Soliton solutions for generalized quasilinear Schrödinger equations (Q1937585)

This article proves the existence of positive solutions \(u\in H^1(\mathbb{R}^N)\) for a certain class of nonlinear Schrödinger equations, of the form: \[ -\Delta u +V(x) u -\Delta(l(u^2))l'(u^2)u = h(u)\,,\quad x\in\mathbb{R}^N, \] for some potential \(V(x)>0\) and nonlinear real functions \(h(s)\), \(l(s)\) satisfying certain assumptions. In the past the above equation has been studied for the case \(l(s)=s\) (an equation appearing in plasma physics) and \(l(s)=(1+s^2)^{1/2}\) (appearing in laser physics). The present paper generalizes the existence results to more arbitrary functions \(l(s)\). The strategy of proof first proceeds by a clever change of the dependent variable: setting \[ v = G(u) = \int_0^u g(t)\,dt,\quad \text{where}\quad g(s) = \sqrt{1+\frac{[d l(s^2)/ds]^2}{2}}\,, \] then the function \(v(x)\) satisfies a semilinear equation. Any solution to the latter must be a critical point for a certain functional \(H^1\ni v\mapsto J(v)\in \;IR\). The existence of a nontrivial critical point is obtained by applying the Ambrosetti-Rabinowitz Mountain Pass theorem, after checking the following geometric properties: on the one hand, for \(\|v\|\) small enough \(J(v)>0\), on the other hand, that there exists \(v\) such that \(J(v)\to -\infty\).