#### Shapour Heidarkhani, Johnny Henderson, Arezoo Solimaninia, Existence results for a class of second order differential equations with nonlinear derivative dependence, Vol. 2018 (2018), Article ID 24, pp. 1-14

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DOI: 10.23952/jnfa.2018.24

Received March 6, 2018; Accepted June 4, 2018; Published July 10, 2018

Abstract. We establish multiplicity results for nontrivial solutions of the quasilinear second order differential equation on a compact interval $[a,b]\subset \mathbb{R}$,
$\left\{\begin{array}{ll} -(p_{i}-1)|u_{i}'(x)|^{p_{i}-2}u_{i}''(x)=\lambda F_{u_{i}}(x,u_{1},\ldots,u_{n})h_{i}(x,u_{i}'),&x\in(a,b),\\ u_{i}(a)=u_{i}(b)=0, \end{array}\right.$
for $1\leq i\leq n$, under appropriate hypotheses. Indeed, using a consequence of the local minimum theorem due to Bonanno and, the mountain pass theorem, we investigate the existence of two solutions for the problem under some algebraic conditions with the classical Ambrosett-Rabinowitz condition on the nonlinear terms. Moreover, by combining two algebraic conditions on the nonlinear terms, and employing two consequences of the local minimum theorem due to Bonanno, we guarantee the existence of two solutions for the scalar case of the problem. Applying the mountain pass theorem given by Pucci and Serrin, we ensure the existence of the third solution for our problem.