Abstract
The nodal solutions of equations are considered to be more difficult than the positive solutions and the ground state solutions. Based on this, this paper intends to study nodal solutions for a kind of Schrödinger-Poisson equation. We consider a class of Schrödinger-Poisson equation with variable potential under weaker conditions in this paper. By introducing some new techniques and using truncated functional, Hardy inequality and Pohožaev identity, we obtain an existence result of a least energy sign-changing solution and a ground state solution for this kind of Schrödinger-Poisson equation. Moreover, the energy of the sign-changing solution is strictly greater than the ground state energy.
Highlights
IntroductionThe following nonlinear Schrödinger-Poisson system will be discussed. (f4) f (t ) is an increasing function of \ {0}
In this paper, the following nonlinear Schrödinger-Poisson system will be discussed−∆u +V ( x)u += λφ ( x)u f (u), x ∈ 3,= −∆φ u2, x ∈ 3, (1.1)where the potential function V : 3 →, λ > 0 is a parameter and ( ) f ∈ C 3
The nodal solutions of equations are considered to be more difficult than the positive solutions and the ground state solutions
Summary
The following nonlinear Schrödinger-Poisson system will be discussed. (f4) f (t ) is an increasing function of \ {0}. The author obtained the existence of solutions for system (1.3) with a general nonlinearity in the critical growth by variational method He did not study the existence of sign-changing. We consider variable potential V ( x) and put some constraints on it, and study the least energy sign-changing solution and ground state solution of the Schrödinger-Poisson Equation (1.1). There exists a positive Υ such that for all λ ∈ (0, Υ) , problem (1.1) has a least energy signchanging solution zλ ∈ λ and a ground solution uλ ∈ λ which is constant sign These two solutions satisfy the following relationship gλ = Kλ ( zλ ) > Kλ (uλ ) = cλ.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
