# American Institute of Mathematical Sciences

May  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 and Continuous Dynamical Systems, 2007, 19 (2) : 447-467. doi: 10.3934/dcds.2007.19.447
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