Abstract
This paper presents sufficient conditions for the existence and nonexistence of eigenvalues for a $p(x)$-biharmonic equation with Navier boundary conditions and weight function on a bounded domain in $\mathbb{R}^N$. Our approach is mainly based on a adequate variational techniques.
Highlights
The interest in analyzing this kind of problems is motivated by some recent advances in the study of fourth order nonlinear eigenvalue problems involving variable exponents in the last few years
Motived by the above-mentioned papers, our purpose in this paper is to extend the results of [5] to a fourth order nonlinear problem with sign-changing potential
We say that λ ∈ R is an eigenvalue of problem 1.1 if the weak solution u defined above is not trivial
Summary
The interest in analyzing this kind of problems is motivated by some recent advances in the study of fourth order nonlinear eigenvalue problems involving variable exponents in the last few years. Motived by the above-mentioned papers, our purpose in this paper is to extend the results of [5] to a fourth order nonlinear problem with sign-changing potential. We analyze the problem (1.1) under the following assumptions:: H(p,q,r) p+ < q− ≤ q+ < p∗2(x) and r(x). We say that λ ∈ R is an eigenvalue of problem 1.1 if the weak solution u defined above is not trivial. In the case of positive weight V , any possible eigenvalue of problem 1.1 is necessarily positive.
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