Electron. J. Diff. Eqns., Vol. 2002(2002), No. 53, pp. 1-17.

Uniqueness theorem for $p$-biharmonic equations

Jiri Benedikt

Abstract:
The goal of this paper is to prove existence and uniqueness of a solution of the initial value problem for the equation
$$ (|u''|^{p-2}u'')''=\lambda |u|^{q-2}u $$
where $\lambda\in{\mathbb{R}}$ and $p,q>1$. We prove the existence for $p\geq q$ 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 $p\leq q$, 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.

An addendum was attached on July 28, 2003. In Remark 4.6, fourth line (page 15) there should be $u(t)=K(t-t_0)^r$ instead of $u(t)=K(H-t)^r$.

Submitted April 15, 2002. Published June 10, 2002.
Math Subject Classifications: 34A12, 34C11, 34L30.
Key Words: p-biharmonic operator, existence and uniqueness of solution, continuous dependence on initial conditions, jumping nonlinearity.

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Jiri Benedikt
Centre of Applied Mathematics
University of West Bohemia
Univerzitni 22, 306 14 Plzen
Czech Republic
e-mail: benedikt@kma.zcu.cz

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