Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 65, pp. 1-16.

Title: Quasilinear biharmonic equations on $R^4$ with exponential subcritical growth

Authors: Antonio de Padua Farias de Souza Filho (Univ. Federal Rural do Semi-Arido, Brazil)

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.