A note on some nonlinear principal eigenvalue problems

  • Mohsen Boujary1Email author and

    Affiliated with

    • Ghasem Alizadeh Afrouzi2

      Affiliated with

      Boundary Value Problems20122012:44

      DOI: 10.1186/1687-2770-2012-44

      Received: 29 November 2011

      Accepted: 16 April 2012

      Published: 16 April 2012

      Abstract

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

      - Δ p u ( x ) = λ g ( x ) 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_Equa_HTML.gif

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

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

      Keywords

      p-Laplacian principal eigenvalue

      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
      - Δ p u ( x ) = λ g ( x ) 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_Equa_HTML.gif
      (1.1)

      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
      - 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_Equ1_HTML.gif
      (2.1)
      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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_Equb_HTML.gif

      When α ≥ 0, it is clear that Sαis bounded below. It is shown by variational arguments that μ(α, λ) = inf Sαand 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 Sαis 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_Equc_HTML.gif

      for all ϕW1,p(Ω).

      Proof. Suppose that the result does not hold. Then ε0 > 0 and sequence {u n } ⊆ W1,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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_Equ2_HTML.gif
      (2.2)

      Suppose first that {Ω |u n | p dx} is unbounded. Let v n = u n u n L p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_IEq1_HTML.gif. Clearly, {υ n } is bounded in W1,p(Ω), and so in L p ( Ω). But Ωn| 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 W1,pand so has a subsequence, which we again denote by {u n }, converging weakly to u in W1,p. Since W1,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 limn→∞ 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_IEq2_HTML.gif, it is easy to deduce from the above result the Sαis bounded below, and it follows exactly as in [3] that μ(α, λ) = inf Sαand 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 ϕW1,p(Ω), λ → Ω |∇ϕ| p dx Ω|ϕ| p dSxΩ 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, ϕ2W1,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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_Equ3_HTML.gif
      (2.3)

      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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_IEq3_HTML.gif and at critical points we must have Ω g u 0 2 d x Ω u 0 2 d x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_IEq4_HTML.gif (see, [2], Lemma 2]).

      But, we cannot generalize it for p ≠ 2. Because, if v ( λ ) = d μ d λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_IEq5_HTML.gif, then we have
      - d d λ Δ p u ( λ ) = - ( p - 1 ) d i v ( v u p - 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_Equd_HTML.gif

      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 ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_IEq6_HTML.gif with respect to the norm
      u V p = Ω u p d x + Ω u p d s 1 p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_Equ4_HTML.gif
      (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 q N p N - 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_IEq7_HTML.gif and V p L q (Ω) compactly for 1 q N p N - 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_IEq7_HTML.gif [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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_Equ5_HTML.gif
      (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 W1,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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_Equ6_HTML.gif
      (2.6)
      and
      j ( u ) = Ω g ( x ) u p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_Equ7_HTML.gif
      (2.7)
      If wV 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-44/MediaObjects/13661_2011_Article_132_Eque_HTML.gif

      holds for any ϕV 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 = {uV p : j(u) = 1} and from Palais-Smale condition satisfied by the functional l on M.

      Declarations

      Authors’ Affiliations

      (1)
      Department of Mathematics, Science and Research Branch, Islamic, Azad University (Iau)
      (2)
      Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran

      References

      1. Diaz JI: Nonlinear Partial Differential Equatians and Free Boundaries. Volume I. Elliptic Equations, London; 1985.
      2. Afrouzi GA, Brown KJ: On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions. Proc Am Math Soc 1999, 127(1):125-130.MATHMathSciNetView Article
      3. Smoller J: Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, Berlin; 1983.MATHView Article
      4. Pao CV: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York, London; 1992.MATH
      5. Drabek P, Schindler I: Positive solutions for p -Laplacian with Robin boundary conditions on irregular domains. Appl Math Lett 2011, 24: 588-591.MATHMathSciNetView Article
      6. Maz'ja VG: Sobolev Spaces in: Springer Series in Soviet Mathematics (translated from the Russian by Shaposhnikova, TO). Springer-Verlag, Berlin; 1985.View Article

      Copyright

      © Boujary and Afrouzi; licensee Springer. 2012

      This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.