Soliton solutions for quasilinear modified Schrödinger equations in applied sciences

In this paper, we prove the existence of nontrivial weak bounded solutions of the quasilinear modified Schrödinger problem \[ \left\{ \begin{array}{ll} -{\rm div}(g^2(u) \nabla u) + g(u) g^{\prime}(u) |\nabla u|^2 + V(x) u = f(x, u) &\hbox{in $\R^3$,}\\ u > 0 &\hbox{in $\R^3$,} \end{array}\right. \] where $V:\R^3\to\R$, $f:\R^3\times\R\to\R$ are ``good'' functions and $g:\R\to\R$ is such that $g^2(u)= 1+\frac{[(l(u^2))^{\prime}]^2}{2}$ for a given $l\in\mathcal{C}^2(\R)$. By means of variational methods and an approximation argument, here we obtain an existence result for the superfluid film equation in Plasma Physics and for the equation which models the self-channelling of a high-power ultrashort laser, which derive from our model problem by taking $l(s)=s$, respectively $l(s)=\sqrt{1+s}$, in the previous definition of $g^2(u)$.