Invariant criteria for existence of solutions to second‐order quasilinear elliptic equations

Existence and multiplicity result for a class of second order elliptic equations

“…One should also mention the paper of Kazdan and Warner [5] in which similar problems for the case where G is not depending on the gradient have been studied. If r(x) §0 in ft but r^O, the basic step in [1] and [5], the construction of a supersolution for a suitable parameter value t does not carry over to our situation. Moreover no multiplicity result is obtained there and no assertion is made for t = t 0 .…”

“…Since u has to be in int(A) we find by the growth condition of y that ||&|| L -(n) = const where the constant is only depending on (f> 1 …”

Maximal and Minimal Solutions to a Class of Elliptic Quasilinear Problems

“…REMARK 1. In the semilinear case a13 = a%3(x), the method of lower and upper solutions has, for instance, been utilized by Kazdan and Kramer [4] and Amann and Crandall [2]. In the former paper, existence of solutions is proven under the additional assumption that a(x, u, z) = a(x, u) + ß(x, u, z) with ß E C°(fi x R x RN); in the latter, existence of maximal and minimal solutions is proven under the additional assumption that a, au and az are continuous in fi x R x RN.…”

“…Notice that the bound on ||it||£,00(n) is now trivialized through the inequalities on the first line of (5), so that our argument does not require any use of either truncation operators or Bony's maximum principle. This allows us to handle rather general assumptions, which are weaker, even for what concerns only the first-order nonlinear term a, than those utilized in [2] and in [4] for semilinear equations (see Remark 1 at the end of the paper).…”

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

“…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]. Most of these are concerned with a-priori estimates, and from those they get existence by using topological methods or sub-super solution techniques.…”