A Class of Semilinear Hypoelliptic Robin Problems at Resonance

ABSTRACT (1) This paper is devoted to the study of a class of semilinear hypoelliptic (degenerate) Robin problems at resonance that include semilinear regular Robin problems at resonance. (2) We give a rigorous proof of main theorems in the framework of Hölder spaces, which is heavily based on the theory of linear elliptic boundary value problems. In point of fact, we develop a modern version of the classical Lyapunov–Schmidt procedure, the global inversion theorem, and the intermediate‐value theorem from scratch in the framework of Sobolev spaces. (3) We extend an earlier theorem due to Landesman and Lazer to the hypoelliptic case. (4) The main purpose of this paper is to understand the essence of a modern version of the classical Lyapunov–Schmidt procedure at resonance through a concrete approach to semilinear hypoelliptic Robin problems.