American Institute of Mathematical Sciences

$iu_t + u_{x x} + |u|^{p-2}u +(|u|^2)_{x x}u = 0,\quad u|_{t=0}=u_0(x),\quad x\in \mathbb R$

for a large class of initial data. The result gives a qualitative description on how small an initial data can ensure the existence of global solutions which sharpen a global existence result with small initial data [7, 10].

$-\epsilon \Delta u+V(x)u-\epsilon k(\Delta(|u|^2))u=g(u), \quad u>0,x \in \mathbb R^N,$

where g has superlinear growth at infinity without any restriction from above on its growth. Mountain pass in a suitable Orlicz space is employed to establish this result. These equations contain strongly singular nonlinearities which include derivatives of the second order which make the situation more complicated. Such equations arise when one seeks for standing wave solutions for the corresponding quasilinear Schrödinger equations. Schrödinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.

limin$f_{x\in\Omega, |x|\to\infty}\frac{u(x)+1}{\dist(x,\partial\Omega)}=0,$

and then, in particular, for strictly sublinear super-solutions in a domain $\Omega$ containing an open cone. In the special case that $\Omega=\mathbb R^N$, or that $F$ is the Bellman operator, we show that the same result holds for the whole class of nonnegative super-solutions.
Our principal assumption on the operator $F$ involves its zero and first order dependence when $|x|\to\infty$. The same kind of assumption was introduced in a recent paper in collaboration with H. Berestycki and F. Hamel [4] to establish a Liouville type result for semilinear equations. The strategy we follow to prove our main results is the same as in [4], even if here we consider fully nonlinear operators, possibly unbounded solutions and more general domains.

$-d i v((1+|\nabla u|^2)^{(p-2)/2} \nabla u)-d i v(c(x) (1+|u|^2)^{(\tau+1)/2}) $

$+b(x) (1+|\nabla u|^2)^{(\sigma+1)/2}=f \ i n \ \mathcal D'(\Omega)\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

$u\in W^{1,p}_0(\Omega)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

where $\Omega$ is a bounded open subset of $\mathbb R^N$ $(N\ge 2)$, $p$ is a real number $\frac{2N}{N+1}< p <+\infty$, the coefficients $c(x)$ and $b(x)$ belong to suitable Lebesgue spaces, $f$ is an element of the dual space $W^{-1,p'}(\Omega)$ and $\tau$ and $\sigma$ are positive constants which belong to suitable intervals specified in Theorem 2.1, Theorem 2.2 and Theorem 2.3.

$i u_t + \Delta u + V(\epsilon x ) |u|^{p-1} u = 0, \quad x \in \mathbf R^n$

with the critical power $ p = 1 + 4/n, n \ge 2, $ under certain conditions on the inhomogeneous term $ V $ with a small $ \epsilon > 0. $ We also demonstrate that these localized standing-waves converge to standing waves of the nonlinear Schrödinger equation with the homogeneous nonlinearity.