Two nontrivial solutions for the nonhomogenous fourth order Kirchhoff equation (Q528828)

Summary: In this paper, we consider the following nonhomogenous fourth order Kirchhoff equation \[ \Delta^2 u - \left( a + b \int_{\mathbb{R}^N} |\nabla u|^2 dx \right) \Delta u + V(x) u = f(x,u) + g(x),\quad x\in \mathbb{R}^N, \] where \(\Delta^2 := \Delta(\Delta)\), constants \(a > 0\), \(b \geq 0\), \(V \in C(\mathbb{R}^N\), \(\mathbb{R})\), \(f \in C(\mathbb{R}^N \times \mathbb{R}, \mathbb{R})\) and \(g \in L^2(\mathbb{R}^N)\). Under more relaxed assumptions on the nonlinear term \(f\) that are much weaker than those in L. Xu and H. Chen, using some new proof techniques especially the verification of the boundedness of Palais-Smale sequence, a new result is obtained.