Multiplicity of nodal solutions for elliptic problems with critical exponential growth on $$\mathbb {S}^2$$

Abstract

We study the existence and multiplicity of sign-changing solutions of the semilinear elliptic equation $$ -\Delta _g u + u = f(u) \quad \text {on } \mathbb {S}^2, $$ where $$(\mathbb {S}^2,g)$$ denotes the two-dimensional unit sphere endowed with a smooth Riemannian metric, $$\Delta _g$$ is the Laplace–Beltrami operator, and f may exhibit critical exponential growth in the sense of the Trudinger-Moser inequality. Our method combines a variational construction on geodesic lunes with a global reflection scheme. We first solve a Dirichlet problem on a fixed lune and show that the corresponding mountain-pass level remains strictly below the Trudinger-Moser threshold. Reflecting this solution across meridians and alternating its sign yields a sequence of global solutions with prescribed nodal patterns and quantized energy levels when f is odd. For small nonsymmetric perturbations of the nonlinearity, we employ a Lyapunov-Schmidt reduction to adjust the resulting multi-lune configurations, thereby proving the persistence of these sign-changing solutions.