Multiple solutions for a class of nonlocal quasilinear elliptic systems in Orlicz\u2013Sobolev spaces

S Heidari,A Razani

doi:10.1186/s13661-021-01496-8

Abstract

In this paper, we study some results on the existence and multiplicity of solutions for a class of nonlocal quasilinear elliptic systems. In fact, we prove the existence of precise intervals of positive parameters such that the problem admits multiple solutions. Our approach is based on variational methods.

Highlights

In this article, we are interested in establishing the existence of multiple solutions to the following Kirchhof-type systems in Orlicz–Sobolev spaces ⎧ ⎨–Mi(i(|∇ui|) dx)( div(αi(|∇ui|)∇ui)) = λFui (x, u1, . . . , un) in, ⎩ui = 0 on ∂, (1.1)for 1 ≤ i ≤ n, where is a bounded domain in RN (N ≥ 3), with smooth boundary ∂ and λ is a positive parameter, F : × Rn → R is a measurable function with respect to x ∈ for every (t1, . . . , tn) ∈ Rn and is C1 with respect to (t1, t2, . . . , tn) ∈ Rn for a.e. x ∈ ; Fti denotes the partial derivative of F with respect to ti
We are interested in establishing the existence of multiple solutions to the following Kirchhof-type systems in Orlicz–Sobolev spaces i(|∇ui|) dx)( div(αi(|∇ui|)∇ui)) = λFui (x, u1, . . . , un) in
For 1 ≤ i ≤ n, where is a bounded domain in RN (N ≥ 3), with smooth boundary ∂ and λ is a positive parameter, F : × Rn → R is a measurable function with respect to x ∈ for every (t1, . . . , tn) ∈ Rn and is C1 with respect to (t1, t2, . . . , tn) ∈ Rn for a.e. x ∈ ; Fti denotes the partial derivative of F with respect to ti

Summary

Introduction

We are interested in establishing the existence of multiple solutions to the following Kirchhof-type systems in Orlicz–Sobolev spaces. For 1 ≤ i ≤ n, where is a bounded domain in RN (N ≥ 3), with smooth boundary ∂ and λ is a positive parameter, F : × Rn → R is a measurable function with respect to x ∈ for every For the functions φi above, let us define i(t) =. ∂2u ∂x2 = 0, where ρ, ρ0, h, E, L are constants, for 0 < x < L, t ≥ 0, and where u = u(x, t) is the lateral displacement at the space coordinate x and time t, E the Young modulus, ρ the mass density, h the cross-section area, L the length, and ρ0 the initial axial tension, proposed by Kirchhoff [17] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings This is an example of a nonlinear problem.

We observe that

If δ

Three weak solutions

For a fixed λ