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A non-periodic indefinite variational problem in ℝN with critical exponent

Part of: Representations of solutions Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  26 June 2023

Gustavo S. do Amaral Costa ,
Giovany M. Figueiredo Open the ORCID record for Giovany M. Figueiredo [Opens in a new window]  and
José Carlos de O. Junior Open the ORCID record for José Carlos de O. Junior [Opens in a new window]
Gustavo S. do Amaral Costa
Affiliation:
Departamento de Matemática, Universidade Federal do Maranhão, São Luís, Maranhão, Brazil (gsa.costa@ufma.br)
Giovany M. Figueiredo
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brasília, Distrito Federal, Brazil (giovany@unb.br)
José Carlos de O. Junior
Affiliation:
Colegiado de Matemática, Universidade Federal do Norte do Tocantins, Araguaína, Tocantins, Brazil (jc.oliveira@uft.edu.br)
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Abstract

We consider the non-linear Schrödinger equation(Pμ)

\begin{equation*}\begin{array}{lc}-\Delta u + V(x) u = \mu f(u) + |u|^{2^*-2}u, &\end{array}\end{equation*}
in $\mathbb{R}^N$, $N\geq3$, where V changes sign and $f(s)/s$, s ≠ 0, is bounded, with V non-periodic in x. The existence of a solution is established employing spectral theory, a general linking theorem due to [12] and interaction between translated solutions of the problem at infinity with some qualitative properties of them.

Keywords

critical exponentabstract linking theoremindefinite problemsspectral theory

MSC classification

Primary: 35A15: Variational methods 35J60: Nonlinear elliptic equations 35D30: Weak solutions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 2 , May 2023 , pp. 579 - 612
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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