Large versus bounded solutions to sublinear elliptic problems

Ewa Damek¹ ; Zeineb Ghardallou²

¹ Institute of Mathematics Wrocław University Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland
² Department of Mathematical Analysis and Applications University Tunis El Manar, LR11ES11 2092 El Manar 1, Tunis, Tunisia

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 67 (2019) no. 1, pp. 69-82.

Résumé

Let $L $ be a second order elliptic operator with smooth coefficients defined on a domain $\varOmega \subset \mathbb {R}^d$ (possibly unbounded), $d\geq 3$. We study nonnegative continuous solutions $u$ to the equation $L u(x) - \varphi (x, u(x))=0$ on $\varOmega $, where $\varphi $ is in the Kato class with respect to the first variable and it grows sublinearly with respect to the second variable. Under fairly general assumptions we prove that if there is a bounded nonzero solution then there is no large solution.

Détail

DOI : 10.4064/ba8180-12-2018

Keywords: second order elliptic operator smooth coefficients defined domain varomega subset mathbb possibly unbounded geq study nonnegative continuous solutions equation varphi varomega where varphi kato class respect first variable grows sublinearly respect second variable under fairly general assumptions prove there bounded nonzero solution there large solution