Multiplicity and concentration of normalized solutions to p-Laplacian equations

Abstract

In this paper, we study a type of p-Laplacian equation

$$\begin{aligned} -\Delta _{p}u =\lambda \left| u\right| ^{p-2}u+\left| u\right| ^{q-2}u,~x \in {\mathbb {R}}^{N}, \end{aligned}$$

with prescribed mass

$$\begin{aligned} \left( \,\int \limits _{{\mathbb {R}}^{N}}|u|^{p}\right) ^{\frac{1}{p}}=c>0, \end{aligned}$$

where $1<p< q< p^{*}: = \frac{pN}{N-p}$, $p < N$, $\lambda \in {\mathbb {R}}$ is a Lagrange multiplier. Firstly, we prove the existence of normalized solutions to p-Laplacian equations and provide accurate descriptions; secondly, we discuss the existence of ground states; finally, we study the radial symmetry of normalized solutions in the mass supercritical case. Besides, we also study normalized solutions to p-Laplacian equation with a potential function V(x)

$$\begin{aligned} -\Delta _{p}u + V(x)\left| u\right| ^{p-2}u =\lambda \left| u\right| ^{p-2}u+\left| u\right| ^{q-2}u,~x \in {\mathbb {R}}^{N}, \end{aligned}$$

under different assumptions on q and the constraint norm c, we prove the existence, nonexistence, concentration phenomenon and exponential decay of normalized solutions.