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Existence of multiple nontrivial solutions for a class of quasilinear Schrödinger equations on \(\mathbb{R}^N\). (English) Zbl 1394.35191

Summary: This paper is concerned with the following fourth-order elliptic equations \[ \triangle^2 u- \Delta u + V(x)u - \frac{\kappa}{2}\Delta(u^2)u=f(x,u),\quad \text{in } \mathbb{R}^N, \] where \(N\leq6\), \(\kappa\geq0\). Under some appropriate assumptions on \(V(x)\) and \(f(x, u)\), we prove the existence and multiplicity of solutions for the above equations via variational methods. Recent results from the literature are extended.

MSC:

35J62 Quasilinear elliptic equations
35J35 Variational methods for higher-order elliptic equations
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Full Text: Euclid