Normalized solutions for $p$-Laplacian equation with critical Sobolev exponent and mixed nonlinearities

TL;DR

This paper investigates the existence, multiplicity, and asymptotic behavior of normalized solutions for a critical p-Laplacian equation with mixed nonlinearities, employing variational methods and concentration compactness.

Contribution

It provides new existence and nonexistence results for solutions under various parameter conditions, including asymptotic analysis as parameters vary.

Findings

Existence of solutions for ta>0 under certain conditions.

Nonexistence of solutions for ta<0.

Infinitely many solutions when p<q<p+ p^2/N.

Abstract

In this paper, we consider the existence and multiplicity of normalized solutions for the following $p$-Laplacian critical equation \begin{align*} \left\{\begin{array}{ll} -\Delta_{p}u=\lambda\lvert u\rvert^{p-2}u+\mu\lvert u\rvert^{q-2}u+\lvert u\rvert^{p^*-1}u&\mbox{in}\ \mathbb{R}^N, \int_{\mathbb{R}^N}\lvert u\rvert^pdx=a^p, \end{array}\right. \end{align*} where $1<p<N$, $2<q<p^*=\frac{Np}{N-p}$, $a>0$, $\mu\in\mathbb{R}$ and $\lambda\in\mathbb{R}$ is a Lagrange multiplier. Using concentration compactness lemma, Schwarz rearrangement, Ekeland variational principle and mini-max theorems, we obtain several existence results under $\mu>0$ and other assumptions. We also analyze the asymptotic behavior of there solutions as $\mu\rightarrow 0$ and $\mu$ goes to its upper bound. Moreover, we show the nonexistence result for $\mu<0$ and get that the $p$-Laplacian equation has infinitely solutions by genus theory when $p<q<p+\frac{p^2}{N}$.