Normalized solutions for some quasilinear elliptic equation with critical Sobolev exponent

Abstract

Consider the equation \begin{equation*} -Δ_p u =λ|u|^{p-2}u+μ|u|^{q-2}u+|u|^{p^\ast-2}u\ \ {\rm in}\ \R^N \end{equation*} under the normalized constraint $${\int }_{{\mathbb{R}}^N}|u{|}^p=c^p,$$ where $-{\Delta }_{p}u=\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)$, $1<p<N$, $p<q<p^{\ast }=\frac{Np}{N-p}$, $c,\mu >0$ and $\lambda \in \mathbb{R}$. 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 $\mu \to 0^{+}$.