Nonlinear elliptic equations having a gradient term with natural growth

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On the connection between two quasilinear elliptic problems with source terms of order 0 or 1 (2008, methodology): “For general p > 1, the problem has been investigated in [59] in the absorption case where β(t) ≦ 0 with measure data, and in [60] with a signed β, with strong growth assumptions on |β|.”
Dirichlet problems with singular and gradient quadratic lower order terms (2008, background): “Puel and to • [4,21]: unbounded solutions and data in the dual space (with the assumption g(x, s, ξ) s ≥ 0); • [9,10,17,27,29]: unbounded solutions and f ∈ L(1)(Ω) (with the assumption (1.”
Some remarks on elliptic problems with critical growth in the gradient (2006, background): “Remark 4.9 The above result should be compared with the existence results by Porretta-Segura [ 43 ] in the case where (s) is a positive function such that lims!+1 (s) = 0. In that paper it is proved that, under this assumption, a solution of (30) exists for all >”
An elliptic problem with a lower order term having singular behaviour (2009, background): “If 0 k 1, we prove the existence of a solution for both the “+” and the “−” signs, while if k ≥ 1, we prove the existence of a solution for the “+” sign only.”
Elliptic equations having a singular quadratic gradient term and a changing sign datum (2012, methodology): “By [18] (or, alternatively, by applying Theorem 2.”