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 a ∈ L∞, 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.
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