Multiplicity and concentration of normalized solutions to p-Laplacian equations

In this paper, we study a type of p-Laplacian equation \(\begin{aligned} -\Delta _{p}u =\lambda \left| u\right| ^{p-2}u+\left| u\right| ^{q-2}u,~x \in {\mathbb {R}}^{N}, \end{aligned}\) $$\begin{array}{c}\hfill -{\mathrm{\Delta }}_{p}u=\lambda {\left|u\right|}^{p-2}u+{\left|u\right|}^{q-2}u,x\in {\mathbb{R}}^N,\end{array}$$ with prescribed mass \(\begin{aligned} \left( \,\int \limits _{{\mathbb {R}}^{N}}|u|^{p}\right) ^{\frac{1}{p}}=c>0, \end{aligned}\) $$\begin{array}{c}\hfill {\left(,\underset{{\mathbb{R}}^N}{\int },{|u|}^p\right)}^{\frac{1}{p}}=c>0,\end{array}$$ where \(1<p< q< p^{*}: = \frac{pN}{N-p}\) $1<p<q<p^{\ast }:=\frac{\mathit{pN}}{N-p}$ , \(p < N\) $p<N$ , \(\lambda \in {\mathbb {R}}\) $\lambda \in \mathbb{R}$ 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) \(\begin{aligned} -\Delta _{p}u + V(x)\left| u\right| ^{p-2}u =\lambda \left| u\right| ^{p-2}u+\left| u\right| ^{q-2}u,~x \in {\mathbb {R}}^{N}, \end{aligned}\) $$\begin{array}{c}\hfill -{\mathrm{\Delta }}_{p}u+V(x){\left|u\right|}^{p-2}u=\lambda {\left|u\right|}^{p-2}u+{\left|u\right|}^{q-2}u,x\in {\mathbb{R}}^N,\end{array}$$ under different assumptions on q and the constraint norm c, we prove the existence, nonexistence, concentration phenomenon and exponential decay of normalized solutions.