A priori estimates for nonlinear fourth order Schr\"odinger type equations

We study the fourth order Schr"odinger type differential inequality $-{\mathrm{\Delta }}^{2}u+\lambda V(x)u\ge a(x)u^q$ with $a,V\in L^1_{loc}($\mathbf{R}$^N)$,bothnonnegative,and$\lambda>0$. We consider nonnegative solutions without making any assumptions about their asymptotic behavior, and choosing appropriate test functions, we find a priori estimates for some type of subharmonic solutions. Finally, applying these a priori estimates, we proof a Liouville type result.