Normalized solutions for the p-Laplacian equation with a trapping potential

Abstract

In this article, we are concerned with normalized solutions for the pp -Laplacian equation with a trapping potential and Lr{L}^{r}-supercritical growth, where r=pr=p or 2.2. The solutions correspond to critical points of the underlying energy functional subject to the Lr{L}^{r}-norm constraint, namely, ∫RN∣u∣rdx=c{\int }_{{{\mathbb{R}}}^{N}}| u{| }^{r}{\rm{d}}x=c for given c>0.c\gt 0. When r=p,r=p, we show that such problem has a ground state with positive energy for cc small enough. When r=2,r=2, we show that such problem has at least two solutions both with positive energy, which one is a ground state and the other one is a high-energy solution.

Keywords

• p-laplacian equation
• normalized solutions
• variational methods
• 35j20
• 35j60
• 35j92