Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient

Abstract

Thanks to a change of unknown we compare two elliptic quasilinear problems with Dirichlet data in a bounded domain of R N . The first one, of the form − p u = β(u)|∇u| p + λf (x), where β is nonnegative, involves a gradient term with natural growth. The second one, of the form − p v = λf (x)(1 + g(v)) p−1 where g is nondecreasing, presents a source term of order 0. The correlation gives new results of existence, nonexistence and multiplicity for the two problems. To cite this article: H.A. Hamid, M.F. Bidaut-Véron, C. R. Acad. Sci. Paris, Ser. I 346 (2008). RésuméCorrélation entre deux problèmes quasilinéaires elliptiques avec terme de source relatif à la fonction ou à son gradient. A l'aide d'un changement d'inconnue nous comparons deux problèmes elliptiques quasilinéaires avec conditions de Dirichlet dans un domaine borné Ω de R N . Le premier, de la forme − p u = β(u)|∇u| p + λf (x), où β est positif, comporte un terme de gradient à croissance critique. Le second, de la forme − p v = λf (x)(1 + g(v)) p−1 où g est croissante, contient un terme de source d'ordre 0. La comparaison donne des résultats nouveaux d'existence, nonexistence et multiplicité pour les deux problèmes. Pour citer cet article : H.A. Hamid, M.F. Bidaut-Véron, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Keller–Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group

“…Generalization of the previous problem to the case of the p-Laplacian has been investigated by [21]. On the other hand, the problem…”

“…In the last part of the paper we show how the techniques introduced can be implemented to study differential inequalities of the form (21) where the functions appearing in the RHS of the above are non-negative. The main results obtained are Theorem 5.3, that is, triviality of the solutions in the general setting under an appropriate KellerOsserman condition, and Theorem 5.7 for the p-Laplace operator, where we show the sharpness of the condition in analogy with Theorem 1.3.…”

“…Section 3 is devoted to the proof of both Theorems 1.1 and 1.4, whereas in Section 4 we show the equivalence in Theorem 1.3. Finally, in Section 5 we study inequality (21) and we give a Liouville theorem under a modified Keller-Osserman condition, together with a companion existence result for the particular case of the p-Laplace operator.…”

“…Differential inequalities of the type of (18) and (21) have been the subject of an increasing interest in the last years, mostly in the Euclidean setting (also for their connection with stochastic control theory, see for instance [25]). Among the literature on this topic we only cite some of the references.…”

“…The aim of this section is to show that the method used so far allows us to treat other types of inequalities; in particular, we focus our attention on (21), that is,…”