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

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