A Liouville-type theorem for the Lane-Emden equation in a half-space

TL;DR

This work addresses the nonexistence of positive solutions to the half-space Dirichlet problem $- abla^2 u=f(u)$ with monotonicity in the normal direction, including the Lane-Emden case $f(u)=u^p$ for $p>1$. The authors develop stability-based $H^1$ bounds, introduce a novel weighted quotient framework $\xi=w/z$ with $z=(1+x_n)u_{x_n}$, and implement a quantified Moser-type iteration to obtain $L^ty$ control of the negative part of $\xi$, which yields $\xi\ge0$ and hence convexity in the normal direction. This convexity, together with a maximum principle and stability, yields a contradiction unless the solution is trivial, establishing a Liouville-type nonexistence result for all convex nonlinearities with $f(0)=0$. The appendix then classifies the case $f(0)>0$, providing a complete Liouville-type picture in that regime. Overall, the paper resolves a long-standing open problem by proving nonexistence of positive solutions monotone in $x_n$ in the half-space and extending the result to general convex nonlinearities, with potential implications for blow-up analysis and symmetry phenomena in semilinear elliptic equations.

Abstract

We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solution which is monotone in the normal direction. As a consequence, this problem does not admit any positive classical solution which is bounded on finite strips. This question has a long history and our result solves a long-standing open problem. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. The result extends to general convex nonlinearities.

Theorems & Definitions (24)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Proposition 2.1
• proof
• Proposition 2.2
• Proposition 2.3