## 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

### Citations:

Zbl 1113.35062; Zbl 0462.35041; Zbl 0358.35032
Full Text:

### References:

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