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Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities

  • Louis JeanjeanEmail author
  • Kazunaga Tanaka
Article

Abstract.

We consider a class of equations of the form \(-\varepsilon^2\Delta u + V(x)u = f(u), \quad u\in H^1({\bf R}^N).\) By variational methods, we show the existence of families of positive solutions concentrating around local minima of the potential V(x), as \(\varepsilon\to 0\). We do not require uniqueness of the ground state solutions of the associated autonomous problems nor the monotonicity of the function \(\xi\mapsto \frac{f(\xi)}{\xi}\). We deal with asymptotically linear as well as superlinear nonlinearities.

Keywords

Local Minimum State Solution Variational Method Potential Versus Elliptic Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  1. 1.Equipe de Mathématiques (UMR CNRS 6623)Université de Franche-ComtéBesançonFrance

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