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DCDS

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.

PROC

In this article, we prove that there are three unstable positive solutions
of a semilinear elliptic equation in a two bumps domain or in a one hole
domain in which one is axially symmetric and the other two are nonaxially
symmetric.

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