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Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities

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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.

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Authors and Affiliations

  1. Equipe de Mathématiques (UMR CNRS 6623), Université de Franche-Comté, 16 Route de Gray, 25030, Besançon, France

    Louis Jeanjean & Kazunaga Tanaka

Authors
  1. Louis Jeanjean
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  2. Kazunaga Tanaka
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Correspondence to Louis Jeanjean.

Additional information

Received: 8 November 2003, Accepted: 18 November 2003, Published online: 2 April 2004

Mathematics Subject Classification (2000):

35B25, 35J65, 58E05

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Jeanjean, L., Tanaka, K. Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Cal Var 21, 287–318 (2004). https://doi.org/10.1007/s00526-003-0261-6

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  • DOI: https://doi.org/10.1007/s00526-003-0261-6

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