Normalized solution for $p$-Laplacian equation in exterior domain

Abstract

We are devoted to the study of the following nonlinear $p$-Laplacian Schrödinger equation with $L^p$-norm constraint \begin{align*} \begin{cases} &-Δ_{p} u=λ|u|^{p-2}u +|u|^{r-2}u\quad\mbox{in}\quadΩ,\\ &u=0\quad\mbox{on}\quad \partialΩ,\\ &\int_Ω|u|^{p}dx=a, \end{cases} \end{align*} where ${\Delta }_{p}u=\text{div}(|\nabla u{|}^{p-2}\nabla u)$, $\Omega \subset {\mathbb{R}}^N$ is an exterior domain with smooth boundary $\partial \Omega \ne \varnothing$ satisfying that ${\mathbb{R}}^N\setminus \Omega$ is bounded, $N\ge 3$, $2\le p<N$, $p<r<p+\frac{p^2}{N}$, $a>0$ and $\lambda \in \mathbb{R}$ is an unknown Lagrange multiplier. First, by using the splitting techniques and the Gagliardo-Nirenberg inequality, the compactness of Palais-Smale sequence of the above problem at higher energy level is established. Then, exploiting barycentric function methods, Brouwer degree and minimax principle, we obtain a solution $(u,\la)$ with $u>0$ in ${\mathbb{R}}^N$ and $\la<0$ when ${\mathbb{R}}^N\setminus \Omega$ is contained in a small ball. Moreover, we give a similar result if we remove the restriction on $\Omega$ and assume $a>0$ small enough. Last, with the symmetric assumption on $\Omega$, we use genus theory to consider infinite many solutions.