Uniqueness Theorem for p-biharmonic Equations
Southwest Texas State University, Department of Mathematics
The goal of this paper is to prove existence and uniqueness of a solution of the initial value problem for the equation (|u''|p-2u'')'' = 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.
p-biharmonic operator, Existence and uniqueness of solution, Continuous dependence on initial conditions, Jumping nonlinearity
Benedikt, J. (2002). Uniqueness theorem for $p$-biharmonic equations. <i>Electronic Journal of Differential Equations, 2002</i>(53), pp. 1-17j.