Partial regularity for weak solutions of nonlinear elliptic equations with supercritical exponents
In the present paper, the author proves a partial regularity result for positive weak solutions of the equation \[ \Delta u+ h_1 u^\alpha+ h_2 u^\beta= 0\quad\text{in }\Omega, \] where \(\alpha\geq {n+2\over n-2}\), \(\alpha+ 1\geq 2\beta> 2\), \(h_i\in C^1(\Omega)\), \(i= 1,2\), \(a_i\leq h_i(x)\leq b_i\), \(0< a_i< b_i\) and \(|\nabla\log h_i(x)|\leq \beta\) for \(x\in\overline\Omega\), \(i= 1,2\). Here \(\Omega\) is an open subset of \(\mathbb R^n\) \((n\geq 3)\). To this end, the author uses the duality of a weighted Hardy space and a weighted BMO.
scientific article; zbMATH DE number 2137246