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}\ \mathbb R^3 , \\ u > 0 &\hbox{in }\ \mathbb R^3 , \end{array} \right. $
where $ V: \mathbb R^3\to \mathbb R $, $ f: \mathbb R^3\times \mathbb R\to \mathbb R $ are "good" functions and $ g: \mathbb R\to \mathbb R $ is such that $ g^2(u) = 1+\frac{[(l(u^2))^{\prime}]^2}{2} $ for a given $ l\in\mathcal{C}^2( \mathbb 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) $.
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