Monotone heteroclinic solutions to semilinear PDEs in cylinders and applications

TL;DR

This paper proves the existence of monotone heteroclinic solutions to semilinear elliptic PDEs in cylinders, with implications for bounded solutions in fluid dynamics and the structure of solutions without critical points.

Contribution

It introduces a new method to construct non-one-dimensional bounded solutions of semilinear PDEs and provides examples relevant to 2D Euler equations without stagnation points.

Findings

Existence of strictly monotone heteroclinic solutions in cylinders.

Construction of non-one-dimensional bounded solutions without critical points.

Examples of stationary solutions for 2D Euler equations without stagnation points.

Abstract

In this paper we show the existence of strictly monotone heteroclinic type solutions of semilinear elliptic equations in cylinders. The motivation of this construction is twofold: first, it implies the existence of an entire bounded solution of a semilinear equation without critical points which is not one-dimensional. Second, this gives an example of a bounded stationary solution for the 2D Euler equations without stagnation points which is not a shear flow, completing previous results of Hamel and Nadirashvili. The proof uses a minimization technique together with a truncation argument, and a limit procedure.