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

-Δpu(x)=λg(x)u(x)p-2u(x),inΩ,Ru=∇u(x)p-2∂u∂ν(x)+αu(x)p-2u(x)=0,on∂Ω.

where Ω ⊆ ℝ^N is 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.

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

- Δ p u ( x ) - λ g ( x ) u ( x ) p - 2 u ( x ) , = μ u ( x ) p - 2 u ( x ) , in Ω , Ru = ∇ u ( x ) p - 2 ∂ u ∂ ν ( x ) + α u ( x ) p - 2 u ( x ) = 0 , on ∂ Ω .

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

S α , λ = ∫ Ω ∇ ϕ p d x + α ∫ ∂ Ω ϕ p d S x - λ ∫ Ω g ϕ p d x : ϕ ∈ W 1 , p ( Ω ) , ∫ Ω ϕ p = 1

When α ≥ 0, it is clear that S_α,λ is bounded below. It is shown by variational arguments that μ(α, λ) = inf S_α,λ and 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,λ is 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

∫ ∂ Ω ϕ p d S x ≤ ε ∫ Ω ∇ ϕ p d x + C ( ε ) ∫ Ω ϕ p d x

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

∫ ∂ Ω u n p d S x ≥ ε 0 + n ∫ Ω u n p d x .

Suppose first that {∫_Ω |u_n|^p dx} is unbounded. Let v n = u n u n L p ( Ω ) . Clearly, {υ_n} is bounded in W^1,p(Ω), and so in L^p(∂Ω). But ∫_∂Ω |υ_n|^p dS_x ≥ n ∫_Ω |υ_n|^p dx = n, which is impossible.

Suppose now that {∫_Ω |u_n|^p dx} is bounded, then {u_n} is bounded in W^1,p and 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|^p dx} 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 ε < 1 α , it is easy to deduce from the above result the S_α,λ is bounded below, and it follows exactly as in 3 that μ(α, λ) = inf S_α,λ and 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|^p dx+α ∫_{∂ Ω} |ϕ|^p dS_x-λ ∫_Ω g|ϕ|^p dx 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|^p dx > 0 and ∫_Ω g|ϕ₂|^p dx < 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

d μ d λ ( α , λ ) = - ∫ Ω g u 0 2 d x ∫ Ω u 0 2 d x .

Therefore, λ → μ(α, λ) is an increasing (decreasing) function, if we have ∫ Ω g u 0 2 d x ∫ Ω u 0 2 d x < 0 ( > 0 ) and at critical points we must have ∫ Ω g u 0 2 d x ∫ Ω u 0 2 d x = 0 (see, 2, Lemma 2]).

But, we cannot generalize it for p ≠ 2. Because, if v ( λ ) = d μ d λ , then we have

- d d λ Δ p u ( λ ) = - ( p - 1 ) d i v ( ∇ v ∇ u p - 2 ) .

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 ( Ω ) ∩ C ( Ω ̄ ) with respect to the norm

u V p = ∫ Ω ∇ u p d x + ∫ ∂ Ω u p d s 1 p .

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 ≤ q ≤ N p N - 1 and V_p ↪ L^q(Ω) compactly for 1≤q≤NpN-1 6.

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

∫ Ω ∇ u p - 2 ∇ u . ∇ ϕ d x + α ∫ ∂ Ω u p - 2 u ϕ d s = ∫ Ω λ g ( x ) u p - 2 u ϕ d x .

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^N is bounded, N > 1, α > 0, and 1 < p < N. We apply variational for (1.1) with λ = 1. We introduce the C¹-functionals

l ( u ) = ∫ Ω ∇ u p + α ∫ ∂ Ω u p d s .

and

j ( u ) = ∫ Ω g ( x ) u p .

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 ∫ Ω ∇ w p - 2 ∇ w . ∇ ϕ d x + p α ∫ ∂ Ω w p - 2 w ϕ d s = λ p ∫ Ω g ( x ) u p - 2 u ϕ d x

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.

The authors declare that they have no competing interests.

Boujary has presented the main purpose of the article and has used Afrouzi contribution due to reaching to conclusions. All authors read and approved the final manuscript.