Departamento de Ciências Exatas e Naturais
Universidade Federal Rural do Semi-Árido
59900-000, Pau dos Ferros-RN, Brazil
email: padua.filho@ufersa.edu.br
Antonio de Padua Farias de Souza Filho
Abstract:
This article studies the fourth-order equation
$$\displaylines{
\Delta^2 u-\Delta u+V(x) u-\frac{1}{2} u
\Delta(u^2)=f(x, u) \quad \hbox{in } R^4, \cr
u \in H^2(R^4),
}$$
where \(\Delta^2 :=\Delta(\Delta)\) is the biharmonic operator,
\(V\in C(R^4,R)\) and
\(f\in C(R^4\times R,R)\) are allowed to
be sign-changing. With some assumptions on \(V\) and \(f\) we prove
existence and multiplicity of nontrivial solutions in
\(H^2(R^4)\), obtained via variational methods.
Three main theorems are proved, the first two assuming that
\(V\) is coercive to obtain compactness, and the third one requires
only that \(V\) be bounded. We work carefully with the sub-criticality of
\(f\) to get a (PS) condition for a related equation.
Submitted August 14, 2024. Published October 29, 2024.
Math Subject Classifications: 35J62, 31B30, 35A15.
Key Words: Biharmonic operator; exponential growth; variational methods; critical groups.
DOI: 10.58997/ejde.2024.65
Show me the PDF file (388 KB), TEX file for this article.
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Antônio de Pádua Farias de Souza Filho Departamento de Ciências Exatas e Naturais Universidade Federal Rural do Semi-Árido 59900-000, Pau dos Ferros-RN, Brazil email: padua.filho@ufersa.edu.br |
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