Uniqueness theorem for p-biharmonic equations

• Main Author: Jiri Benedikt
• Format: Article
• Language: English
• Published: Texas State University 2002-06-01
• Series: Electronic Journal of Differential Equations
• Subjects: p-biharmonic operator; existence and uniqueness of solution; continuous dependence on initial conditions; jumping nonlinearity.
• Online Access: http://ejde.math.txstate.edu/Volumes/2002/53/abstr.html

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 $lambdain{mathbb{R}}$ 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.