Ground state sign-changing solutions for a class of Schr\"{o}dinger-Poisson-Kirchhoff type problems with a critical nonlinearity in $\mathbb{R}^{3}$
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In the present paper, we are concerned with the existence of ground state sign-changing solutions for the following Schr\"{o}dinger-Poisson-Kirchhoff system $$\left\{ \aligned &-(1\!+\!b\int_{\mathbb R^3}|\nabla u|^{2}dx)\triangle u \!+\! V(x)u \!+\! k(x) \phi u\!=\! \lambda f(x)u \!+\! |u|^{4}u, &\hbox{in}\ \ \mathbb{R}^{3},\\ & -\triangle \phi \!=\! k(x) u^{2}, & \hbox{in}\ \ \mathbb{R}^{3},\endaligned\right.$$ where $b>0$, $V(x)$, $k(x)$ and $f(x)$ are positive continuous smooth functions; $0<\lambda<\lambda_{1}$ and $\lambda_{1}$ is the first eigenvalue of the problem $-\triangle u + V(x)u = \lambda f(x)u$ in $H$. With the help of the constraint variational method, we obtain that the Schr\"{o}dinger-Poisson-Kirchhoff type system possesses at least one ground state sign-changing solution for all $b>0$ and $0<\lambda<\lambda_{1}$. Moreover, we prove that its energy is strictly larger than twice that of the ground state solutions of Nehari type.
Abstract
In the present paper, we are concerned with the existence of ground state sign-changing solutions for the following Schr\"{o}dinger-Poisson-Kirchhoff system $$\left\{ \aligned &-(1\!+\!b\int_{\mathbb R^3}|\nabla u|^{2}dx)\triangle u \!+\! V(x)u \!+\! k(x) \phi u\!=\! \lambda f(x)u \!+\! |u|^{4}u, &\hbox{in}\ \ \mathbb{R}^{3},\\ & -\triangle \phi \!=\! k(x) u^{2}, & \hbox{in}\ \ \mathbb{R}^{3},\endaligned\right.$$ where $b>0$, $V(x)$, $k(x)$ and $f(x)$ are positive continuous smooth functions; $0<\lambda<\lambda_{1}$ and $\lambda_{1}$ is the first eigenvalue of the problem $-\triangle u + V(x)u = \lambda f(x)u$ in $H$. With the help of the constraint variational method, we obtain that the Schr\"{o}dinger-Poisson-Kirchhoff type system possesses at least one ground state sign-changing solution for all $b>0$ and $0<\lambda<\lambda_{1}$. Moreover, we prove that its energy is strictly larger than twice that of the ground state solutions of Nehari type.
참고문헌 (26)
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