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| Uniqueness for the Semilinear Elliptic Problems |
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| KeyWord:Elliptic, reaction-diffusion equation, uniqueness |
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| Abstract: |
| In this paper, we study the positive solutions of the semilinear elliptic equation
\begin{equation*}
\begin{cases}
{L}u+g(x,u)u=0 &\text{ in } \Omega,\Bu=0 &\text{ on } \partial\Omega,
\end{cases}
\end{equation*}
where $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain, $L$ is an elliptic operator, $B$ is a general boundary operator and $g(\cdot,\cdot)$ is a continuous function. This is a general problem proposed by Amann [\emph{Arch. Rational Mech. Anal.} 44 (1972)], Cac [\emph{J. London Math. Soc.} 25 (1982)] and Hess [\emph{Math. Z.} 154 (1977)]. We obtain various uniqueness results when the nonlinearity function $g$ satisfies some additional conditions. |
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