Sign-changing solutions for elliptic equations with fast increasing weight and concave-convex nonlinearities

Abstract

In this article, we study the problem $$ -\operatorname{div}(K(x)\nabla u)=a(x)K(x)|u|^{q-2}u+b(x)K(x)|u|^{2^{\ast}-2}u, \quad x\in \mathbb{R}^N, $$ where $2^{\ast}=2N/(N-2)$, $N\geq3$, $1<q<2$, $K(x)=\exp({|x|^{\alpha}/4})$ with $\alpha\geq2$. Under some assumptions on the potentials a(x) and b(x), we obtain a pair of sign-changing solutions of the problem via variational methods and certain estimates.

Keywords

• Sign-changing solutions
• variational method
• critical problem