Differential and Integral Equations

Monotonicity of the solutions of some quasilinear elliptic equations in the half-plane, and applications

L. Damascelli and B. Sciunzi

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Abstract

We consider weak positive solutions of the equation $-\Delta_m u=f(u)$ in the half-plane with zero Dirichlet boundary conditions. Assuming that the nonlinearity $f$ is locally Lipschitz continuous and $f(s)>0$ for $s>0$, we prove that any solution is monotone. Some Liouville-type theorems follow in the case of Lane-Emden-Fowler-type equations. Assuming also that $|\nabla u|$ is globally bounded, our result implies that solutions are one dimensional, and the level sets are flat.

Article information

Source
Differential Integral Equations, Volume 23, Number 5/6 (2010), 419-434.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356019303

Mathematical Reviews number (MathSciNet)
MR2654242

Zentralblatt MATH identifier
1240.35208

Subjects
Primary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B65: Smoothness and regularity of solutions 35J70: Degenerate elliptic equations

Citation

Damascelli, L.; Sciunzi, B. Monotonicity of the solutions of some quasilinear elliptic equations in the half-plane, and applications. Differential Integral Equations 23 (2010), no. 5/6, 419--434. https://projecteuclid.org/euclid.die/1356019303


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