Multiple positive solutions for a singular elliptic equation with Neumann boundary condition in two dimensions

Let $Omegasubset mathbb{R}^2$ be a bounded domain with $C^2$ boundary. In this paper, we are interested in the problem $$displaylines{ -Delta u+u = h(x,u) e^{u^2}/|x|^eta,quad u>0 quad ext{in } Omega, cr frac{partial u}{partial u}= lambda psi u^q quad ext{on }partial Omega, }$$ where $0in partial Omega$, $etain [0,2)$, $lambda>0$, $qin [0,1)$ and $psige 0$ is a H"older continuous function on $overline{Omega}$. Here $h(x,u)$ is a $C^{1}(overline{Omega} imes mathbb{R})$ having superlinear growth at infinity. Using variational methods we show that there exists $0Lambda$ and at least one solution when $lambda = Lambda$.