On a class of non-uniformly elliptic equations

Abstract

In this paper we study the existence and uniqueness of both weak solutions and entropy solutions for the Dirichlet boundary value problem of a class of non-uniformly elliptic equations. A comparison result is also discussed. Some well-known elliptic equations are the special cases of this equation.Mathematics Subject Classification (2010). Primary 35D05; Secondary 35D10.

Leray–Lions operators with logarithmic growth

“…We recall that variational integrals of nearly linear growth (but different from the one in (1.4)) were introduced in [13] for the study of non-Newtonian fluids of Prandtl-Eyring type, and studied in [18,2,12,21,24,6,14,15]. In particular, in [18] regularity results for the gradients of minima are proved (with respect to the regularity of the data).…”

Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth

“…is in such way that W 1,Φ 0 (Ω) is not reflexive. The problem (1.5) have been studied by many authors during the last years, see Boccardo et al [2,3], Esposito et al [11], Passarelli [24], Fuchs [12,13], Zhang et al [27] and references therein. For further results on Orlicz and Orlicz-Sobolev framework in refer the reader to [1,14,15,17,18,22].…”

Bounded very weak solutions for some non-uniformly elliptic equation with L^1 datum

“…Our work can be seen as a natural outgrowth of the results in [3] to the more general quasilinear problem (1.1). To this aim, we first employ a unifying method developed in [17] (see [7] for the parabolic case) to prove the existence of weak solutions for problem (1.1) under the integrability conditions that f ∈ L N (Ω) and a ∈ L ∞ (Ω). It is worth pointing out that we do not assume polynomial or exponential growth for function Φ as in [1,8,14].…”