Sign-changing solutions of critical quasilinear Kirchhoff-Schrödinger-Poisson system with logarithmic nonlinearity

Abstract

<abstract><p>In the present paper, we study the following Kirchhoff-Schrödinger-Poisson system with logarithmic and critical nonlinearity:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align} \begin{array}{ll} \left \{ \begin{array}{ll} - \Bigr(a+b\int_\Omega|\nabla u|^2{\mathrm{d}}x \Bigr)\Delta u+V(x)u-\frac{1}{2}u\Delta (u^2)+\phi u = \lambda |u|^{q-2}u\ln|u|^2+|u|^4u, &amp;x\in \Omega, \\ -\Delta \phi = u^2,&amp; x\in \Omega, \\ u = \phi = 0,&amp; x\in \partial\Omega, \end{array} \right . \end{array} \end{align} $\end{document} </tex-math></disp-formula></p> <p>where $ \lambda, b &gt; 0, a &gt; \frac{1}{4}, 4 &lt; q &lt; 6, $ $ V(x) $ is a smooth potential function and $ \Omega $ is a bounded domain in $ \mathbb{R}^3 $ with Lipschitz boundary. Combining constraint variational method and perturbation method, we prove that the above problem has a least energy sign-changing solution $ u_0 $ which has precisely two nodal domains. Moreover, we show that the energy of $ u_0 $ is strictly larger than twice the ground state energy.</p></abstract>