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

Source 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.