Electron. J. Differential Equations,
Vol. 2019 (2019), No. 41, pp. 123.
Multiplicity and concentration of nontrivial solutions for
generalized extensible beam equations in R^N
Juntao Sun, TsungFang Wu
Abstract:
In this article, we study a class of generalized extensible beam equations
with a superlinear nonlinearity
where
,
with
and
,
is a parameter,
and
.
Unlike most other papers on this problem, we allow the constant
to be
nonpositive, which has the physical significance. Under some suitable
assumptions on
and
, when
is small and
is
large enough, we prove the existence of two nontrivial solutions
and
,
one of which will blow up as the nonlocal term vanishes. Moreover,
and
strongly in
as
,
where
are two
nontrivial solutions of Dirichlet BVPs on the bounded domain
.
Also, the nonexistence of nontrivial solutions is also obtained for
large enough.
Submitted December 1, 2018. Published March 19, 2019.
Math Subject Classifications: 35J30, 35J35.
Key Words: Extensible beam equations; nontrivial solution; multiplicity;
concentration of solutions.
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Juntao Sun
School of Mathematics and Statistics
Shandong University of Technology
Zibo 255049, China
email: jtsun@sdut.edu.cn


TsungFang Wu
Department of Applied Mathematics
National University of Kaohsiung
Kaohsiung 811, Taiwan
email: tfwu@go.nuk.edu.tw

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