Existence results for some elliptic problems in RN including variable exponents above the critical growth

Abstract

We establish existence results for the following class of equations involving variable exponents $-\Delta u+u={\left|u\left(x\right)\right|}^{p\left(\left|x\right|\right)-1}u\left(x\right)+\lambda {\left|u\left(x\right)\right|}^{q\left(\left|x\right|\right)-1}u\left(x\right),x\in R^N,$where $\lambda \ge 0$, $N\ge 3$ and $p,q:[0,+\infty )\to (1,+\infty )$ are radial continuous functions which satisfy suitable conditions. For this purpose, it is sufficient to consider either subcriticality or criticality within a small region near the origin. Surprisingly, outside this region, the nonlinearity may oscillate between subcritical, critical, and supercritical growth in the Sobolev sense. Our approach enables the use of the variational methods to tackle problems with variable exponents in $R^N$ without imposing restrictions outside of a neighborhood of zero.