Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities


DOI: 10.1007/s00526-003-0261-6

Cite this article as:
Jeanjean, L. & Tanaka, K. Cal Var (2004) 21: 287. doi:10.1007/s00526-003-0261-6


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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© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  1. 1.Equipe de Mathématiques (UMR CNRS 6623)Université de Franche-ComtéBesançonFrance
  2. 2.Department of MathematicsSchool of Science and Engineering, Waseda UniversityTokyoJapan