Existence and Nonexistence of Solutions for Quasilinear Elliptic Equations

A priori estimates and existence of positive solutions for strongly nonlinear problems

“…Hence, we can use the regularity result [10,21] (which is possible thanks to (16)) to conclude that jjz n jj C 1;t oC in Bð0; RÞ for certain C independent of n: Therefore, z n converges in the C 1 norm (up to a subsequence) to a certain function z 0 : Observe that z 0 ð0Þ ¼ 1: Applying (17) with jxjo % M À1 n d=2; we obtain that:…”

“…Another related paper is [16], where problem (1) is also considered, but under different hypotheses on the nonlinearity f :…”

Elliptic equations involving the $p$-Laplacian and a gradient term having natural growth

“…The change of variables introduced for p = 2 by Kazdan-Kramer [20] has been used, for instance, by Montenegro-Montenegro [25], Iturriaga-Lorca-Ubilla [17], and Iturriaga-Lorca-Sánchez [16]. In [25] the authors obtain existence of solutions for some specific functions g, for instance g constant or g such that lim s→∞ g(s) = 0, cf. Examples 2.3, 3.1 and 4.1 in [25].…”

“…In [25] the authors obtain existence of solutions for some specific functions g, for instance g constant or g such that lim s→∞ g(s) = 0, cf. Examples 2.3, 3.1 and 4.1 in [25]. In [17] the authors prove the existence of a solution when g is a constant and f (x, s) has at most an exponential growth (compare with our example (i)).…”

Existence and uniqueness of positive solution of a logistic equation with nonlinear gradient term

“…In each step, the authors use [17], Theorem 8.3, page 301, and so q ≤ 2 must be imposed. Some other papers consider similar problems with critical growth (q = 2), in which a convenient change of variables works (see, for instance, [20,21] and [23]). Generally speaking, equations of the form (1) −∆u = f (x, u, ∇u) in Ω, u = 0 on ∂Ω, have been very studied in the literature, see the classical works [3,5,6,8,10,15,22].…”