Abstract
Keywords:
pLaplacian; principal eigenvalue1. Introduction
Mathematical models described by nonlinear partial differential equations have become more common recently. In particular, the pLaplacian operator appears in subjects such as filtration problem, powerlow materials, nonNewtonian fluids, reactiondiffusion 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
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.
2. Main result
Our analysis is based on a method used by Afrouzi and Brown [2]. Consider, for fixed λ, the eigenvalue problem
We denote the lowest eigenvalue of (2.1) by μ(α, λ). Let
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
for all ϕ ∈ W^{1,p}(Ω).
Proof. Suppose that the result does not hold. Then ε_{0 }> 0 and sequence {u_{n}} ⊆ W^{1,p}(Ω) such that ∫_{Ω}∇u_{n}^{p }= 1 and
Suppose first that {∫_{Ω }u_{n}^{p }dx} is unbounded. Let . 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,p}is 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 , 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 }gϕ^{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 ϕ_{1}, ϕ_{2 }∈ W^{1,p}(Ω) such that ∫_{Ω }g ϕ_{1}^{p }dx > 0 and ∫_{Ω }gϕ_{2}^{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 u_{0 }is eigenfunction of (2.1) corresponding to principal eigenvalue μ(α, λ), then
Therefore, λ → μ(α, λ) is an increasing (decreasing) function, if we have and at critical points we must have (see, [2], Lemma 2]).
But, we cannot generalize it for p ≠ 2. Because, if , then we have
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 with respect to the norm
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 and V_{p }↪ L^{q}(Ω) compactly for [6].
We say that u ∈ V_{p }is a weak solution to (1.1) if for all ϕ ∈ V_{p }we have
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 C^{1}functionals
and
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.,
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 PalaisSmale condition satisfied by the functional l on M.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
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.
References

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