A discontinuous problem involving the p-Laplacian operator and critical exponent in $mathbb{R}^N$

Abstract

Using convex analysis, we establish the existence of at least two nonnegative solutions for the quasilinear problem $$ -Delta_{p}u=H(u-a)u^{p^*-1} +lambda h(x)quadhbox{in }mathbb{R}^N $$ where $Delta_{p}u$ is the $p$-Laplacian operator, $H$ is the Heaviside function, $p^*$ is the Sobolev critical exponent, and $h$ is a positive function.

Keywords

• Variational methods
• discontinuous nonlinearities
• critical exponents.