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

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

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 $-\Delta$.
  Article Metrics


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.

Copyright Info: © 2016, Diane Denny, licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved