Electron. J. Diff. Eqns., Vol. 2005(2005), No. 57, pp. 1-12.

### A multiplicity result for quasilinear problems with convex and concave nonlinearities and nonlinear boundary conditions in unbounded domains Dimitrios A. Kandilakis

Abstract:
We study the following quasilinear problem with nonlinear boundary conditions

where is an unbounded domain in with a noncompact and smooth boundary , denotes the unit outward normal vector on , is the -Laplacian, , , and are nonnegative essentially bounded functions, and . The properties of the first eigenvalue and the associated eigenvectors of the related eigenvalue problem are examined. Then it is shown that if , the original problem admits an infinite number of solutions one of which is nonnegative, while if it admits at least one nonnegative solution. Our approach is variational in character.

Submitted September 27, 2004. Published May 31, 2005.
Math Subject Classifications: 35J20, 35J60.
Key Words: Variational method; fibering method; Palais-Smale condition; genus.

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 Dimitrios A. Kandilakis Department of Sciences Technical University Of Crete Chania, Crete, 73100 Greece email: dkan@science.tuc.gr

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