J. Korean Math. Soc. 2017; 54(5): 1573-1594
Online first article May 29, 2017 Printed September 1, 2017
https://doi.org/10.4134/JKMS.j160569
Copyright © The Korean Mathematical Society.
Jifeng Chu, Shapour Heidarkhani, Kit Ian Kou, and Amjad Salari
Shanghai Normal University, Razi University, University of Macau, Razi University
This paper deals with the existence and energy estimates of solutions for a class of degenerate nonlocal problems involving sub-linear nonlinearities, while the nonlinear part of the problem admits some hypotheses on the behavior at origin or perturbation property. In particular, for a precise localization of the parameter, the existence of a non-zero solution is established requiring the sublinearity of nonlinear part at origin and infinity. We also consider the existence of solutions for our problem under algebraic conditions with the classical Ambrosetti-Rabinowitz. In what follows, by combining two algebraic conditions on the nonlinear term which guarantees the existence of two solutions as well as applying the mountain pass theorem given by Pucci and Serrin, we establish the existence of the third solution for our problem. Moreover, concrete examples of applications are provided.
Keywords: $p$-Laplacian operator, nonlocal problem, singularity, multiple solutions, critical point theory
MSC numbers: Primary 35J92, 35J75, 34B10, 58E05
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