A priori estimates for nonlinear fourth order Schr'odinger type equations

摘要

We study the fourth order Schr\"odinger type differential inequality$-\Delta^2 u + \lambda V(x)u \geq a(x)u^q$ with $a,V\inL^1_{loc}(\mathbf{R}^N)$, both nonnegative, and $\lambda>0$. We considernonnegative solutions without making any assumptions about their asymptoticbehavior, and choosing appropriate test functions, we find a priori estimatesfor some type of subharmonic solutions. Finally, applying these a prioriestimates, we proof a Liouville type result.