Existence of Solutions for Sublinear Kirchhoff Problems with Sublinear Growth

Wei Yang,Zhuo Yao,Zhan Wang

doi:10.1155/2019/5364134

Abstract

In this paper, we consider the following sublinear Kirchhoff problems -a+b∫RN∇u2dxΔu+V(x)u=f(x,u), in RN, where a>0 and b≥0 with N≥3. A new sublinear growth condition is given. When f(x,u) is not odd in u and not integrable in x, we obtain the existence of solutions for the above problem.

Highlights

Introduction and Main ResultsIn this paper, we consider the following nonlinear Kirchhoff type problems:− (a + b ∫ |∇u|2 dx) Δu + V (x) u = f (x, u), RN [1]in RN, where a, b ≥ 0, f(x, u) ∈ C(RN × R, R)
Mathematical Problems in Engineering condition, Wang and Han considered a class of sublinear nonlinearities for problem [1] and showed the existence of infinitely many solutions when f(x, t) is odd in t which is the following theorem
Motivated by the above papers, we consider problem [1] with some new sublinear nonlinearities and, before we state our result, we assume that Γ is a continuous function space such that, for any δ(s) ∈ Γ, there exists a constant l0 > 0 such that (i) δ(s) > 0 for all s > 0; (ii) ∫ll0(1/sδ(s))ds 󳨀→ +∞ as l 󳨀→ +∞

Summary

Introduction and Main Results

We consider the following nonlinear Kirchhoff type problems:. in RN, where a, b ≥ 0, f(x, u) ∈ C(RN × R, R). We consider the following nonlinear Kirchhoff type problems:. Mathematical Problems in Engineering condition, Wang and Han considered a class of sublinear nonlinearities for problem [1] and showed the existence of infinitely many solutions when f(x, t) is odd in t which is the following theorem. With coercive condition (V), the authors obtained a new compact embedding theorem, stated as follows. Motivated by the above papers, we consider problem [1] with some new sublinear nonlinearities and, before we state our result, we assume that Γ is a continuous function space such that, for any δ(s) ∈ Γ, there exists a constant l0 > 0 such that (i) δ(s) > 0 for all s > 0; (ii) ∫ll0(1/sδ(s))ds 󳨀→ +∞ as l 󳨀→ +∞. This is the first time to use such condition on the existence of solutions for sublinear Kirchhoff equations

Preliminaries

Proof of Theorem 1

Conflicts of Interest