Existence and regularity for critical anisotropic equations with critical directions (Q536519)

The author establishes existence and regularity results for doubly critical anisotropic equations of the type \[ -\Delta_{\vec p}=\lambda |u|^{p^*-2}u\quad\text{in }\Omega\subset \mathbb R^n,\qquad u\in D^{1,\vec p}(\Omega), \] where \(\lambda>0\) is a real number, \(p^*\) is the critical Sobolev exponent, \(\vec p=(p_1,\dots,p_n),\) \(\|u\|_{D^{1,\vec p}}=\sum_{i=1}^n \|\partial u/\partial x_i\|_{L^{p_i}(\Omega)},\) and \(\Delta_{\vec p}\) is the anisotropic Laplace operator defined by \[ \Delta_{\vec p}u=\sum_{i=1}^n {\partial \over {\partial x_i}}\nabla^{p_i}_{x_i}u. \]