕ. We have E(λ̂k(2)) ⊆ C01(Ω¯) and we have the following orthogonal direct sum decomposition

H01(Ω)=kNE(λ^k(2))¯.

Each eigenspace E(λ̂k(2)) has the so-called Unique Continuation Property (UCP for short) which says that, if uE(λ̂k(2)) vanishes on a set of positive Lebesgue measure, then u ≡ 0.

The eigenvalues {λ̂k(2)}k∈ℕ have the following properties:

  • λ̂1(2) > 0 is simple (that is, dim E(λ̂1(2)) = 1).

  • λ^1(2)=infDu22u22:uH01(Ω),u0(2.2)

  • λ^m(2)=supDu22u22:uk=1mE(λ^k),u0=infDu22u22:ukmE(λ^k),u0(2.3)

In (2.2) the infimum is realized on E(λ̂1(2)).

In (2.3) both the supremum and the infimum are realized on E(λ̂m(2)).

The above properties imply that the elements of E(λ̂1) have constant sign. On the other hand the elements of E(λ̂k(2)), k ≥ 2, are nodal (that is, sign-changing). Moreover, if by û1(2) we denote the L2-normalized (that is, ∥û1(2)∥2 = 1) positive eigenfunction corresponding to λ̂1(2), then the strong maximum principle implies that û1(2) ∈ int C+.

The following useful inequalities are easy consequences of the above properties.

Proposition 2.4

  1. If m ∈ ℕ, ηL(Ω), η(z) ≤ λ̂m(2) for a.a. zΩ, ηλ̂m(2), then

    Du22Ωη(z)u2dzc1Du22

    for some c1 > 0, all ukmE(λ^k(2))¯.

  2. If m ∈ ℕ, ηL(Ω), η(z) ≥ λ̂m(2) for a.a. zΩ, ηλ̂m(2), then

    Du22Ωη(z)u2dzc2Du22

    for some c2 > 0, all uk=1mE(λ^k(2)).

We also consider the corresponding nonlinear eigenvalue problem for the p-Laplacian

Δpu(z)=λ^|u(z)|p2u(z)in Ω,u|Ω=0.

This problem has a smallest eigenvalue λ̂1(p) > 0 which is isolated (that is, there exists ϵ > 0 such that (λ̂1(p), λ̂1(p) + ϵ) contains no eigenvalues), simple (that is, if û, are eigenfunctions corresponding to λ̂1(p) > 0, then û = ξ for some ξ ∈ ℝ ∖ {0}) and admits the following variational characterization

λ^1(p)=infDuppupp:uW01,p(Ω),u0.(2.4)

The infimum in (2.4) is realized on the corresponding one dimensional eigenspace, the elements of which are in C01(Ω¯) (nonlinear regularity theory, see Lieberman [14]) and have fixed sign. Using (2.4) and these properties, we obtain

Proposition 2.5

If ηL(Ω), η(z) ≤ λ̂1(p) for a.a. zΩ, ηλ̂1(p), then there exists c3 > 0 such that

DuppΩη(z)|u|pdzc3DuppforalluW01,p(Ω).

Next we recall some basic definitions and facts concerning critical groups.

So, let X be a Banach space, φC1(X, ℝ), c ∈ ℝ. We introduce the following sets

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

For a topological pair (Y1, Y2) such that Y2Y1X and every k ∈ ℕ0 by Hk(Y1, Y2) we denote the kth-relative singular homology group with integer coefficients. Given uKφc isolated, the critical groups of φ at u, are defined by

Ck(φ,u)=Hk(φcU,φcU{u}),

with 𝓤 being a neighborhood of u such that Kφφc ∩ 𝓤 = {u}. The excision property of singular homology, implies that the above definition is independent of the particular choice of the neighborhood 𝓤.

Suppose that φC1(X, ℝ) satisfies the C-condition and inf φ(Kφ) > −∞. Let c < inf φ(Kφ). Then the critical groups of φ at infinity, are defined by

Ck(φ,)=Hk(X,φc)for all kN0.

This definition is independent of the choice of the level c < inf φ(Kφ). Indeed, if c′ < c < inf φ(Kφ), then by the second deformation theorem (see [18], p. 628), we know that φc′ is a strong deformation retract of φc. Therefore

Hk(X,φc)=Hk(X,φc)for all kN0

(see Motreanu-Motreanu-Papageorgiou [17], p. 145).

Suppose that Kφ is finite. We define the following items:

M(t,u)=kN0rankCk(φ,u)tkfor alltR,alluKφ,P(t,)=kN0rankCk(φ,)tkfor alltR.

The Morse relation says that

uKφM(t,u)=P(t,)+(1+t)Q(t)for all tR,(2.5)

where Q(t)=k0βktk is a formal series in t ∈ ℝ with nonnegative integer coefficients.

Finally, let us fix our notation. For x ∈ ℝ, 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.

By |⋅|N we denote the Lebesgue measure on ℝN and by |⋅| the norm of ℝN as well as the absolute value in ℝ. By (⋅, ⋅)N we denote the inner product in ℝN. Given u, vW01,p(Ω), uv, then the order interval in W01,p(Ω) determined by u and v is defined by

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

By intC01(Ω¯)[u,v] we denote the interior in the C01(Ω¯)-norm topology of [u, v] ∩ C01(Ω¯). By [u) we denote the half-line in W01,p(Ω) defined by

[u)={yW01,p(Ω):u(z)y(z)for a.a. zΩ}.

Finally, by δk,m, k, m ∈ ℕ0, we denote the Kronecker symbol, that is,

δk,m=1if k=m0if km.

3 Constant sign solutions

In this section we produce constant sign solutions and we investigate how the sets of positive and negative solutions of (Pλ) depend on the parameter λ > 0.

The hypotheses on the perturbation f(z, x) are the following:

H(f): f : Ω × ℝ → ℝ is a Caratheodory function such that f(z, 0) = 0 for a.a. zΩ and

  1. |f(z, x)| ≤ a(z)(1 + |x|r−1) for a.a. zΩ, all x ∈ ℝ, with aL(Ω) and

    p<r<p=NpNpif p<N+if pN(the critical Sobolev exponent corresponding top);

  2. If F(z,x)=0xf(z,s)ds, then limx±F(z,x)|x|p=+ uniformly for a.a. zΩ;

  3. there exist η̂ > 0 and q(rp)maxNp,1,p such that

    0<η^lim infx±f(z,x)xpF(z,x)|x|quniformly for a.a. zΩ;

  4. there exist m ∈ ℕ, m ≥ 2, and functions ϑ, ϑ̂L(Ω) such that

    λ^m(2)ϑ(z)ϑ^(z)λ^m+1(2)for a.a. zΩ,

    ϑλ^m(2),ϑ^λ^m+1(2),

    ϑ(z)lim infx0f(z,x)xlim supx0f(z,x)xϑ^(z)uniformly for a.a. zΩ;

  5. for every ρ > 0, there exists ξ̂ρ > 0 such that for a.a. zΩ the function

    xf(z,x)+ξ^ρ|x|p2x

    is nondecreasing on [−ρ, ρ].

Remarks

Hypotheses H(f)(ii), (iii) imply that

limx±f(z,x)|x|p2x=+uniformlyfora.a.zΩ.

So, the perturbation term is (p − 1)-superlinear. However, we do not use the usual in such cases AR-condition. Recall that the AR-condition says that there exist q > p and M > 0 such that

0<qF(z,x)f(z,x)xfora.a.zΩ,all|x|Mand0<essinfΩF(,±M).(3.1)

Integrating, we obtain the following weaker condition

c4|x|qF(z,x)fora.a.zΩ,all|x|M,withc4>0.(3.2)

From (3.1) and (3.2) it follows that for a.a. zΩ, f(z, ⋅) has at least (q − 1)-polynomial growth near ± ∞. So, the AR-condition although very convenient in verifying the C-condition, it is rather restrictive (see the Examples below). For this reason we employ hypothesis H(f)(iii) which is more general. Indeed, suppose that the AR-condition holds. We may assume that q>(rp)maxNp,1. Then

f(z,x)xpF(z,x)|x|q=f(z,x)xqF(z,x)|x|q+(qp)F(z,x)|x|q=(qp)F(z,x)|x|q (see (3.1))=(qp)c4>0 (see (3.2)),

lim infx±f(z,x)xpF(z,x)|x|q(qp)c4>0 uniformly for a.a.zΩ.

So, hypothesis H(f)(iii) is verified. Near zero, for a.a. zΩ, f(z, ⋅) is nonuniformly nonresonant with respect to the spectral interval [λ̂m(2), λ̂m+1(2)].

Examples

The following functions satisfy hypotheses H(f). For the sake of simplicity, we drop the z-dependence:

f1(x)=ϑx+|x|τ2xif|x|1ϑ|x|r2x|x|q2xif|x|>1,

with ϑ ∈ (λ̂m(2), λ̂m+1(2)) for some m ∈ ℕ, m ≥ 2 and 2 < τ < ∞, pq < r,

f2(x)=ϑ(x|x|τ2x)if|x|1ϑ|x|p2xln|x|if|x|>1,

with ϑ ∈ (λ̂m(2), λ̂m+1(2)) for some m ∈ ℕ, m ≥ 2 and τ > 2.

Note that f1 satisfies the AR-condition, while f2 does not.

We introduce the following sets:

L+={λ>0: problem(Pλ)has a positive solution},Sλ+=set of positive solutions of (Pλ).

Similarly, we define,

L={λ>0: problem(Pλ) has a negative solution},Sλ=set of negative solutions of(Pλ).

We start by establishing the nonemptiness of ℒ+ and ℒ and we locate the set Sλ+andSλ.

Proposition 3.1

If hypotheses H(f) hold, then+, ℒ ≠ ∅ and Sλ+intC+,SλintC+.

Proof

We do the proof for the pair (L+,Sλ+), the proof for the pair (L,Sλ) being similar.

So, we consider the C1-functional ψλ+:W01,p(Ω) → ℝ defined by

ψλ+(u)=1pDupp+12Du22λpu+ppΩF(z,u+)dz,for all uW01,p(Ω).

Evidently if τ ∈ (1, 2), hypothesis H(f)(iv) implies that

limx0+f(z,x)xτ1=0uniformly for a.a. zΩ.

So, given ϵ > 0, we can find c5 = c5(ϵ, τ) > 0 such that

F(z,x)ϵ|x|τ+c5|x|rfor a.a.zΩ,allxR.(3.3)

Then we have

ψλ+(u)1pDupp+1pDu+ppλu+ppϵc6uτc7urfor somec6>0,c7>0(see(3.3)).

If λ ∈ (0, λ̂1(p)), then using Proposition 2.5 we obtain

ψλ+(u)c8upϵc6uτ+c7urfor somec8>0=c8ϵc6uτp+c7urpup.(3.4)

We consider the function

ξ(t)=ϵc6tτp+c7trp,t>0.

Evidently ξC1(0, +∞). Moreover, since τ < 2 < p < r, we see that

ξ(t)+ as t0+ and as t+.

So, we can find t0 ∈ (0, +∞) such that

ξ(t0)=infξ(t):t>0,ξ(t0)=0,t0=t0(ϵ)=ϵc6(pτ)c7(rp)pτrp.

Note that ξ(t0) → 0+ as ϵ → 0+. Therefore we can find ϵ0 > 0 such that

ξ(t0)<c8for all ϵ(0,ϵ0),infψλ+(u):u=t0=mλ+>0(see (3.4))(3.5)

Hypothesis H(f)(ii) implies that if u ∈ int C+, then

ψλ+(tu) ast+.(3.6)

Claim. For every λ > 0, the functional ψλ+ satisfies the C-condition.

Let {un}n≥1W01,p(Ω) be a sequence such that

|ψλ+(un)|M1for someM1>0,allnN,(3.7)

(1+un)(ψλ+)(un)0inW1,p(Ω)asn.(3.8)

From (3.8) we have

Ap(un),h+A(un),hλΩ(un+)p1hdzΩf(z,un+)hdzϵnh1+un(3.9)

for all hW01,p(Ω), with ϵn → 0+.

In (3.9) we choose h=unW01,p(Ω). Then

Dunpp+Dun22ϵnfor allnN,un0 inW01,p(Ω)asn.(3.10)

From (3.7) and (3.10), we have

Dun+pp+p2Dun+22Ωλ(un+)p+pF(z,un+)dzM2(3.11)

for some M2 > 0, all n ∈ ℕ.

Also from (3.9) with h=un+W01,p(Ω), we obtain

Dun+ppDun+22+Ωλ(un+)p+f(z,un+)un+dzϵnfor allnN.(3.12)

We add (3.11) and (3.12) and obtain

Ωf(z,un+)un+pF(z,un+)dzM3for all M3>0,allnN,(recallp>2).(3.13)

Hypotheses H(f)(i), H(f)(iii) imply that we can find η̂0 ∈ (0, η̂) and c9 > 0 such that

η^0|x|qc9f(z,x)xpF(z,x)for a.a.zΩ,allxR.

Using this in (3.13), we obtain that

un+n1Lq(Ω) is bounded,(3.14)

First suppose that Np. From hypothesis H(f)(iii) it is clear that we can have q < r < p* (recall that if Np, then p* = +∞). So, we can find t ∈ (0, 1) such that

1r=1tq+tp.

Invoking the interpolation inequality (see, for example, Gasiński-Papageorgiou [18], p. 905), we have

un+run+q1tun+pt,un+rrc10un+trfor somec10>0,allnN,(see (3.4) and recall thatW01,p(Ω)Lp(Ω)).(3.15)

In (3.9) let h=un+W01,p(Ω). Then

Dun+pp+Dun+22Ωλ(un+)p+f(z,un+)un+dzϵn for allnN,

un+pc111+un+rrfor some c11=c11(λ)>0,allnN(see hypothesisH(f)(i)and recall thatr>p)un+pc121+un+trfor some c12>0,allnN,(see(3.15)).(3.16)

Hypothesis H(f)(iii) implies that tr < p. So, from (3.16) it follows that

un+n1W01,p(Ω) is bounded,unn1W01,p(Ω) is bounded (see(3.10)).(3.17)

Now suppose that N = p. In this case p* = +∞ and W01,pLs(Ω) for all s ∈ [1, +∞). Let s > r > q and as before pick t ∈ (0, 1) such that

1r=1tq+ts,tr=s(rq)sq.

We see that

s(rq)sqrq as sp=+.

By hypothesis H(f)(iii) we have

rq<p,tr=s(rq)sq<p fors>rbig.

Therefore in this case too, we conclude that (3.17) holds.

assing to a subsequence if necessary, we have

unwu in W01,p(Ω)andunu in Lr(Ω).(3.18)

In (3.9) we choose h = unuW01,p(Ω), pass to the limit as n → ∞ and use (3.18). Then

limnAp(un),unu+A(un),unu=0,lim supnAp(un),unu+A(u),unu0(sinceA()is monotone)lim supnAp(un),unu0,unuinW01,p(Ω)(see Proposition 2.3)).

Therefore ψλ+ satisfies the C-condition. This proves the Claim.

Then with λ ∈ (0, λ̂1(p)), from (3.5), (3.6) and the Claim, we see that we can apply Theorem 1 (the mountain pass theorem) and find uλW01,p(Ω) such that

uλKψλ+andψλ+(0)=0<mλ+ψλ+(uλ)(see (3.5)).

Therefore uλ ≠ 0 and we have

Ap(uλ),h+A(uλ),h=Ωλ(uλ+)p1+f(z,uλ+)hdzfor all hW01,p(Ω).

Choosing h=uλW01,p(Ω), we obtain

uλ0,uλ0.

From (3.9) we have

Δpuλ(z)Δuλ(z)=λuλ(z)p1+f(z,uλ(z))for a.a.zΩ,uλ|Ω=0.(3.19)

From (3.19) and Corollary 6.8, p. 208, of Motreanu-Motreanu-Papageorgiou [17], we have that uλL(Ω). Then Theorem 1 of Lieberman [14], implies that

uλC+{0}.

Let ρ = ∥uλ and let ξ̂ρ > 0 be as postulated by hypothesis H(f)(v). Then from (3.19) we have

Δpuλ(z)Δuλ(z)+ξ^ρuλ(z)p10for a.a.zΩ,uλintC+ (see Pucci-Serrin [19], pp. 111,120).

Therefore (0, λ̂1(p)) ⊆ 𝓛+ and Sλ+ ⊆ int C+. Similarly we show that 𝓛 ≠ ∅ and that Sλ ⊆ − int C+. □

Next we show that both 𝓛+ and 𝓛 are intervals.

Proposition 3.2

If hypotheses H(f) hold, λ ∈ 𝓛+ (resp. λ ∈ 𝓛) and 0 < ϑ < λ, then ϑ ∈ 𝓛+ (resp. ϑ ∈ 𝓛).

Proof

We do the proof for 𝓛+, the proof for 𝓛 being similar.

Let λ ∈ 𝓛+. We can find uλSλ+ ⊆ int C+. Then we introduce the following truncation of the reaction in problem (Pϑ):

eϑ(z,x)=0ifx<0ϑxp1+f(z,x)if 0xuλ(z)ϑuλ(z)p1+f(z,uλ(z))if uλ(z)<x.(3.20)

This is a Caratheodory function. We set Eϑ(z,x)=0xeϑ(z,s)ds and consider the C1-functional ψ^ϑ+:W01,p(Ω)R defined by

ψ^ϑ+(u)=1pDupp+12Du22ΩEϑ(z,u)dzfor all uW01,p(Ω).

From (3.20) it is clear that ψ^ϑ+() is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can fin uϑW01,p(Ω) such that

ψ^ϑ+(uϑ)=infψ^ϑ+(u):uW01,p(Ω).(3.21)

On account of hypothesis H(f)(iv), we see that given ϵ > 0, we can find δ > 0 such that

F(z,x)12ϑ(z)ϵx2for a.a.zΩ,all|x|δ.(3.22)

Let uE(λ̂m(2)) ⊆ C01(Ω¯) and choose t ∈ (0, 1) small such that

0tu(z)δfor all zΩ¯.(3.23)

Then we have

ψ^ϑ+(tu)tppDupp+t22Du22t22Ωϑ(z)u2dz+ϵ2t2u22(see (3.22), (3.23))=tppDupp+t22Du22Ωϑ(z)u2dz+ϵ2t2u22tppDupp+t22(c13+ϵ)u22for some c13>0(see Proposition 2.4).

Choosing ϵ ∈ (0, c13), we have that

ψ^ϑ+(tu)tppDuppt22c14u22.

Since p > 2, choosing t ∈ (0, 1) even smaller, we have

ψ^ϑ+(tu)<0,ψ^ϑ+(uϑ)<0=ψ^ϑ+(0)(see (3.21)),uϑ0.

From (3.21), we have

ψ^ϑ+(uϑ)=0Ap(uϑ),h+A(uϑ),h=Ωeϑ(z,uϑ)hdzfor all hW01,p(Ω).(3.24)

In (3.24) we choose h=uϑW01,p(Ω). Then

Duϑpp+Duϑ22=0(see (3.20)),uϑ0,uϑ0.

Also, in (3.24) we choose h = (uϑuλ)+W01,p(Ω). Then

Ap(uϑ),(uϑuλ)++A(uϑ),(uϑuλ)+=Ωϑuλp1+f(z,uλ)(uϑuλ)+dz(see (3.20))Ωλuλp1+f(z,uλ)(uϑuλ)+dz(sinceϑ<λ)=Ap(uλ),(uϑuλ)++A(uλ),(uϑuλ)+(sinceuλSλ),uϑuλ.

So, we have proved that

uϑ[0,uλ],uϑ0.(3.25)

From (3.24), (3.25) and (3.20), we conclude that

Δpuϑ(z)Δuϑ(z)=ϑuϑ(z)p1+f(z,uϑ(z))for a.a.zΩ,uϑ|Ω=0,ϑL+ and uϑSϑ+intC+.

Similarly for 𝓛. □

The following Corollary is a useful byproduct of the above proof.

Corollary 3.1

If hypotheses H(f) hold, then

  1. if 0 < ϑ < λ ∈ ℒ+ and uλSλ+, then ϑ ∈ ℒ+ and we can find uϑSϑ+ ⊆ int C+ such that

    uλuϑC+{0};

  2. if 0 < ϑ < λ ∈ ℒ and vλSλ, then ϑ ∈ ℒ and we can find vϑSϑ ⊆ −int C+ such that

    vϑvλC+{0}.

We can improve this corollary.

Proposition 3.3

If hypotheses H(f) hold, then

  1. if 0 < ϑ < λ ∈ ℒ+ and uλSλ+, then ϑ ∈ ℒ+ and we can find uϑSϑ+ ⊆ int C+ such that

    uλuϑintC+;

  2. if 0 < ϑ < λ ∈ ℒ and vλSλ, then ϑ ∈ ℒ and we can find vϑSϑ ⊆ −int C+ such that

    vϑvλintC+.

Proof

  1. From Corollary 3.1, we already know that ϑ ∈ ℒ+ and we can find uϑSϑ+ ⊆ int C+ such that

    uλuϑC+{0}.(3.26)

    Let ρ = ∥uλ and let ξ̂ρ > 0 be as postulated by hypothesis H(f)(v). Then

    ΔpuϑΔuϑ+ξ^ρuϑp1=ϑuϑp1+f(z,uϑ)+ξ^ρuϑp1=λuϑp1+f(z,uϑ)+ξ^ρuϑp1(λϑ)uϑp1λuλp1+f(z,uλ)+ξ^ρuλp1(see (3.26), hypothesisH(f)(v)and recall thatϑ<λ)=ΔpuλΔuλ+ξ^ρuλp1(sinceuλSλ+).(3.27)

    Let

    h1(z)=ϑuϑp1+f(z,uϑ)+ξ^ρuϑp1,h2(z)=λuλp1+f(z,uλ)+ξ^ρuλp1.

    Evidently h1, h2L(Ω) and we have

    h2(z)h1(z)(λϑ)uϑ(z)p1for a.a zΩ.

    Since uϑ ∈ int C+ we see that h1h2. Invoking Proposition 2.2, from (3.27) we conclude that uλuϑ ∈ int C+.

  2. The proof is similar, using this time part (b) of Corollary 3.1. □

We set λ+ = sup ℒ+ and λ = sup ℒ.

Proposition 3.4

If hypotheses H(f) hold, then λ+<+andλ<+.

Proof

We do the proof for λ+, the proof for λ being similar. On account of hypotheses H(f)(i), (ii), (iii), we can find λ͠ > 0 big such that

λ~xp1+f(z,x)0for a.a.zΩ,allx0.(3.28)

Let λ > λ͠ and suppose that λ ∈ ℒ+. We can find uλSλ ⊆ int C+. So, we have

uλn|Ω<0.

Therefore we can find δ > 0 such that, if ∂ Ωδ = {zΩ : d(z, ∂ Ω) = δ}, then

uλn|Ωδ<0.(3.29)

Consider the open set Ωδ = {zΩ : d(z, ∂ Ω) > δ} and set mδ=minΩδ¯uλ>0 (recall that uλ ∈ int C+). For ϵ > 0, we set mδϵ = mδ + ϵ and for ρ = ∥uλ let ξ̂ρ > 0 be as postulated by hypothesis H(f)(v). We have

ΔpmδϵΔmδϵ+ξ^ρ(mδϵ)p1ξ^ρmδp1+μ(ϵ)withμ(ϵ)0+asϵ0+λ~mδp1+f(z,mδ)+ξ^ρmδp1+μ(ϵ)(see (3.28))=λmδp1+f(z,mδ)+ξ^ρmδp1(λλ~)mδp1+μ(ϵ)(see (3.28))λmδp1+f(z,mδ)+ξ^ρmδp1forϵ>0smallλuλp1+f(z,uλ)+ξ^ρuλp1(recall thatmδuλonΩδ¯)=ΔpuλΔuλ+ξ^ρuλp1for a.a.zΩδ.(3.30)

Then from (3.29), (3.30) and Proposition 2.10 of Papageorgiou-Rădulescu-Repovš [20], we have

uλmδϵintC+(Ωδ¯)forϵ>0small,

which contradicts the definition of mδ. Therefore λ ∉ ℒ+ and so

λ+λ~<+.

Similarly we show that λ < +∞. □

Hypotheses H(f)(i), (iv), imply that given ϵ > 0, we can find c15 > 0 such that

λ|x|p+f(z,x)x[ϑ(z)ϵ]x2c15|x|rfor a.a.zΩ,allxR,allλ>0.(3.31)

This unilateral growth restriction on the reaction of (Pλ), leads to the following auxiliary (p, 2)-equation:

Δpu(z)Δu(z)=[ϑ(z)ϵ]u(z)c15|u(z)|r2u(z)in Ωu|Ω=0(3.32)

Proposition 3.5

For all ϵ > 0 small, problem (3.32) has a unique positive solution uλ ∈ int C+ and, since (3.32) is odd, vλ=uλ ∈ −int C+ is the unique solution of (3.32).

Proof

Consider the C1-functional σ : W01,p(Ω) → ℝ defined by

σ(u)=1pDupp+12Du22+c15ru+rr12Ω[ϑ(z)ϵ](u+)2dzfor all uW01,p(Ω).

Evidently σ(⋅) is coercive (recall that p > 2). Also, it is sequentially weakly lower semicontinuous. So, we can find uλW01,p(Ω) such that

σ(uλ)=infσ(u):uW01,p(Ω).(3.33)

As in the proof of Proposition 3.2, for ϵ > 0 small we have

σ(uλ)<0=σ(0),uλ0.

From (3.33) we have

σ(uλ)=0,

Ap(uλ),h+A(uλ),h=Ω[ϑ(z)ϵ](uλ)+hdzλΩ((uλ)+)r1hdzfor all hW01,p(Ω).(3.34)

In (3.34) we choose h=(uλ)W01,p(Ω). Then

D(uλ)pp+D(uλ)22=0,uλ0,uλ0.

So, from (3.34) we have that uλ is a positive solution of (3.32) and the nonlinear regularity theory (see [14]) implies that uλC+ ∖ {0}. We have

Δpuλ+Δuλc15uλrp(uλ)p1for a.a.zΩ,uλintC+(see Pucci-Serrin [19], pp. 111, 120).

Next we show the uniqueness of this positive solution. To this end we consider the integral functional j : L1(Ω) → ℝ = ℝ ∪ {+∞} defined by

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

Let dom j = {uL1(Ω) : j(u) < +∞} (the effective domain of j(⋅)).

From Lemma 1 of Diaz-Saá [21], we have that

j() is convex.

Suppose that uλ,u~λ are two positive solutions of (3.32). We have

uλ,u~λintC+

Then, for hC01(Ω¯) and for |t| < 1 small, we have

(uλ)2+thdomjand(u~λ)2+thdomj.

It is easy to see that j(⋅) is Gateaux differentiable at (uλ)2 and at (u~λ)2 in the direction h. Moreover, using the chain rule and the nonlinear Green’s identity (see Gasiński-Papageorgiou [18], p. 211), we have

j(uλ)2(h)=12ΩΔpuλΔuλuλhdzj(u~λ)2(h)=12ΩΔpu~λΔu~λu~λhdz

for all hC01(Ω¯).

The convexity of j(⋅) implies the monotonicity of j′(⋅). Therefore

0ΩΔpuλΔuλuλΔpu~λΔu~λu~λ(uλu~λ)dz=Ωc15(u~λ)r2(uλ)r2(uλu~λ)dz0,

uλ=u~λ.

This proves the uniqueness of the positive solution of problem (3.32).

Since problem (3.32) is odd, it follows that

vλ=uλintC+,

is the unique negative solution of (3.32). □

These solutions provide bounds of the elements of Sλ+and ofSλ.

Proposition 3.6

If hypotheses H(f) hold, then

  1. uλu for all uSλ+, λ ∈ ℒ+;

  2. vvλ for all vSλ, λ ∈ ℒ.

Proof

  1. Let λ ∈ ℒ+ and uSλ+ ⊆ int C+. With ϵ > 0 small as dictated by Proposition 3.5, we introduce the following Caratheodory function:

    k+(z,x)=0ifx<0[ϑ(z)ϵ]xc15xr1if0xu(x)[ϑ(z)ϵ]u(z)c15u(z)r1ifu(z)<x.(3.35)

    We set K+(z, x) = 0x k+(z, s) ds and consider the C1-functional τ+ : W01,p(Ω) → ℝ defined by

    τ+(u)=1pDupp+12Du22ΩK+(z,u)dzfor alluW01,p(Ω).

    Evidently τ+(⋅) is coercive (see (3.35)) and sequentially weakly lower semicontinuous. So, we can find u^λW01,p(Ω) such that

    τ+(u^λ)=infτ+(u):uW01,p(Ω).(3.36)

    As before we have

    τ+(u^λ)<0=τ+(0)u^λ0.

    From (3.36) we have

    τ+(u^λ)=0,Ap(u^λ),h+A(u^λ),h=Ωk+(z,u^λ)hdzfor allhW01,p(Ω).(3.37)

    In (3.37) first we choose h=(u^λ)W01,p(Ω). Then

    D(u^λ)pp+D(u^λ)22=0(see (3.35)),u^λ0,u^λ0.

    Next in (3.37) we choose (u^λu)+W01,p(Ω). Then

    Ap(u^λ),(u^λu)++A(u^λ),(u^λu)+=Ω(ϑ(z)ϵ)uc15ur1(u^λu)+dz(see (3.35))Ωλup1+f(z,u)(u^λu)+dz(see (3.31))=Ap(u),(u^λu)++A(u),(u^λu)+(sinceuSλ+),

    u^λu.

    So, we have proved that

    u^λ[0,u],u^λ0.(3.38)

    From (3.37) and (3.38) it follows that u^λ is a positive solution of problem (3.32). Hence Proposition 3.5 implies that

    u^λ=uλintC+,uλu for alluSλ+(see(3.38)).

  2. Let λ ∈ ℒ and vSλ. We introduce the Caratheodory function k(z, x) defined by

    k(z,x)=[ϑ(z)ϵ]v(z)c15|v(z)|r2v(z)ifx<v(z)[ϑ(z)ϵ]xc15|x|r2xifv(z)x00if0<x.(3.39)

    We set K(z, x) = 0x k(z, s) ds and consider the C1-functional τ : W01,p(Ω) → ℝ defined by

    τ(u)=1pDupp+12Du22ΩK(z,u)dzfor alluW01,p(Ω).

    Working as in part (a), using this time the functional τ(⋅) and (3.39) we show that

    vvλfor allvSλ.

Using these bounds, we can produce extremal constant sign solutions, that is, a smallest positive solution and a biggest negative solution.

Proposition 3.7

If hypotheses H(f) hold, then

  1. for every λ ∈ ℒ+ problem (Pλ) has a smallest positive solution uλSλ+ ⊆ int C+, that is,

    u¯λuforalluSλ+;

  2. for every λ ∈ ℒ problem (Pλ) has a biggest negative solution vλSλ ⊆ –int C+, that is,

    vv¯λforallvSλ.

Proof

  1. From Filippakis-Papageorgiou [22], we know that Sλ+ is downward directed (that is, if u1, u2Sλ+, then we can find uSλ+ such that uu1, uu2). Hence using Lemma 3.10, p. 178, of Hu-Papageorgiou [23], we can find {un}n≥1Sλ+ decreasing such that

    infSλ+=infn1un.

    We have

    Ap(un),h+A(un),h=Ωλunp1+f(z,un)hdzfor allhW01,p(Ω),allnN,(3.40)

    0unu1for allnN.(3.41)

    In (3.40) we choose h = unW01,p(Ω). Then on account of (3.41) and hypothesis H(f)(i), we obtain

    Dunpp+Dun22c16for somec16>0,allnN,{un}n1W01,p(Ω) is bounded.

    So, by passing to a subsequence if necessary, we have

    unwu¯λ in W01,p(Ω)andunu¯λ in Lp(Ω).(3.42)

    If in (3.40) we choose h = unuλW01,p(Ω), pass to the limit as n → ∞, use (3.42) and reason as in the proof of Proposition 3.1 (see the Claim), we obtain

    unu¯λ in W01,p(Ω).(3.43)

    So, if in (3.40) we pass to the limit as n → ∞ and use (3.43), then

    Ap(u¯λ),h+A(u¯λ),h=Ωλu¯λp1+f(z,u¯λ)hdzfor allhW01,p(Ω).(3.44)

    From Proposition 3.6, we know that

    uλunfor allnN,uλu¯λ(see(3.43)).(3.45)

    From (3.44) and (3.45) we conclude that

    u¯λSλ+intC+andu¯λ=infSλ+.

  2. From Filippakis-Papageorgiou [22], we know that Sλ is upward directed (that is, if v1, v2Sλ, then we can find vSλ such that v1v, v2v). So, in this case we can find {vn}n≥1Sλ increasing such that

    supSλ=supn1vn.

    Reasoning as in part (a), we obtain

    v¯λSλintC+andv¯λ=supSλ.

We examine the maps λuλ from ℒ+ into C+C01(Ω) and of λvλ from ℒ into –C+C01(Ω).

Proposition 3.8

If hypotheses H(f) hold, then

  1. the map λuλ from+ into C+ is

    • strictly increasing (that is, if 0 < ϑ < λ ∈ ℒ+, then uλuϑ ∈ int C+);

    • left continuous;

  2. the map λvλ from intoC+ is

    • strictly decreasing (that is, if 0 < ϑ < λ ∈ ℒ, then uϑuλ ∈ int C+);

    • left continuous.

Proof

  1. From Proposition 3.3(a) we know that we can find uϑSϑ+ ⊆ int C+ such that

    u¯λuϑintC+,u¯λu¯ϑintC+.

    Also let {λn}n≥1 ⊆ ℒ+ such that λn(λ+). We set un = uλnSλn+ ⊆ int C+ for all n ∈ ℕ. Then

    Ap(u¯n),h+A(u¯n),h=Ωλn(u¯n)p1+f(z,u¯n)hdz for allhW01,p(Ω),allnN,(3.46)

    0u¯nu¯λ+for allnN(from the monotonicity ofλu¯λ).(3.47)

    Then (3.46) and (3.47) imply that

    u¯nn1W01,p(Ω)is bounded.(3.48)

    From (3.48) and Corollary 8.6, p. 208, of Motreanu-Motreanu-Papageorgiou [17], we know that we can find c17 > 0 such that

    u¯nc17for allnN.(3.49)

    Using (3.49) and Theorem 1 of Lieberman [14], we can find α ∈ (0, 1) and c18 > 0 such that

    u¯nC01,α(Ω¯)andu¯nC01,α(Ω¯)c18for allnN.

    The compact embedding of C01,α(Ω) into C01(Ω), implies that at least for a subsequence we have

    u¯nu~λ+in C01(Ω¯),u~λ+Sλ++.(3.50)

    We claim that u~λ+=u¯λ+. Arguing by contradiction, suppose that u~λ+u¯λ+. So, we can find z0Ω such that

    u¯λ+(z0)<u~λ+(z0)u¯λ+(z0)<u¯n(z0)=u¯λn(z0)for allnn0,

    which contradicts the strict monotonicity of λuλ. Hence u~λ+=u¯λ+ and for the original sequence we have

    u¯nu¯λ+in C01(Ω¯)as n,λu¯λis left continuous.

  2. In this case Proposition 3.5(b) implies that λvλ is strictly decreasing from ℒ into C01(Ω). Also, reasoning as in part (a) and using the maximality of vλ, we establish the left continuity of λvλ from ℒ into –C+.□

So far we know that

(0,λ+)L+(0,λ+],(0,λ)L(0,λ].

It is natural to ask whether the critical parameter values λ+ and λ are admissible. In the next proposition we show that λ+,λ are not admissible and so

L+=(0,λ+)andL=(0,λ).

Proposition 3.9

If hypotheses H(f) hold, then λ+ ∉ ℒ+ and λ ∉ ℒ.

Proof

We do the proof for λ+, the proof for λ being similar.

We argue indirectly. So, suppose that λ+ ∈ ℒ+. From Proposition 3.7, we know that problem (Pλ+) admits a minimal positive solution u* = u¯λ+ ∈ int C+. Let ϑ < λ+ < λ. We know that u*uϑ ∈ int C+. So, we can define the following Caratheodory function:

β^λ(z,x)=λu¯ϑ(z)p1+f(z,u¯ϑ(z))ifx<u¯ϑ(z)λxp1+f(z,x)ifu¯ϑ(z)xu¯(z)λu¯(z)p1+f(z,u¯(z))ifu¯(z)<x.

Let λ(z, x) = 0x β̂λ(z, s) ds and consider the C1-functional ŷλ : W01,p(Ω) → ℝ defined by

y^λ(u)=1pDupp+12Du22ΩB^λ(z,u)dz,for alluW01,p(Ω).

Evidently ŷλ(⋅) is coercive and sequentially lower semicontinuous. So, we can find ûλW01,p(Ω) such that

y^λ(u^λ)=infy^λ(u):uW01,p(Ω),y^λ(u^λ)=0,Ap(u^λ),h+A(u^λ),h=Ωβ^λ(z,u^λ)hdzfor allhW01,p(Ω).

First we choose h = (uϑûλ)+W01,p(Ω). Then

Ap(u^λ),(u¯ϑu^λ)++A(u^λ),(u¯ϑu^λ)+=Ωλu¯ϑp1+f(z,u¯ϑ)(u¯ϑu^λ)+dzΩϑu¯ϑp1+f(z,u¯ϑ)(u¯ϑu^λ)+dz(sinceϑ<λ)=Ap(u¯ϑ),(u¯ϑu^λ)++A(u¯ϑ),(u¯ϑu^λ)+(sinceu¯ϑSϑ+),

u¯ϑu^λ.

Similarly, choosing h = (ûλu*)+W01,p(Ω), we obtain

u^λu¯.

So, we have proved that

u^λ[u¯ϑ,u¯],λL+, a contradiction sinceλ>λ+.

This means that λ+ ∉ ℒ+.

Similarly we show that λ ∉ ℒ.□

Remark

It is worth pointing out that when we have a concave-convex problem (that is, when the parametric term in the reaction, is λu(z)q–1 with 1 < q < 2 < p), then λ+ ∈ ℒ+ and λ ∈ ℒ– (see Papageorgiou-Rădulescu [24]).

So, we have

L+=(0,λ+)andL=(0,λ).

Now we show that for all λ ∈ ℒ+ (resp. all λ ∈ ℒ), we have at least two positive (resp. two negative) solutions.

Proposition 3.10

If hypotheses H(f) hold, then

  1. for all λ ∈ ℒ+ = (0, λ+) problem (Pλ) has at least two positive solutions uλ, ûλ ∈ int C+, uλûλ, uλûλ;

  2. for all λ ∈ ℒ = (0, λ) problem (Pλ) has at least two negative solutions vλ, λ ∈ int C+, λvλ, vλλ.

Proof

  1. Since λ ∈ ℒ+, we can find uλSλ+ ⊆ int C+. Using uλ ∈ int C+ to truncate the reaction of problem (Pλ), we introduce the Caratheodory function gλ+(z, x) defined by

    gλ+(z,x)=λuλ(z)p1+f(z,uλ(z))ifxuλ(z)λxp1+f(z,x)ifuλ(z)<x.(3.51)

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

    φ^λ+(u)=1pDupp+12Du22ΩGλ+(z,u)dzfor alluW01,p(Ω).

    Let η ∈ (λ, λ+) and uηSη ⊆ int C+ such that uηuλ ∈ int C+. Consider the Caratheodory function

    g~λ+(z,x)=gλ+(z,x)ifxuη(z)gλ+(z,uη(z))ifuη(z)<x.(3.52)

    We set G~λ+(z,x)=0xg~λ+(z,s) ds and consider the C1-functional φ~λ+:W01,p(Ω)R defined by

    φ~λ+(u)=1pDupp+12Du22ΩG~λ+(z,u)dzfor alluW01,p(Ω).

    As before we can check that

    Kφ^λ+[uλ)intC+andKφ~λ+[uλ,uη]intC+.(3.53)

    Moreover, since φ~λ+ is coercive and sequentially weakly lower semicontinuous, we can find λW01,p(Ω) such that

    φ~λ+(u~λ)=infφ~λ+(u):uW01,p(Ω),u~λKφ~λ+[uλ,uη]intC+(see (3.53)).(3.54)

    We may assume that λ = uλ or otherwise we already have a second positive solution of (Pλ) (see (3.51), (3.52)). Note that

    φ~λ+|[0,uη]=φ^λ+|[0,uη](see (3.51), (3.52)).(3.55)

    Since uηuλ ∈ int C+ and uλ ∈ int C+, from (3.55) we infer that

    uλis a localC01(Ω¯)minimizer ofφ^λ+,uλis a localW01,p(Ω)minimizer ofφ^λ+(see Proposition2.1).(3.56)

    On account of (3.53) we may assume that

    Kφ^λ+ is finite.(3.57)

    Otherwise we already have an infinity of positive solutions of problem (Pλ), all bigger than uλ and so we are done. Therefore (3.56) and (3.57) imply that there exists ρ ∈ (0, 1) small such that

    φ^λ+(uλ)<infφ^λ+(u):uuλ=ρ=mλ+(3.58)

    (see Aizicovici-Papageorgiou-Staicu [25], proof of Proposition 29).

    Hypothesis H(f)(ii) implies that if u ∈ int C+, then

    φ^λ+(tu)ast+.(3.59)

    Finally as in the proof of Proposition 3.1 (see the Claim), we show that

    φ^λ+()satisfies the C-condition.(3.60)

    Then (3.58), (3.59), (3.60) permit the use of Theorem 1 (the mountain pass theorem). So, we can find ûλW01,p(Ω) such that

    u^λKφ^λ+[uλ)intC+ (see (3.53))andmλ+φ^λ+(u^λ).(3.61)

    From (3.58) and (3.61) we conclude that

    u^λintC+is a solution of(Pλ),uλu^λ,uλu^λ.

  2. In this case, let vλSλ ⊆ –int C+ and consider the Caratheodory function gλ(z, x) defined by

    gλ(z,x)=λ|x|p2x+f(z,x)ifxvλ(z)λ|vλ(z)|p2vλ(z)+f(z,vλ(z))ifx>vλ(z).(3.62)

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

    φ^λ(u)=1pDupp+12Du22ΩGλ(z,u)dzfor alluW01,p(Ω).

    Working as in part (a) this time using (3.62) and the functional φ^λ, we produce a second positive solution λ ∈ –int C+ such that λvλ, vλλ.□

So, summarizing the situation concerning the solutions of constant sign for problem (Pλ), we can state the following theorem.

Theorem 3.1

If hypotheses H(f) hold, then

  1. there exists λ+ ∈ (0, +∞) such that

    • for all λ > λ+ problem (Pλ) has no positive solutions;

    • for all λ ∈ (0, λ+) problem (Pλ) has at least two positive solutions uλ, ûλ ∈ int C+, uλûλ, uλûλ;

    • for all λ ∈ (0, λ+) problem (Pλ) has a smallest positive solution uλ ∈ int C+ and the map λuλ from+ = (0, λ+) into C+ is strictly increasing and left continuous;

  2. there exists λ ∈ (0, +∞) such that

    • for all λ > λ problem (Pλ) has no negative solutions;

    • for all λ ∈ (0, λ) problem (Pλ) has at least two negative solutions vλ, ∈ –int C+, λvλ, vλλ;

    • for all λ ∈ (0, λ) problem (Pλ) has a biggest negative solution vλ ∈ –int C+ and the map λvλ from = (0, λ) intoC+ is strictly decreasing and left continuous.

4 Nodal solutions

In this section we look for nodal (that is, sign changing) solutions for problem (Pλ).

To this end, we need to strengthen the conditions on the perturbation f(z, ⋅). The new hypotheses on f(z, x) are the following:

H(f)′ : f : Ω × ℝ → ℝ is a Caratheodory function such that f(z, 0) = 0 for a.a. zΩ, f(z, ⋅) ∈ C1(ℝ) and

  1. |fx(z, x)|≤ a(z)(1 + |x|r–2) for a.a. zΩ, all x ∈ ℝ, with aL(Ω), p < r < p*.

  2. If F(z, x) = 0x f(z, s) ds, then limx±F(z,x)|x|p=+ uniformly for a.a. zΩ;

  3. there exist η̂ > 0 and q(rp)maxNp,1,p such that

    0<η^lim infx±f(z,x)xpF(z,x)|x|quniformly for a.a. zΩ;

  4. there exist m ∈ ℕ, m ≥ 2, such that

    λ^m(2)fx(z,0)=limx0f(z,x)xλ^m+1(2)uniformly for a.a. zΩ,fx(,0)λ^m(2),fx(,0)λ^m+1(2).

Remark

Note that in this case hypothesis H(f)(v) is automatically satisfied.

Let λ=min{λ+,λ}>0. Also, for λ > 0, let φλW01,p(Ω) → ℝ be the energy (Euler) functional for problem (Pλ) defined by

φλ(u)=1pDupp+12Du22λpuppΩF(z,u)dzfor alluW01,p(Ω).

We know that φλC2(W01,p(Ω), ℝ) for all λ > 0.

Lemma 4.1

If hypotheses H(f) hold and λ > 0, then Ck(φλ, 0) = δk,dmfor all k ∈ ℕ0 with dm=k=1mE(λ^k(2)).

Proof

Let ζ̂λ : H01(Ω) → ℝ be the C2-functional defined by

ζ^λ(u)=12Du22λpuppΩF(z,u)dzfor alluH01(Ω)

We consider the following orthogonal direct sum decomposition of the space H01(Ω):

H01(Ω)=H¯mH^m+1,(4.1)

with

H¯m=k=1mE(λ^k(2))andH^m+1=km+1E(λ^k(2))¯.

Hypothesis H(f)(iv) implies that given ϵ > 0, we can find δ > 0 such that

12[ϑ(z)ϵ]x2F(z,x)12[ϑ^(z)+ϵ]x2for a.a.zΩ,all|x|δ.(4.2)

The subspace Hm is finite dimensional. So, all norms on Hm are equivalent. Therefore, we can find ρ1 ∈ (0, 1) small such that

uH¯m,uH01(Ω)ρ1|u(z)|δ for allzΩ¯(see(4.2)).(4.3)

Therefore for uHm with uH01(Ω)ρ1, we have

ζ^λ(u)12Du2212Ωϑ(z)u2dz+ϵ2uH01(Ω)2(see (4.3))12[c2+ϵ]uH01(Ω)2(see Proposition 2.4(b)).

Choosing ϵ ∈ (0, c2), we obtain

ζ^λ(u)0for alluH¯mwithuH01(Ω)ρ1.(4.4)

On the other hand from (4.2) and hypothesis H(f)(i), we have

F(z,x)12[ϑ^(z)+ϵ]x2+c19|x|rfor a.a.zΩ,allxR(4.5)

with c19 > 0. For uĤm+1 we have

ζ^λ(u)12Du22λpupp12Ωϑ^(z)u2dzϵ2uH01(Ω)2c19urr(see(4.5))12[c1ϵ]uH01(Ω)2c20λuH01(Ω)p+uH01(Ω)rfor somec20>0.

Choosing ϵ ∈ (0, c1) and assuming that uH01(Ω)1, we have

ζ^λ(u)c21uH01(Ω)2c22uH01(Ω)pfor alluH01(Ω)and withc21>0,c22=c22(λ)>0.

Since p > 2, we can find ρ2 ∈ (0, 1) small such that

ζ^λ(u)>0for alluH^m+1,0<uH01(Ω)ρ2.(4.6)

Let ρ = min{ρ1, ρ2} > 0. From (4.4) and (4.6) it follows that ζ̂λ(⋅) has a local linking at the origin with respect to the decomposition (4.1). Since ζ̂λC2(H01(Ω), ℝ), we can apply Proposition 2.3 of Su [26] and infer that

Ck(ζ^λ,0)=δk,dmZfor allkN0.(4.7)

Let ζλ=ζ^λ|W01,p(Ω). Since W01,p(Ω) is dense in H01(Ω), from (4.7) we have

Ck(ζλ,0)=Ck(ζ^λ,0)for allkN0(see[10]),Ck(ζλ,0)=δk,dmZfor allkN0(see(4.7)).(4.8)

Note that

|φλ(u)ζλ(u)|=1pup(4.9)

and

|φλ(u)ζλ(u),h|=|Ap(u),h|Dupp1hφλ(u)ζλ(u)up1.(4.10)

From (4.9), (4.10) and the C1-continuity of critical groups (see Gasiński-Papageorgiou [27], Theorem 5.126, p. 836), we have

Ck(ζλ,0)=Ck(φλ,0)for allkN0,Ck(φλ,0)=δk,dmZfor allkN0(see(4.8)).

We can use this lemma to produce multiple nodal solutions.

Proposition 4.1

If hypotheses H(f)′ hold and λ ∈ (0, λ*), then problem (Pλ) admits at least three nodal solutions

y0,y^,y~C01(Ω¯).

Proof

According to Proposition 3.7, we have two extremal constant sign solutions

u¯λintC+andv¯λintC+.

We consider the Caratheodory function wλ(z, x) defined by

wλ(z,x)=λ|v¯λ(z)|p2v¯λ(z)+f(z,v¯λ(z))ifx<v¯λ(z)λ|x|p2x+f(z,x)ifv¯λ(z)xu¯λ(z)λu¯λ(z)p1+f(z,u¯λ(z))ifu¯λ(z)<x.(4.11)

We set Wλ(z, x) = 0x wλ(z, s) ds and consider the C1-functional τ̂λ : W01,p(Ω) → ℝ defined by

τ^λ(u)=1pDupp+12Du22ΩWλ(z,u)dzfor alluW01,p(Ω).

Also, let τ^λ± be the positive and negative truncations of τ̂λ, that is,

τ^λ±(u)=1pDupp+12Du22ΩWλ(z,±u±)dzfor alluW01,p(Ω).

As before, using (4.11) we can show that

Kτ^λ[v¯λ,u¯λ]C01(Ω¯),Kτ^λ+[0,u¯λ]C+,Kτ^λ[v¯λ,0](C+).

The extremality of uλ and vλ implies that

Kτ^λ[v¯λ,u¯λ]C01(Ω¯),Kτ^λ+={0,u¯λ},Kτ^λ{v¯λ,0}.(4.12)

On account of (4.12) we see that we may assume that

Kτ^λ is finite.(4.13)

Otherwise from (4.11) and the extremality of uλ and vλ, we see that we already have an infinity of smooth nodal solutions.

Claim. uλ ∈ int C+ and vλ ∈ –int C+ are local minimizers of τ̂λ.

Evidently τ^λ+ is coercive (see (4.11)) and sequentially weakly lower semicontinuous. So, we can find λW01,p(Ω) such that

τ^λ+(u~λ)=infτ^λ+:uW01,p(Ω).(4.14)

As in the proof of Proposition 3.2, exploiting hypothesis H(f)(iv) we see that

τ^λ+(u~λ)<0=τ^λ+(0),u~λ0.(4.15)

From (4.14) we have

u~λKτ^λ+={0,u¯λ}(see (4.12))u~λ=u¯λintC+(see (4.15))

Note that

τ^λ+|C+=τ^λ|C+.

So, it follows that

u¯λintC+is a localC01(Ω¯)minimizer ofτ^λ,u¯λintC+is a localW01,p(Ω)minimizer ofτ^λ(see Proposition2.1).

Similarly for vλ ∈ –int C+, using this time the functional τ^λ.

This proves the Claim.

Without any loss of generality, we assume that

τ^λ(v¯λ)τ^λ(u¯λ).

The reasoning is similar if the opposite inequality holds. From (4.13) and the Claim it follows that there exists ρ ∈ (0, 1) small such that

τ^λ(v¯λ)τ^λ(u¯λ)<infτ^λ(u):uu¯λ=ρ=m^λ,v¯λu¯λ>ρ.(4.16)

The functional τ̂λ is coercive, hence

τ^λ satisfies the C-condition.(4.17)

Then (4.16) and (4.17) permit the use of Theorem 2.1 (the mountain pass theorem). So, there exists y0W01,p(Ω) such that

y0Kτ^λ[v¯λ,u¯λ]C01(Ω¯)(see (4.12)),m^λτ^λ(y0)(see (4.16)).(4.18)

From (4.16) and (4.18) we see that

y0{u¯λ,v¯λ}.(4.19)

We consider the homotopy

h^(t,u)=(1t)τ^λ(u)+tφλ(u)for all(t,u)[0,1]×W01,p(Ω).

Suppose we could find {tn}n≥1 ⊆ [0, 1] and {un}n≥1W01,p(Ω) such that

tntin[0,1],un0inW01,p(Ω),h^u(tn,un)=0 for allnN.(4.20)

From the equality in (4.20), we have

Ap(un),h+A(un),h=(1tn)Ωwλ(t,un)hdz+tnΩλ|un|p2unhdz+tnΩf(z,un)hdzfor allhW01,p(Ω),allnN.(4.21)

In (4.21) we choose h = unW01,p(Ω) and we infer that

{un}n1W01,p(Ω) is bounded.

Invoking Corollary 6.8, p. 208, of Motreanu-Motreanu-Papageorgiou [17], we see that we can find α ∈ (0, 1) and c23 > 0 such that

unC01,α(Ω¯)andunC01,α(Ω¯)c23for allnN.(4.22)

From (4.20) and the compact embedding of C01,α(Ω) into C01(Ω), we have

un0inC01(Ω¯),un[v¯λ,u¯λ]for allnn0,{un}nn0Kτ^λ(see (4.11)).

This contradicts (4.13). Therefore (4.20) can not occur and so from the homotopy invariance of critical groups (see Gasiński-Papageorgiou [27], Theorem 5.125, p. 836), we have that

Ck(τ^λ,0)=Ck(φλ,0)for allkN0,Ck(τ^λ,0)=δk,dmZfor allkN0.(4.23)

Recall that y0 is a critical point of τ̂λ of mountain pass type. Therefore

C1(τ^λ,y0)0(4.24)

(see Motreanu-Motreanu-Papageorgiou [17], Proposition 6.100, p. 176).

Comparing (4.23) and (4.24), we infer that

y0{0,u¯λ,v¯λ}(see (3.61)).

Then (4.18), (4.11) and the extremality of uλ and vλ, imply that y0C01(Ω) is a nodal solution of (Pλ).

Let a : ℝN → ℝN be defined by

a(y)=|y|p2y+yfor allyRN.

Note that aC1(ℝN, ℝN) (recall that p > 2) and

diva(Du)=Δpu+Δufor alluW01,p(Ω).

We have

a(y)=|y|p2I+yy|y|2+Ifor allyRna(y)ξ,ξRN|ξ|2for ally,ξRN.

So, applying the tangency principle of Pucci-Serrin [19] (Theorem 2.5.2, p. 35), we obtain

v¯λ(z)<y0(z)<u¯λ(z)for allzΩ.(4.25)

Let ρ = max{∥uλ, ∥vλ}. The differentiability of f(z, ⋅) and hypothesis H(f)′(i) imply that we can find ξ̂ρ > 0 such that for a.a. zΩ, the function

xf(z,x)+ξ^ρ|x|p2x

is nondecreasing on [–ρ, ρ]. Then we have

Δpy0(z)Δy0(z)+ξ^ρ|y0(z)|p2y0(z)=λ|y0(z)|p2y0(z)+f(z,y0(z))+ξ^ρ|y0(z)|p2y0(z)λu¯λ(z)p1+f(z,u¯λ(z))+ξ^ρu¯λ(z)p1=Δpu¯λ(z)Δu¯λ(z)+ξ^ρu¯λ(z)p1for a.a.zΩ.(4.26)

We set

h1(z)=λ|y0(z)|p2y0(z)+f(z,y0(z))+ξ^ρ|y0(z)|p2y0(z),h2(z)=λu¯λ(z)p1+f(z,u¯λ(z))+ξ^ρu¯λ(z)p1.

Evidently h1, h2L(Ω) and we have

λu¯λ(z)p1|y0(z)|p2y0(z)h2(z)h1(z)for a.a.zΩ,h1h2(see (4.25)).

Then from (4.26) and invoking Proposition 2.2, we infer that

u¯λy0intC+.

In a similar fashion, we show that

y0v¯λintC+,y0intC01(Ω¯)[v¯λ,u¯λ].(4.27)

Consider the homotopy

h~(t,u)=(1t)τ^λ(u)+tφλ(u)for all(t,u)[0,1]×W01,p(Ω).

Suppose we could find {tn}n≥1 ⊆ [0, 1] and {un}n≥1W01,p(Ω) such that

tntin[0,1],uny0inW01,p(Ω),h~u(tn,un)=0 for allnN.(4.28)

Then reasoning as before, via the nonlinear regularity theory, we obtain

uny0inC01(Ω¯) as n,un[v¯λ,u¯λ]for allnn0(see (4.27)){un}nn0Kτ^λ(see (4.11)),

which contradicts (4.13). So, (4.28) can not be true and we have

Ck(τ^λ,y0)=Ck(φλ,y0)for allkN0,(4.29)

C1(φλ,y0)0(see (4.24)).(4.30)

But φλC2(W01,p(Ω), ℝ). So, from (4.29) and Proposition 3.5, Claim 3, in Papageorgiou-Rădulescu [9], we have

Ck(φλ,y0)=δk,1Zfor allkN0,(4.31)

Ck(τ^λ,y0)=δk,1Zfor allkN0(see (4.29)).(4.32)

From the Claim in the beginning of the proof, we know that uλ and vλ are local minimizers of τ̂λ. Hence

Ck(τ^λ,u¯λ)=Ck(τ^λ,v¯λ)=δk,0Zfor allkN0.(4.33)

From (4.23) we have

Ck(τ^λ,0)=δk,dmZfor allkN0.(4.34)

We know that τ̂λ is coercive (see (4.11)). Therefore

Ck(τ^λ,)=δk,0Zfor allkN0.(4.35)

Suppose that Kτ̂λ = {0, uλ, vλ, y0}. Then using (4.34), (4.33), (4.31), (4.35) and the Morse relation with t = –1 (see (2.5)), we obtain

(1)dm+2(1)0+(1)1=(1)0,(1)dm=0,a contradiction.

So, there exists ŷKτ̂λ, ŷ ∉ {0, uλ, vλ, y0}. From (4.12) it follows that ŷC01(Ω) is nodal. Moreover, as for y0, using Proposition 2.2, we show that

y^intC01(Ω¯)[v¯λ,u¯λ].(4.36)

Finally, from Proposition 10 of He-Guo-Huang-Lei [8], we know that (Pλ) has a nodal solution C01(Ω) such that

y~intC01(Ω¯)[v¯λ,u¯λ],y~C01(Ω¯)is the third nodal solution of(Pλ).

So, we can state the following multiplicity theorem for problem (Pλ).

Theorem 4.2

If hypotheses H(f)′ hold, then there exists λ* > 0 such that for all λ ∈ (0, λ*) problem (Pλ) has at least seven nontrivial solutions

uλ,u^λintC+,uλu^λ,uλu^λ,vλ,v^λintC+,v^λvλ,vλv^λ,y0,y^,y~C01(Ω¯)nodalwithy0,y^intC01(Ω¯)[v¯λ,u¯λ].

Acknowledgement

The authors wish to express their gratitude to the anonymous referee for his/her useful remarks.

This research was supported by Piano della Ricerca 2016-2018 - Linea di intervento 2: “Metodi variazionali ed equazioni differenziali”.

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About the article

Received: 2018-10-04

Accepted: 2018-12-24

Published Online: 2019-06-06

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 449–478, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0009.

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© 2020 N. S. Papageorgiou and A. Scapellato, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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