Longge Shi | Searcharxiv

arXiv2024

On the existence of positive solution for a Neumann problem with double critical exponents in half-space

In this paper, we consider the existence and nonexistence of positive solution for a Neumann problem with double critical exponents and fast increasing weighted in half-space. This problem is closely related to the study of self-similar solutions for nonlinear heat equation. By applying the Mountain Pass Theorem without (PS) condition and the delicate estimates for the Mountain Pass level, we obtain the existence of a positive solution under different assumptions. Meanwhile, some nonexistence results for this problem is also obtained by an improved Pohozaev identity and Hardy inequality according to the value of the parameters. Particularly, we give the best lower bound of the parameter for the existence of a positive solution of this problem if dimension N=4.

math.AP↗

arXiv2024

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

In this paper, we consider the existence and multiplicity of solutions for the critical Neumann problem \begin{equation}\label{1.1ab} \left\{ \begin{aligned} -\Delta {u}-\frac{1}{2}(x \cdot{\nabla u})&= \lambda{|u|^{{2}^{*}-2}u}+{\mu {|u|^{p-2}u}}& \ \ \mbox{in} \ \ \ {{\mathbb{R}}^{N}_{+}}, \frac{{\partial u}}{{\partial n}}&=\sqrt{\lambda}|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\}$, $N\geq3$, $\lambda>0$, $\mu\in \mathbb{R}$, $2< p <{2}^{*}$, $n$ 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 $\mu \le 0$; 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 $\lambda >0$, $N\ge 4$, $2 0$. Moreover, the existence of multiple solutions for the problem is also obtained by dual variational principle for all $\mu>0$ and suitable $\lambda$.

math.AP↗