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

Abstract

In the present paper, we study the following Kirchhoff-Schrödinger-Poisson system with logarithmic and critical nonlinearity: $ \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, &x\in \Omega, \\ -\Delta \phi = u^2,& x\in \Omega, \\ u = \phi = 0,& x\in \partial\Omega, \end{array} \right . \end{array} \end{align} $ where $ \lambda, b > 0, a > \frac{1}{4}, 4 < q < 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.

Keywords

• quasilinear kirchhoff-schrödinger-poisson
• critical problem
• logarithmic nonlinearity