Global regularity for a singular equation and local \(H^ 1\) minimizers of a nondifferentiable functional. (English) Zbl 1048.35022

Dávila, Juan

Commun. Contemp. Math. 6, No. 1, 165-193 (2004).

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^d\). The author is mainly interested in nonnegative solutions to the equation \[ \begin{cases} -\Delta u+ u^{-\beta}=\lambda f(x,u)&\text{in }\Omega,\\ u= 0 &\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(0<\beta< 1\), \(\lambda> 0\) and \(f: \Omega\times \mathbb{R}^+\to \mathbb{R}^+\) is a nonnegative function. Under some suitable assumptions on \(f\), the author proves optimal Hölder estimates up to the boundary for the maximal solution of (1).

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

35J60	Nonlinear elliptic equations
35B65	Smoothness and regularity of solutions to PDEs
35J25	Boundary value problems for second-order elliptic equations

Keywords:

singular elliptic equation; gradient estimates; maximal solution

References:

[1]	Brezis H., C. R. Acad. Sci. Paris, Série I 317 pp 465–
[2]	Dávila J., J. Anal. Math.
[3]	M. Giaquinta, Annals of Math. Stud., Multiple integrals in the calculus of variations and nonlinear elliptic systems (Princeton University Press, 1983) p. 105.
[4]	DOI: 10.1007/BF01389324 · Zbl 0513.49003 · doi:10.1007/BF01389324
[5]	Giaquinta M., Boll. Un. Mat. Ital. A (6) 3 pp 239–
[6]	DOI: 10.1017/S030821050002970X · Zbl 0805.35032 · doi:10.1017/S030821050002970X
[7]	DOI: 10.1512/iumj.1983.32.32001 · Zbl 0545.35013 · doi:10.1512/iumj.1983.32.32001