Uniqueness Theorem for p-biharmonic Equations

Abstract

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")" = λ|u| q-2 u where λ ∈ ℝ and p, q > 1. We prove the existence for p ≥ 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 p ≤ 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.