Nonlinear elliptic equations with variable exponents involving singular nonlinearity

Abstract

In this paper, we prove the existence and regularity of weak positive solutions for a class of nonlinear elliptic equations with a singular nonlinearity, lower order terms and L1 datum in the setting of Sobolev spaces with variable exponents. We will prove that the lower order term has some regularizing effects on the solutions. This work generalizes some results given in [1–3].

Degenerate elliptic problem with singular gradient lower order term and variable exponents

“…In some cases, they provide realistic models for the study of natural phenomena in electro-rheological fluids and important applications are related to image processing. We cite some papers that have dealt with the equation (1) or similar problems, we refer the reader to [1][2][3][4] and the references therein.…”

Existence and Uniqueness Results for a Singular Nonlinear Elliptic Equations with Variable Exponents

“…Moreover, by proceeding as in [4,42], we deduce u n ∈ L ∞ (Ω) because the righthand side of (4.2) is in L ∞ (Ω). Now, we will closely follow the proof of [ Lemma 2.1, Lemma 2.2, [6]], [Lemma 2.1, [20]], [Lemma 4.2, [24] ] and of [ Lemma 2, [28] ] hence we will omit the details, giving only a sketch of the passages. By (1.5), (1.8) and the fact that 0 ≤ f n ≤ f n+1 , we can prove that the sequence u n is increasing with respect to n. knowing that, for n ∈ N fixed, u n ∈ L ∞ (Ω).…”

“…In recent years, boundary value problems with variable exponents has received a lot of attention, reader can look at the nice surveys books [16,18,22,39] and the references therein, while the study of the elliptic and parabolic problems involving singular nonlinearities with variable exponents is still developing with slowness, because there are a very limited number of results exist on this topic, reader can have the opportunity to look at the new marvellous works [8,24,28,41]. We mention also that much attention has been devoted to nonlinear elliptic equations with singularities because of their wide application to physical models such as Non-Newtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogenous, etc.…”

“…They proved the existence and regularity results of problem (1.2) by adapting some techniques used in [14], nevertheless it was a new spirit of idea different from that used in [12], ( when γ(•) = γ ≥ 1, see [25] for a similar paper ). In [28], the authors generalized the work [2] by considering nonlinear elliptic equations with a singular nonlinearity, lower order terms and L 1 datum in the setting of Sobolev spaces with variable exponents. They proved that the lower order term has some regularizing effects on the solutions.…”

“…Distinctly, this motivated us to propose the study of the problem (1.2) and get new uniquely results which were not considered in the literature. Inspired by [34], we will prove the existence, uniqueness and regularity of nonnegative weak solution for the problem (1.2) by applying the method of approximations, Schauder fixed point theorem and adopting some techniques used in [28], make sure we will show that how the nonlinear singular term has regularizing effects on the solutions of the problem (1.2). More precisely, we will prove (Theorems 3.3-3.6-3.11 below).…”