Normalized solutions for p-Laplacian equations with potential

TL;DR

This paper establishes the existence of normalized solutions for a class of p-Laplacian equations with potential, revealing conditions under which solutions with positive or negative energy exist, including multiple solutions under certain bounds.

Contribution

It introduces new existence results for normalized solutions of p-Laplacian equations with potential, including multiple solutions and energy sign distinctions, under minimal assumptions.

Findings

Existence of a mountain pass solution with positive energy under small potential assumptions.

Existence of two solutions when the mass has an upper bound, including a local minimizer with negative energy.

No solutions with negative energy under certain conditions.

Abstract

In this paper, we consider the existence of normalized solutions for the following $p$-Laplacian equation \begin{equation*} \left\{\begin{array}{ll} -\Delta_{p}u-V(x)\lvert u\rvert^{p-2}u+\lambda\lvert u\rvert^{p-2}u=\lvert u\rvert^{q-2}u&\mbox{in}\ \mathbb{R}^N, \int_{\mathbb{R}^N}\lvert u\rvert^pdx=a^p, \end{array}\right. \end{equation*} where $N\geqslant 1$, $p>1$, $p+\frac{p^2}{N}<q<p^*=\frac{Np}{N-p}$(if $N\leqslant p$, then $p^*=+\infty$), $a>0$ and $\lambda\in\mathbb{R}$ is a Lagrange multiplier which appears due to the mass constraint. Firstly, under some smallness assumptions on $V$, but no any assumptions on $a$, we obtain a mountain pass solution with positive energy, while no solution with negative energy. Secondly, assuming that the mass $a$ has an upper bound depending on $V$, we obtain two solutions, one is a local minimizer with negative energy, the other is a mountain pass solution with positive energy.