Normalized Solution for p-Kirchhoff Equation with a L²-supercritical Growth

Abstract

In this paper, we investigate the following p-Kirchhoff equation

$$\left\{ {\matrix{{(a + b\,\int_{{\mathbb{R}^N}} {(|\nabla u{|^p} + |u{|^p})dx)\,( - {\Delta _p}u + |u{|^{p - 2}}u) = |u{|^{s - 2}}u + \mu u,\,\,x \in {\mathbb{R}^N},} } \hfill \cr {\int_{{\mathbb{R}^N}} {|u{|^2}dx = \rho ,} } \hfill \cr } } \right.$$

where a > 0, b ≥ 0, ρ > 0 are constants, ${p^ * } = {{Np} \over {N - p}}$ is the critical Sobolev exponent, μ is a Lagrange multiplier, $- {\Delta _p}u = - {\rm{div}}(|\nabla u{|^{p - 2}}\nabla u)$, $2 < p < N < 2p,\,\,\,\mu \in \mathbb{R}$ and $s \in (2{{N + 2} \over N}p - 2,\,\,\,{p^ * })$. We demonstrate that the p-Kirchhoff equation has a normalized solution using the mountain pass lemma and some analysis techniques.