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Qian, Aixia,Zhang, Mingming Korean Mathematical Society 2021 대한수학회지 Vol.58 No.5
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 < λ<sub>1</sub> and λ<sub>1</sub> 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 < λ<sub>1</sub>. Moreover, we prove that its energy is strictly larger than twice that of the ground state solutions of Nehari type.
Positive solution and ground state solution for a Kirchhoff type equation with critical growth
Caixia Chen,Aixia Qian 대한수학회 2022 대한수학회보 Vol.59 No.4
In this paper, we consider the following Kirchhoff type equation on the whole space $$\left\{\aligned &-(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\triangle u = u^{5} + \lambda k(x)g(u), \ x\in \mathbb{R}^{3},\\ & u\in\mathcal{D}^{1,2}(\mathbb{R}^{3}),\endaligned\right.$$ where $\lambda>0$ is a real number and $k, g$ satisfy some conditions. We mainly investigate the existence of ground state solution via variational method and concentration-compactness principle.