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MULTIPLE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH NONLINEARITY HAVING ARBITRARY GROWTH

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  13 March 2017

MARCELO F. FURTADO Open the ORCID record for MARCELO F. FURTADO [Opens in a new window]  and
HENRIQUE R. ZANATA
MARCELO F. FURTADO*
Affiliation:
Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília-DF, Brazil email mfurtado@unb.br
HENRIQUE R. ZANATA
Affiliation:
Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília-DF, Brazil email h.r.zanata@mat.unb.br
*
✉mfurtado@unb.br
Rights & Permissions [Opens in a new window]

Abstract

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We prove the existence of infinitely many solutions $u\in W_{0}^{1,2}(\unicode[STIX]{x1D6FA})$ for the Kirchhoff equation

$$\begin{eqnarray}\displaystyle -\biggl(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}\int _{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}u|^{2}\,dx\biggr)\unicode[STIX]{x1D6E5}u=a(x)|u|^{q-1}u+\unicode[STIX]{x1D707}f(x,u)\quad \text{in }\unicode[STIX]{x1D6FA}, & & \displaystyle \nonumber\end{eqnarray}$$
where $\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{N}$ is a bounded smooth domain, $a(x)$ is a (possibly) sign-changing potential, $0<q<1$, $\unicode[STIX]{x1D6FC}>0$, $\unicode[STIX]{x1D6FD}\geq 0$, $\unicode[STIX]{x1D707}>0$ and the function $f$ has arbitrary growth at infinity. In the proof, we apply variational methods together with a truncation argument.

Keywords

Kirchhoff equationinfinitely many solutionsvariational methodtruncation argument

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J25: Boundary value problems for second-order elliptic equations 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 1 , August 2017 , pp. 98 - 109
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author was partially supported by the National Council for Scientific and Technological Development (CNPq), Brazil.

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