15 March 2005 Optimal regularity conditions for elliptic problems via $L^p_\delta$-spaces
Philippe Souplet
Duke Math. J. 127(1): 175-192 (15 March 2005). DOI: 10.1215/S0012-7094-04-12715-0


We consider positive solutions of the Dirichlet problem for nonlinear elliptic equations, where the nonlinearity is assumed to satisfy the polynomial upper growth condition f(x,u) ≤ C(1 + up). It is known from a classical work of Brezis and Turner that the condition p < pBT := (N + 1)/(N − 1) implies a uniform a~priori bound for all solutions. Yet this exponent appeared to be technical since Gidas and Spruck later showed that, for nonlinearities with precise power behavior like f(x,u) ~ up, the critical exponent is given by the Sobolev number pS := (N + 2)/(N − 2).

Surprisingly, we show that the exponent pBT is, however, sharp: whenever p > pBT, for a suitable nonlinearity f(x,u) = a(x)up, with a(x) ≥ 0 and aL, we prove the existence of an unbounded weak solution.

We next consider the case of systems and show that the polynomial growth conditions for a priori estimates recently obtained by Quittner and the author are also optimal.

Our results are strongly connected with the regularity theory of the Laplace operator in the spaces $L^p_\delta(\Omega)$, the Lebesgue spaces weighted by the distance to the boundary. As a by-product, we in turn establish the optimality of the known linear $L^p_\delta$-regularity estimates. Our proofs are based on the construction of a solution of the Laplace equation with a suitable boundary singularity, with conical support, and we use recent results on the boundary behavior of heat kernels.


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Philippe Souplet. "Optimal regularity conditions for elliptic problems via $L^p_\delta$-spaces." Duke Math. J. 127 (1) 175 - 192, 15 March 2005. https://doi.org/10.1215/S0012-7094-04-12715-0


Published: 15 March 2005
First available in Project Euclid: 4 March 2005

zbMATH: 1130.35057
MathSciNet: MR2126499
Digital Object Identifier: 10.1215/S0012-7094-04-12715-0

Primary: 35B65 , 35J25 , 35J55 , 35J60 , 46E30
Secondary: 35B45

Rights: Copyright © 2005 Duke University Press


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Vol.127 • No. 1 • 15 March 2005
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