On the set of positive solutions for resonant Robin (p, q)-equations

1 Introduction

In this paper we study the following nonlinear Robin problem driven by the (p, q)-Laplacian:

In this problem, Ω ⊆ ℝ^N is a bounded domain with a C²-boundary ∂Ω. For every r ∈ (1, + ∞), we denote by Δ_r the r-Laplace differential operator defined by

In problem (P_λ) we have the sum of two such operators and so the differential operator (left-hand side) in problem (P_λ) is not homogeneous. There is also a potential term ξ(z)u^p–1 with ξ ∈ L^∞(Ω), ξ ⩾ 0. In the reaction (right-hand side) of problem (P_λ), we have the combined effects of two distinct terms with different asymptotic behavior as x → + ∞. The first term, is the parametric (p – 1)-sublinear (that is, “concave”) nonlinearity u ↦ λu^τ–1(1 < τ < q < p) and the second term is a Carathéodory perturbation f(z, x) (that is, for all x ∈ ℝ, the mapping z ↦ f(z, x) is measurable and for a.a. z ∈ Ω, the function x ↦ f(z, x) is continuous), which exhibits (p – 1)-linear growth as x → + ∞ and is resonant with respect to a nonprincipal variational eigenvalue of the operator u ↦ –Δ_pu + ξ(z) |u|^p–2 u with Robin boundary condition. In the Robin boundary condition, ∂u∂npq denotes the conormal derivative corresponding to the (p, q)-Laplacian. This derivative is understood via the nonlinear Green’s identity (see Papageorgiou, Rădulescu & Repovš [20, pp. 33-35]). If u ∈ C¹(Ω), then ∂u∂npq=|Du|p−2+|Du|q−2∂u∂n, with n(⋅) being the outward unit normal on ∂Ω. Therefore, problem (P_λ) is a nonlinear extension to Robin boundary value problems, of the classical “concave-convex” problem, but in our case the “convex” ((p – 1)-superlinear) perturbation, is replaced by a (p – 1)-linear and resonant one.

Our aim in this paper is to describe the changes in the set of positive solutions of problem (P_λ) as the parameter λ moves on the open semiaxis ℝ̊₊ = (0, +∞) and also determine the continuity properties of the solution multifunction λ ↦ S_λ, with S_λ being the set of positive solutions of problem (P_λ). Our work here complements the recent one by Marano, Marino & Papageorgiou [14], which deals with an equation driven by the Dirichlet (p, q)-Laplacian and with a reaction of the form u ↦ u^τ–1 + λf(z, u) with f(z, ⋅) being (p – 1)-superlinear. So, the equation in [14] is a “concave-convex” problem with the parameter λ > 0 multiplying the “convex” perturbation. They prove a bifurcation-type theorem (see [14, Theorem 3.10]), which describes the changes in the set of positive solutions as λ moves in ℝ̊₊. Having as their starting point the work in [14], Zeng, Gasiński, Nguyen & Bai [30], proved some continuity properties for the solution multifunction. We also mention the related works of Garcia Azorero, Manfredi & Peral Alonso [5], Gasiński, Papageorgiou & Winowski [9], Guo & Zhang [10], Papageorgiou & Scapellato [21], Vetro [29] (p-Laplacian equations with competition phenomena), Papageorgiou, Rădulescu & Repovš [19] (anisotropic p-Laplacian equations with competition phenomena), Papageorgiou, Vetro & Vetro [22], Papageorgiou & Zhang [25] (Dirichlet (p, 2)-equations), Papageorgiou, Vetro & Vetro [24] (weighted (p, q)-equations) and Papageorgiou, Vetro & Vetro [23] (non-homogeneous Robin equations). We also mention the works of Bai, Motreanu & Zeng [2] on the continuity properties of the solution multifunction for a parametric singular problem driven by the Dirichlet p-Laplacian and of Papageorgiou & Zhang [26], where the “concave” contribution is in the boundary condition.

We mention that equations driven by the sum of two differential operators of different nature (such as (p, q)-equations), arise in many mathematical models of physical processes. We refer to the survey papers of Marano & Mosconi [15], Rădulescu [1] and the references therein.

2 Mathematical background-hypotheses

In the analysis of problem (P_λ), the main spaces are the Sobolev space W^1,p(Ω) and the Banach space C¹(Ω). We denote by ∥⋅∥ the norm of W^1,p(Ω) defined by

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

This cone has a nonempty interior given by

We will also use another open cone in C¹(Ω)

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 spaces” L^r(∂Ω), 1 ⩽ r ⩽ +∞. There exists a unique continuous linear map γ̂₀ : W^1,p(Ω) ↦ L^p(∂Ω) known as the “trace map” such that γ̂₀(u) = u|_∂Ω 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 im γ^0=W−1p′,p(∂Ω), kerγ^0=W01,p(Ω)1p+1p′=1. Also γ̂₀(⋅) is compact into L^r(∂Ω) for 1⩽r<(N−1)pN−p if p < N and into L^r(Ω) for 1 ⩽ r < +∞ if p ⩾ N. In the sequel, for the sake of notational economy, we drop the use of the map γ̂₀(⋅). All restrictions on ∂Ω of the Sobolev functions, are understood in the sense of traces.

Given r ∈ (1, +∞), we denote by A_r : W^1,r(Ω) ↦ W^1,r(Ω)^* the nonlinear operator defined by

The next proposition summarizes the well-known properties of this operator (see, for example, Gasiński & Papageorgiou [7, p. 279]).

Given u, v : Ω ↦ ℝ two measurable functions with u(z) ⩽ v(z) for a.a. z ∈ Ω, we introduce the following order intervals in W^1,p(Ω):

Also, if u, v ∈ W^1,p(Ω), we define u^± = max{±u, 0}. We know that u^± ∈ W^1,p(Ω) and u = u⁺ – u^–, |u| = u⁺ + u^–.

If X is a Banach space and φ ∈ C¹(X), then we define

Also we say that φ(⋅) satisfies the “C-condition”, if it has the following property:

This is a compactness-type condition on the functional φ which compensates for the fact that the ambient space X is not in general locally compact (being infinite dimensional in most cases). Using this condition, one can prove a deformation theorem from which follow the minimax theorems of critical theory (see [20, Section 5.4]).

We introduce our hypotheses on the potential function ξ(⋅) and the boundary coefficient β(⋅).

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

The next two lemmata are useful in estimating the growth of the functionals that arise in the study of problem (P_λ). The first lemma can be found in Papageorgiou, Rădulescu & Repovš [18, Lemma 2.8].

The second lemma can be found in Gasiński & Papageorgiou [8, Proposition 2.4].

Let γ_p : W^1,p(Ω) ↦ ℝ be the C¹-functional defined by

for all u ∈ W^1,p(Ω).

On account of Lemmata 2 and 3, we see that if hypotheses H₀ hold, then we have

We consider the following nonlinear eigenvalue problem

This problem was studied by Fragnelli, Mugnai & Papageorgiou [4], who proved that there exists a smallest eigenvalue λ̂₁ which is simple, isolated and has the following variational characterization

Therefore if hypotheses H₀ hold, then from (2.1) above we see that we have λ̂₁ > 0. The infimum in (2.2) is realized on the corresponding one-dimensional eigenspace, the elements of which have constant sign. The eigenfunctions corresponding to an eigenvalue λ̂ ≠ λ̂₁ are all nodal (that is, sign-changing). Moreover, the nonlinear regularity theory of Lieberman [12], implies that all eigenfunctions belong in C¹(Ω). Finally, the Ljusternik-Schnirelmann minimax scheme generates a whole sequence {λ̂_m}_m∈ℕ of distinct eigenvalues, known as “variational eigenvalues”, such that λ̂_m → +∞ as m → ∞.

Now let us recall some basic continuity notations for multifunctions. So, let (Y, d) be a metric space. We introduce the following hyperspaces

Given B, C ⊆ Y, we set

where C_ε = {y ∈ Y : d(y, C) < ε} (the “ε-enlargement of C”). Then the “Hausdorff distance” between B, C is defined by

It is easy to see that h(B, C) = 0 if and only B = C. So, h(⋅, ⋅) is a metric on P_f(Y) and if (Y, d) is complete, then so is (P_f(Y), h). Moreover, if Y is separable, then P_f(Y) is an h-closed subset of P_f(Y).

Let D be a metric space and S : D ↦ P_f(Y) a multifunction. We introduce the following continuity notions for S(⋅).

1. We say that S(⋅) is “upper semicontinuous” (usc for short), if for all U ⊆ Y open, the set S⁺(U) = {v ∈ D : S(v) ⊆ U} is open.

   We say that S(⋅) is “lower semicontinuous” (lsc for short), if for all C ⊆ Y closed, the set S⁺(C) = {v ∈ D : S(v) ⊆ C} is closed.

   If S(⋅) is both usc and lsc, then we say that S(⋅) is “continuous” (or “Vietoris continuous”).

2. We say that S(⋅) is “h-usc”, if for all y ∈ Y, z ↦ h^*(S(z), S(y)) is continuous.

   We say that S(⋅) is “h-lsc”, if for all y ∈ Y, z ↦ h^*(S(y), S(z)) is continuous.

   If S(⋅) is both h-usc and h-lsc, then we say that S(⋅) is “h-continuous” (or “Hausdorff continuous”).

   In general, the above continuity notations are distinct and we have:

   usc⟹h-usc and h-lsc⟹lsc.

However, if S(⋅) is P_k(Y)-valued, then

For details, we refer to Hu & Papageorgiou [11].

Next, let us introduce the hypotheses on the perturbation f(z, x).

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

1. for every ρ > 0, there exists a_ρ ∈ L^∞(Ω) such that

   |f(z,x)|⩽aρ(z) for a.a. z∈Ω, all 0⩽x⩽ρ;

2. for m ⩾ 2 such that λ̂_m > ∥ξ∥_∞, we have

   limx→+∞f(z,x)xp−1=λ^m uniformly for a.a. z∈Ω;

3. if F(z,x)=∫0xf(z,s)ds, then there exists η ∈ (q, p) such that

   0<c1⩽lim infx→+∞pF(z,x)−f(z,x)xxη uniformly for a.a. z∈Ω;

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

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

   x↦f(z,x)+ξ^ρxp−1

   is nondecreasing on [0, ρ].

The following function satisfies the above hypotheses. For the sake of simplicity, we drop the z-dependence:

with c > λ̂_m, τ < θ, 1 < s < p. Note that f(⋅) is sign-changing.

3 Bifurcation-type theorem

In this section we prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter λ moves on ℝ̊₊ = (0, +∞).

So, we introduce the following two sets:

Next we show that 𝔏 is an interval.

We set λ^* = sup 𝔏.

According to the above proposition, we have

Next, we will produce a lower bound for the elements of S_λ. This will be useful in producing a minimal positive solution for problem (P_λ) and in the study of the continuity properties of the solution multifunction.

Hypotheses H₁(i), (ii), (iv), imply that given ε > 0, we can find c_ε > 0 such that

Since ε > 0 is arbitrary, we choose ε ∈ (0, λ) and we have

Motivated by this unilateral growth restriction, we introduce the following auxiliary Robin (p, q)-equation:

This solution u_λ is a lower bound for the solution set S_λ.

Using this lower bound, we can show that for λ ∈ 𝔏, S_λ has a smallest element (minimal positive solution).