On a nonlinear Dirichlet problem with a singularity along the boundary (Q1850179)

The authors of this paper prove the existence and uniqueness of strong solutions \(u\in W^{2,p}_{\text{loc}}(\Omega) \cap C(\overline\Omega)\) for the nonlinear Dirichlet problem \[ -\Delta u+\lambda u=Ku^{-\alpha}+f\text{ in }\Omega,\quad u=0\text{ on }\partial\Omega, \] where \(\Omega\) is a \(C^{1,1}\) and bounded domain in \(\mathbb{R}^n\), with \(n\geq 2\), \(\lambda\) and \(\alpha\) are nonnegative real numbers, \(K\geq 0\), \(f\geq 0\) and \(K\), \(f\in L^p(\Omega)\) for some \(p>n/2\). A similar result is stated in the particular case \(\lambda=0\), \(f=0\), \(\alpha>0\), under the hypothesis \(p>\frac{(\alpha^2+1)n}{2\alpha^2+n}\). In this case it is introduced the set \(M_\alpha=\{u\in W^{2,p}_{\text{loc}}(\Omega):0\leq u\leq w^{\frac 1{\alpha+1}}\) for some nonnegative \(w\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)\}\). The set \(M_\alpha\) contains the functions which vanish in some sense on the boundary. It is proved that there exists \(u\in W^{2,p}_{\text{loc}}(\Omega)\cap M_\alpha\) satisfying \[ -\Delta u=K u^{-\alpha}\text{ on }\Omega. \]