A MULTIPLICITY RESULT FOR QUASILINEAR PROBLEMS WITH CONVEX AND CONCAVE NONLINEARITIES AND NONLINEAR BOUNDARY CONDITIONS IN UNBOUNDED DOMAINS

Dimitrios Kandilakis
2005 Electronic Journal of Differential Equations   unpublished
We study the following quasilinear problem with nonlinear boundary conditions −∆pu = λa(x)|u| p−2 u + k(x)|u| q−2 u − h(x)|u| s−2 u, in Ω, ||u| p−2 u · η + b(x)|u| p−2 u = 0 on ∂Ω, where Ω is an unbounded domain in R N with a noncompact and smooth boundary ∂Ω, η denotes the unit outward normal vector on ∂Ω, ∆pu = div(||u| p−2 u) is the p-Laplacian, a, k, h and b are nonnegative essentially bounded functions, q < p < s and p * < s. The properties of the first eigen-value λ 1 and the associated
more » ... nd the associated eigenvectors of the related eigenvalue problem are examined. Then it is shown that if λ < λ 1 , the original problem admits an infinite number of solutions one of which is nonnegative, while if λ = λ 1 it admits at least one nonnegative solution. Our approach is variational in character.
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