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AIMS Mathematics

2016, Volume 1, Issue 3: 208-211. doi: 10.3934/Math.2016.3.208
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Research article

Existence of a solution to a semilinear elliptic equation

  • Department of Mathematics and Statistics, Texas A&M University - Corpus Christi, TX 78412 USA
  • Received: 21 August 2016 Accepted: 26 August 2016 Published: 30 August 2016

  • 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 $-\Delta$.

    Citation: Diane Denny. Existence of a solution to a semilinear elliptic equation[J]. AIMS Mathematics, 2016, 1(3): 208-211. doi: 10.3934/Math.2016.3.208

    Related Papers:

  • 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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    [2] L. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, Rhode Island, 1998.
    [3] J.P. Gossez and P. Omari, A necessary and su cient condition of nonresonance for a semilinear Neumann problem, Proceedings of the American Mathematical Society 114 (1992), no. 2, 433-442.
    [4] Chaitan P. Gupta, Perturbations of second order linear elliptic problems by unbounded nonlinearities,Nonlinear Analysis: Theory, Methods & Applications 6 (1982), no. 9, 919-933.
    [5] P.L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Review 24(1982), no. 4, 441-467.
    [6] Jason R. Looker, Semilinear elliptic Neumann problems with rapid growth in the nonlinearity, Bull.Austral. Math. Soc. 74 (2006), 161-175.
    [7] M. Renardy and R. Rogers, An Introduction to Partial Di erential Equations, Springer-Verlag:New York, 1993.
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  • © 2016 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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