Electron. J. Diff. Equ.,
Vol. 2016 (2016), No. 217, pp. 115.
Multiplicity of solutions for equations involving a nonlocal term
and the biharmonic operator
Giovany M. Figueiredo, Rubia G. Nascimento
Abstract:
In this work we study the existence and multiplicity result of solutions
to the equation
where
is a bounded smooth domain of
,
,
or
,
is a continuous function. Since there is a competition between the
function M and the critical exponent, we need to make a truncation on
the function M. This truncation allows to define an auxiliary problem.
We show that, for
large, exists one solution and for
small there are infinitely many solutions for the auxiliary problem.
Here we use arguments due to BrezisNiremberg [12] to show
the existence result and genus theory due to Krasnolselskii [29] to
show the multiplicity result. Using the size of
,
we show that
each solution of the auxiliary problem is a solution of the original problem.
Submitted April 20, 2016. Published August 16, 2016.
Math Subject Classifications: 34B15, 34B18. 35J35, 35G30
Key Words: Beam equation; Berger equation; critical exponent;
variational methods.
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Giovany M. Figueiredo
Universidade Federal do Pará
Faculdade de Matemática
CEP: 66075110 Belém  Pa, Brazil
email: giovany@ufpa.br


Rúbia G. Nascimento
Universidade Federal do Pará
Faculdade de Matemática
CEP: 66075110 Belém  Pa, Brazil
email: rubia@ufpa.br

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