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

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

References

• 1. Haim Brezis and Walter A. Strauss, Semi-linear second-order elliptic equations in L1, J. Math.Soc. Japan 25 (1973), no. 4, 565-590.
• 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.