Multiple solutions for Schrödinger equations on Riemannian manifolds via ∇-theorems

2022/03/16 by Appolloni, Luigi, Bisci, Giovanni Molica, Secchi, Simone · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2203.08482

Abstract

We consider a smooth, complete and non-compact Riemannian manifold (M,g) of dimension d ≥ 3, and we look for positive solutions to the semilinear elliptic equation -Δ_g w + V w = αf(w) + λw \hboxin M. The potential V \colon M → ℝ is a continuous function which is coercive in a suitable sense, while the nonlinearity f has a subcritical growth in the sense of Sobolev embeddings. By means of ∇-Theorems introduced by Marino and Saccon, we prove that at least three solution exists as soon as the parameter λ is sufficiently close to an eigenvalue of the operator -Δ_g.

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