Singular quasilinear elliptic equations and Hölder regularity
The authors prove that solutions of the boundary value problem NEWLINE\[NEWLINE -\Delta_ pu = K(x)u^{-\delta}+g(x) \text{ in } \Omega, \;u=0 \text{ on } \partial\Omega NEWLINE\]NEWLINE are Hölder continuous when \(\Omega\) is an open, bounded domain in \(\mathbb R^ n\) with sufficiently smooth boundary, \(K\) satisfies the inequality \(0\leq K(x)\leq Cd(x)^{-\omega}\), \(g\) is nonnegative and bounded, and the positive constants \(\delta\) and \(\omega\) satisfy the inequality \(\omega<1+(1-\delta)(1-1/p)\). In addition, the solution \(u\) is assumed to be positive in \(\Omega\). When \(\omega+\delta<1\), the authors showed in a previous work [Ann. Sci. Norm. Sup. Pisa 6, 117--158 (2007; Zbl 1181.35116)] that \(u\) actually has Hölder continuous gradient, and the present paper is concerned with the case \(1\leq\omega+\delta\).NEWLINENEWLINE There is one peculiarity in the authors' Theorem 2.1: the statement of that theorem requires the exponent \(\beta\) to be less than the parameter \(\alpha\) but this restriction is not present in the proof of the theorem.
scientific article; zbMATH DE number 6042967
scientific article