Normalized solutions for some quasilinear elliptic equation with critical Sobolev exponent

Abstract

Consider the equation \begin{equation*} -\Delta_p u =\lambda |u|^{p-2}u+\mu|u|^{q-2}u+|u|^{p^\ast-2}u\ \ {\rm in}\ \R^N \end{equation*} under the normalized constraint $$\int_{ \R^N}|u|^p=c^p,$$ where $-\Delta_pu={\rm div} (|\nabla u|^{p-2}\nabla u)$, $1<p<N$, $p<q<p^\ast=\frac{Np}{N-p}$, $c,\mu>0$ and $\lambda\in\R$. In the purely $L^p$-subcritical case, we obtain the existence of ground state solution by virtue of truncation technique, and obtain multiplicity of normalized solutions. In the purely $L^p$-critical and supercritical case, we drive the existence of positive ground state solution, respectively. Finally, we investigate the asymptotic behavior of ground state solutions obtained above as $\mu\to0^+$.