This article establishes the existence of a ground state and infinitely many solutions for the modified fourth-order elliptic equation: \[ \begin{ aligned} \left\{ \begin{ array}{ ll} \Delta^2 u - \Delta u + u - \frac{ 1}{ 2}u\Delta(u^2) = f(u), & \text{ in } \mathbb{ R}^N, u \in H^2(\mathbb{ R}^N), \end{ array} \right. \end{ aligned} \] where $4 < N \leq 6$ and$f:\mathbb{ R}\rightarrow\mathbb{ R}$ is a nonlinearity of Berestycki-Lions type. For the ground state solution, we develop a novel approach that combines Jeanjean's technique with a Pohozaev-Palais-Smale sequence construction. When $f$ is odd, we prove infinite multiplicity of radially symmetric solutions via minimax methods on a topologically constrained comparison functional. This work resolves the lack of results for this autonomous problem under almost the weakest nonlinearity conditions.
Categories: math.AP