Existence of solutions for critical Neumann problem with superlinear perturbation in the half-space

Abstract

In this paper, we consider the existence and multiplicity of solutions for the critical Neumann problem \begin{equation}\label{1.1ab} \left\{ \begin{aligned} -Δ{u}-\frac{1}{2}(x \cdot{\nabla u})&= λ{|u|^{{2}^{*}-2}u}+{μ{|u|^{p-2}u}}& \ \ \mbox{in} \ \ \ {\mathbb{R}^{N}_{+}}, \frac{\partial u}{\partial n}&=\sqrtλ|u|^{{2}_{*}-2}u \ & \mbox{on}\ {\partial {{\mathbb{R}^{N}_{+}}}}, \end{aligned} \right. \end{equation} where ${\mathbb{R}}_{+}^N=\{(x{}^{\prime },x_N):x{}^{\prime }\in {\mathbb{R}}^{N-1},x_N>0\}$, $N\ge 3$, $\lambda >0$, $\mu \in \mathbb{R}$, $2<p<2^{\ast }$, $n$ is the outward normal vector at the boundary $\partial {\mathbb{R}}_{+}^N$, $2^{\ast }=\frac{2N}{N-2}$ is the usual critical exponent for the Sobolev embedding $D^{1,2}({\mathbb{R}}_{+}^N)\hookrightarrow L^{2^{\ast }}({\mathbb{R}}_{+}^N)$ and $2_{\ast }=\frac{2(N-1)}{N-2}$ is the critical exponent for the Sobolev trace embedding $D^{1,2}({\mathbb{R}}_{+}^N)\hookrightarrow L^{2_{\ast }}(\partial {\mathbb{R}}_{+}^N)$. By establishing an improved Pohozaev identity, we show that the problem has no nontrivial solution if $\mu \le 0$; By applying the Mountain Pass Theorem without $(PS)$ condition and the delicate estimates for Mountain Pass level, we obtain the existence of a positive solution for all $\lambda >0$ and the different values of the parameters $p$ and $\mu >0$. Particularly, for $\lambda >0$, $N\ge 4$, $2<p<2^{\ast }$, we prove that the problem has a positive solution if and only if $\mu >0$. Moreover, the existence of multiple solutions for the problem is also obtained by dual variational principle for all $\mu >0$ and suitable $\lambda$.