Details

Autor(en) / Beteiligte

• Hamour, Boussad

Titel

Singular Nonlinear Problems with Natural Growth in the Gradient

Ist Teil von

• Mathematical modelling and analysis, 2024-03, Vol.29 (2), p.367-386

Ort / Verlag

Vilnius: Vilnius Gediminas Technical University

Erscheinungsjahr

2024

Link zum Volltext

Quelle

EZB Electronic Journals Library

Beschreibungen/Notizen

• In this paper, we consider the equation--div [a(x, u, Du)=H(x, u, Du) + [[a.sub.0](x)/[|u|.sup.[theta]]] + [[chi].sub.{u[not equal to]0}] f (x) in [OMEGA], with boundary conditions u = 0 on [partial derivative][OMEGA], where [OMEGA] is an open bounded subset of [R.sup.N], 1 < p < N, -div(a(x, u, Du)) is a Leray-Lions operator defined on [W.sup.1,p.sub.0]([OMEGA]), [a.sub.0] [member of] [L.sup.N/p]([OMEGA]), [a.sub.0] > 0, 0 < [theta] [less than or equal to] 1, [[chi].sub.{u=0}] is a characteristic function, f [member of] [L.sup.N/p]([ohm]) and H(x,s,[xi]) is a Caratheodory function such that -[c.sub.0] a(x,s,[xi])[xi] [less than or equal to] H(x,s,[xi]) sign(s) [less than or equal to] [gamma]a(x,s,[xi])[xi] a.e. x [member of] [ohm], [for all]s [member of] R, [for all][xi] [member of] [R.sup.N]. For [||[a.sub.0]||.sub.N/p] and [||f||.sub.N/p] sufficiently small, we prove the existence of at least one solution u of this problem which is moreover such that the function exp([delta]|u|) - 1 belongs to [W.sub.1,p.sub.0]([ohm]) for some [delta] [greater than or equal to] [gamma]. This solution satisfies some a priori estimates in [W.sub.1,p.sub.0]([ohm]). Keywords: nonlinear problems, existence, singularity. AMS Subject Classification: 35J60; 35J75.