Existence of nontrivial solution for a nonlocal problem with subcritical nonlinearity

In this paper, we consider the following new nonlocal Dirichlet boundary value problem: $$ \textstyle\begin{cases} -(a-b\int_{\Omega} \vert \nabla u \vert ^{2}\,dx)\Delta u=\lambda u+g(x,u),& x\in \Omega, \\ u=0,& x\in\partial\Omega, \end{cases} $$ (0.1) where a and b are positive, λ is a positive parameter, \(0\leq\lambda< a\lambda_{1}\), \(\lambda_{1}\) is the first eigenvalue of operator −Δ. Under appropriate assumptions on the function g which is of subcritical growth, we obtain a nontrivial solution.