A class of semilinear hypoelliptic Robin problems and Morse theory

Abstract

(1) Background: This paper is devoted to the study of a class of semilinear elliptic boundary value problems with hypoelliptic (degenerate) Robin condition that includes as particular cases the Dirichlet, Neumann and regular Robin problems. (2) Methods: We give a rigorous proof of main theorem, which is based heavily on the theory of linear elliptic boundary value problems in the framework of \(L^{p}\) Sobolev spaces. (3) Results: We extend earlier theorems due to Ambrosetti–Lupo and Struwe to the hypoelliptic Robin case via Morse theory. (4) Conclusions: The main purpose of this paper is to understand the essence of a modern version of the classical Lyapunov–Schmidt procedure through a concrete approach to semilinear hypoelliptic Robin problems.