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Large versus bounded solutions to sublinear elliptic problems

Volume 67 / 2019

Ewa Damek, Zeineb Ghardallou Bulletin Polish Acad. Sci. Math. 67 (2019), 69-82 MSC: Primary 31D05, 35J08, 35J61; Secondary 31C05. DOI: 10.4064/ba8180-12-2018 Published online: 15 March 2019

Abstract

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.

Authors

  • Ewa DamekInstitute of Mathematics
    Wrocław University
    Pl. Grunwaldzki 2/4
    50-384 Wrocław, Poland
    e-mail
  • Zeineb GhardallouDepartment of Mathematical
    Analysis and Applications
    University Tunis El Manar, LR11ES11
    2092 El Manar 1, Tunis, Tunisia
    e-mail

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