Ground states for a modified capillary surface equation in weighted Orlicz-Sobolev space

Abstract

In this article, we prove a compact embedding theorem for the weighted Orlicz-Sobolev space of radially symmetric functions. Using the embedding theorem and critical points theory, we prove the existence of multiple radial solutions and radial ground states for the following modified capillary surface equation $$\displaylines{ -\operatorname{div}\Big(\frac{|\nabla u|^{2p-2}\nabla u} {\sqrt{1+|\nabla u|^{2p}}}\Big) +T(|x|)|u|^{\alpha-2}u=K(|x|)|u|^{s-2}u,\quad u>0,\; x\in\mathbb{R}^{N},\cr u(|x|)\to 0,\quad\text{as } |x|\to \infty, }$$ where $N\geq3$, $1<\alpha<p<2p<N$, $s$ satisfies some suitable conditions, $K(|x|)$ and $T(|x|)$ are continuous, nonnegative functions.

Keywords

• Compact theorem
• modified capillary surface equation
• weighted Orlicz-Sobolev space
• ground state