(Universidade Federal da Paraíba)
On a class of quasilinear Schrödinger equations
with superlinear or asymptotically linear terms
Abstract:
We will talk about existence and nonexistence of nonzero solutions for
the following class of quasilinear Schrödinger equations:
`-\Delta u+V(x)u+
\frac{\kappa}{2}[\Delta(u^{2})]u=h(u), \quad x \in
\mathbb{R}^N,`
where `\kappa` is a real parameter, `N\geq3`,
`V(x)` and `h(t)` are continuous
functions satisfying additional conditions. In order to prove our
existence result we use minimax techniques together with careful
`L^{\infty}-`estimates. Moreover, we show a Pohozaev identity which
justifies that `2^\ast=2N``/``(N-2)` is the critical exponent
for this class of problems when `\kappa` is positive, in contrast to
`22^\ast=4N``/``(N-2)` for `\kappa` negative and it is also used to
show nonexistence results.