Uniqueness theorem for p-biharmonic equations - Jiri Benedikt (Electronic Journal of Differential Equations, 2002)
The goal of this paper is to prove existence and uniqueness of a solution of the initial value problem for the equation $$ (|u''|^{p-2}u'')''=lambda |u|^{q-2}u $$ where $lambdainmathbbR$ and $p,q>1$. We prove the existence for $pgeq q$ only, and give a counterexample which shows that for $p<q$ there need not exist a global solution (blow-up of the solution can occur). On the other hand, we prove the uniqueness for $pleq q$, and show that for $p>q$ the uniqueness does not hold true (we give a corresponding counterexample again). Moreover, we deal with continuous dependence of the solution on the initial conditions and parameters.