A note on some nonlinear principal eigenvalue problems

Abstract

We investigate the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) the boundary value problem

$$\left\{\begin{array}{cc}-{\Delta }_{p}u\left(x\right)=\lambda g\left(x\right){\left|u\left(x\right)\right|}^{p-2}u\left(x\right),\hfill & \mathsf{\text{in}}\Omega \mathsf{\text{,}}\hfill \\ \mathsf{\text{Ru}}={\left|\nabla u\left(x\right)\right|}^{p-2}\frac{\partial u}{\partial \nu }\left(x\right)+\alpha {\left|u\left(x\right)\right|}^{p-2}u\left(x\right)=0,\hfill & \mathsf{\text{on}}\partial \Omega \mathsf{\text{.}}\hfill \end{array}\right.$$

where Ω ⊆ ℝ^Nis a bounded domain, 1 < p < ∞ and α is a real number.

AMS Subject Classification: 35J60; 35B30; 35B40.

1. Introduction

Mathematical models described by nonlinear partial differential equations have become more common recently. In particular, the p-Laplacian operator appears in subjects such as filtration problem, power-low materials, non-Newtonian fluids, reaction-diffusion problems, nonlinear elasticity, petroleum extraction, etc., see,[1]. The nonlinear boundary condition describes the flux through the boundary ∂ Ω which depends on the solution itself.

The purpose of this study is to discuss the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem

$$\left\{\begin{array}{cc}-{\Delta }_{p}u\left(x\right)=\lambda g\left(x\right){\left|u\left(x\right)\right|}^{p-2}u\left(x\right),\hfill & \mathsf{\text{in}}\Omega \mathsf{\text{,}}\hfill \\ \mathsf{\text{Ru}}={\left|\nabla u\left(x\right)\right|}^{p-2}\frac{\partial u}{\partial \nu }\left(x\right)+\alpha {\left|u\left(x\right)\right|}^{p-2}u\left(x\right)=0,\hfill & \mathsf{\text{on}}\partial \Omega \mathsf{\text{.}}\hfill \end{array}\right.$$

(1.1)

where Ω ⊆ ℝ^Nis a bounded domain, 1 < p < ∞ and α is a real number. Attention has been confined mainly to the cases of Dirichlet and Neumann boundary conditions but we have the Robin boundary in (1.1).

We discuss about to exist principal eigenvalue for (1.1). In the case 0 < α < ∞, We shall show that there has exactly two principal eigenvalues, one positive and one negative.

2. Main result

Our analysis is based on a method used by Afrouzi and Brown [2]. Consider, for fixed λ, the eigenvalue problem

$$\left\{\begin{array}{cc}-{\Delta }_{p}u\left(x\right)-\lambda g\left(x\right){\left|u\left(x\right)\right|}^{p-2}u\left(x\right),=\mu {\left|u\left(x\right)\right|}^{p-2}u\left(x\right),\hfill & \mathsf{\text{in}}\Omega \mathsf{\text{,}}\hfill \\ \mathsf{\text{Ru}}={\left|\nabla u\left(x\right)\right|}^{p-2}\frac{\partial u}{\partial \nu }\left(x\right)+\alpha {\left|u\left(x\right)\right|}^{p-2}u\left(x\right)=0,\hfill & \mathsf{\text{on}}\partial \Omega \mathsf{\text{.}}\hfill \end{array}\right.$$

(2.1)

We denote the lowest eigenvalue of (2.1) by μ(α, λ). Let

$$S_{\alpha ,\lambda }=\left\{\underset{\Omega }{\int }{\left|\nabla \varphi \right|}^{p}dx+\alpha \underset{\partial \Omega }{\int }{\left|\varphi \right|}^{p}d{S}_x-\lambda \underset{\Omega }{\int }g{\left|\varphi \right|}^{p}dx:\varphi \in W^{1,p}\left(\Omega \right),\underset{\Omega }{\int }{\left|\varphi \right|}^p=1\right\}$$

When α ≥ 0, it is clear that S_α,λis bounded below. It is shown by variational arguments that μ(α, λ) = inf S_α,λand that an eigenfunction corresponding to μ(α, λ) does not change sign on Ω [3]. Thus, clearly, λ is a principal eigenvalue of (1.1) if and only if μ(α, λ) = 0.

When α < 0, the boundedness below of S_α,λis not obvious, but is a consequence of the following lemma.

Lemma 2.1. For every ε > 0 there exists a constant C(ε) such that

$$\underset{\partial \Omega }{\int }{\left|\varphi \right|}^{p}d{S}_x\le \epsilon \underset{\Omega }{\int }{\left|\nabla \varphi \right|}^{p}dx+C\left(\epsilon \right)\underset{\Omega }{\int }{\left|\varphi \right|}^{p}dx$$

for all ϕ ∈ W^1,p(Ω).

Proof. Suppose that the result does not hold. Then ε₀ > 0 and sequence {u _n } ⊆ W^1,p(Ω) such that ∫_Ω|∇u _n |^p= 1 and

$$\underset{\partial \Omega }{\int }{\left|u_n\right|}^{p}d{S}_x\ge {\epsilon }_0+n\underset{\Omega }{\int }{\left|u_n\right|}^{p}dx.$$

(2.2)

Suppose first that {∫_Ω |u _n |^p dx} is unbounded. Let $v_n=\frac{u_n}{{∥u_n∥}_{L^p\left(\Omega \right)}}$. Clearly, {υ _n } is bounded in W^1,p(Ω), and so in L^p(∂ Ω). But ∫_{∂ Ω}|υ_n|^pdS _x ≥ n ∫_Ω |υ _n |^pdx = n, which is impossible.

Suppose now that {∫_Ω |u _n |^pdx} is bounded, then {u _n } is bounded in W^1,pand so has a subsequence, which we again denote by {u _n }, converging weakly to u in W^1,p. Since W^1,pis compactly embedded in L^p(∂ Ω) and in L^p(Ω), it follows that {u _n } converges to some function u in L^p(∂ Ω) and in L^p(Ω). Thus {∫ _∂Ω |u _n |^pdx} is bounded, and so it follows from (2.2) that lim_n→∞∫ Ω |u _n |^p dx = 0, i.e.,, {u _n } converges to zero in L^p(Ω). Hence {u _n } converges to zero in L^p(∂ Ω), and this is impossible because (2.2).

Choosing $\epsilon <\frac{1}{\alpha }$, it is easy to deduce from the above result the S_α,λis bounded below, and it follows exactly as in [3] that μ(α, λ) = inf S_α,λand that an eigenfunction corresponding to μ(α, λ) does not change sign on Ω. Thus it is again λ is a principal eigenvalue of (1.1) if and only if μ(α, λ) = 0.

For fixed ϕ ∈ W^1,p(Ω), λ → ∫_Ω |∇ϕ|^pdx+α ∫_{∂ Ω}|ϕ|^pdS_x-λ ∫_Ω g|ϕ|^pdx is an affine and so concave function. As the infimum of any collection of concave functions is concave, it follows that λ → μ(α, λ) is concave. Also, by considering test functions ϕ₁, ϕ₂ ∈ W^1,p(Ω) such that ∫_Ω g |ϕ₁|^pdx > 0 and ∫_Ω g|ϕ₂|^pdx < 0, it is easy to see that μ(α, λ) → -∞ as λ → ± ∞. Thus λ → μ(α, λ) is an increasing function until it attains its maximum, and is an decreasing function thereafter.

It is natural that the flux across the boundary should be outwards if there is a positive concentration at the boundary. This motivates the fact that the sign of α > 0. For a physical motivation of such conditions, see for example [4]. Suppose that 0 < α < ∞, i.e., we have the Robin boundary condition. Then, as can be seen from the the variational characterization of μ(α, λ) or -Δ _p has a positive principal eigenvalue, μ(α, 0) > 0 and so λ → μ(α, λ) must has exactly two zero. Thus in this case (1.1) exactly two principal eigenvalues, one positive and one negative.

Our results may be summarized in the following theorem.

Theorem 2.2. If 0 < α < ∞, then (1.1) exactly two principal eigenvalues, one positive and one negative.

However, for α < 0 we have μ(α, 0) ≤ 0. For p = 2, if u₀ is eigenfunction of (2.1) corresponding to principal eigenvalue μ(α, λ), then

$$\frac{d\mu }{d\lambda }\left(\alpha ,\lambda \right)=-\frac{{\int }_{\Omega }g{u}_0^{2}dx}{{\int }_{\Omega }{u}_0^{2}dx}.$$

(2.3)

Therefore, λ → μ(α, λ) is an increasing (decreasing) function, if we have $\frac{{\int }_{\Omega }g{u}_0^{2}dx}{{\int }_{\Omega }{u}_0^{2}dx}<0\left(>0\right)$ and at critical points we must have $\frac{{\int }_{\Omega }g{u}_0^{2}dx}{{\int }_{\Omega }{u}_0^{2}dx}=0$ (see, [2], Lemma 2]).

But, we cannot generalize it for p ≠ 2. Because, if $v\left(\lambda \right)=\frac{d\mu }{d\lambda }$, then we have

$$-\frac{d}{d\lambda }{\Delta }_{p}u\left(\lambda \right)=-\left(p-1\right)div\left(\nabla v{\left|\nabla u\right|}^{p-2}\right).$$

So, we cannot get a similar result (2.3).

Now our analysis is based by Drabek and Schindler [5]. We define the space V _p as completion of $W^{1,p}\left(\Omega \right)\cap C\left(\bar{\Omega }\right)$ with respect to the norm

$${{∥u∥}_V}_p={\left(\underset{\Omega }{\int }{\left|\nabla u\right|}^{p}dx+\underset{\partial \Omega }{\int }{\left|u\right|}^{p}ds\right)}^{\frac{1}{p}}.$$

(2.4)

The spaces equivalent to V _p were introduced in [6]. In particular, V _p is a uniformly convex (and hence a reflexive) Banach space, V _p ↪ L^q(Ω) continuously for $1\le q\le \frac{Np}{N-1}$ and V _p ↪ L^q(Ω) compactly for $1\le q\le \frac{Np}{N-1}$ [6].

We say that u ∈ V _p is a weak solution to (1.1) if for all ϕ ∈ V _p we have

$$\underset{\Omega }{\int }{\left|\nabla u\right|}^{p-2}\nabla u.\nabla \varphi dx+\alpha \underset{\partial \Omega }{\int }{\left|u\right|}^{p-2}u\varphi ds=\underset{\Omega }{\int }\lambda g\left(x\right){\left|u\right|}^{p-2}u\varphi dx.$$

(2.5)

In fact there are domains Ω for which the embedding V _p ↪ L^p (Ω) is not injective. This is to the influence of the wildness of the boundary ∂ Ω. The domains for which the above embedding is injective are then called admissible. Ω is called admissible irregular domain for which W^1,p(Ω) is not subset L^q(Ω) for all p > q.

We assume that the domain Ω ⊂ ℝ^Nis bounded, N > 1, α > 0, and 1 < p < N. We apply variational for (1.1) with λ = 1. We introduce the C¹-functionals

$$l\left(u\right)=\underset{\Omega }{\int }{\left|\nabla u\right|}^p+\alpha \underset{\partial \Omega }{\int }{\left|u\right|}^{p}ds.$$

(2.6)

and

$$j\left(u\right)=\underset{\Omega }{\int }g\left(x\right){\left|u\right|}^p.$$

(2.7)

If w ∈ V _p be a global minimizer of l subject to the constraint j(w) = 1, then the Lagrange multiplier method yields a λ ∈ ℝ such that l'(u) = λj'(u), i.e.,

$$p\underset{\Omega }{\int }{\left|\nabla w\right|}^{p-2}\nabla w.\nabla \varphi dx+p\alpha \underset{\partial \Omega }{\int }{\left|w\right|}^{p-2}w\varphi ds=\lambda p\underset{\Omega }{\int }g\left(x\right){\left|u\right|}^{p-2}u\varphi dx$$

holds for any ϕ ∈ V _p . Then w is a weak solution (1.1). The existence of a minimizer follows from the fact that l(u) is bounded from below on the manifold M = {u ∈ V _p : j(u) = 1} and from Palais-Smale condition satisfied by the functional l on M.