Electron. J. Diff. Equ., Vol. 2013 (2013), No. 145, pp. 1-24.

Existence of multiple solutions to elliptic equations satisfying a global eigenvalue-crossing condition

Duong Minh Duc, Loc Hoang Nguyen, Luc Nguyen

We study the multiplicity of solutions to the elliptic equation $\Delta u+ f(x,u)=0$, under the assumption that f(x,u)/u crosses globally but not pointwise any eigenvalue for every x in a part of the domain, when u varies from $-\infty$ to $\infty$. Also we relax the conditions on uniform convergence of f(x,s)/s, which are essential in many results on multiplicity for asymptotically linear problems.

Submitted April 10, 2013. Published June 25, 2013.
Math Subject Classifications: 47H11, 55M25, 35J20.
Key Words: Index of critical points; mountain pass type; nonlinear elliptic equations; multiplicity of solutions.

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Duong Minh Duc
University of Sciences - Hochiminh City, Vietnam
email: dmduc@hcmuns.edu.vn
Loc Hoang Nguyen
Department of Mathematics, Ecole Normale Suprieure
Paris, France
email: lnguyen@dma.ens.fr
  lu Nguyen
Department of Mathematics, Princeton University
Princeton, New Jersey 08544, USA
email: llnguyen@princeton.edu

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