Multiple solutions for quasilinear elliptic problems with concave-convex nonlinearities in Orlicz\u2013Sobolev spaces

Shujun Wu

doi:10.1186/s13661-019-1256-3

Abstract

Using variational arguments, we establish the existence of two nontrivial solutions for quasilinear elliptic problems in Orlicz–Sobolev spaces, where the nonlinear terms exhibit the combined effects of concave and convex without the Ambrosetti–Rabinowitz type condition.

Highlights

IntroductionWe investigate a class of nonlinear problems in the Orlicz–Sobolev setting:
Papageorgiou and Rocha [10] considered a p-Laplacian problem with nonlinearities of the form m(x)|u|r–2u + f (x, u) with 1 < r < p < ∞ when f is (p – 1) superlinear near infinity but does not satisfy the AR-condition. They employed variational approach and the Ekeland variational principle [11] to show the existence of two nontrivial solutions
In this paper, motivated by [12,13,14, 16,17,18], we investigate a class of quasilinear elliptic problems (1.1) with concave and convex nonlinearities which do not satisfy the AR-condition in Orlicz–Sobolev spaces

Summary

Introduction

We investigate a class of nonlinear problems in the Orlicz–Sobolev setting:. Papageorgiou and Rocha [10] considered a p-Laplacian problem with nonlinearities of the form m(x)|u|r–2u + f (x, u) with 1 < r < p < ∞ when f is (p – 1) superlinear near infinity but does not satisfy the AR-condition. They employed variational approach and the Ekeland variational principle [11] to show the existence of two nontrivial solutions. In the case of λ = 0, Chung [27], Carvalho et al [28] studied problem (1.1) when f is (φ0 –1) superlinear near infinity without the AR-condition. Our results complement and extend previous studies such as [10, 27, 28]

Preliminaries

Main results

Findings

Conclusions