On a quasilinear Schrödinger equation

Andrzej Szulkin (Stockholm University), Jueves 26 de septiembre de 2013, 11:00 hrs.
Cuándo 26/09/2013
de 11:00 a 12:00
Dónde Graciela Salicrup
Agregar evento al calendario vCal

We consider the semilinear Schrödinger equation \[-\Delta u+V(x)u-\Delta(u^{2})u=g(x,u), \quad x\in \mathbb{R}^{N},\] where \(g\) and \(V\) are periodic in \(x_1,\ldots,x_N, V>0, g\) is odd in \(u\) and of subcritical growth in the sense that \(|g(x,u)|\leq a(1+|u|^{p-1})\), where \(4<p<2\cdot 2^*\). We show that this equation has infinitely many geometrically distinct solutions in each of the following two cases:

(i)  \(g(x,u)=o(u)\) as \(u\to 0\), \(G(x,u)/u^4\to\infty\) as \(|u|\to\infty\), where \(G\) is the primitive of \(g\), and \(u\mapsto g(x,u)/u^3\) is positive for \(u\ne 0\), nonincreasing on \((-\infty, 0)\) and nondecreasing on \((0, \infty)\).

(ii) \(g(x,u)=q(x)u^3\), where \(q>0\).  

The argument uses the Nehari manifold technique. A special feature here is that the Nehari manifold is not likely to be of class \(C^1\).


This is joint work with Xiangdong Fang.