Monotonicity for quasilinear elliptic problems with a sign-changing nonlinearity in half-planes

Hieu Thanh Nguyen; Phuong Le; Thanh Chi Vo

doi:10.5802/crmath.700

Article de recherche - Équations aux dérivées partielles, Géométrie et Topologie

Monotonicity for quasilinear elliptic problems with a sign-changing nonlinearity in half-planes
[Monotonie des solutions de certains problèmes elliptiques quasi-linéaires dans le demi-plan avec non-linéarité changeant de signe]

Hieu Thanh Nguyen ¹ ; Phuong Le ^2,³ ; Thanh Chi Vo ^3,⁴

¹ Department of Mathematics, University of Wisconsin-Madison, Madison, WI, USA
² Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh City, Vietnam
³ Vietnam National University, Ho Chi Minh City, Vietnam
⁴ Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 69-87.

Résumé
Abstract

Dans cet article, nous étudions la monotonie des solutions positives $u$ du problème

Δ_{p} u + a (u) {| \nabla u |}^{q} = f (u) dans ℝ_{+}^{2}, u = 0 sur \partial ℝ_{+}^{2},

où $p > \frac{3}{2}$ , $q \geq max \{p - 1, 1\}$ et $a, f$ sont des fonctions localement Lipschitz. Nous considérons des non-linéarités qui changent de signe dans le cas $\frac{3}{2} < p < 2$ , respectivement positives dans le cas $p > 2$ . Sans aucune hypothèse sur le caractère borné de $u$ ou de $| \nabla u |$ , nous montrons que $u$ est croissante par rapport à la direction orthogonale à la frontière. Ceci améliore un résultat récent d’Esposito et al. [10], où $| \nabla u |$ est supposé être borné dans chaque bande. Notre preuve combine les techniques géométriques dans le plan avec les célèbres méthodes du plan glissant et du plan mobile. Certains outils analytiques sont également développés pour traiter l’absence de principes de comparaison forte et de maximum fort lorsque $f$ change de signe.

In this paper, we study the monotonicity of positive solutions $u$ to the problem

- Δ_{p} u + a (u) {| \nabla u |}^{q} = f (u) in ℝ_{+}^{2}, u = 0 on \partial ℝ_{+}^{2},

where $p > \frac{3}{2}$ , $q \geq max \{p - 1, 1\}$ and $a, f$ are locally Lipschitz continuous functions. We consider sign-changing nonlinearities in the case $\frac{3}{2} < p < 2$ and positive nonlinearities in the case $p > 2$ . Without any assumptions on the boundedness of $u$ or $| \nabla u |$ , we show that $u$ is monotone increasing with respect to the direction orthogonal to the boundary. This improves a recent result by Esposito et al. [10], where $| \nabla u |$ is assumed to be bounded in strips. Our proof combines the geometric techniques in the plane with the celebrated sliding and moving plane methods. Some analytic tools are also developed to deal with the lack of strong comparison and strong maximum principles when $f$ changes sign.

Reçu le : 2024-08-18
Accepté le : 2024-11-05
Publié le : 2025-03-06

DOI : 10.5802/crmath.700

Classification : 35J92, 35B06, 35B51
Keywords: $p$-Laplacian, quasilinear elliptic equation, strong comparison principle, monotonicity of solutions
Mots-clés : $p$-Laplacien, équation elliptique quasi-linéaire, principe de comparaison forte, monotonie des solutions

Affiliations des auteurs :

Hieu Thanh Nguyen ¹ ; Phuong Le ^{2, 3} ; Thanh Chi Vo ^{3, 4}

¹ Department of Mathematics, University of Wisconsin-Madison, Madison, WI, USA
² Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh City, Vietnam
³ Vietnam National University, Ho Chi Minh City, Vietnam
⁴ Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Hieu Thanh Nguyen and Phuong Le and Thanh Chi Vo},
     title = {Monotonicity for quasilinear elliptic problems with a sign-changing nonlinearity in half-planes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {69--87},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.700},
     language = {en},
}

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Hieu Thanh Nguyen; Phuong Le; Thanh Chi Vo. Monotonicity for quasilinear elliptic problems with a sign-changing nonlinearity in half-planes. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 69-87. doi : 10.5802/crmath.700. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.700/

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