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High energy solutions of modified quasilinear fourth-order elliptic equations with sign-changing potential. (English) Zbl 1368.35111

Summary: In the present paper, we consider the following modified quasilinear fourth-order elliptic equation \[ \begin{cases} \Delta^2u-\Delta u+V(x)u-\frac{1}{2}\Delta(u^2)u=f(x,u),\quad\text{ for }x\in\mathbb R^N, \\ u(x)\in H^2(\mathbb R^N),\end{cases}\tag{1.1} \] where \(\Delta^2:=\Delta(\Delta)\) is the biharmonic operator, \(V\in C(\mathbb R^N, (\mathbb R)\) and \(f\in C(\mathbb R^N\times (\mathbb R, (\mathbb R)\), \(N \leq 5\), are allowed to be sign-changing. Two main theorems on the existence of nontrivial solutions and infinitely many high energy solutions for Eq. (1.1) are obtained via variational methods.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
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