2007, 19(2): 447-467. doi: 10.3934/dcds.2007.19.447

On domains and their indexes with applications to semilinear elliptic equations

1. 

Department of Applied Mathematics, Hsuan Chuang University, Hsinchu

Received  May 2005 Revised  October 2005 Published  July 2007

Let $\Omega$ be a domain in $\mathbb{R}^{N}$, $N\geq1$, and $2^$∗$=\infty$ if $N=1,2$, $2^$∗$=\frac{2N}{N-2}$ if $N>2$, $2 < p < 2^$∗. Consider the semilinear elliptic equation $ -\Delta u+u=|u|^{p-2}u\text{ in }\Omega; u\in H_{0}^{1}(\Omega). $ The existence, the nonexistence, and the multiplicity of positive solutions of the equation are affected by the geometry and the topology of the domain $\Omega$. In the article, we first present various analyses and use them to characterize which domain $\Omega$ is a ground state domain or a non-ground state domain. Secondly, for a $y$-symmetric domain $\Omega$, we study their index $\alpha(\Omega)$ and $y$-symmetric index $\alpha_{s}(\Omega)$. We determine whether $\alpha(\Omega)=\alpha_{s}(\Omega)$ or $\alpha (\Omega)<\alpha_{s}(\Omega)$. In case that $\alpha(\Omega)<\alpha_{s}(\Omega)$ and that both $\alpha(\Omega)$ and $\alpha_{s}(\Omega)$ admits ground state solutions, then we obtain that in $\Omega$, the equation has three positive solutions, of which one is $y$-symmetric and other two are not $y$-symmetric.
Citation: Hwai-Chiuan Wang. On domains and their indexes with applications to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 447-467. doi: 10.3934/dcds.2007.19.447
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