Existence of solutions for critical Neumann problem with superlinear perturbation in the half-space
Yinbin Deng, Longge Shi, Xinyue Zhang
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{'}, x_{N}): x{'}\in {\mathbb{R}}^{N-1}, x_{N}>0\}$, $N\geq3$, $λ>0$, $μ\in \mathbb{R}$, $2< p <{2}^{*}$, $n$ is the outward normal vector at the boundary ${\partial {{\mathbb{R}^{N}_{+}}}}$, $2^{*}=\frac{2N}{N-2}$ is the usual critical exponent for the Sobolev embedding $D^{1,2}({\mathbb{R}}^{N}_{+})\hookrightarrow {L^{{2}^{*}}}({\mathbb{R}}^{N}_{+})$ and ${2}_{*}=\frac{2(N-1)}{N-2}$ is the critical exponent for the Sobolev trace embedding $D^{1,2}({\mathbb{R}}^{N}_{+})\hookrightarrow {L^{{2}_{*}}}(\partial \mathbb{R}^{N}_{+})$. By establishing an improved Pohozaev identity, we show that the problem has no nontrivial solution if $μ\le 0$; 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 $λ>0$ and the different values of the parameters $p$ and $μ>0$. Particularly, for $λ>0$, $N\ge 4$, $2<p<2^*$, we prove that the problem has a positive solution if and only if $μ>0$. Moreover, the existence of multiple solutions for the problem is also obtained by dual variational principle for all $μ>0$ and suitable $λ$.