Electron. J. Diff. Equ., Vol. 2015 (2015), No. 85, pp. 1-18.

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

Guoqing Zhang, Huiling Fu

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
$$\displaylines{ -\hbox{div}\Big(\frac{|\nabla u|^{2p-2}\nabla u} {\sqrt{1+|\nabla u|^{2p}}}\Big) +T(|x|)|u|^{\alpha-2}u=K(|x|)|u|^{s-2}u,\quad u>0,\; x\in\mathbb{R}^{N},\cr u(|x|)\to 0,\quad\text{as } |x|\to \infty, }$$
where $N\geq3$, $1<\alpha<p<2p<N$, $s$ satisfies some suitable conditions, $K(|x|)$ and $T(|x|)$ are continuous, nonnegative functions.

Submitted August 6, 2014. Published March 7, 2015.
Math Subject Classifications: 35J65, 35J70.
Key Words: Compact theorem; modified capillary surface equation; weighted Orlicz-Sobolev space; ground state.

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Guoqing Zhang
College of Sciences
University of Shanghai for Science and Technology
Shanghai 200093, China
email: shzhangguoqing@126.com
Huiling Fu
College of Sciences
University of Shanghai for Science and Technology
Shanghai 200093, China
email: fuhuiliing80@163.com

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