1 results
Existence and multiplicity of positive solutions for a fourth-order elliptic equation
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 2 / April 2020
- Published online by Cambridge University Press:
- 28 January 2019, pp. 1053-1069
- Print publication:
- April 2020
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- Article
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We prove existence and multiplicity of solutions for the problem
where
$$\left\{ {\matrix{ {\Delta ^2u + \lambda \Delta u = \vert u \vert ^{2*-2u},{\rm in }\Omega ,} \hfill \hfill \hfill \hfill \cr {u,-\Delta u > 0,\quad {\rm in}\;\Omega ,\quad u = \Delta u = 0,\quad {\rm on}\;\partial \Omega ,} \cr } } \right.$$
$\Omega \subset {\open R}^N$, 
$N \ges 5$, is a bounded regular domain, 
$\lambda >0$ and 
$2^*=2N/(N-4)$ is the critical Sobolev exponent for the embedding of 
$W^{2,2}(\Omega )$ into the Lebesgue spaces.
