Existence results to a nonlinearp(x)-Laplace equation with degenerate coercivity and zero-order term: renormalized and entropy solutions

The obstacle problem for non-coercive equations with lower order term and $L^{1}$-data

“…To overcome these difficulties, "cutting" the non-coercivity term and using the technique of approximation, a pseudomonotone and coercive differential operator on W 1,p 0 (Ω) can be applied to establish a priori estimates on approximating solutions. As a result, existence of solutions, or entropy solutions, can be obtained by taking limitation for f ∈ L m (Ω), m ≥ 1, and b > 0 due to the almost everywhere convergence of gradients of the approximating solutions, see, e.g., [4,6,[9][10][11]15] (see also [1,2,7,12,13,16] for b = 0). However, there is little literature that considers regularities for entropy solutions of obstacle problems governed by (1) and functions (f , ψ, g) with f ∈ L 1 (Ω), except [19], in which the authors considered the obstacle problem (7) with b = 0 and L 1 -data.…”

Nonlinear Degenerate Anisotropic Elliptic Equations with Variable Exponents and L¹ Data

“…Eq., 33 (2020), pp. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] Carathéodory functions such that for almost every x in Ω and for every (σ,ξ) ∈ R×R N the following assumptions are satisfied for all i = 1,..., N…”

“…The nonlinear term g : Ω×R×R N → R is a Carathéodory function such that for a.e. x ∈ Ω and all (σ,ξ) ∈ R×R N , we have where b : R + → R + is a continuous and increasing function with finite values, c ∈ L 1 (Ω) and ∃ρ > 0 such that:…”

“…, ∀σ such that |σ| > ρ. (1.8) In [1], the authors obtain the existence of renormalized and entropy solutions for the nonlinear elliptic equation with degenerate coercivity of the type −div[a(x,u)|∇u| p(x)−2 ∇u]+g(x,u) = f ∈ L 1 (Ω).…”

The obstacle problem for nonlinear degenerate equations with $L^{1}$-data

“…Existence of solutions (including distributional solutions (W 1,1regular) and entropy solutions) was established in [5,8,15]. Recently, problems like (1.3) were extended to variable Sobolev space in [19], obtaining renormalized and entropy solutions with f ∈ L 1 (Ω).…”