Abstract

PARDO, Rosa. On the existence of a priori bounds for positive solutions of elliptic problems, I. Integración - UIS [online]. 2019, vol.37, n.1, pp.77-111. ISSN 0120-419X.  https://doi.org/10.18273/revint.v37n1-2019005.

This paper gives a survey over the existence of uniform L°° a priori bounds for positive solutions of subcritical elliptic equations widening the known ranges of subcritical nonlinearities for which positive solutions are a-priori bounded. Our arguments rely on the moving planes method, a Pohozaev identity, W 1,q regularity for q > N, and Morrey's Theorem. In this part I, when p = 2, we show that there exists a-priori bounds for classical, positive solutions of (P)2 with f (u) = u2*-1/[ln(e + u)]α, with 2* = 2N/(N - 2), and α > 2/(N - 2). Appealing to the Kelvin transform, we cover non-convex domains.

In a forthcoming paper containing part II, we extend our results for Hamil-tonian elliptic systems (see [22]), and for the p-Laplacian (see [10]). We also study the asymptotic behavior of radially symmetric solutions u α = u α (r) of (see [24]).

MSC2010: 35B45, 35J92, 35B33, 35J47, 35J60, 35J61.

Keywords : A priori estimates; subcritical nonlinearity; moving planes method; Pohozaev identity; critical Sobolev hyperbola; biparameter bifurcation.

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