The article is concerned with the existence of positive solutions of a semi-linear elliptic system defined in a cylinder Ω = Ω ′× (0 , a) ⊂ Rn , where Ω ′⊂ Rn-1 is a bounded and smooth domain. The system couples a superlinear equation defined in the whole cylinder Ω with another superlinear (or linear) equation defined at the bottom of the cylinder Ω ′× { 0 } . Possible applications for such systems are interacting substances (gas in the cylinder and fluid at the bottom) or competing species in a cylindrical habitat (insects in the air and plants on the ground). We provide a priori L∞ bounds for all positive solutions of the system when the nonlinear terms satisfy certain growth conditions. It is interesting that due to the structure of the system our growth restrictions are weaker than those of the pioneering result by Brezis–Turner for a single equation. Using the a priori bounds and topological arguments, we prove the existence of positive solutions for these particular semi-linear elliptic systems.
A system of superlinear elliptic equations in a cylinder / Y. Guo, B. Ruf. - In: NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS. - ISSN 1021-9722. - 30:5(2023 Sep), pp. 62.1-62.29. [10.1007/s00030-023-00869-5]
A system of superlinear elliptic equations in a cylinder
Y. Guo
;B. Ruf
2023
Abstract
The article is concerned with the existence of positive solutions of a semi-linear elliptic system defined in a cylinder Ω = Ω ′× (0 , a) ⊂ Rn , where Ω ′⊂ Rn-1 is a bounded and smooth domain. The system couples a superlinear equation defined in the whole cylinder Ω with another superlinear (or linear) equation defined at the bottom of the cylinder Ω ′× { 0 } . Possible applications for such systems are interacting substances (gas in the cylinder and fluid at the bottom) or competing species in a cylindrical habitat (insects in the air and plants on the ground). We provide a priori L∞ bounds for all positive solutions of the system when the nonlinear terms satisfy certain growth conditions. It is interesting that due to the structure of the system our growth restrictions are weaker than those of the pioneering result by Brezis–Turner for a single equation. Using the a priori bounds and topological arguments, we prove the existence of positive solutions for these particular semi-linear elliptic systems.File | Dimensione | Formato | |
---|---|---|---|
s00030-023-00869-5.pdf
accesso aperto
Tipologia:
Publisher's version/PDF
Dimensione
538.16 kB
Formato
Adobe PDF
|
538.16 kB | Adobe PDF | Visualizza/Apri |
Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.