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http://dx.doi.org/10.4134/JKMS.j200497

GROUND STATE SIGN-CHANGING SOLUTIONS FOR A CLASS OF SCHRÖDINGER-POISSON-KIRCHHOFF TYPEPROBLEMS WITH A CRITICAL NONLINEARITY IN ℝ3  

Qian, Aixia (School of Mathematical Sciences Qufu Normal University)
Zhang, Mingming (School of Mathematical Sciences Qufu Normal University)
Publication Information
Journal of the Korean Mathematical Society / v.58, no.5, 2021 , pp. 1181-1209 More about this Journal
Abstract
In the present paper, we are concerned with the existence of ground state sign-changing solutions for the following Schrödinger-Poisson-Kirchhoff system $$\;\{\begin{array}{lll}-(1+b{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{{\mathbb{R}}^3}}}{\mid}{\nabla}u{\mid}^2dx){\Delta}u+V(x)u+k(x){\phi}u={\lambda}f(x)u+{\mid}u{\mid}^4u,&&\text{in }{\mathbb{R}}^3,\\-{\Delta}{\phi}=k(x)u^2,&&\text{in }{\mathbb{R}}^3,\end{array}$$ where b > 0, V (x), k(x) and f(x) are positive continuous smooth functions; 0 < λ < λ1 and λ1 is the first eigenvalue of the problem -∆u + V(x)u = λf(x)u in H. With the help of the constraint variational method, we obtain that the Schrödinger-Poisson-Kirchhoff type system possesses at least one ground state sign-changing solution for all b > 0 and 0 < λ < λ1. Moreover, we prove that its energy is strictly larger than twice that of the ground state solutions of Nehari type.
Keywords
Schrodinger-Poisson-Kirchhoff system; ground state sign-changing solution; critical nonlinearity; nonlocal term;
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