Existence of normalized solutions to a class of non-autonomous (p, q)-Laplacian equations

Abstract

We study the multiplicity of normalized solutions of the following (p, q)-Laplacian equation

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _p u-\Delta _q u=\lambda |u|^{p-2}u+V(\epsilon x)f(u)\ \ \text {in}\ \ \mathbb {R}^N,\\ \int _{\mathbb {R}^N}|u|^pdx=a^p,\\ \end{array}\right. \end{aligned}$$

where \(1<p<q<N\), a, \(\epsilon >0\), \(\Delta _lu:=\hbox {div}(|\nabla u|^{l-2}\nabla u)\) with \(l\in \{p,q\}\), stands for the l-Laplacian operator. \(\lambda \in \mathbb {R}\) is an unknown parameter that appears as a Lagrange multiplier. \(V:\mathbb {R}^N\rightarrow \mathbb {R}\) is a continuous function with some proper assumptions. f is a continuous function with \(L^p\)-mass subcritical growth. By using variational methods, we prove that the equation has multiple normalized solutions, as \(\epsilon \) is small enough. Precisely, the number of normalized solutions is at least twice that of the global maximum points of V.