Abstract
In this paper, we considered with the following Schr?dinger-Kirchhoff type problem: -(a+b ∫RN|∇u|2dx)Δu + V(x)u = f(x,u) in RN. We put forward general assumptions on the nonlinearity f with the subcritical growth and we find a ground state solution being a minimizer of the energy functional associated with a Nehari-Pankov manifold by using a linking theorem.
Highlights
Consider the following nonlinear Schrödinger-Kirchhoff type problem:( ) ∫ − a + b RN ∇u 2 dx ∆u + V ( x)u =f ( x,u) in RN (1.1)( ) ( ) where constants a > 0,b ≥ 0, V ∈ RN, R and f ∈ RN × R, R satisfy some assumptions
We put forward general assumptions on the nonlinearity f with the subcritical growth and we find a ground state solution being a minimizer of the energy functional associated with a Nehari-Pankov manifold by using a linking theorem
Our goal is to find a ground state solution of Equation (1.1) i.e., a critical point being a minimizer of I on the Nehari-Pankov manifold defined as follows:
Summary
Consider the following nonlinear Schrödinger-Kirchhoff type problem:. ( ) ( ) where constants a > 0,b ≥ 0 , V ∈ RN , R and f ∈ RN × R, R satisfy some assumptions. Consider the following nonlinear Schrödinger-Kirchhoff type problem:. If V ( x) ≡ 0 and RN are replaced by a smooth bounded domain Ω ∈ RN , Equation (1.1) is a Dirichlet problem of Kirchhoff type [5]:. Our aim is to study ground state solutions to (1.1) with a class of nonlinearities. Our goal is to find a ground state solution of Equation (1.1) i.e., a critical point being a minimizer of I on the Nehari-Pankov manifold defined as follows:. Since contains all critical points of I, a ground state is a energy solution. To conquer the linking geometry of I and to structure , we give the following assumptions:. There is a nontrivial critical point of I, such that.
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