95-200 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) Apr 20, 95

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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.

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