Positive solutions for concave-convex elliptic problems involving $p(x)$-Laplacian

Abstract

We study the existence and nonexistence of positive solutions of the nonlinear equation -\Delta_{p(x)} u = \lambda k(x) u^q \pm h(x) u^r \text{in} \Omega,\quad u=0 \text{on} \partial\Omega, \tag{\rm Q} where $\Omega\subset\mathbb{R}^N$, $N\geq2$, is a regular bounded open domain in $\mathbb{R}^N$ and the $p(x)$-Laplacian \Delta_{p(x)} u := div( |\nabla u|^{p(x)-2} \nabla u) is introduced for a continuous function $p(x)>1$ defined on $\Omega$. The positive parameter $\lambda$ induces the bifurcation phenomena. The study of the equation (Q) needs generalized Lebesgue and Sobolev spaces. In this paper, under suitable assumptions, we show that some variational methods still work. We use them to prove the existence of positive solutions to the problem (Q) in $W_0^{1,p(x)}(\Omega)$. When we prove the existence of minimal solution, we use the sub-super solutions method.

Keywords

• variable exponent sobolev space
• $p(x)$-laplace operator
• concave-convex nonlinearities
• variational method