A nonlinear elliptic PDE with the inverse square potential.
This paper concerns positive solutions to the boundary value problem \[ \begin{aligned} -\Delta u-\frac{c} {| x|^2} & u=u^p+tf\text{ in }\Omega,\\ & u=0\text{ on }\partial\Omega.\end{aligned} \] Here \(\Omega\) in a bounded and open set in \(\mathbb{R}^N\) \((N\geq 3)\) containing the origin; \(p>1\), \(t>0\) and \(0<c\leq\frac{(N-2)^2}{4}\) are constants, and \(f\) is a smooth bounded nonnegative function. The author introduces the notions of weak solutions, \(H^1_0\)-solutions and strong solutions, which coincide for solutions \(u\) with \(0\leq u(x)\leq\text{const}\,| x|^{-\alpha}\), \(\alpha:=\frac {N-2-\sqrt{(N-2)^2-4c}} {4}\). The main results are as follows: If \(1<p<1+\alpha\), then there exists \(t_0>0\) such that for all \(0<t<t_0\) there exists a minimal strong solution (it is stable and the only stable \(H^1_0\)-solution), for \(t=t_0\) there exists a minimal weak solution, and for \(t>t_0\) there is even no weak solution. Moreover, results about so-called complete blow up are proved. In the proofs sub- and supersolution techniques and monotone iteration are used.
scientific article; zbMATH DE number 1802591