 95200 F. Bernis, J. GarciaAzorero, I. Peral
 Existence and Multiplicity of Nontrivial Solutions
in Semilinear Critical Problems of Fourth Order
(55K, Plain TeX with AMS macros)
Apr 20, 95

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Abstract. In this paper we consider the equation
$\bil u = \l u^{q2} u + u^{\pss2} u\equiv f(u)$ 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 $\pss= 2N/(N4)$.
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.
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