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

TL;DR

This paper studies a normalized p-Laplacian Schrödinger equation in exterior domains, establishing existence, multiplicity, and properties of solutions using variational and topological methods.

Contribution

It introduces new existence and multiplicity results for solutions of the p-Laplacian equation with norm constraints in exterior domains, employing advanced analytical techniques.

Findings

Existence of positive solutions with negative Lagrange multiplier in small exterior domains.

Multiple solutions obtained via genus theory under symmetry assumptions.

Compactness of Palais-Smale sequences at higher energy levels.

Abstract

We are devoted to the study of the following nonlinear $p$-Laplacian Schr\"odinger equation with $L^{p}$-norm constraint \begin{align*} \begin{cases} &-\Delta_{p} u=\lambda |u|^{p-2}u +|u|^{r-2}u\quad\mbox{in}\quad\Omega,\\ &u=0\quad\mbox{on}\quad \partial\Omega,\\ &\int_{\Omega}|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 $\partial\Omega\neq\emptyset$ satisfying that $\R^{N}\setminus\Omega$ is bounded, $N\geq3$, $2\leq p<N$, $p<r<p+\frac{p^{2}}{N}$, $a>0$ and $\lambda\in\R$ 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 $\R^{N}$ and $\la<0$ when $\R^{N}\setminus\Omega$ 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.