Abstract
This paper deals with the mathematical analysis of a class of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator. We are concerned both with the coercive and the noncoercive (and nonresonant) cases, which are in relationship with two associated Rayleigh quotients. The proof combines critical point theory arguments and the dual variational principle. The arguments developed in this paper can be extended to other classes of nonlinear eigenvalue problems with nonstandard growth.
Highlights
In a pioneering paper, Tolksdorf [1] studied Dirichlet problems involving the quasilinear elliptic second order differential operatorAu := div a |∇u|2 ∇u, where the potential a : (0, +∞) → (0, +∞) is of class C1 and satisfies the following ellipticity and growth conditions of Leray–Lions type: There are constants γ, Γ > 0, κ ∈ [0, 1], and p ∈ (1, +∞) such that, for every t > 0, γ tp–2 ≤ a t2 ≤ Γ (κ + t)p–2 (1) and 1 γ–a(t) ≤ ta (t) ≤ Γ a(t). (2)we can define the function A : [0, +∞) → [0, +∞) by A(t) = t 0 a(s) ds
Our purpose in the present paper is to use variational methods for the qualitative analysis of a class of nonlinear eigenvalue problems driven by the differential operator div(a(|∇u|2)∇u)
The aim of the present paper is to study the following nonlinear Dirichlet problem driven by a nonhomogeneous differential operator:
Summary
Our purpose in the present paper is to use variational methods for the qualitative analysis of a class of nonlinear eigenvalue problems driven by the differential operator div(a(|∇u|2)∇u). By the mountain pass theorem (see Ambrosetti and Rabinowitz [10]), problem (5) has a solution for all λ < λ1, where λ1 denotes the first eigenvalue of (– ) in H01(Ω).
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