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arXiv 1606.04892math.AP
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有界区域上具有伪相对论算子的半线性椭圆型方程

Semilinear elliptic equations with the pseudo-relativistic operator on a bounded domain

Woocheol Choi, Younghun Hong, Jinmyoung Seok

备注：
24 pages. In this version, we improved the previous version of this paper by obtaining nontrivial solutions for the super-critical case
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摘要：

研究了有界区域上含伪相对论算子的半线性方程（\sqrt{-Δ+ m^2} - m）u = |u| ^{p-1} uPDMATHABMO \textrm{in} ~Ω的Dirichlet问题，其Dirichlet边界条件为$\partial Ω$上的$u=0$.这里，$p \in (1,\infty)$和算子$(\sqrt{-Δ+ m^2} - m)$根据谱分解来定义。在本文中，我们调查的存在性和非平凡的解决方案，根据$p$，$m$和$Ω$的选择。准确地说，我们表明，如果$p$不是$H^1$亚临界，则$(i)$（$p \geq \frac{n+2}{n-2}$）和$Ω$是星形的，则方程对所有$m > 0$都没有非平凡解;如果$p$不是$H^{1/2}$超临界，则为$(ii)$（$1 <p \leq \frac{n+1}{n-1}$），则存在所有$m>0$和任何有界域$Ω$的最小能量解; $(iii)$最后，在中间范围内（$\frac{n+1}{n-1}<p<\frac{n+2}{n-2}$），该问题有一个非平凡解，条件是$m$足够大且问题-Δu = |u| ^{p-1}u \quad \textrm{in} ~Ω，\qquad u =0 \quad \textrm{on} ~ \partial Ω存在非退化非平凡解，例如，当$Ω$是球或环时。

英文摘要：

We study the Dirichlet problem for the semilinear equations involving the pseudo-relativistic operator on a bounded domain, (\sqrt{-Δ+ m^2} - m)u =|u|^{p-1}u \quad \textrm{in}~Ω, with the Dirichlet boundary condition $u=0$ on $\partial Ω$. Here, $p \in (1,\infty)$ and the operator $(\sqrt{-Δ+ m^2} - m)$ is defined in terms of spectral decomposition. In this paper, we investigate existence and nonexistence of a nontrivial solution, depending on the choice of $p$, $m$ and $Ω$. Precisely, we show that $(i)$ if $p$ is not $H^1$ subcritical ($p \geq \frac{n+2}{n-2}$) and $Ω$ is star-shaped, the equation has no nontrivial solution for all $m > 0$; $(ii)$ if $p$ is not $H^{1/2}$ supercritical ($1 <p \leq \frac{n+1}{n-1}$), then there exists a least energy solution for all $m>0$ and any bounded domain $Ω$; $(iii)$ finally, in the intermediate range ($\frac{n+1}{n-1}<p<\frac{n+2}{n-2}$), the problem has a nontrivial solution, provided that $m$ is sufficiently large and the problem -Δu = |u|^{p-1}u \quad \textrm{in}~Ω, \qquad u =0\quad \textrm{on}~\partial Ωadmits a non-degenerate nontrivial solution, for example, when $Ω$ is a ball or an annulus.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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