Multiple solutions for an elliptic system with indefinite Robin boundary conditions

1 Introduction

This work is devoted to the study of existence and multiplicity of vector solutions u ⁢ ( x ) = ( u 1 ⁢ ( x ) , … , u N ⁢ ( x ) ) of the problem

with the boundary condition

Here L denotes the vector Laplacian given by L ⁢ u := ( Δ ⁢ u 1 , … , Δ ⁢ u N ) , Ω ⊂ ℝ n is a smooth bounded domain and g 0 : Ω ¯ × ℝ N → ℝ N is a smooth function. Without loss of generality, it shall be assumed throughout the paper that g 0 ⁢ ( x , 0 ) = 0 . We shall also assume that p ∈ C ⁢ ( Ω ¯ , ℝ N ) and γ ∈ C 1 ⁢ ( Ω ¯ ) . It is remarked that, unlike in the most standard Robin condition, we do not assume that γ ≤ 0 .

The problem arises in many different applications; for instance (see [9] and the references therein) for some particular g 0 , it can be seen as the steady state of the population density of some species, governed by a parabolic problem. The boundary condition corresponds to the law of the population flux, given on the boundary by γ. Thus, the inflow/outflow of population to the region occurs when γ ⁢ ( x ) > 0 or γ ⁢ ( x ) < 0 , respectively. Our direct motivation for the present work can be found in [4], where a Painlevé II (ordinary) scalar equation was considered, namely

under the radiation boundary conditions u ′ ⁢ ( 0 ) = a 0 ⁢ u ⁢ ( 0 ) , u ′ ⁢ ( 1 ) = a 1 ⁢ u ⁢ ( 1 ) with a 0 , a 1 > 0 . It was shown that a 1 > 0 is compensated by the effect of the superlinear term u 3 2 in such a way that the associated functional is still coercive, giving rise to a unique solution when a 0 ≥ a 1 and exactly three solutions when a 0 < a 1 , provided that the positive parameter p is small. It is worth noticing that, in this model, δ is a linear function depending on a 0 and a 1 ; thus, the uniqueness/multiplicity of solutions can be understood as a consequence of the interaction between the nonlinear term of the equation and the Robin coefficients. The previous results have been extended in [3] and [2] for the equation u ′′ ⁢ ( x ) = g 0 ⁢ ( x , u ⁢ ( x ) ) + p ⁢ ( x ) , where g 0 is a general superlinear function, that is,

uniformly on x. Indeed, it was shown that the interaction between g , a 0 and a 1 can be expressed in a very precise way in terms of the spectrum of the associated linear operator - u ′′ under the Robin conditions: if

for all x and all u, then the problem has a unique solution, and if

for some odd k, then the problem has at least three solutions when ∥ p ∥ ∞ is small. Furthermore, if (1.4) holds with k even, then the problem has at least five solutions for small p, provided that λ 1 < 0 .

The main purpose of this paper consists in obtaining suitable extensions of the results mentioned above in two directions: in the first place, we shall consider a system instead of a scalar equation and in the second place, we shall not be longer dealing with ODEs but with a PDE system. We shall study system (1.1) in terms of the eigenvalues λ 1 < λ 2 ≤ λ 3 ≤ ⋯ → + ∞ of the scalar problem - Δ ⁢ φ = λ ⁢ φ associated to the Robin condition ∂ ⁡ φ ∂ ⁡ ν | ∂ ⁡ Ω = γ ⁢ φ and λ 0 := - ∞ . At first sight, it seems reasonable to assume that g 0 is superlinear, that is,

uniformly on x, where 〈 ⋅ , ⋅ 〉 and | ⋅ | denote the usual inner product and norm of ℝ N , respectively. However, we shall employ a weaker condition, namely, that there exists a constant μ ≥ - λ 1 depending only on Ω and γ such that, if

uniformly on x, then the problem has a solution (see Theorem 3.3). This constant can be estimated: for example, it will be shown that it always suffices to take

where φ 1 > 0 is the (unique) principal eigenfunction such that ∥ φ 1 ∥ L 2 = 1 . For the ODE case n = 1 or when g grows like | u | q for some q ≤ n + 2 n - 2 , it can be seen that the condition μ = - λ 1 is optimal for our purposes.

Extra solutions will be obtained when p is small, under an appropriate condition on the interaction between D u ⁢ g 0 ⁢ ( x , 0 ) and the spectrum of the linear associated operator. In more precise terms, if for simplicity we assume that g 0 = ∇ u ⁡ G 0 for some smooth function G 0 : Ω ¯ × ℝ N → ℝ , then our main hypothesis is that the Hessian matrix at u = 0 satisfies

where A , B ∈ ℝ N × N are symmetric matrices with eigenvalues a 1 ≤ ⋯ ≤ a N and b 1 ≤ ⋯ ≤ b N such that

for some ν k ∈ ℕ 0 and all k = 1 , … , N . Here, the ordering ≤ over the set of symmetric functions is understood in the usual sense that Y ≤ Z if and only if Z - Y is positive semi-definite or equivalently, 〈 Y ⁢ u , u 〉 ≤ 〈 Z ⁢ u , u 〉 for all u ∈ ℝ N .

It shall be seen that if ν 1 + ⋯ + ν N is an odd number, then the problem has at least two solutions (and generically three), provided that ∥ p ∥ L 2 is small enough. In fact, the result holds for a general function g 0 , with the role of the Hessian matrix now played by the Jacobian matrix D u ⁢ g 0 ⁢ ( x , 0 ) , provided it is symmetric. In contrast with the case g 0 = ∇ u ⁡ G 0 , the general situation does not have variational structure and is presented in our main result through the Leray–Schauder degree method (see Theorem 5.2 below). To our knowledge, there are no previous similar results in the literature concerning the multiplicity of solutions for an elliptic system with indefinite Robin conditions.

The paper is organised as follows. In the next section, we recall some basic facts concerning the spectrum of the associated linear operator. In Section 3, we prove the existence of at least one solution by means of an appropriate space-dependent Hartman condition. In Section 4, we obtain H 1 bounds for all solutions, provided that μ ≥ - λ 1 . Furthermore, enlarging μ if necessary, we also obtain L ∞ bounds, which shall be the key for proving our multiplicity result. This task is accomplished in Section 5, where we firstly prove a lemma that helps to compute the degree of certain linear operators and then apply the lemma for the computation of the degree of I - K , where K is an appropriate compact fixed point operator. Roughly speaking, for p = 0 we shall prove that the degree is equal to 1 over a large ball and equal to -1 over a small neighbourhood of the origin. This guarantees, when p is small, the existence of a solution close to the origin and typically two ‘large’ solutions.

2 Preliminaries

For convenience, throughout the paper we shall employ the notation g ⁢ ( x , u ) := g 0 ⁢ ( x , u ) + p ⁢ ( x ) . It is clear that if g 0 satisfies (1.5), then so does g for arbitrary p ∈ C ⁢ ( Ω ¯ ) . Sometimes we shall express the condition in the following equivalent form: there exist C μ ≥ 0 and ε > 0 such that

Before establishing our main results, we need to recall some basic facts concerning the spectrum of the associated linear operator. Let us observe, in the first place, that the scalar operator - Δ is symmetric for the Robin boundary condition. Thus, by standard arguments (see e.g. [5]) one can deduce the existence of a sequence of eigenvalues λ 1 ≤ λ 2 ≤ λ 3 ≤ ⋯ → + ∞ and an associated orthonormal basis of L 2 ⁢ ( Ω ) consisting of eigenfunctions { φ j } j ∈ ℕ . For convenience, we shall denote λ 0 := - ∞ .

It is verified (see [1]) that λ 1 is simple and the associated eigenfunctions do not vanish on Ω ¯ . From now on, we shall always assume that φ 1 is such that φ 1 > 0 . Moreover, φ 1 can be obtained as a global minimiser of the functional

subject to the restriction ∥ φ ∥ L 2 = 1 . It follows that

for any smooth function φ satisfying the Robin condition. In particular, adding ( η - λ 1 ) ⁢ ∥ φ ∥ L 2 2 to both sides for some η > 0 , the Cauchy–Schwarz inequality yields the following estimate:

In order to understand the dependence of φ 1 with respect to γ, observe that if ℐ ~ is the functional associated to some γ ~ and if λ ~ 1 , φ ~ 1 are the respective eigenvalue and eigenfunction, then, by virtue of the trace embedding H 1 ⁢ ( Ω ) ↪ L 2 ⁢ ( ∂ ⁡ Ω ) , we deduce that

In particular,

and

We conclude that if γ ~ → γ uniformly, then λ ~ 1 → λ 1 . Furthermore, since - Δ ⁢ φ ~ 1 = λ ~ 1 ⁢ φ ~ 1 we deduce that the family { φ ~ 1 } is uniformly bounded in H 2 ⁢ ( Ω ) , and by a bootstrapping argument, we conclude that it is uniformly bounded in C 2 ⁢ ( Ω ¯ ) . We claim that φ ~ 1 → φ 1 for the C 1 norm. Indeed, otherwise we may suppose that φ ~ 1 converges for the C 1 norm to some φ ≠ φ 1 . Observe that ∥ φ ∥ L 2 = 1 and φ ≥ 0 ; moreover, since Δ ⁢ φ ~ 1 → - λ 1 ⁢ φ 1 , it is readily seen that Δ ⁢ φ = - λ 1 ⁢ φ and hence φ = φ 1 , a contradiction.

3 A space-dependent Hartman condition

Our general existence result is based on the fact that g satisfies the following Hartman type condition (the original condition was formulated in [6]):

1. There exists a smooth function R : Ω ¯ → ( 0 , + ∞ ) such that

   (3.1) 〈 g ⁢ ( x , u ) , u 〉 ≥ R ⁢ ( x ) ⁢ Δ ⁢ R ⁢ ( x ) + | ∇ ⁡ R ⁢ ( x ) | 2

   for all x ∈ Ω and all u ∈ ℝ N such that | u | = R ⁢ ( x ) .

In order to establish the existence of solutions, let us observe that, without loss of generality, we may assume that Ω = F - 1 ⁢ ( - ∞ , 0 ) for some C 2 mapping F : Ω ¯ → ℝ such that 0 is a regular value of F. Thus, the outer normal at x ∈ ∂ ⁡ Ω is simply computed as ν ⁢ ( x ) = ∇ ⁡ F ⁢ ( x ) | ∇ ⁡ F ⁢ ( x ) | . For convenience, let us fix the following constants:

4 A priori bounds for the solutions

5 Multiplicity