Show Summary Details
More options …

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

IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

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

Mathematical Citation Quotient (MCQ) 2018: 3.18

Open Access
Online
ISSN
2191-950X
See all formats and pricing
More options …
Volume 9, Issue 1

# 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
• Email
• Other articles by this author:
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:

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

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

$Δqu=div|Du|q−2Dufor all u∈W01,q(Ω).$

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, u$\begin{array}{}{W}_{0}^{1,p}\end{array}$(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

${φ(un)}n≥1⊆R is bounded,$

$(1+∥un∥)φ′(un)→0 in X∗ as n→∞,$

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)}

and

$c=infy∈Γmax0≤t≤1φ(y(t))with Γ={y∈C([0,1],X):y(0)=u0,y(1)=u1},$

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,

$Kφ={u∈X:φ′(u)=0}.$

The main spaces in the analysis of problem (Pλ), are the Sobolev spaces $\begin{array}{}{W}_{0}^{1,p}\left(\mathit{\Omega }\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{H}_{0}^{1}\left(\mathit{\Omega }\right)\end{array}$ and the Banach space $\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)=\left\{u\in {C}^{1}\left(\overline{\mathit{\Omega }}\right):u{|}_{\mathrm{\partial }\mathit{\Omega }}=0\right\}\end{array}$.

We have

$C01(Ω¯)⊆W01,p(Ω)⊆H01(Ω)(recall that p>2)$

and the space $\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$ is dense in both $\begin{array}{}{W}_{0}^{1,p}\left(\mathit{\Omega }\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{H}_{0}^{1}\left(\mathit{\Omega }\right)\end{array}$. By ∥⋅∥ we denote the norm of the Sobolev space $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). On account of the Poincaré inequality, we have

$∥u∥=∥Du∥pfor all u∈W01,p(Ω).$

The space $\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$ is an ordered Banach space with positive (order) cone

$C+={u∈C01(Ω¯):u(z)≥0 for all z∈Ω¯}.$

This cone has a nonempty interior given by

$intC+=u∈C+:u(z)>0 for all z∈Ω and ∂u∂n|∂Ω<0.$

Here $\begin{array}{}\frac{\mathrm{\partial }u}{\mathrm{\partial }n}=\left(Du,n{\right)}_{{\mathbb{R}}^{N}}\end{array}$ 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

$|f0(z,x)|≤a0(z)1+|x|r−1for a.a. z∈Ω, all x∈R$

with a0L(Ω) and

$1

We set $\begin{array}{}{F}_{0}\left(z,x\right)={\int }_{0}^{x}{f}_{0}\left(z,s\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}s\end{array}$ and consider the C1-functional φ0 : $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) → ℝ defined by

$φ0(u)=1p∥Du∥pp+12∥Du∥22−∫ΩF0(z,u)dzfor all u∈W01,p(Ω).$

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 u0$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) is a local $\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$-minimizer of φ0, that is, there exists ρ0 > 0 such that

$φ0(u0)≤φ0(u0+h)for all ∥h∥C01(Ω¯)≤ρ0,$

then $\begin{array}{}{u}_{0}\in {C}_{0}^{1,\alpha }\left(\overline{\mathit{\Omega }}\right)={C}^{1,\alpha }\left(\overline{\mathit{\Omega }}\right)\cap {C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$ and it is also a local $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω)-minimizer of φ0, that is, there exists ρ1 > 0 such that

$φ0(u0)≤φ0(u0+h)for all ∥h∥≤ρ1.$

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 u$\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$ ∖ {0}, v ∈ int C+, uv satisfy

$−Δpu(z)−Δu(z)+ξ(z)|u(z)|p−2u(z)=h1(z),−Δpv(z)−Δv(z)+ξ(z)v(z)p−1=h2(z)$

for a.a. zΩ, then vu ∈ int C+.

For q ∈ (1, +∞), let $\begin{array}{}{A}_{q}:{W}_{0}^{1,q}\left(\mathit{\Omega }\right)\to {W}^{-1,{q}^{\prime }}\left(\mathit{\Omega }\right)={W}_{0}^{1,q}\left(\mathit{\Omega }{\right)}^{\ast }\left(\frac{1}{q}+\frac{1}{{q}^{\prime }}=1\right)\end{array}$ be the nonlinear map defined by

$〈Aq(u),h〉=∫Ω|Du|q−2(Du,Dh)RNdzfor all u,h∈W01,q(Ω).$

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 $\begin{array}{}\stackrel{w}{\to }\end{array}$ u in $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) and $\begin{array}{}\underset{n\to +\mathrm{\infty }}{lim sup}〈A\left({u}_{n}\right),{u}_{n}-u〉\le 0\end{array}$, then unu in $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω)).

If q = 2, then A2 = A$\begin{array}{}\mathcal{L}\left({H}_{0}^{1}\left(\mathit{\Omega }\right),{H}^{-1}\left(\mathit{\Omega }\right)\right)\end{array}$.

We will need some basic facts about the spectrum of $\begin{array}{}\left(-\mathit{\Delta },{H}_{0}^{1}\left(\mathit{\Omega }\right)\right)\end{array}$. 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, $\begin{array}{}{H}_{0}^{1}\left(\mathit{\Omega }\right)\end{array}$), if problem (2.1) admits a nontrivial solution û$\begin{array}{}{H}_{0}^{1}\left(\mathit{\Omega }\right)\end{array}$ 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∈ℕ$\begin{array}{}{H}_{0}^{1}\left(\mathit{\Omega }\right)\end{array}$ of eigenfunctions of (2.1), forms an orthonormal basis of $\begin{array}{}{H}_{0}^{1}\left(\mathit{\Omega }\right)\end{array}$ and an orthogonal basis of L2(Ω). Standard regularity theory implies that {ûn(2)}n∈ℕ$\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$. By E(λ̂k(2)) we denote the eigenspace corresponding to the eigenvalue λ̂k(2), k ∈ ℕ. We have E(λ̂k(2)) ⊆ $\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$ and we have the following orthogonal direct sum decomposition

$H01(Ω)=⨁k∈NE(λ^k(2))¯.$

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).

• $λ^1(2)=inf∥Du∥22∥u∥22:u∈H01(Ω),u≠0$(2.2)

• $λ^m(2)=sup∥Du∥22∥u∥22:u∈⨁k=1mE(λ^k),u≠0=inf∥Du∥22∥u∥22:u∈⨁k≥mE(λ^k),u≠0$(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

$∥Du∥22−∫Ωη(z)u2dz≥c1∥Du∥22$

for some c1 > 0, all $\begin{array}{}u\in \overline{\underset{k\ge m}{⨁}E\left({\stackrel{^}{\lambda }}_{k}\left(2\right)\right)}\end{array}$.

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

$∥Du∥22−∫Ωη(z)u2dz≤−c2∥Du∥22$

for some c2 > 0, all $\begin{array}{}u\in \underset{k=1}{\overset{m}{⨁}}E\left({\stackrel{^}{\lambda }}_{k}\left(2\right)\right)\end{array}$.

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

$−Δpu(z)=λ^|u(z)|p−2u(z)in Ω,u|∂Ω=0.$

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

$λ^1(p)=inf∥Du∥pp∥u∥pp:u∈W01,p(Ω),u≠0.$(2.4)

The infimum in (2.4) is realized on the corresponding one dimensional eigenspace, the elements of which are in $\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$ (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

$∥Du∥pp−∫Ωη(z)|u|pdz≥c3∥Du∥ppfor all u∈W01,p(Ω).$

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

$φc={x∈X:φ(u)≤c},Kφ={u∈X:φ′(u)=0}(the critical set of φ),Kφc={u∈Kφ:φ(u)=c}.$

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 u$\begin{array}{}{K}_{\phi }^{c}\end{array}$ isolated, the critical groups of φ at u, are defined by

$Ck(φ,u)=Hk(φc∩U,φc∩U∖{u}),$

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

$Ck(φ,∞)=Hk(X,φc)for all k∈N0.$

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

$Hk(X,φc)=Hk(X,φc′)for all k∈N0$

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

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

$M(t,u)=∑k∈N0rankCk(φ,u)tkfor all t∈R, all u∈Kφ,P(t,∞)=∑k∈N0rankCk(φ,∞)tkfor all t∈R.$

The Morse relation says that

$∑u∈KφM(t,u)=P(t,∞)+(1+t)Q(t)for all t∈R,$(2.5)

where $\begin{array}{}Q\left(t\right)=\sum _{k\ge 0}{\beta }_{k}{t}^{k}\end{array}$ 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 u$\begin{array}{}{W}_{0}^{1,p}\end{array}$(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, v$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω), uv, then the order interval in $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) determined by u and v is defined by

$[u,v]={y∈W01,p(Ω):u(z)≤y(z)≤v(z)for a.a. z∈Ω}.$

By $\begin{array}{}{\mathrm{i}\mathrm{n}\mathrm{t}}_{{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)}\left[u,v\right]\end{array}$ we denote the interior in the $\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$-norm topology of [u, v] ∩ $\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$. By [u) we denote the half-line in $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) defined by

$[u)={y∈W01,p(Ω):u(z)≤y(z)for a.a. z∈Ω}.$

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

$δk,m=1if k=m0if k≠m.$

## 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

2. If $\begin{array}{}F\left(z,x\right)={\int }_{0}^{x}f\left(z,s\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}s\end{array}$, then $\begin{array}{}\underset{x\to ±\mathrm{\infty }}{lim}\frac{F\left(z,x\right)}{|x{|}^{p}}=+\mathrm{\infty }\end{array}$ uniformly for a.a. zΩ;

3. there exist η̂ > 0 and $\begin{array}{}q\in \left(\left(r-p\right)max\left\{\frac{N}{p},1\right\},{p}^{\ast }\right)\end{array}$ such that

$0<η^≤lim infx→±∞f(z,x)x−pF(z,x)|x|quniformly for a.a. z∈Ω;$

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

$λ^m(2)≤ϑ(z)≤ϑ^(z)≤λ^m+1(2)for a.a. z∈Ω,$

$ϑ≢λ^m(2),ϑ^≢λ^m+1(2),$

$ϑ(z)≤lim infx→0f(z,x)x≤lim supx→0f(z,x)x≤ϑ^(z)uniformly for a.a. z∈Ω;$

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

$x↦f(z,x)+ξ^ρ|x|p−2x$

is nondecreasing on [−ρ, ρ].

#### Remarks

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

$limx→±∞f(z,x)|x|p−2x=+∞uniformly for a.a. z∈Ω.$

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(3.1)

Integrating, we obtain the following weaker condition

$c4|x|q≤F(z,x)for a.a. z∈Ω, all |x|≥M, with c4>0.$(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 $\begin{array}{}q>\left(r-p\right)max\left\{\frac{N}{p},1\right\}\end{array}$. Then

$f(z,x)x−pF(z,x)|x|q=f(z,x)x−qF(z,x)|x|q+(q−p)F(z,x)|x|q=(q−p)F(z,x)|x|q (see (3.1))=(q−p)c4>0 (see (3.2)),$

$⇒lim infx→±∞f(z,x)x−pF(z,x)|x|q≥(q−p)c4>0 uniformly for a.a.z∈Ω.$

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:

$f1(x)=ϑx+|x|τ−2xif |x|≤1ϑ|x|r−2x−|x|q−2xif |x|>1,$

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

$f2(x)=ϑ(x−|x|τ−2x)if |x|≤1ϑ|x|p−2xln⁡|x|if |x|>1,$

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 $\begin{array}{}{S}_{\lambda }^{+}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{S}_{\lambda }^{-}\end{array}$.

#### Proposition 3.1

If hypotheses H(f) hold, then+, ℒ ≠ ∅ and $\begin{array}{}{S}_{\lambda }^{+}\subseteq \mathrm{i}\mathrm{n}\mathrm{t}\phantom{\rule{thinmathspace}{0ex}}{C}_{+},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{S}_{\lambda }^{-}\subseteq -\mathrm{i}\mathrm{n}\mathrm{t}\phantom{\rule{thinmathspace}{0ex}}{C}_{+}\end{array}$.

#### Proof

We do the proof for the pair $\begin{array}{}\left({\mathcal{L}}^{+},{S}_{\lambda }^{+}\right)\end{array}$, the proof for the pair $\begin{array}{}\left({\mathcal{L}}^{-},{S}_{\lambda }^{-}\right)\end{array}$ being similar.

So, we consider the C1-functional $\begin{array}{}{\psi }_{\lambda }^{+}:{W}_{0}^{1,p}\end{array}$(Ω) → ℝ defined by

$ψλ+(u)=1p∥Du∥pp+12∥Du∥22−λp∥u+∥pp−∫ΩF(z,u+)dz,for all u∈W01,p(Ω).$

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

$limx→0+f(z,x)xτ−1=0uniformly for a.a. z∈Ω.$

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

$F(z,x)≤ϵ|x|τ+c5|x|rfor a.a. z∈Ω, all x∈R.$(3.3)

Then we have

$ψλ+(u)≥1p∥Du−∥pp+1p∥Du+∥pp−λ∥u+∥pp−ϵc6∥u∥τ−c7∥u∥rfor some c6>0, c7>0(see (3.3)).$

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

$ψλ+(u)≥c8∥u∥p−ϵc6∥u∥τ+c7∥u∥rfor some c8>0=c8−ϵc6∥u∥τ−p+c7∥u∥r−p∥u∥p.$(3.4)

We consider the function

$ξ(t)=ϵc6tτ−p+c7tr−p,t>0.$

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

$ξ(t)→+∞ as t→0+ and as t→+∞.$

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

$ξ(t0)=infξ(t):t>0,⇒ξ′(t0)=0,⇒t0=t0(ϵ)=ϵc6(p−τ)c7(r−p)p−τr−p.$

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

$ξ(t0)0(see (3.4))$(3.5)

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

$ψλ+(tu)→−∞ as t→+∞.$(3.6)

Claim. For every λ > 0, the functional $\begin{array}{}{\psi }_{\lambda }^{+}\end{array}$ satisfies the C-condition.

Let {un}n≥1$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) be a sequence such that

$|ψλ+(un)|≤M1for some M1>0,all n∈N,$(3.7)

$(1+∥un∥)(ψλ+)′(un)→0in W−1,p′(Ω) as n→∞.$(3.8)

From (3.8) we have

$〈Ap(un),h〉+〈A(un),h〉−λ∫Ω(un+)p−1hdz−∫Ωf(z,un+)hdz≤ϵn∥h∥1+∥un∥$(3.9)

for all h$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω), with ϵn → 0+.

In (3.9) we choose $\begin{array}{}h=-{u}_{n}^{-}\in {W}_{0}^{1,p}\left(\mathit{\Omega }\right)\end{array}$. Then

$∥Dun−∥pp+∥Dun−∥22≤ϵnfor all n∈N,⇒un−→0 in W01,p(Ω) as n→∞.$(3.10)

From (3.7) and (3.10), we have

$∥Dun+∥pp+p2∥Dun+∥22−∫Ωλ(un+)p+pF(z,un+)dz≤M2$(3.11)

for some M2 > 0, all n ∈ ℕ.

Also from (3.9) with $\begin{array}{}h={u}_{n}^{+}\in {W}_{0}^{1,p}\left(\mathit{\Omega }\right)\end{array}$, we obtain

$−∥Dun+∥pp−∥Dun+∥22+∫Ωλ(un+)p+f(z,un+)un+dz≤ϵnfor all n∈N.$(3.12)

We add (3.11) and (3.12) and obtain

$∫Ωf(z,un+)un+−pF(z,un+)dz≤M3for all M3>0, all n∈N, (recall p>2).$(3.13)

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

$η^0|x|q−c9≤f(z,x)x−pF(z,x)for a.a. z∈Ω, all x∈R.$

Using this in (3.13), we obtain that

$un+n≥1⊆Lq(Ω) is bounded,$(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

$1r=1−tq+tp∗.$

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

$∥un+∥r≤∥un+∥q1−t∥un+∥p∗t,⇒∥un+∥rr≤c10∥un+∥trfor some c10>0, all n∈N,(see (3.4) and recall that W01,p(Ω)↪Lp∗(Ω)).$(3.15)

In (3.9) let $\begin{array}{}h={u}_{n}^{+}\in {W}_{0}^{1,p}\left(\mathit{\Omega }\right)\end{array}$. Then

$∥Dun+∥pp+∥Dun+∥22−∫Ωλ(un+)p+f(z,un+)un+dz≤ϵn for all n∈N,$

$⇒∥un+∥p≤c111+∥un+∥rrfor some c11=c11(λ)>0, all n∈N(see hypothesisH(f) (i)and recall that r>p)⇒∥un+∥p≤c121+∥un+∥trfor some c12>0, all n∈N,(see (3.15)).$(3.16)

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

$un+n≥1⊆W01,p(Ω) is bounded,⇒unn≥1⊆W01,p(Ω) is bounded (see (3.10)).$(3.17)

Now suppose that N = p. In this case p* = +∞ and $\begin{array}{}{W}_{0}^{1,p}\end{array}$Ls(Ω) for all s ∈ [1, +∞). Let s > r > q and as before pick t ∈ (0, 1) such that

$1r=1−tq+ts,⇒tr=s(r−q)s−q.$

We see that

$s(r−q)s−q→r−q as s→p∗=+∞.$

By hypothesis H(f)(iii) we have

$r−qr big.$

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

assing to a subsequence if necessary, we have

$un →w u in W01,p(Ω)andun→u in Lr(Ω).$(3.18)

In (3.9) we choose h = unu$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω), pass to the limit as n → ∞ and use (3.18). Then

$limn→∞〈Ap(un),un−u〉+〈A(un),un−u〉=0,⇒lim supn→∞〈Ap(un),un−u〉+〈A(u),un−u〉≤0(sinceA(⋅) is monotone)⇒lim supn→∞〈Ap(un),un−u〉≤0,⇒un→uinW01,p(Ω) (see Proposition 2.3)).$

Therefore $\begin{array}{}{\psi }_{\lambda }^{+}\end{array}$ 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 uλ$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) such that

$uλ∈Kψλ+andψλ+(0)=0

Therefore uλ ≠ 0 and we have

$〈Ap(uλ),h〉+〈A(uλ),h〉=∫Ωλ(uλ+)p−1+f(z,uλ+)hdzfor all h∈W01,p(Ω).$

Choosing $\begin{array}{}h=-{u}_{\lambda }^{-}\in {W}_{0}^{1,p}\left(\mathit{\Omega }\right)\end{array}$, we obtain

$uλ≥0,uλ≠0.$

From (3.9) we have

$−Δpuλ(z)−Δuλ(z)=λuλ(z)p−1+f(z,uλ(z))for a.a. z∈Ω,uλ|∂Ω=0.$(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

$uλ∈C+∖{0}.$

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

$−Δpuλ(z)−Δuλ(z)+ξ^ρuλ(z)p−1≥0for a.a. z∈Ω,⇒uλ∈intC+ (see Pucci-Serrin [19], pp. 111,120).$

Therefore (0, λ̂1(p)) ⊆ 𝓛+ and $\begin{array}{}{S}_{\lambda }^{+}\end{array}$ ⊆ int C+. Similarly we show that 𝓛 ≠ ∅ and that $\begin{array}{}{S}_{\lambda }^{-}\end{array}$ ⊆ − 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 uλ$\begin{array}{}{S}_{\lambda }^{+}\end{array}$ ⊆ int C+. Then we introduce the following truncation of the reaction in problem (Pϑ):

$eϑ(z,x)=0if x<0ϑxp−1+f(z,x)if 0≤x≤uλ(z)ϑuλ(z)p−1+f(z,uλ(z))if uλ(z)(3.20)

This is a Caratheodory function. We set $\begin{array}{}{E}_{\vartheta }\left(z,x\right)={\int }_{0}^{x}{e}_{\vartheta }\left(z,s\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}s\end{array}$ and consider the C1-functional $\begin{array}{}{\stackrel{^}{\psi }}_{\vartheta }^{+}:{W}_{0}^{1,p}\left(\mathit{\Omega }\right)\to \mathbb{R}\end{array}$ defined by

$ψ^ϑ+(u)=1p∥Du∥pp+12∥Du∥22−∫ΩEϑ(z,u)dzfor all u∈W01,p(Ω).$

From (3.20) it is clear that $\begin{array}{}{\stackrel{^}{\psi }}_{\vartheta }^{+}\left(\cdot \right)\end{array}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can fin uϑ$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) such that

$ψ^ϑ+(uϑ)=infψ^ϑ+(u):u∈W01,p(Ω).$(3.21)

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

$F(z,x)≥12ϑ(z)−ϵx2for a.a. z∈Ω, all |x|≤δ.$(3.22)

Let uE(λ̂m(2)) ⊆ $\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$ and choose t ∈ (0, 1) small such that

$0≤tu(z)≤δfor all z∈Ω¯.$(3.23)

Then we have

$ψ^ϑ+(tu)≤tpp∥Du∥pp+t22∥Du∥22−t22∫Ωϑ(z)u2dz+ϵ2t2∥u∥22(see (3.22), (3.23))=tpp∥Du∥pp+t22∥Du∥22−∫Ωϑ(z)u2dz+ϵ2t2∥u∥22≤tpp∥Du∥pp+t22(−c13+ϵ)∥u∥22for some c13>0 (see Proposition 2.4).$

Choosing ϵ ∈ (0, c13), we have that

$ψ^ϑ+(tu)≤tpp∥Du∥pp−t22c14∥u∥22.$

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

$ψ^ϑ+(tu)<0,⇒ψ^ϑ+(uϑ)<0=ψ^ϑ+(0)(see (3.21)),⇒uϑ≠0.$

From (3.21), we have

$ψ^ϑ+′(uϑ)=0⇒〈Ap(uϑ),h〉+〈A(uϑ),h〉=∫Ωeϑ(z,uϑ)hdzfor all h∈W01,p(Ω).$(3.24)

In (3.24) we choose $\begin{array}{}h=-{u}_{\vartheta }^{-}\in {W}_{0}^{1,p}\left(\mathit{\Omega }\right)\end{array}$. Then

$∥Duϑ−∥pp+∥Duϑ−∥22=0(see (3.20)),⇒uϑ≥0,uϑ≠0.$

Also, in (3.24) we choose h = (uϑuλ)+$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). Then

$〈Ap(uϑ),(uϑ−uλ)+〉+〈A(uϑ),(uϑ−uλ)+〉 =∫Ωϑuλp−1+f(z,uλ)(uϑ−uλ)+dz(see (3.20)) ≤∫Ωλuλp−1+f(z,uλ)(uϑ−uλ)+dz(since ϑ<λ) =〈Ap(uλ),(uϑ−uλ)+〉+〈A(uλ),(uϑ−uλ)+〉(since uλ∈Sλ), ⇒uϑ≤uλ.$

So, we have proved that

$uϑ∈[0,uλ],uϑ≠0.$(3.25)

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

$−Δpuϑ(z)−Δuϑ(z)=ϑuϑ(z)p−1+f(z,uϑ(z))for a.a. z∈Ω,uϑ|∂Ω=0,⇒ϑ∈L+ and uϑ∈Sϑ+⊆intC+.$

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 uλ$\begin{array}{}{S}_{\lambda }^{+}\end{array}$, then ϑ ∈ ℒ+ and we can find uϑ$\begin{array}{}{S}_{\vartheta }^{+}\end{array}$ ⊆ int C+ such that

$uλ−uϑ∈C+∖{0};$

2. if 0 < ϑ < λ ∈ ℒ and vλ$\begin{array}{}{S}_{\lambda }^{-}\end{array}$, then ϑ ∈ ℒ and we can find vϑ$\begin{array}{}{S}_{\vartheta }^{-}\end{array}$ ⊆ −int C+ such that

$vϑ−vλ∈C+∖{0}.$

We can improve this corollary.

#### Proposition 3.3

If hypotheses H(f) hold, then

1. if 0 < ϑ < λ ∈ ℒ+ and uλ$\begin{array}{}{S}_{\lambda }^{+}\end{array}$, then ϑ ∈ ℒ+ and we can find uϑ$\begin{array}{}{S}_{\vartheta }^{+}\end{array}$ ⊆ int C+ such that

$uλ−uϑ∈intC+;$

2. if 0 < ϑ < λ ∈ ℒ and vλ$\begin{array}{}{S}_{\lambda }^{-}\end{array}$, then ϑ ∈ ℒ and we can find vϑ$\begin{array}{}{S}_{\vartheta }^{-}\end{array}$ ⊆ −int C+ such that

$vϑ−vλ∈intC+.$

#### Proof

1. From Corollary 3.1, we already know that ϑ ∈ ℒ+ and we can find uϑ$\begin{array}{}{S}_{\vartheta }^{+}\end{array}$ ⊆ int C+ such that

$uλ−uϑ∈C+∖{0}.$(3.26)

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

$−Δpuϑ−Δuϑ+ξ^ρuϑp−1=ϑuϑp−1+f(z,uϑ)+ξ^ρuϑp−1=λuϑp−1+f(z,uϑ)+ξ^ρuϑp−1−(λ−ϑ)uϑp−1≤λuλp−1+f(z,uλ)+ξ^ρuλp−1(see (3.26), hypothesisH(f)(v) and recall that ϑ<λ)=−Δpuλ−Δuλ+ξ^ρuλp−1(since uλ∈Sλ+).$(3.27)

Let

$h1(z)=ϑuϑp−1+f(z,uϑ)+ξ^ρuϑp−1,h2(z)=λuλp−1+f(z,uλ)+ξ^ρuλp−1.$

Evidently h1, h2L(Ω) and we have

$h2(z)−h1(z)≥(λ−ϑ)uϑ(z)p−1for a.a z∈Ω.$

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 $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$ = sup ℒ+ and $\begin{array}{}{\lambda }_{\ast }^{-}\end{array}$ = sup ℒ.

#### Proposition 3.4

If hypotheses H(f) hold, then $\begin{array}{}{\lambda }_{\ast }^{+}<+\mathrm{\infty }\text{\hspace{0.17em}}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\lambda }_{\ast }^{-}<+\mathrm{\infty }\end{array}$.

#### Proof

We do the proof for $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$, the proof for $\begin{array}{}{\lambda }_{\ast }^{-}\end{array}$ being similar. On account of hypotheses H(f)(i), (ii), (iii), we can find λ͠ > 0 big such that

$λ~xp−1+f(z,x)≥0for a.a. z∈Ω, all x≥0.$(3.28)

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

$∂uλ∂n|∂Ω<0.$

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

$∂uλ∂n|∂Ωδ<0.$(3.29)

Consider the open set Ωδ = {zΩ : d(z, ∂ Ω) > δ} and set $\begin{array}{}{m}_{\delta }=\underset{\overline{{\mathit{\Omega }}_{\delta }}}{min}{u}_{\lambda }>0\end{array}$ (recall that uλ ∈ int C+). For ϵ > 0, we set $\begin{array}{}{m}_{\delta }^{ϵ}\end{array}$ = mδ + ϵ and for ρ = ∥uλ let ξ̂ρ > 0 be as postulated by hypothesis H(f)(v). We have

$−Δpmδϵ−Δmδϵ+ξ^ρ(mδϵ)p−1≤ξ^ρmδp−1+μ(ϵ)with μ(ϵ)→0+ as ϵ→0+≤λ~mδp−1+f(z,mδ)+ξ^ρmδp−1+μ(ϵ)(see (3.28))=λmδp−1+f(z,mδ)+ξ^ρmδp−1−(λ−λ~)mδp−1+μ(ϵ)(see (3.28))≤λmδp−1+f(z,mδ)+ξ^ρmδp−1for ϵ>0 small≤λuλp−1+f(z,uλ)+ξ^ρuλp−1(recall that mδ≤uλ on Ωδ¯)=−Δpuλ−Δuλ+ξ^ρuλp−1for a.a. z∈Ωδ.$(3.30)

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

$uλ−mδϵ∈intC+(Ωδ¯)for ϵ>0 small,$

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

$λ∗+≤λ~<+∞.$

Similarly we show that $\begin{array}{}{\lambda }_{\ast }^{-}\end{array}$ < +∞. □

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

$λ|x|p+f(z,x)x≥[ϑ(z)−ϵ]x2−c15|x|rfor a.a. z∈Ω, all x∈R, all λ>0.$(3.31)

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

$−Δpu(z)−Δu(z)=[ϑ(z)−ϵ]u(z)−c15|u(z)|r−2u(z)in Ωu|∂Ω=0$(3.32)

#### Proposition 3.5

For all ϵ > 0 small, problem (3.32) has a unique positive solution $\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$ ∈ int C+ and, since (3.32) is odd, $\begin{array}{}{v}_{\lambda }^{\ast }=-{u}_{\lambda }^{\ast }\end{array}$ ∈ −int C+ is the unique solution of (3.32).

#### Proof

Consider the C1-functional σ : $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) → ℝ defined by

$σ(u)=1p∥Du∥pp+12∥Du∥22+c15r∥u+∥rr−12∫Ω[ϑ(z)−ϵ](u+)2dzfor all u∈W01,p(Ω).$

Evidently σ(⋅) is coercive (recall that p > 2). Also, it is sequentially weakly lower semicontinuous. So, we can find $\begin{array}{}{u}_{\lambda }^{\ast }\in {W}_{0}^{1,p}\left(\mathit{\Omega }\right)\end{array}$ such that

$σ(uλ∗)=infσ(u):u∈W01,p(Ω).$(3.33)

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

$σ(uλ∗)<0=σ(0),⇒uλ∗≠0.$

From (3.33) we have

$σ′(uλ∗)=0,$

$⇒〈Ap(uλ∗),h〉+〈A(uλ∗),h〉=∫Ω[ϑ(z)−ϵ](uλ∗)+hdz−λ∫Ω((uλ∗)+)r−1hdzfor all h∈W01,p(Ω).$(3.34)

In (3.34) we choose $\begin{array}{}h=-\left({u}_{\lambda }^{\ast }{\right)}^{-}\in {W}_{0}^{1,p}\left(\mathit{\Omega }\right)\end{array}$. Then

$∥D(uλ∗)−∥pp+∥D(uλ∗)−∥22=0,⇒uλ∗≥0,uλ∗≠0.$

So, from (3.34) we have that $\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$ is a positive solution of (3.32) and the nonlinear regularity theory (see [14]) implies that $\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$C+ ∖ {0}. We have

$Δpuλ∗+Δuλ∗≤c15∥uλ∗∥∞r−p(uλ∗)p−1for a.a. z∈Ω,⇒uλ∗∈intC+(see Pucci-Serrin [19], pp. 111, 120).$

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

$j(u)=1p∥Du1/2∥pp+12∥Du1/2∥22if u≥0, u1/2∈W01,p(Ω)+∞otherwise.$

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 $\begin{array}{}{u}_{\lambda }^{\ast },{\stackrel{~}{\phantom{\rule{thinmathspace}{0ex}}u}}_{\lambda }^{\ast }\end{array}$ are two positive solutions of (3.32). We have

$uλ∗,u~λ∗∈intC+$

Then, for h$\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$ and for |t| < 1 small, we have

$(uλ∗)2+th∈domjand(u~λ∗)2+th∈domj.$

It is easy to see that j(⋅) is Gateaux differentiable at $\begin{array}{}\left({u}_{\lambda }^{\ast }{\right)}^{2}\end{array}$ and at $\begin{array}{}\left({\stackrel{~}{\phantom{\rule{thinmathspace}{0ex}}u}}_{\lambda }^{\ast }{\right)}^{2}\end{array}$ in the direction h. Moreover, using the chain rule and the nonlinear Green’s identity (see Gasiński-Papageorgiou [18], p. 211), we have

$j′(uλ∗)2(h)=12∫Ω−Δpuλ∗−Δuλ∗uλ∗hdzj′(u~λ∗)2(h)=12∫Ω−Δpu~λ∗−Δu~λ∗u~λ∗hdz$

for all h$\begin{array}{}{C}_{0}^{1}\left(\overline{\mathit{\Omega }}\right)\end{array}$.

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

$0≤∫Ω−Δpuλ∗−Δuλ∗uλ∗−−Δpu~λ∗−Δu~λ∗u~λ∗(uλ∗−u~λ∗)dz=∫Ωc15(u~λ∗)r−2−(uλ∗)r−2(uλ∗−u~λ∗)dz≤0,$

$⇒uλ∗=u~λ∗.$

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

Since problem (3.32) is odd, it follows that

$vλ∗=−uλ∗∈−intC+,$

is the unique negative solution of (3.32). □

These solutions provide bounds of the elements of $\begin{array}{}{S}_{\lambda }^{+}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and of}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{S}_{\lambda }^{-}\end{array}$.

#### Proposition 3.6

If hypotheses H(f) hold, then

1. $\begin{array}{}{u}_{\lambda }^{\ast }\end{array}$u for all u$\begin{array}{}{S}_{\lambda }^{+}\end{array}$, λ ∈ ℒ+;

2. v$\begin{array}{}{v}_{\lambda }^{\ast }\end{array}$ for all v$\begin{array}{}{S}_{\lambda }^{-}\end{array}$, λ ∈ ℒ.

#### Proof

1. Let λ ∈ ℒ+ and u$\begin{array}{}{S}_{\lambda }^{+}\end{array}$ ⊆ int C+. With ϵ > 0 small as dictated by Proposition 3.5, we introduce the following Caratheodory function:

$k+(z,x)=0if x<0[ϑ(z)−ϵ]x−c15xr−1if 0≤x≤u(x)[ϑ(z)−ϵ]u(z)−c15u(z)r−1if u(z)(3.35)

We set K+(z, x) = $\begin{array}{}{\int }_{0}^{x}\end{array}$ k+(z, s) ds and consider the C1-functional τ+ : $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) → ℝ defined by

$τ+(u)=1p∥Du∥pp+12∥Du∥22−∫ΩK+(z,u)dzfor all u∈W01,p(Ω).$

Evidently τ+(⋅) is coercive (see (3.35)) and sequentially weakly lower semicontinuous. So, we can find $\begin{array}{}{\stackrel{^}{u}}_{\lambda }^{\ast }\in {W}_{0}^{1,p}\left(\mathit{\Omega }\right)\end{array}$ such that

$τ+(u^λ∗)=infτ+(u):u∈W01,p(Ω).$(3.36)

As before we have

$τ+(u^λ∗)<0=τ+(0)⇒u^λ∗≠0.$

From (3.36) we have

$τ+′(u^λ∗)=0,⇒〈Ap(u^λ∗),h〉+〈A(u^λ∗),h〉=∫Ωk+(z,u^λ∗)hdzfor all h∈W01,p(Ω).$(3.37)

In (3.37) first we choose $\begin{array}{}h=-\left({\stackrel{^}{u}}_{\lambda }^{\ast }{\right)}^{-}\in {W}_{0}^{1,p}\left(\mathit{\Omega }\right).\end{array}$ Then

$∥D(u^λ∗)−∥pp+∥D(u^λ∗)−∥22=0(see (3.35)),⇒u^λ∗≥0,u^λ∗≠0.$

Next in (3.37) we choose $\begin{array}{}\left({\stackrel{^}{u}}_{\lambda }^{\ast }-u{\right)}^{+}\in {W}_{0}^{1,p}\left(\mathit{\Omega }\right).\end{array}$ Then

$〈Ap(u^λ∗),(u^λ∗−u)+〉+〈A(u^λ∗),(u^λ∗−u)+〉=∫Ω(ϑ(z)−ϵ)u−c15ur−1(u^λ∗−u)+dz(see (3.35))≤∫Ωλup−1+f(z,u)(u^λ∗−u)+dz(see (3.31))=〈Ap(u),(u^λ∗−u)+〉+〈A(u),(u^λ∗−u)+〉(since u∈Sλ+),$

$⇒u^λ∗≤u.$

So, we have proved that

$u^λ∗∈[0,u],u^λ∗≠0.$(3.38)

From (3.37) and (3.38) it follows that $\begin{array}{}{\stackrel{^}{u}}_{\lambda }^{\ast }\end{array}$ is a positive solution of problem (3.32). Hence Proposition 3.5 implies that

$u^λ∗=uλ∗∈intC+,⇒uλ∗≤u for all u∈Sλ+(see (3.38)).$

2. Let λ ∈ ℒ and v$\begin{array}{}{S}_{\lambda }^{-}\end{array}$. We introduce the Caratheodory function k(z, x) defined by

$k−(z,x)=[ϑ(z)−ϵ]v(z)−c15|v(z)|r−2v(z)if x(3.39)

We set K(z, x) = $\begin{array}{}{\int }_{0}^{x}\end{array}$ k(z, s) ds and consider the C1-functional τ : $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) → ℝ defined by

$τ−(u)=1p∥Du∥pp+12∥Du∥22−∫ΩK−(z,u)dzfor all u∈W01,p(Ω).$

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

$v≤vλ∗for all v∈Sλ−.$

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 uλ$\begin{array}{}{S}_{\lambda }^{+}\end{array}$ ⊆ int C+, that is,

$u¯λ≤ufor all u∈Sλ+;$

2. for every λ ∈ ℒ problem (Pλ) has a biggest negative solution vλ$\begin{array}{}{S}_{\lambda }^{-}\end{array}$ ⊆ –int C+, that is,

$v≤v¯λfor all v∈Sλ−.$

#### Proof

1. From Filippakis-Papageorgiou [22], we know that $\begin{array}{}{S}_{\lambda }^{+}\end{array}$ is downward directed (that is, if u1, u2$\begin{array}{}{S}_{\lambda }^{+}\end{array}$, then we can find u$\begin{array}{}{S}_{\lambda }^{+}\end{array}$ such that uu1, uu2). Hence using Lemma 3.10, p. 178, of Hu-Papageorgiou [23], we can find {un}n≥1$\begin{array}{}{S}_{\lambda }^{+}\end{array}$ decreasing such that

$infSλ+=infn≥1un.$

We have

$〈Ap(un),h〉+〈A(un),h〉=∫Ωλunp−1+f(z,un)hdzfor all h∈W01,p(Ω), all n∈N,$(3.40)

$0≤un≤u1for all n∈N.$(3.41)

In (3.40) we choose h = un$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). Then on account of (3.41) and hypothesis H(f)(i), we obtain

$∥Dun∥pp+∥Dun∥22≤c16for some c16>0, all n∈N,⇒{un}n≥1⊆W01,p(Ω) is bounded.$

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

$un →w u¯λ in W01,p(Ω)andun→u¯λ in Lp(Ω).$(3.42)

If in (3.40) we choose h = unuλ$\begin{array}{}{W}_{0}^{1,p}\end{array}$(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

$un→u¯λ in W01,p(Ω).$(3.43)

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

$〈Ap(u¯λ),h〉+〈A(u¯λ),h〉=∫Ωλu¯λp−1+f(z,u¯λ)hdzfor all h∈W01,p(Ω).$(3.44)

From Proposition 3.6, we know that

$uλ∗≤unfor all n∈N,⇒uλ∗≤u¯λ(see(3.43)).$(3.45)

From (3.44) and (3.45) we conclude that

$u¯λ∈Sλ+⊆intC+andu¯λ=infSλ+.$

2. From Filippakis-Papageorgiou [22], we know that $\begin{array}{}{S}_{\lambda }^{-}\end{array}$ is upward directed (that is, if v1, v2$\begin{array}{}{S}_{\lambda }^{-}\end{array}$, then we can find v$\begin{array}{}{S}_{\lambda }^{-}\end{array}$ such that v1v, v2v). So, in this case we can find {vn}n≥1$\begin{array}{}{S}_{\lambda }^{-}\end{array}$ increasing such that

$supSλ−=supn≥1vn.$

Reasoning as in part (a), we obtain

$v¯λ∈Sλ−⊆−intC+andv¯λ=supSλ−.$

We examine the maps λuλ from ℒ+ into C+$\begin{array}{}{C}_{0}^{1}\end{array}$(Ω) and of λvλ from ℒ into –C+$\begin{array}{}{C}_{0}^{1}\end{array}$(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 uϑ$\begin{array}{}{S}_{\vartheta }^{+}\end{array}$ ⊆ int C+ such that

$u¯λ−uϑ∈intC+,⇒u¯λ−u¯ϑ∈intC+.$

Also let {λn}n≥1 ⊆ ℒ+ such that λn$\begin{array}{}\left({\lambda }_{\ast }^{+}{\right)}^{-}\end{array}$. We set un = uλn$\begin{array}{}{S}_{{\lambda }_{n}}^{+}\end{array}$ ⊆ int C+ for all n ∈ ℕ. Then

$〈Ap(u¯n),h〉+〈A(u¯n),h〉=∫Ωλn(u¯n)p−1+f(z,u¯n)hdz for all h∈W01,p(Ω), all n∈N,$(3.46)

$0≤u¯n≤u¯λ∗+for all n∈N(from the monotonicity of λ↦u¯λ).$(3.47)

Then (3.46) and (3.47) imply that

$u¯nn≥1⊆W01,p(Ω)is bounded.$(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

$∥u¯n∥∞≤c17for all n∈N.$(3.49)

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

$u¯n∈C01,α(Ω¯)and∥u¯n∥C01,α(Ω¯)≤c18for all n∈N.$

The compact embedding of $\begin{array}{}{C}_{0}^{1,\alpha }\end{array}$(Ω) into $\begin{array}{}{C}_{0}^{1}\end{array}$(Ω), implies that at least for a subsequence we have

$u¯n→u~λ∗+in C01(Ω¯),u~λ∗+∈Sλ∗++.$(3.50)

We claim that $\begin{array}{}{\stackrel{~}{u}}_{{\lambda }_{\ast }^{+}}={\overline{u}}_{{\lambda }_{\ast }^{+}}\end{array}$. Arguing by contradiction, suppose that $\begin{array}{}{\stackrel{~}{u}}_{{\lambda }_{\ast }^{+}}\ne {\overline{u}}_{{\lambda }_{\ast }^{+}}\end{array}$. So, we can find z0Ω such that

$u¯λ∗+(z0)

which contradicts the strict monotonicity of λuλ. Hence $\begin{array}{}{\stackrel{~}{u}}_{{\lambda }_{\ast }^{+}}={\overline{u}}_{{\lambda }_{\ast }^{+}}\end{array}$ and for the original sequence we have

$u¯n→u¯λ∗+in C01(Ω¯)as n→∞,⇒λ↦u¯λis left continuous.$

2. In this case Proposition 3.5(b) implies that λvλ is strictly decreasing from ℒ into $\begin{array}{}{C}_{0}^{1}\end{array}$(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 $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$ and $\begin{array}{}{\lambda }_{\ast }^{-}\end{array}$ are admissible. In the next proposition we show that $\begin{array}{}{\lambda }_{\ast }^{+},{\lambda }_{\ast }^{-}\end{array}$ are not admissible and so

$L+=(0,λ∗+)andL−=(0,λ∗−).$

#### Proposition 3.9

If hypotheses H(f) hold, then $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$ ∉ ℒ+ and $\begin{array}{}{\lambda }_{\ast }^{-}\end{array}$ ∉ ℒ.

#### Proof

We do the proof for $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$, the proof for $\begin{array}{}{\lambda }_{\ast }^{-}\end{array}$ being similar.

We argue indirectly. So, suppose that $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$ ∈ ℒ+. From Proposition 3.7, we know that problem $\begin{array}{}\left({\mathrm{P}}_{{\lambda }_{\ast }^{+}}\right)\end{array}$ admits a minimal positive solution u* = $\begin{array}{}{\overline{u}}_{{\lambda }_{\ast }^{+}}\end{array}$ ∈ int C+. Let ϑ < $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$ < λ. We know that u*uϑ ∈ int C+. So, we can define the following Caratheodory function:

$β^λ(z,x)=λu¯ϑ(z)p−1+f(z,u¯ϑ(z))if x

Let λ(z, x) = $\begin{array}{}{\int }_{0}^{x}\end{array}$ β̂λ(z, s) ds and consider the C1-functional ŷλ : $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) → ℝ defined by

$y^λ(u)=1p∥Du∥pp+12∥Du∥22−∫ΩB^λ(z,u)dz,for all u∈W01,p(Ω).$

Evidently ŷλ(⋅) is coercive and sequentially lower semicontinuous. So, we can find ûλ$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) such that

$y^λ(u^λ)=infy^λ(u):u∈W01,p(Ω),⇒y^λ′(u^λ)=0,⇒〈Ap(u^λ),h〉+〈A(u^λ),h〉=∫Ωβ^λ(z,u^λ)hdzfor all h∈W01,p(Ω).$

First we choose h = (uϑûλ)+$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω). Then

$〈Ap(u^λ),(u¯ϑ−u^λ)+〉+〈A(u^λ),(u¯ϑ−u^λ)+〉=∫Ωλu¯ϑp−1+f(z,u¯ϑ)(u¯ϑ−u^λ)+dz≥∫Ωϑu¯ϑp−1+f(z,u¯ϑ)(u¯ϑ−u^λ)+dz(since ϑ<λ)=〈Ap(u¯ϑ),(u¯ϑ−u^λ)+〉+〈A(u¯ϑ),(u¯ϑ−u^λ)+〉(since u¯ϑ∈Sϑ+),$

$⇒u¯ϑ≤u^λ.$

Similarly, choosing h = (ûλu*)+$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω), we obtain

$u^λ≤u¯∗.$

So, we have proved that

$u^λ∈[u¯ϑ,u¯∗],⇒λ∈L+, a contradiction since λ>λ∗+.$

This means that $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$ ∉ ℒ+.

Similarly we show that $\begin{array}{}{\lambda }_{\ast }^{-}\end{array}$ ∉ 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 $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$ ∈ ℒ+ and $\begin{array}{}{\lambda }_{\ast }^{-}\end{array}$ ∈ 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, $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$) problem (Pλ) has at least two positive solutions uλ, ûλ ∈ int C+, uλûλ, uλûλ;

2. for all λ ∈ ℒ = (0, $\begin{array}{}{\lambda }_{\ast }^{-}\end{array}$) problem (Pλ) has at least two negative solutions vλ, λ ∈ int C+, λvλ, vλλ.

#### Proof

1. Since λ ∈ ℒ+, we can find uλ$\begin{array}{}{S}_{\lambda }^{+}\end{array}$ ⊆ int C+. Using uλ ∈ int C+ to truncate the reaction of problem (Pλ), we introduce the Caratheodory function $\begin{array}{}{g}_{\lambda }^{+}\end{array}$(z, x) defined by

$gλ+(z,x)=λuλ(z)p−1+f(z,uλ(z))if x≤uλ(z)λxp−1+f(z,x)if uλ(z)(3.51)

We set $\begin{array}{}{G}_{\lambda }^{+}\end{array}$(z, x) = $\begin{array}{}{\int }_{0}^{x}{g}_{\lambda }^{+}\left(z,s\right)\end{array}$ ds and consider the C1-functional $\begin{array}{}{\stackrel{^}{\phi }}_{\lambda }^{+}:{W}_{0}^{1,p}\left(\mathit{\Omega }\right)\to \mathbb{R}\end{array}$ defined by

$φ^λ+(u)=1p∥Du∥pp+12∥Du∥22−∫ΩGλ+(z,u)dzfor all u∈W01,p(Ω).$

Let η ∈ (λ, $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$) and uηSη ⊆ int C+ such that uηuλ ∈ int C+. Consider the Caratheodory function

$g~λ+(z,x)=gλ+(z,x)if x≤uη(z)gλ+(z,uη(z))if uη(z)(3.52)

We set $\begin{array}{}{\stackrel{~}{G}}_{\lambda }^{+}\left(z,x\right)={\int }_{0}^{x}{\stackrel{~}{g}}_{\lambda }^{+}\left(z,s\right)\end{array}$ ds and consider the C1-functional $\begin{array}{}{\stackrel{~}{\phi }}_{\lambda }^{+}:{W}_{0}^{1,p}\left(\mathit{\Omega }\right)\to \mathbb{R}\end{array}$ defined by

$φ~λ+(u)=1p∥Du∥pp+12∥Du∥22−∫ΩG~λ+(z,u)dzfor all u∈W01,p(Ω).$

As before we can check that

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

Moreover, since $\begin{array}{}{\stackrel{~}{\phi }}_{\lambda }^{+}\end{array}$ is coercive and sequentially weakly lower semicontinuous, we can find λ$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) such that

$φ~λ+(u~λ)=infφ~λ+(u):u∈W01,p(Ω),⇒u~λ∈Kφ~λ+⊆[uλ,uη]∩intC+(see (3.53)).$(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

$uλ is a local C01(Ω¯)−minimizer of φ^λ+,⇒uλ is a local W01,p(Ω)−minimizer of φ^λ+(see Proposition 2.1).$(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)(3.58)

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

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

$φ^λ+(tu)→−∞as t→+∞.$(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 ûλ$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) such that

$u^λ∈Kφ^λ+⊆[uλ)∩intC+ (see (3.53))andmλ+≤φ^λ+(u^λ).$(3.61)

From (3.58) and (3.61) we conclude that

$u^λ∈intC+is a solution of (Pλ),uλ≤u^λ,uλ≠u^λ.$

2. In this case, let vλSλ ⊆ –int C+ and consider the Caratheodory function $\begin{array}{}{g}_{\lambda }^{-}\end{array}$(z, x) defined by

$gλ−(z,x)=λ|x|p−2x+f(z,x)if x≤vλ(z)λ|vλ(z)|p−2vλ(z)+f(z,vλ(z))if x>vλ(z).$(3.62)

We set $\begin{array}{}{G}_{\lambda }^{-}\left(z,x\right)={\int }_{0}^{x}{g}_{\lambda }^{-}\left(z,s\right)\end{array}$ ds and consider the C1-functional $\begin{array}{}{\stackrel{^}{\phi }}_{\lambda }^{-}:{W}_{0}^{1,p}\left(\mathit{\Omega }\right)\to \mathbb{R}\end{array}$ defined by

$φ^λ−(u)=1p∥Du∥pp+12∥Du∥22−∫ΩGλ−(z,u)dzfor all u∈W01,p(Ω).$

Working as in part (a) this time using (3.62) and the functional $\begin{array}{}{\stackrel{^}{\phi }}_{\lambda }^{-}\end{array}$, 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 $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$ ∈ (0, +∞) such that

• for all λ > $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$ problem (Pλ) has no positive solutions;

• for all λ ∈ (0, $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$) problem (Pλ) has at least two positive solutions uλ, ûλ ∈ int C+, uλûλ, uλûλ;

• for all λ ∈ (0, $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$) problem (Pλ) has a smallest positive solution uλ ∈ int C+ and the map λuλ from+ = (0, $\begin{array}{}{\lambda }_{\ast }^{+}\end{array}$) into C+ is strictly increasing and left continuous;

2. there exists $\begin{array}{}{\lambda }_{\ast }^{-}\end{array}$ ∈ (0, +∞) such that

• for all λ > $\begin{array}{}{\lambda }_{\ast }^{-}\end{array}$ problem (Pλ) has no negative solutions;

• for all λ ∈ (0, $\begin{array}{}{\lambda }_{\ast }^{-}\end{array}$) problem (Pλ) has at least two negative solutions vλ, ∈ –int C+, λvλ, vλλ;

• for all λ ∈ (0, $\begin{array}{}{\lambda }_{\ast }^{-}\end{array}$) problem (Pλ) has a biggest negative solution vλ ∈ –int C+ and the map λvλ from = (0, $\begin{array}{}{\lambda }_{\ast }^{-}\end{array}$) 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. |$\begin{array}{}{f}_{x}^{\prime }\end{array}$(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) = $\begin{array}{}{\int }_{0}^{x}\end{array}$ f(z, s) ds, then $\begin{array}{}\underset{x\to ±\mathrm{\infty }}{lim}\frac{F\left(z,x\right)}{|x{|}^{p}}=+\mathrm{\infty }\end{array}$ uniformly for a.a. zΩ;

3. there exist η̂ > 0 and $\begin{array}{}q\in \left(\left(r-p\right)max\left\{\frac{N}{p},1\right\},{p}^{\ast }\right)\end{array}$ such that

$0<η^≤lim infx→±∞f(z,x)x−pF(z,x)|x|quniformly for a.a. z∈Ω;$

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

$λ^m(2)≤fx′(z,0)=limx→0f(z,x)x≤λ^m+1(2)uniformly for a.a. z∈Ω,fx′(⋅,0)≢λ^m(2),fx′(⋅,0)≢λ^m+1(2).$

#### Remark

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

Let $\begin{array}{}{\lambda }_{\ast }=min\left\{{\lambda }_{\ast }^{+},{\lambda }_{\ast }^{-}\right\}>0.\end{array}$ Also, for λ > 0, let φλ$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) → ℝ be the energy (Euler) functional for problem (Pλ) defined by

$φλ(u)=1p∥Du∥pp+12∥Du∥22−λp∥u∥pp−∫ΩF(z,u)dzfor all u∈W01,p(Ω).$

We know that φλC2($\begin{array}{}{W}_{0}^{1,p}\end{array}$(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 $\begin{array}{}{d}_{m}=\underset{k=1}{\overset{m}{⨁}}E\left({\stackrel{^}{\lambda }}_{k}\left(2\right)\right).\end{array}$

#### Proof

Let ζ̂λ : $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) → ℝ be the C2-functional defined by

$ζ^λ(u)=12∥Du∥22−λp∥u∥pp−∫ΩF(z,u)dzfor all u∈H01(Ω)$

We consider the following orthogonal direct sum decomposition of the space $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω):

$H01(Ω)=H¯m⊕H^m+1,$(4.1)

with

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

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

$12[ϑ(z)−ϵ]x2≤F(z,x)≤12[ϑ^(z)+ϵ]x2for a.a. z∈Ω, all |x|≤δ.$(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

$u∈H¯m,∥u∥H01(Ω)≤ρ1⇒|u(z)|≤δ for all z∈Ω¯ (see (4.2)).$(4.3)

Therefore for uHm with $\begin{array}{}\parallel u{\parallel }_{{H}_{0}^{1}\left(\mathit{\Omega }\right)}\le {\rho }_{1},\end{array}$ we have

$ζ^λ(u)≤12∥Du∥22−12∫Ωϑ(z)u2dz+ϵ2∥u∥H01(Ω)2(see (4.3))≤12[−c2+ϵ]∥u∥H01(Ω)2(see Proposition 2.4(b)).$

Choosing ϵ ∈ (0, c2), we obtain

$ζ^λ(u)≤0for all u∈H¯mwith ∥u∥H01(Ω)≤ρ1.$(4.4)

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

$F(z,x)≤12[ϑ^(z)+ϵ]x2+c19|x|rfor a.a. z∈Ω,all x∈R$(4.5)

with c19 > 0. For uĤm+1 we have

$ζ^λ(u)≥12∥Du∥22−λp∥u∥pp−12∫Ωϑ^(z)u2dz−ϵ2∥u∥H01(Ω)2−c19∥u∥rr(see(4.5))≥12[c1−ϵ]∥u∥H01(Ω)2−c20λ∥u∥H01(Ω)p+∥u∥H01(Ω)rfor some c20>0.$

Choosing ϵ ∈ (0, c1) and assuming that $\begin{array}{}\parallel u{\parallel }_{{H}_{0}^{1}\left(\mathit{\Omega }\right)}\le 1,\end{array}$ we have

$ζ^λ(u)≥c21∥u∥H01(Ω)2−c22∥u∥H01(Ω)pfor all u∈H01(Ω) and with c21>0,c22=c22(λ)>0.$

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

$ζ^λ(u)>0for all u∈H^m+1,0<∥u∥H01(Ω)≤ρ2.$(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($\begin{array}{}{H}_{0}^{1}\end{array}$(Ω), ℝ), we can apply Proposition 2.3 of Su [26] and infer that

$Ck(ζ^λ,0)=δk,dmZfor all k∈N0.$(4.7)

Let $\begin{array}{}{\zeta }_{\lambda }={\stackrel{^}{\zeta }}_{\lambda }{|}_{{W}_{0}^{1,p}\left(\mathit{\Omega }\right)}.\end{array}$ Since $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) is dense in $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω), from (4.7) we have

$Ck(ζλ,0)=Ck(ζ^λ,0)for all k∈N0 (see [10]),⇒Ck(ζλ,0)=δk,dmZfor all k∈N0 (see (4.7)).$(4.8)

Note that

$|φλ(u)−ζλ(u)|=1p∥u∥p$(4.9)

and

$|〈φλ′(u)−ζλ′(u),h〉|=|〈Ap(u),h〉|≤∥Du∥pp−1∥h∥⇒∥φλ′(u)−ζλ′(u)∥∗≤∥u∥p−1.$(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

$Ck(ζλ,0)=Ck(φλ,0)for all k∈N0,⇒Ck(φλ,0)=δk,dmZfor all k∈N0(see (4.8)).$

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

$u¯λ∈intC+andv¯λ∈−intC+.$

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

$wλ(z,x)=λ|v¯λ(z)|p−2v¯λ(z)+f(z,v¯λ(z))if x(4.11)

We set Wλ(z, x) = $\begin{array}{}{\int }_{0}^{x}\end{array}$ wλ(z, s) ds and consider the C1-functional τ̂λ : $\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) → ℝ defined by

$τ^λ(u)=1p∥Du∥pp+12∥Du∥22−∫ΩWλ(z,u)dzfor all u∈W01,p(Ω).$

Also, let $\begin{array}{}{\stackrel{^}{\tau }}_{\lambda }^{±}\end{array}$ be the positive and negative truncations of τ̂λ, that is,

$τ^λ±(u)=1p∥Du∥pp+12∥Du∥22−∫ΩWλ(z,±u±)dzfor all u∈W01,p(Ω).$

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 $\begin{array}{}{\stackrel{^}{\tau }}_{\lambda }^{+}\end{array}$ is coercive (see (4.11)) and sequentially weakly lower semicontinuous. So, we can find λ$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) such that

$τ^λ+(u~λ)=infτ^λ+:u∈W01,p(Ω).$(4.14)

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

$τ^λ+(u~λ)<0=τ^λ+(0),⇒u~λ≠0.$(4.15)

From (4.14) we have

$u~λ∈Kτ^λ+={0,u¯λ}(see (4.12))⇒u~λ=u¯λ∈intC+(see (4.15))$

Note that

$τ^λ+|C+=τ^λ|C+.$

So, it follows that

$u¯λ∈intC+is a local C01(Ω¯)−minimizer of τ^λ,⇒u¯λ∈intC+is a local W01,p(Ω)−minimizer of τ^λ(see Proposition 2.1).$

Similarly for vλ ∈ –int C+, using this time the functional $\begin{array}{}{\stackrel{^}{\tau }}_{\lambda }^{-}\end{array}$.

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

$τ^λ(v¯λ)≤τ^λ(u¯λ)ρ.$(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 y0$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) such that

$y0∈Kτ^λ⊆[v¯λ,u¯λ]∩C01(Ω¯)(see (4.12)),m^λ≤τ^λ(y0)(see (4.16)).$(4.18)

From (4.16) and (4.18) we see that

$y0∉{u¯λ,v¯λ}.$(4.19)

We consider the homotopy

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

Suppose we could find {tn}n≥1 ⊆ [0, 1] and {un}n≥1$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) such that

$tn→tin[0,1],un→0inW01,p(Ω),h^u′(tn,un)=0 for all n∈N.$(4.20)

From the equality in (4.20), we have

$〈Ap(un),h〉+〈A(un),h〉=(1−tn)∫Ωwλ(t,un)hdz+tn∫Ωλ|un|p−2unhdz+tn∫Ωf(z,un)hdzfor all h∈W01,p(Ω),all n∈N.$(4.21)

In (4.21) we choose h = un$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) and we infer that

${un}n≥1⊆W01,p(Ω) is bounded.$

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

$un∈C01,α(Ω¯)and∥un∥C01,α(Ω¯)≤c23for all n∈N.$(4.22)

From (4.20) and the compact embedding of $\begin{array}{}{C}_{0}^{1,\alpha }\end{array}$(Ω) into $\begin{array}{}{C}_{0}^{1}\end{array}$(Ω), we have

$un→0inC01(Ω¯),⇒un∈[v¯λ,u¯λ]for all n≥n0,⇒{un}n≥n0⊆Kτ^λ(see (4.11)).$

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

$Ck(τ^λ,0)=Ck(φλ,0)for all k∈N0,⇒Ck(τ^λ,0)=δk,dmZfor all k∈N0.$(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 y0$\begin{array}{}{C}_{0}^{1}\end{array}$(Ω) is a nodal solution of (Pλ).

Let a : ℝN → ℝN be defined by

$a(y)=|y|p−2y+yfor all y∈RN.$

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

$div a(Du)=Δpu+Δufor all u∈W01,p(Ω).$

We have

$∇a(y)=|y|p−2I+y⊗y|y|2+Ifor all y∈Rn⇒∇a(y)ξ,ξRN≥|ξ|2for all y,ξ∈RN.$

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

$v¯λ(z)(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

$x↦f(z,x)+ξ^ρ|x|p−2x$

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

$−Δpy0(z)−Δy0(z)+ξ^ρ|y0(z)|p−2y0(z)=λ|y0(z)|p−2y0(z)+f(z,y0(z))+ξ^ρ|y0(z)|p−2y0(z)≤λu¯λ(z)p−1+f(z,u¯λ(z))+ξ^ρu¯λ(z)p−1=−Δpu¯λ(z)−Δu¯λ(z)+ξ^ρu¯λ(z)p−1for a.a. z∈Ω.$(4.26)

We set

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

Evidently h1, h2L(Ω) and we have

$λu¯λ(z)p−1−|y0(z)|p−2y0(z)≤h2(z)−h1(z)for a.a.z∈Ω,⇒h1≺h2(see (4.25)).$

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

$u¯λ−y0∈intC+.$

In a similar fashion, we show that

$y0−v¯λ∈intC+,⇒y0∈intC01(Ω¯)[v¯λ,u¯λ].$(4.27)

Consider the homotopy

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

Suppose we could find {tn}n≥1 ⊆ [0, 1] and {un}n≥1$\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω) such that

$tn→tin[0,1],un→y0inW01,p(Ω),h~u′(tn,un)=0 for all n∈N.$(4.28)

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

$un→y0inC01(Ω¯) as n→∞,⇒un∈[v¯λ,u¯λ]for all n≥n0(see (4.27))⇒{un}n≥n0⊆Kτ^λ(see (4.11)),$

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

$Ck(τ^λ,y0)=Ck(φλ,y0)for all k∈N0,$(4.29)

$⇒C1(φλ,y0)≠0(see (4.24)).$(4.30)

But φλC2($\begin{array}{}{W}_{0}^{1,p}\end{array}$(Ω), ℝ). So, from (4.29) and Proposition 3.5, Claim 3, in Papageorgiou-Rădulescu [9], we have

$Ck(φλ,y0)=δk,1Zfor all k∈N0,$(4.31)

$⇒Ck(τ^λ,y0)=δk,1Zfor all k∈N0(see (4.29)).$(4.32)

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

$Ck(τ^λ,u¯λ)=Ck(τ^λ,v¯λ)=δk,0Zfor all k∈N0.$(4.33)

From (4.23) we have

$Ck(τ^λ,0)=δk,dmZfor all k∈N0.$(4.34)

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

$Ck(τ^λ,∞)=δk,0Zfor all k∈N0.$(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 ŷ$\begin{array}{}{C}_{0}^{1}\end{array}$(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 $\begin{array}{}{C}_{0}^{1}\end{array}$(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

$uλ,u^λ∈intC+,uλ≤u^λ,uλ≠u^λ,vλ,v^λ∈−intC+,v^λ≤vλ,vλ≠v^λ,y0,y^,y~∈C01(Ω¯)nodal with y0,y^∈intC01(Ω¯)[v¯λ,u¯λ].$

## 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”.

## References

• [1]

V. Benci, P. D’Avenia, D. Fortunato, L. Pisani, Solitons in several dimensions: Derrick’s problem and infinitely many solutions, Arch. Rat. Mech. Anal. 154 (2000), 297–324.

• [2]

V.V. Zhikov, Averaging functionals of the calculus of variations and elasticity theory, Math. USSR-Izvestiya 29 (1987), 33–66.

• [3]

S. Aizicovici, N.S. Papageorgiou, V. Staicu, Nodal solutions for (p, 2)-equations, Trans. Amer. Math. Soc. 367 (2015), 7343–7372. Google Scholar

• [4]

S. Aizicovici, N.S. Papageorgiou, V. Staicu, Multiple solutions with sign information for (p, 2)-equations with asymmetric resonant reaction, Pure Appl. Funct. Anal., in press. Google Scholar

• [5]

S. Cingolani, M. Degiovanni, Nontrivial solutions for p-Laplace equations with right hand side having p-linear growth, Comm. Partial Diff. Equ. 30 (2005), 1191–1203.

• [6]

L. Gasiński, N.S. Papageorgiou, Multiplicity of positive solutions for eigenvalue problems of (p, 2)-equations, Bound. Value Probl. 152 (2012), 1–17.

• [7]

L. Gasiński, N.S. Papageorgiou, Nonlinear elliptic equations with a jumping reaction, J. Math. Anal. Appl. 443 (2016), 1033–1070.

• [8]

T. He, P. Guo, Y. Huang, Y. Lei, Multiple nodal solutions for nonlinear nonhomogeneous elliptic problems with a superlinear reaction, Nonlin. Anal. - RWA 42 (2018), 207–219.

• [9]

N.S. Papageorgiou, V.D. Rădulescu, Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim. 69 (2014), 393–430.

• [10]

N.S. Papageorgiou, V.D. Rădulescu, Noncoercive resonant (p, 2)-equations, Appl. Math. Optim. 76 (2017), 621–639.

• [11]

N.S. Papageorgiou, V.D. Rădulescu, D. Repovš, On a class of parametric (p, 2)-equations, Appl. Math. Optim. 75 (2017), 193–228.

• [12]

M. Sun, Multiplicity of solutions for a class of quasilinear elliptic equations at resonance, J. Math. Anal. Appl. 386 (2012), 661–668.

• [13]

M. Sun, M. Zhang, J. Su, Critical groups at zero and multiple solutions for quasilinear elliptic equations, J. Math. Anal. Appl. 428 (2015), 696–712.

• [14]

G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203-1219.

• [15]

N.S. Papageorgiou, V.D. Rădulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction, Adv. Nonlin. Studies 16 (2016), 737-764. Google Scholar

• [16]

L. Gasiński, N.S. Papageorgiou, Positive solutions for the Robin p-Laplacian problem with competing nonlinearities, Adv. Calc. Var., 12 (2019), 31-56.

• [17]

D. Motreanu, V. Motreanu, N.S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York (2014). Google Scholar

• [18]

L. Gasiński, N.S. Papageorgiou, Nonlinear Analysis, Chapman & Hall / CRC, Boca Raton, Fl. (2006) Google Scholar

• [19]

P. Pucci, J. Serrin, The Maximum Principle, Birkhäuser, Basel (2007). Google Scholar

• [20]

N.S. Papageorgiou, V.D. Rădulescu, D. Repovš, Positive solutions for nonlinear nonhomogeneous parametric Robin problems, Forum Math. 30 (2018), 553–580.

• [21]

J.I. Diaz, J.E. Saá, Existence and unicité de solutions positives pour certaines equations elliptiques quasilineaires, CRAS Paris 305 (1987), 521–524. Google Scholar

• [22]

M. Filippakis, N.S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear equations with the p-Laplacian, J. Differential Equation 245 (2008), 1883–1922.

• [23]

S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, Kluwer Academic Publishers, Dordrecht, The Netherlands (1997). Google Scholar

• [24]

N.S. Papageorgiou, V.D. Rădulescu, Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities, Discr. Cont. Dyn. System - A 35 (2015), 5008–5036. Google Scholar

• [25]

S. Aizicovici, N.S. Papageorgiou, V. Staicu, Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints, Memoirs Amer. Math. Soc., Vol. 196, No. 915, (2008), pp. 70. Google Scholar

• [26]

J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlin. Anal. 48 (2002), 881–895.

• [27]

L. Gasiński, N.S. Papageorgiou, Exercises in Analysis. Part 2: Nonlinear Analysis, Springer, Cham (2016). Google Scholar

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,

Export Citation