A Priori Estimates and Continuation Methods for Positive Solutions of p-Laplace Equations

Superlinear Equations Involving Nonlinearities Limited by Asymptotically Homogeneous Functions

“…Thus our result is an improvement because we do not impose either the regularity condition on the function f (as in [6]) or condition (1.2) for all s ≥ 0 (as in [5,7]). Also, we note that we do not assume any convex assumption on Ω.…”

“…We mention recent work on existence of solutions of problem (1.1) where a combination of blowup arguments and nonexistence results for R N is used. Azizieh and Clément [5] studied the case 1 < p ≤ 2. It is assumed that the domain Ω is strictly convex and that there exist positive constants C 1 , C 2 , and q, where…”

“…Following ideas of [5][6][7], we obtain the existence of a solution of problem (1.1). (See Theorem 4.1.…”

Nonexistence of Positive Solution for Quasilinear Elliptic Problems in the Half-Space

“…This is particularly useful if the problem under consideration is nonvariational (see, e.g., [2][3][4] and the references therein). Usually these a priori estimates are obtained by using a blow up technique.…”

“…To avoid the case of the half-space, it is assumed in [3] that Ω is convex, f does not depend on x, and 1 < m ≤ 2. These assumptions together with the moving plane method allow to obtain a positive solution of −Δ m u ≥ u p in R N , which is a contradiction with the Liouville result in [1].…”