Infinitely many solutions of dirichlet problem for p-mean curvature operator

Abstract

The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator: \(\left\{ \begin{gathered} div((1 + \left| {\nabla u} \right|^2 )^{\frac{{P - 2}}{2}} \nabla u) = f(x,u), x \in \Omega , \hfill \\ u \in W_0^{1P} (\Omega ), \hfill \\ \end{gathered} \right.\) is considered, where Θ is a bounded domain in R n (n>p>1) with smooth boundary ∂Θ. Under some natural conditions together with some conditions weaker than (AR) condition, we prove that the above problem has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if \(\frac{{f(x,u)}}{{\left| u \right|^{p - 2} u}} \to + \infty as u \to \infty \).