Ground state and multiple solutions for modified autonomous fourth-order elliptic equations with Berestycki-Lions type conditions

This article establishes the existence of a ground state and infinitely many solutions for the modified fourth-order elliptic equation: $$\begin{array}{r}{\displaystyle \{\begin{array}{ll}{\mathrm{\Delta }}^{2}u-\mathrm{\Delta }u+u-\frac{1}{2}u\mathrm{\Delta }(u^2)=f(u),& \text{in }{\mathbb{R}}^N,u\in H^2({\mathbb{R}}^N),\end{array}}\end{array}$$ where $4<N\le 6$ and$f:\mathbb{R}\to \mathbb{R}$ 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.