Double phase anisotropic variational problems involving critical growth

Abstract

In this study, we investigate some existence results for double phase anisotropic variational problems involving critical growth. We first establish a Lions-type concentration-compactness principle and its variant at infinity for the solution space, which are our independent interests. Using these results, we obtain a nontrivial nonnegative solution to problems of generalized concave-convex type. We also obtain infinitely many solutions when the nonlinear term is symmetric. Our results are new even for the p(⋅)p\left(\cdot )-Laplace equations.

Keywords

• variable exponent elliptic operator
• variable exponent orlicz-sobolev spaces
• critical growth
• concentration-compactness principle
• variational methods
• 35j20
• 35j60
• 35j70
• 47j10
• 46e35