| Abstract: |
| We give some existence results for weak bounded solutions of the generalized
quasilinear Schr{\{o}}dinger equation
\[
{-{\rm div}(a(x,u,\nabla u))+A_{t}(x,u,\nabla u)+V(x){|u|}^{p-2}u=g(x,u)\qquad \mbox{ in } {\R}}^{N}
\]
with $p>1,$ $N\geq 2$ and $V:{\R}^{N}\rightarrow {\R}$ suitable measurable positive function.
Here, $A:{\R}^{N}\times {\R} \times {\R}^{N}\rightarrow {\R}$ is a ${\mathcal{C}}^{1}$-Carath\`{e}odory function which grows as ${|\xi|}^{p}$ and has partial derivatives
\[
A_{t}(x,t,\xi)=\ \frac{\partial A}{\partial t}(x,t,\xi),\quad a(x,t,\xi)=\left(\frac{\partial A}{\partial {\xi_{1}}}(x,t,\xi),...,\frac{\partial A}{\partial {\xi_{N}}}(x,t,\xi )\right)
\]
and $g:{\R}^{N}\times {\R}\rightarrow {\R}$ is a given Carath\`{e}odory function with super$-p-$linear but subcritical growth or with
sub$-p-$linear growth.
Since the principal term depends on $u$, other than on its gradient ${\nabla u}$ and on the spatial variable $x$, we study the interaction of two different norms in a suitable Banach space with the aim of obtaining a good variational approach.
These results are part of joint works with A.M. Candela, F. Mennuni, G. Palmieri and C. Sportelli. |
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