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.


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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