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Optimal regularity conditions for elliptic problems via \(L^p_\delta\)-spaces. (English) Zbl 1130.35057
Summary: We consider positive solutions of the Dirichlet problem for nonlinear elliptic equations, where the nonlinearity is assumed to satisfy the polynomial upper growth condition \(f(x,u) \leq C(1 + u^p)\). It is known from a classical work of H. Brézis and R. E. L. Turner [Commun. Partial Differ. Equations 2, 601-614 (1977; Zbl 0358.35032)] that the condition \(p < p_{\text{BT}}:= (N + 1)/(N-1)\) implies a uniform a priori bound for all solutions. Yet this exponent appeared to be technical since B. Gidas and J. Spruck [Commun. Partial Differ. Equ. 6, 883–901 (1981; Zbl 0462.35041)] later showed that, for nonlinearities with precise power behavior like \(f(x,u) \sim u^p\), the critical exponent is given by the Sobolev number \(p_S := (N + 2)/(N-2)\).
Surprisingly, we show that the exponent \(p_{\text{BT}}\) is, however, sharp: whenever \(p > p_{\text{BT}}\), for a suitable nonlinearity \(f(x,u) = a(x)u^p\), with \(a(x) \geq 0\) and \(a \in L^\infty\), 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 P. Quittner and the author [Arch. Ration. Mech. Anal. 174, No. 1, 49–81 (2004; Zbl 1113.35062)] 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.

MSC:
35J60 Nonlinear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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