论文标题
半空间中的非线性schrödinger方程
The nonlinear Schrödinger equation in the half-space
论文作者
论文摘要
本文与半个空间dirichlet问题\ begin {equation} \ tag {$ p_c $} \ label {Quards-abstract}-ΔV + v = | v | v |^{p-1} v，\ \ \ \ \ \ \ mbox {in} \ partial \ Mathbb {r}^n _ {+}，\ \ qquad \ lim_ {x_n \ to \ infty} v（x'，x'，x_n）= 0 \ mbox {均匀in} x' $ \ mathbb {r}^n _ {+}：= \ {\，x \ in \ mathbb {r}^n：x_n：x_n> 0 \} $，对于某些$ n \ geq 1 $和$ p> 1 $，$ p> 1 $，$ c> 0 $是常量。我们分析了有限的阳性解决方案对\ eqref {argie-abstract}的存在，不存在和多样性。我们证明，有界的正解决方案的存在和多样性以\ eqref {argie-abstract}的形式取决于$ c> 0 $的值以及尺寸$ n $。我们在（1，\ sqrt {e}）$中找到一个明确的数字$ c_p \，仅取决于$ p $，这决定了存在和不存在之间的阈值。特别是，在$ n \ geq 2 $中，我们证明，对于$ 0 <c <c_p $，问题\ eqref {argie-abstract}承认，无限的许多有限的积极解决方案，而对于$ c> c_p $，对\ eqref {comefef {areage-aighate-abscrats}}}}}}}。
The present paper is concerned with the half-space Dirichlet problem \begin{equation} \tag{$P_c$} \label{problem-abstract} -Δv + v = |v|^{p-1}v,\ \mbox{ in } \mathbb{R}^N_{+}, \qquad v = c,\ \mbox{ on } \partial \mathbb{R}^N_{+},\ \qquad \lim_{x_N \to \infty} v(x',x_N) = 0 \mbox{ uniformly in }x' \in \mathbb{R}^{N-1}, \end{equation} where $\mathbb{R}^N_{+} := \{\,x \in \mathbb{R}^N: x_N > 0\, \}$ for some $N \geq 1$ and $p > 1$, $c > 0$ are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to \eqref{problem-abstract}. We prove that the existence and multiplicity of bounded positive solutions to \eqref{problem-abstract} depend in a striking way on the value of $c > 0$ and also on the dimension $N$. We find an explicit number $c_p \in (1,\sqrt{e})$, depending only on $p$, which determines the threshold between existence and non-existence. In particular, in dimensions $N \geq 2$, we prove that, for $0 < c < c_p$, problem \eqref{problem-abstract} admits infinitely many bounded positive solutions, whereas, for $c > c_p$, there are no bounded positive solutions to \eqref{problem-abstract}.
