Multiple solutions for quasi-linear elliptic equations with Berestycki-Lions type nonlinearity

Abstract

<abstract><p>We studied the modified nonlinear Schrödinger equation</p> <p><disp-formula> <label>0.1</label> <tex-math id="E0.1"> \begin{document}$ \begin{equation} -\Delta u-\frac12\Delta(u^2)u = g(u)+h(x), \quad u\in H^1({\mathbb{R}}^N), \end{equation} $\end{document} </tex-math></disp-formula></p> <p>where $ N\geq3 $, $ g\in C({\mathbb{R}}, {\mathbb{R}}) $ is a nonlinear function of Berestycki-Lions type, and $ h\not\equiv 0 $ is a nonnegative function. When $ \|h\|_{L^2({\mathbb{R}}^N)} $ is suitably small, we proved that (0.1) possesses at least two positive solutions by variational approach, one of which is a ground state while the other is of mountain pass type.</p></abstract>