## On a nonlinear Dirichlet problem with a singularity along the boundary.(English)Zbl 1021.35029

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 $-\Delta u=K u^{-\alpha}\text{ on }\Omega.$

### MSC:

 35J25 Boundary value problems for second-order elliptic equations 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs