Infinitely many solutions for nonlocal problems with variable exponent and nonhomogeneous Neumann conditions

Shapour Heidarkhani,Amjad Salari,Anderson Luis Albuquerque De Araujo

doi:10.5269/bspm.v38i4.41664

Abstract

In this article we will provide new multiplicity results of the solutions for nonlocal problems with variable exponent and nonhomogeneous Neumann conditions. We investigate the existence of infinitely many solutions for perturbed nonlocal problems with variable exponent and nonhomogeneous Neumann conditions. The approach is based on variational methods and critical point theory.

Highlights

Problems like (Pgf,h) are usually called nonlocal problems because of the presence of the integral over the entire domain, and this implies that the first equation in (Pgf,h) is no longer a pointwise identity
Kirchhoff-type boundary value problems model several physical and biological systems where u describes a process which depends on the average of itself, for example the population density
Viasi in [56] used variational techniques to prove an eigenvalue theorem for a stationary p(x)-Kirchhoff problem, and provided an estimate for the range of such eigenvalues. He employed a specific family of test functions in variable exponent Sobolev spaces

Summary

If δ

1 δ (c1) there is a global minimum of Φ which is a local minimum of Iλ, or (c2) there is a sequence of pairwise distinct critical points (local minima) of. And in the sequel, meas(Ω) denotes the Lebesgue measure of the set Ω, and we assume that p ∈ C(Ω ) verifies the following condition:. ∂Ω η where dσ is the surface measure on ∂Ω It is well known (see [29]) that, in view of (2.1), both Lp(x)(Ω) and W1,p(x)(Ω), with the respective norms, are separable, reflexive and uniformly convex Banach spaces. Where D = diam(Ω) and meas(Ω) is the Lebesgue measure of Ω (see [8, Remark 1]). Let meas(Ω) = ∂Ω dσ = 1, f : R −→ R be a continuous function and put F (t) =. M0 hbn the problem (Pgf,h) has an unbounded sequence of solutions in W1,p(x)(Ω). Proof: Fix λ ∈ (λ1, λ2) and let h be a function satisfying the condition (3.1)

If hbn

Now put

Let us show that