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

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 $p<q<p+\frac{p^2}{N}$.