The existence results for a class of generalized quasilinear Schrödinger equation with nonlocal term

Abstract

<abstract><p>In this paper, we discuss the generalized quasilinear Schrödinger equation with nonlocal term:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document} $ \begin{align} -\mathrm{div}(g^{2}(u)\nabla u)+g(u)g'(u)|\nabla u|^{2}+V(x)u = \left(|x|^{-\mu}\ast F(u)\right)f( u),\; \; x\in \mathbb{R}^{N}, \;\;\;\;\;\;\;\;({{\rm{P}}})\end{align} $ \end{document} </tex-math> </disp-formula></p> <p>where $ N\geq 3 $, $ \mu\in(0, N) $, $ g\in \mathbb{C}^{1}(\mathbb{R}, \mathbb{R}^{+}) $, $ V\in \mathbb{C}^{1}(\mathbb{R}^N, \mathbb{R}) $ and $ f\in \mathbb{C}(\mathbb{R}, \mathbb{R}) $. Under some "Berestycki-Lions type conditions" on the nonlinearity $ f $ which are almost necessary, we prove that problem $ (\rm P) $ has a nontrivial solution $ \bar{u}\in H^{1}(\mathbb{R}^{N}) $ such that $ \bar{v} = G(\bar{u}) $ is a ground state solution of the following problem</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document} $ \begin{align} - \Delta v+V(x)\frac{G^{-1}(v)}{g(G^{-1}(v))} = \left(|x|^{-\mu}\ast F(G^{-1}(v))\right)f( G^{-1}(v)),\; \; x\in \mathbb{R}^{N}, \;\;\;\;\;\;\;\;({{\rm{\bar P}}})\end{align} $ \end{document} </tex-math> </disp-formula></p> <p>where $ G(t): = \int_{0}^{t} g(s) ds $. We also give a minimax characterization for the ground state solution $ \bar{v} $.</p></abstract>