The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds
Let \((M, g)\) be a smooth compact Riemannian \(N\)-manifold, \(N \geq 2,\) and \(p > 2\) if \(N = 2\) and \({2 < p < 2^{*} = {2N \over N-2}}\) if \(N \geq 3.\) The authors show that the positive solutions of the problem \[ -\varepsilon^2\Delta_g u + u = u^{p-1}\quad \text{in}\;M \] are generated by stable critical points of the scalar curvature of \(g,\) if \(\varepsilon\) is small enough.
scientific article; zbMATH DE number 5532488
scientific article