AIMS Mathematics, 2016, 1(3): 208-211. doi: 10.3934/Math.2016.3.208

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Existence of a solution to a semilinear elliptic equation

Department of Mathematics and Statistics, Texas A&M University - Corpus Christi, TX 78412 USA

We consider the equation $-\Delta u =f(u)-\frac{1}{|\Omega|}\int_{\Omega} f(u)d\mathbf{x}$, where the domain $\Omega= \mathbb{T}^N$, the $N$-dimensional torus, with $N=2$ or $N=3$. And $f$ is a given smooth function of $u$ for$u(\mathbf{x}) \in G \subset \mathbb{R}$. We prove that there exists a solution $u$ to this equation which is unique if $|\frac{df}{du}(u_0)|$ is sufficiently small, where $u_0 \in G$ is a given constant. And we prove that the solution $u$ is not unique if $\frac{df}{du}(u_0) $ is a simple eigenvalue of $-\Delta$.
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Copyright Info: © 2016, Diane Denny, licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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