Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions

Ghasem A Afrouzi,Nguyen Thanh Chung,Z Naghizadeh

doi:10.5269/bspm.44144

Abstract

In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using $(S_+)$ mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provied that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.

Highlights

Kirchhoff type problems have been studied by many authors and many important and interesting results are established, we refer to [2,6,7,8,9,10,11,13] for the problem with Dirichlet aboundary condition
In [11], Fan firstly considered a class of bi-nonlocal p(x)-Kirchhoff type problems with Dirichlet boundary conditions of the form
Under suitable conditions on a, b and the Ambrosetti-Rabinowitz type condition on the nonlinear term f, the author proved the existence of at least a non-trivial solution or the existence of infinitely many solutions for problem (1.4) by using variational methods. We notice that it follows from the Ambrosetti-Rabinowitz type condition that the nonlinear term is superlinear at infinity

Summary

For any p

For all (N −1)p N −p denote by Sq,∂Ω the best constant in the embedding W 1,p(Ω) ֒→ Lq(∂Ω), i.e. We say that u is a weak solution of problem (1.1) if and only if J ′(u), v = λ I′(u), v + μ ψ′(u), v for any v ∈ X. There exist λ∗, μ∗ > 0 such that problem (1.1) has at least two distinct, nonnegative, nontrivial weak solutions, provided that λ ≥ λ∗ and μ ≥ μ∗. Weak solutions of problem (1.1) correspond to the critical points of Eλ,μ. The functionals L1, given by (1.5) is sequentially weakly lower semicontinuous. By Corollary III. in [5], it is enough to show that L1 is sequentially lower semicontinuous. For this purpose, we fix u ∈ X and ǫ > 0.

The proof of

Findings

Cλ dx