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Homoclinic solutions for fourth order differential equations with superlinear nonlinearities
Keywords:Homoclinic solutions; Critical point; Variational methods; Fountain Theorem
Abstract:
      In this paper we investigate the existence of homoclinic solutions for the following fourth order differential equations $$ u^{(4)}+ w u'+a(x)u=f(x,u), \eqno(\mbox{FDE}) $$ where $w$ is a constant, $a\in C(\R,\R)$ and $f\in C(\R\times\R,\R)$. The novelty of this paper is that, assuming $a(x)$ is not required to be either nonnegative or coercive, and $f(x,u)$ is supposed to satisfy some new superlinear conditions, we establish one new criterion to guarantee the existence of infinitely many homoclinic solutions of (FDE). Recent results in the literature are generalized and significantly improved.