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

#### Kandylakis Dimitrios

Πλήρης Εγγραφή

URI: http://purl.tuc.gr/dl/dias/509DC46F-5781-4063-8C94-E02B73A72C77
Έτος 2005
Τύπος Δημοσίευση σε Περιοδικό με Κριτές
Άδεια Χρήσης Λεπτομέρειες
Βιβλιογραφική Αναφορά D.A. Kandilakis, "A multiplicity result for quasilinear problems with convex and concave nonlinearities and nonlinear boundary conditions in unbounded domain,"Electronic Journal of Differential Equations, vol. 2005, no. 57, pp. 1–12, 2005.
Εμφανίζεται στις Συλλογές

## Περίληψη

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