Nonlinear nonhomogeneous Robin problems with dependence on the gradient

Abstract

We consider a nonlinear elliptic equation driven by a nonhomogeneous partial differential operator with Robin boundary condition and a convection term. Using a topological approach based on the Leray-Schauder alternative principle, together with truncation and comparison techniques, we show the existence of a smooth positive solution without imposing any global growth condition on the reaction term. MSC: 35J92; 35P30

Nonlinear nonhomogeneous Robin problems with convection

“…In these two works the gradient term is not decoupled from the perturbation. This leads to different hypotheses which do not cover the present setting (see hypotheses H(f )(ii) and (iii) in [1] and H(f ) (iii) in [17]). Moreover, in [17] the differential operator is the p-Laplacian.…”

“…However, it is a mild requirement and it is satisfied in all cases of interest. Similar conditions were also used in Bai-Gasinski-Papageorgiou [1].…”

“…For Robin problems, there are the works of Bai-Gasinski-Papageorgiou [1] and Papageorgiou-Rădulescu-Repovs [17]. In these two works the gradient term is not decoupled from the perturbation.…”

Positive solutions for anisotropic singular $$\varvec{(p,q)}$$-equations

“…Nonparametric isotropic singular problems were considered by Bai-Gasiński-Papageorgiou [2], Papageorgiou-Rȃdulescu-Repovš [25] and Papageorgiou-Vetro-Vetro [34]. Papers [2,25] deal with equations driven by the p-Laplacian and in [2] the perturbation f (z, •) is (p − 1)-superlinear, while in [25] the perturbation f (z, •) is (p − 1)-linear and resonant. In [34], the authors consider a (p, 2)-equation with…”

Existence of positive solutions for nonlinear Robin problems with gradient dependence

“…In this direction we mention the works of de Figueiredo-Girardi-Matzeu [6], Girardi-Matzeu [12] (semilinear problems) and Bai [2], Faraci-Motreanu-Puglisi [4], Faria-Miyagaki-Motreanu-Tanaka [5], García Melián-Sabina de Lis-Takáč [7], Huy-Quan-Khanh [14], Ruiz [26], Tanaka [27] (nonlinear problems) together with Motreanu-Vetro-Vetro [16,17] (nonlinear system problems). For Neumann and Robin problems, there are the recent works of Gasinski-Papageorgiou [10], Papageorgiou-Radulescu-Repovs [24] (semilinear Neumann problems with a differential operator of the form div(a(u)Du), u ∈ W 1,p (Ω)) and Bai-Gasinski-Papageorgiou [3], Papageorgiou-Radulescu-Repovs [25] (nonlinear Robin problems). In these papers the hypotheses on the reaction are different and in many respects more restrictive.…”