Constant sign and nodal solutions for superlinear (p, q)–equations with indefinite potential and a concave boundary term

1 Introduction

In this paper we study the following parametric (p, q)–equation:

In this problem Ω ⊆ ℝ^N is a bounded domain with a C²–boundary ∂ Ω. For every 1 < r < ∞ by Δ_r we denote the r–Laplace differential operator defined by

for all u ∈ W^1,r(Ω).

In problem (P_λ) we have the sum of two such operators. Therefore the differential operator driving the equation, is not homogeneous. In addition there is a potential term u → ξ(z)|u|^p−2 u which is indefinite since the potential function ξ ∈ L^∞(Ω) is in general sign–changing. So, the left hand side of problem (P_λ) is not coercive. The source ( reaction ) term f(z, x) is a Carathéodory function ( that is, for all x ∈ ℝ, z → f(z, x) is measurable and for a.a. z ∈ Ω, x → f(z, x) is continuous ). We assume that f(z, ⋅) exhibits (p − 1)–superlinear growth as x → ± ∞. In fact we assume that f(z, ⋅) satisfies the well–known Ambrosetti–Rabinowitz condition ( the AR–condition ) which is common in the literature for superlinear problems. It is an interesting open problem if we can relax the AR–condition and use a more general one, as for example in Mugnai-Papageorgiou [12] or alternatively in Papageorgiou-Rădulescu [13]. Near zero we assume that f(z, ⋅) has a kind of oscillatory behavior. In the boundary condition, ∂u∂npq denotes conormal derivative corresponding to the (p, q)–Laplace differential operator. We interpret this directional derivative using the nonlinear Green's identity ( see Corollary 1.5.17 of Papageorgiou-Rădulescu-Repovš [19, p. 35] ). We know that for all u ∈ C¹(Ω),

with n(⋅) being the outward unit normal on ∂Ω. The boundary coefficient β(⋅) is Hölder continuous on ∂Ω and strictly positive, and λ > 0 is a parameter. Since 1 < τ < q < p, we see that the boundary condition contributes a concave term in the energy functional of the problem. Therefore, in problem we have the competing effects of two items which are of different nature. One is the “convex ( superlinear ) source term f(z, ⋅) and the other is the parametric “concave” ( sublinear ) boundary term. By restricting appropriately the parameter, we will be able to balance the different effects of these two terms. Problem (P_λ) is a variant of the classical “concave–convex” problem. The different feature of our problem here is that the concave contribution comes from the boundary condition.

For the classical concave–convex problem, where the parametric concave term is part of the reaction of the problem, we refer to the works of Ambrosetti-Brezis-Cerami [1], Garcia Azorero-Manfredi-Peral Alonso [4], Guo-Zhang [6], Leonardi-Papagergiou [9], Marano-Marino-Papagergiou [11], Papageorgiou-Rădulescu-Repovš [18]. Analogous nonlinear parametric problems but with different competition phenomena, can be found in Bai-Motreanu-Zeng [2], Papageorgiou-Scapellato [20], Papageorgiou-Zhang [22].

Problems with a superlinear source term and a concave boundary condition were studied by Sabina de Lis-Segura de León [24], Hu-Papageorgiou [8], Papageorgiou-Rădulescu [15], Papageorgiou-Scapellato [21]. All these works deal with parametric problems, with the parameter appearing in the boundary term. They focus on the existence of positive solutions and they prove bifurcation–type results describing the changes in the set of positive solutions as the parameter λ > 0 varies. We mention that in all these works the potential coefficient ξ(⋅) is strictly positive, making the left side of the equation coercive. In particular Sabina de Lis-Segura de León [24] consider equations driven by the p–Laplacian plus a potential term with ξ ≡ 1. So, in their equation the left hand side is both homogeneous and coercive and these strong properties are exploited in their proof. In Hu-Papageorgiou [8], the problem is semilinear driven by the Laplacian plus an indefinite potential term. In Papageorgiou-Rădulescu [14], the equation is driven by the p–Laplacian and only existence of positive solutions is proved. Finally, in the recent work of Papageorgiou-Scapellato [21], the authors deal with equations driven by the (p, 2)–Laplacian plus a positive potential term and β ≡ 1. As we already mentioned, all the aforementioned works focus on the existence and multiplicity of positive solutions as the parameter λ > 0 varies.

In the present paper, using the variational tools from the critical point theory, together with suitable truncation and perturbation techniques and the use of critical groups ( Morse theory ), we show that for all small values of the parameter λ > 0, problem (P_λ) has at least five nontrivial smooth solutions which can be ordered linearly and we can provide sign information for all of them. More precisely, we produce two positive solutions, two negative solutions and a fifth one which is nodal ( sign–changing ). This is the first result on the existence of nodal solutions for problems with concave boundary condition.

2 Auxiliary results and hypotheses

The main spaces which will be used in the study of problem (P_λ) are the Sobolev space W^1,p(Ω), the Banach space C¹(Ω) and the boundary Lebesgue spaces L ^s(∂Ω), 1 ≤ s ≤ ∞.

By ∥ ⋅ ∥ we denote the norm of the Sobolev space W^1,p(Ω). We know that

for all u ∈ W^1,p(Ω).

The Banach space C¹(Ω) is ordered with positive (order) cone

This cone has a nonempty interior given by

On ∂Ω we consider the (N − 1)–dimensional Hausdorff ( surface ) measure σ(⋅). Using this measure, we can define in the usual way the boundary Lebesgue space L^s(∂Ω). From the theory of Sobolev spaces (see [19]), we know that there exists a unique continuous linear map y₀ : W^1,p(Ω) → L^p(∂Ω), known as the “trace map”, such that

for all u ∈ W^1,p(Ω) ∩ C(Ω).

So, the trace map extends the notion of boundary values to all Sobolev functions. We know that the trace map y₀ (⋅) is compact into L^s(∂Ω) for all s∈[1,(N−1)pN−p) if p < N and into L^s(∂Ω) for all 1 ≤ s < ∞ if N ≤ p. Moreover, we have

In the sequel for the sake of notational simplicity, we drop the use of the trace map y₀ (⋅). All restrictions of the Sobolev functions on ∂Ω, are understood in the sense of traces.

For x ∈ ℝ, we set x^± = max{± x, 0}. Then given u ∈ W^1,p(Ω), we define u^±(z) = u(z)^±for all z ∈ Ω. We have

Given u, v ∈ W^1,p(Ω) with u ≤ v, we define the following sets

Given a set S ⊆ W^1,p(Ω), we say that S is “downward directed” ( resp. “upward directed” ), if for every pair u₁, u₂ ∈ S, we can find u ∈ S such that u ≤ u₁, u ≤ u₂ ( resp. for every v₁, v₂ ∈ S, we can find v ∈ S such that v₁ ≤ v, v₂ ≤ v ).

Let 1 < r < ∞ and consider the map A_r : W^1,r(Ω) → W^1,r(Ω)^* defined by

for all u, h ∈ W^1,r(Ω).

From Problem 2.192 of Gasinski-Papageorgiou [5, p. 279], we have:

If X is a Banach space and φ ∈ C¹(X, ℝ), we say that φ(⋅) satisfies the “C–condition”, if it has the following property:

This is essentially a compactness–type condition on the function φ(⋅). It compensates for the fact that the ambient space X is not in general locally compact ( since in almost all situations of interest, X is infinite dimensional ). Using this property, one can prove a deformation theorem, from which follows the minimax theory of the critical values of φ ( see [19], Chapter 5 ).

We define

For (Y₁, Y₂) a topological pair such Y₂ ⊆ Y₁ ⊆ X. For every k ∈ ℕ, by H_k(Y₁, Y₂) we denote the k-th relative singular homology group with coefficients in ℝ. Then the relative singular homology groups H_k (Y₁, Y₂) are real vector spaces. Let u ∈ K_φ isolated and c = φ(u). The “critical groups” of φ(⋅) at u, are defined by

for all k ∈ ℕ, with U a neighborhood of u such that K_φ ∩ φ^c ∩ U = {u}. The excision property of singular homology implies that the above definition is independent of the choice the isolating neighborhood U.

Now we will introduce our hypotheses on the data of problem (P_λ):

H₀: ξ ∈ L^∞(Ω), β ∈ C^0,α(∂Ω) with α ∈ (0, 1) and β(z) > 0 for all z ∈ ∂ Ω.

H₁: f : Ω × ℝ → ℝ is a Carathéodory function such that f(z, 0) = 0 for a.a. z ∈ Ω and

1. |f(z, x)| ≤ a(z)[1 + |x|^r−1] for a.a. z ∈ Ω, all x ∈ ℝ, with a ∈ L^∞(Ω), p < r < p^* with p^* being the critical Sobolev exponent for p > 1, that is,

   p∗=NpN−pif p<N+∞if N≤p;

2. If F(z, x) = ∫0x f(z, s) ds, then there exist μ > p and M > 0 such that

   0<μF(z,x)≤f(z,x)x for a.a. z∈Ω, all |x|≥M,0<essinfΩ⁡F(⋅,±M);

3. there exist constants θ₋ < 0 < θ₊ such that

   f(z,θ+)−ξ(z)θ+p−1≤−c+<0<c−≤f(z,θ−)+ξ(z)|θ−|p−1 for a.a. z∈Ω;

4. there exists δ₀ > 0 and d ∈ [τ, q] such that

   0<f(z,x)x≤dF(z,x) for a.a. z∈Ω, all |x|≤δ0

   and for every ρ > 0, we can find ξ̂_ρ > 0 such that for a.a. z ∈ Ω, the function

   x→f(z,x)+ξ^ρ|x|p−2x

   is nondecreasing on [−ρ, ρ].

3 Constant sign solutions

We start by proving the existence of two constant sign solutions located in the order intervals [0, θ₊] and [θ₋, 0] respectively. To do this we do not need all the conditions in hypotheses H₁. More precisely we do not need hypothesis H₁(ii) which describes the asymptotic behavior as x → ± ∞ of the source term. Using truncation and perturbation techniques we focus on the intervals [0, θ₊] and [θ₋, 0] and so the behavior of f(z, ⋅) near ± ∞ becomes irrelevant.

Next we will produce two more sign smooth solutions, localized with respect to u₀ and v₀ respectively ( see Proposition 3 ). With η > ∥ξ∥_∞ as before, we consider the following truncation perturbations of the reaction

Also we consider the following truncations of the boundary term

We set

and introduce the C¹–functional ψ±λ : W^1,p(Ω) → ℝ defined by

for all u ∈ W^1,p(Ω), all 0 < λ ≤ λ^*.

Also we consider the following truncations of g_± (z, ⋅) and b_±(z, ⋅)