A priori bounds for positive solutions of subcritical elliptic equations

Abstract

ABSTRACT. We provide a-priori L ∞ bounds for positive solutions to a class of subcritical elliptic problems in bounded C 2 domains. Our arguments rely on the moving planes method applied on the Kelvin transform of solutions. We prove that locally the image through the inversion map of a neighborhood of the boundary contains a convex neighborhood; applying the moving planes method, we prove that the transformed functions have no extremal point in a neighborhood of the boundary of the inverted domain. Retrieving the original solution u, the maximum of any positive solution in the domain Ω, is bounded above by a constant multiplied by the maximum on an open subset strongly contained in Ω. The constant and the open subset depend only on geometric properties of Ω, and are independent of the non-linearity and on the solution u. Our analysis answers a longstanding open problem.

A priori estimates for some elliptic equations involving the p-Laplacian

“…Nevertheless, we also prove another result (see Theorem 1.4 that follows), weakening the hypotheses needed for the result (except for a further technical hypothesis, H 5 ), which is satisfied for a general class of nonlinearities) and extends the class of nonlinearities allowed, including functions f more general than subcritical powers, and can be seen as the counterpart for p = 2 to the results in [13] in case p = 2.…”

A-priori bounds for quasilinear problems in critical dimension

“…The main assumption therein was that the growth at ∞ of f is controlled by a suitable power with exponent 2 * − 1 = N +2 N −2 , 2 * being the critical Sobolev exponent. We also point out the more recent work of Castro and Pardo [7] which enlarges the class of nonlinearities involved.…”

“…Originally, most of these results concerned the case 1 < m ≤ 2 due to some symmetry and monotonicity arguments for solutions to m-laplacian equations which were at that time available only for that range of m; however, those techniques have been later extended also to the case m > 2 in the papers [10,11]. See also the recent work of Damascelli and Pardo [9], which extends further the class of nonlinearities for which the a-priori bound applies, in the spirit of [12,7].…”

On the existence of a priori bounds for positive solutions of elliptic problems, I

“…This Theorem is in fact a Corollary of Theorem 2.1 (see Subsection 2.3 for a proof of Theorem 2.1; see also [5,Corollary 2.2]). The ideas of the proof of Theorem 2.1 lie on the following arguments:…”