Article Contents

2014, Volume 13, Issue 6: 2141-2154. Doi: 10.3934/cpaa.2014.13.2141

This issue Previous Article Next Article Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators

Nonlinear Biharmonic Problems with Singular Potentials

1.
Departamento de Matematica, Universidade Federal de Minas Gerais, 31270-010 Belo Horizonte-MG
2.
Departamento de Física e Matemática, Universidade Federal Fluminense, Campus de Rio das Ostras, Rio das Ostras, RJ 28890-000
3.
Departmento de Matemática, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG

Received: November 2011

Revised: May 2014

Published: November 2014

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Abstract

We deal with the problem \begin{eqnarray} \Delta^2 u +V(|x|)u = f(u), u\in D^{2,2}(R^N) \end{eqnarray} where $\Delta^2$ is biharmonic operator and the potential $V > 0 $ is measurable, singular at the origin and may also have a continuous set of singularities. The nonlinearity is continuous and has a super-linear power-like behaviour; both sub-critical and super-critical cases are considered. We prove the existence of nontrivial radial solutions. If $f$ is odd, we show that the problem has infinitely many radial solutions.

Keywords:

Mathematics Subject Classification: Primary: 35J30, 35J35; Secondary: 35J70, 35J91.

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