Normalized solutions for p-Laplacian equations with potential

Abstract

In this paper, we consider the existence of normalized solutions for the following $p$-Laplacian equation \begin{equation*} \left\{\begin{array}{ll} -Δ_{p}u-V(x)\lvert u\rvert^{p-2}u+λ\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 $N⩾1$, $p>1$, $p+\frac{p^2}{N}<q<p^{\ast }=\frac{Np}{N-p}$(if $N⩽p$, then $p^{\ast }=+\infty$), $a>0$ 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.