In this paper we consider the following problem $$\begin{ cases} -\Delta_{ g}u+V(x)u=\lambda\alpha(x)f(u), & \mbox{ in }M\\ u\geq0, & \mbox{ in }M\\ u\to0, & \mbox{ as }d_{ g}(x_{ 0},x)\to\infty \end{ cases}$$where $(M,g)$ is a $N$-dimensional ($N\geq3)$, non-compact Riemannian manifold with asymptotically non-negative Ricci curvature, $\lambda$ is a real parameter, $V$ is a positive coercive potential, $\alpha$ is a bounded function and $f$ is a suitable nonlinearity. By using variational methods we prove a characterization result for existence of solutions for our problem.
Categories: math.AP