Abstract
In the present paper, our purpose is to construct the Fučik spectrum C_{l}^{pm} (cf. (2.13) and (2.17) below) with different weights for the p-Laplacian. As an application, we will discuss the existence of nontrivial solutions to the p-Laplace equations with resonance on the Fučik spectrum by making use of variational methods and Morse theory.
Highlights
1 Introduction In this paper, we consider the existence of nontrivial solutions to the Dirichlet boundary value problem
It is well known that the value of (λ, μ) plays an important role in the study of the solvability of ( . )
The approach here requires the preliminary study of weighted asymmetric eigenvalue problems of the form
Summary
We consider the existence of nontrivial solutions to the Dirichlet boundary value problem. The purpose of this paper is to discuss the general resonance case (λ, μ) ∈ p(a, b), with nonconstant and different weights a(x) and b(x). The usual Morse theory supposes that the functional (u) satisfies the Palais-Smale condition ((P.S.) condition for short), which is not clear in the case where the nonlinear term g(x, u) is asymptotic to |u|p– u at infinity. To overcome this difficulty, Cerami introduced a weaker compactness condition (see [ ]). Replacing the usual (P.S.) condition by Cerami’s weaker compactness condition ((Cc) condition for short), we still have the deformation lemma (see [ ])
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