Generalized eigenvalue problems of nonhomogeneous elliptic operators and their application
Dumitru Motreanu, Mieko Tanaka · Pacific Journal of Mathematics · 2013
We consider the equation -div(a(x, |∇u|)∇u) = λ|u| p-2 u (whose special case a(x, t) = t p-2 is the p-Laplace equation) on a bounded domain ⊂ ޒ N with C 2 boundary, with null boundary condition.We prove that there are λ ∈ ޒ for which the equation has a nontrivial solution.As an application, by variational methods, we present the existence of a positive solution to -div(a(x, |∇u|)∇u) = f (x, u) in , where f is asymptotically ( p-1)linear near zero and ∞, considering the nonresonant, resonant, and doubly resonant cases.We show that, generally, the spectrum of the operator -div(a(x, |∇u|)∇u) on W 1, p 0 ( ) is not discrete.