Positive solutions for asymptotically linear Schrödinger-Kirchhoff-type equations

Abstract

In this paper, we focus on the Schrödinger–Kirchhoff‐type equation MathClass-bin−()aMathClass-bin+bMathClass-op∫double-struckR3MathClass-rel|MathClass-rel∇uMathClass-rel|2dxMathClass-rel△uMathClass-bin+V(x)uMathClass-rel=f(xMathClass-punc,u)MathClass-punc,indouble-struckR3 where a,b > 0 are constants, VMathClass-punc:double-struckR3MathClass-rel→double-struckR may not be radially symmetric, and f(x,u) is asymptotically linear with respect to u at infinity. Under some technical assumptions on V and f, we prove that the problem (SK) has a positive solution. Copyright © 2013 John Wiley & Sons, Ltd.

Multiple Solutions for the Asymptotically Linear Kirchhoff Type Equations onRN

“…Very recently, some authors had studied the Kirchhoff type equation on the whole space R . Many solvability conditions with near zero and infinity for problem (2) have been considered, such as the superlinear case (see [19][20][21][22][23][24][25][26][27][28]); the asymptotically linear case (see [29,30]); the sublinear case (see [31][32][33]). Particularly, the following Kirchhoff type problem has been studied widely by some authors under various conditions on and :…”

Existence of positive ground state solutions for Kirchhoff type problems

“…See, for example, [32,18,15,30,28,24,25]. In particular, Wu [31] obtained a sequence of high energy solutions for problem (1.3) with ϵ = 1 via a symmetric Mountain Pass Theorem under the condition that f (x, u) is 4-suplinear at infinity and satisfies that (AR) There exists θ > 4 such that 0 < θF (x, u) ≤ f (x, u)u for all u ̸ = 0, where F (x, u) =  u 0 f (x, t)dt, and that the potential…”

Positive solutions of Kirchhoff type elliptic equations in with critical growth

“…In recent years, the following nonlocal elliptic problem in whole space has been extensively studied by many researchers, see , , , , , , , , , and the references therein, Among them, some existence results are obtained via the variational and topological methods even for the critical case. It seems that there is no result except for , , , when the dimension is higher than three.…”

“…In recent years, the following nonlocal elliptic problem in whole space has been extensively studied by many researchers, see [3], [5], [13], [16], [21], [24], [25], [32], [33], [35] and the references therein,…”