F. Bernis, J. Garcia-Azorero, I. Peral
Existence and Multiplicity of Nontrivial Solutions 
in  Semilinear Critical Problems of Fourth Order
(55K, Plain TeX with AMS macros)

ABSTRACT.  In this paper we consider the equation
$\bil u = \l |u|^{q-2} u + |u|^{\pss-2} u\equiv f(u)$ in a smooth
bounded domain $\O\subset\ren$ with boundary conditions either
$u|_{\p \O} =\frac{\p u}{\p n}|_{\p \O}=0$ or
$u|_{\p \O}=\D u|_{\p \O}=0$,
where $N>4$, $ 1<q<2, \,\l >0$ and  $\pss= 2N/(N-4)$.
We prove the existence of $\l_0$ such that for
$0<\l<\l_0$  the above problems have  infinitely many solutions.
For the problem  with the second boundary conditions, we prove the
existence of a positive solution also in the supercritical case,i.e.
when we have an exponent larger than $ \pss $. Moreover, in the
critical case, we show the existence of at least two positive solutions.