A characterization related to Schrödinger equations on Riemannian manifolds

Abstract

In this paper we consider the following problem, non-compact Riemannian manifold with asymptotically non-negative Ricci curvature, λ is a real parameter, V is a positive coercive potential, α is a bounded function and f is a suitable nonlinearity. By using variational methods we prove a characterization result for existence of solutions for (P λ ).

Multiple solutions for Schrödinger equations on Riemannian manifolds via $\nabla$-theorems

“…Remark 1. The boundedness of our solutions and their decay at infinity (1) follow from [7,Theorem 3.1]. This remark applies to the eigenfunctions considered in Section 2 as well.…”

“…Secondly, the problem we want to investigate is settled in a non compact Riemannian manifold and, as far as we know, results as the one we are going to prove are not present in literature. One of the first contribute for the Nonlinear Schrödinger equation on Riemannian manifold was given in [7], where Faraci and Farkas established a necessary and sufficient condition for the existence of non trivial solutions with hypothesis on the manifold equal to the ones we will assume. More recently, Molica Bisci and Secchi in [16] showed the existence of at least two solutions for (2) requiring large enough under our assumptions on .…”

Multiple solutions for Schrödinger equations on Riemannian manifolds via $$\nabla $$-theorems

“…We recall that under the assumptions we made on the potential and the manifold, the embedding H 1 V (M) → L q (M) is continuous for any q ∈ 2, 2 * . Furthermore, as a result of the Hypotheses V 1 and V 2 , we also have the following Lemma, whose proof can be found in [10,Lemma 2.1].…”

A note on the Nonlinear Schrödinger Equation on Cartan‐Hadamard manifolds with unbounded and vanishing potentials

“…Although the Schrödinger equation in $$$ {\mathrm{\mathbb{R}}}^N $$$ has been extensively studied, there is a surprising lack of understanding when it comes to looking for solutions for the equation on non‐Euclidean spaces such as Riemannian manifolds. One of the first contributions in this direction is given in the papers [11] and [12] where the authors proved the existence of solutions for the Schrödinger equation or for the Schrödinger‐Maxwell system requiring suitable bounds on the Ricci or sectional curvature. More recently, Appolloni, Molica Bisci, and Secchi proved, respectively, in [13] and [14] the existence of three solutions for the Schrödinger equation on a manifold with asymptotically non‐negative Ricci curvature with a coercive potential and the existence of infinitely many solutions on a Cartan‐Hadamard manifold with a constant potential and an oscillatory nonlinearity.…”