Quasilinear biharmonic equations on R^4 with exponential subcritical growth

• Published in: Electronic journal of differential equations Vol. 2024; no. 1-??; p. 65
• Main Author: Souza Filho, Antonio de Padua Farias de
• Format: Journal Article
• Language: English
• Published: 29-10-2024
• ISSN: 1072-6691; 1072-6691
• DOI: 10.58997/ejde.2024.65

   This article studies the fourth-order equation $$\displaylines{ \Delta^2 u-\Delta u+V(x)  u-\frac{1}{2} u  \Delta(u^2)=f(x, u) \quad \text{in } \mathbb{R}^4, \cr u \in H^2(\mathbb{R}^4), }$$ where \(\Delta^2 :=\Delta(\Delta)\) is the biharmonic operator, \(V\in C(R^4,\mathbb{R})\) 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  \(H^2(\mathbb{R}^4)\), 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.  For more information see https://ejde.math.txstate.edu/Volumes/2024/65/abstr.html