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A multiplicity result for quasilinear problems with convex andconcave nonlinearities and nonlinear boundary conditions in unbounded domains

Kandylakis Dimitrios

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Summary

We study the following quasilinear problem with nonlinear boundaryconditions−∆pu = λa(x)|u|p−2u + k(x)|u|q−2u − h(x)|u|s−2u, in Ω,|∇u|p−2∇u · η + b(x)|u|p−2u = 0 on ∂Ω,where Ω is an unbounded domain in RN with a noncompact and smoothboundary ∂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 essentiallybounded functions, q < p < s and p∗ < s. The properties of the first eigenvalueλ1 and the associated eigenvectors of the related eigenvalue problem areexamined. Then it is shown that if λ < λ1, the original problem admits an infi-nite number of solutions one of which is nonnegative, while if λ = λ1 it admitsat least one nonnegative solution. Our approach is variational in character

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