Existence and Non-Existence Results for Quasilinear Elliptic Exterior Problems with Nonlinear Boundary Conditions

Abstract

Existence and non-existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition between the different growths of the non-linearities.

Sublinear eigenvalue problems associated to the Laplace operator revisited

“…Theorem 1: Assume that f is given by relation (5) and conditions (H1), (H2) and (H3) are fulfilled. Then λ 1 defined in (3) is an isolated eigenvalue of problem (1) and the corresponding set of eigenvectors is a cone.…”

“…In what concerns the eigenvalue problems involving quasilinear operators we remember, in the case of homogeneous elliptic operators, the contributions of Anane [1], de Thélin [14], Lindqvist [7] and Filippucci-Pucci-Rȃdulescu [5], while in the case of nonhomogeneous elliptic operators we point out the recent advances of Fan-Zhang-Zhao [4], Mihȃilescu-Rȃdulescu [10,11,12], Mihȃilescu-Pucci-Rȃdulescu [8,9] and Fan [3].…”

Solutions to p(x)-Laplace type equations via nonvariational techniques

“…Therefore, (a4) is consistent with the nonlinear growth condition (a2) which is accepted as a natural growth of order p(x). We would also like to mention that condition (a1) has been previously assumed in papers [20,30].…”

Continuous spectrum of Steklov nonhomogeneous elliptic problem

“…Inspired by the works of Mihăilescu and Rădulescu [10,14,15,17], we study (1.1) in three distinct situations.…”