We study the boundary value problem in , u =0 on , where is a smooth bounded domain in and is a -Laplace type operator, with . We prove that if λ is large enough then there exist at least two non-negative weak solutions. Our approach relies on the variable exponent theory of generalized Lebesgue–Sobolev spaces, combined with adequate variational methods and a variant of the Mountain Pass lemma.

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

We study a nonlinear Neumann boundary value problem associated to a nonhomogeneous differential operator. Taking into account the competition between the nonlinearity and the bifurcation parameter, we establish sufficient conditions for the existence of nontrivial solutions in a related Orlicz–Sobolev space.

In this paper, we prove the existence of a continuous spectrum for a family of discrete boundary value problems. The main existence results are obtained by using critical point theory. The equations studied in the paper represent a discrete variant of some recent anisotropic variable exponent problems, which deserve as models in different fields of mathematical physics.