On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

Top Cited Papers
Publisher Website
Google Scholar
Abstract
We consider the nonlinear eigenvalue problem \[ − div ( | ∇ u | p ( x ) − 2 ∇ u ) = λ | u | q ( x ) − 2 u -\textrm {div}\left (|\nabla u|^{p(x)-2}\nabla u\right )=\lambda |u|^{q(x)-2}u \] in Ω \Omega , u = 0 u=0 on ∂ Ω \partial \Omega , where Ω \Omega is a bounded open set in R N \mathbb R^N with smooth boundary and p p , q q are continuous functions on Ω ¯ \overline \Omega such that 1 > inf Ω q > inf Ω p > sup Ω q 1>\inf _\Omega q> \inf _\Omega p>\sup _\Omega q , sup Ω p > N \sup _\Omega p>N , and q ( x ) > N p ( x ) / ( N − p ( x ) ) q(x)>Np(x)/\left (N-p(x)\right ) for all x ∈ Ω ¯ x\in \overline \Omega . The main result of this paper establishes that any λ > 0 \lambda >0 sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland’s variational principle.

This publication has 23 references indexed in Scilit: