On existence, non-existence and blow-up results for a singular semilinear Laplacian problem
This article addresses the solvability of Hénon problems with perturbed Hardy and source terms of the form \[ -\Delta u = \lambda\frac{u}{|x|^2} + |x|^{\alpha} u^p + f(x) \text{ in }\Omega \tag{1} \] with Dirichlet data, \[ u = 0 \text{ on } \partial \Omega. \] Here \(N \geq 3\), \(\alpha\), \(\lambda > 0\), \(p > 1\), \(\Omega \subset \mathbb{R}^N\) is a bounded and open smooth domain containing the origin, and \(f\) is a non-negative measurable function. The authors identify the critical threshold \(p = p_{+}(\lambda,\alpha)\) which guarantees the existence of a positive solution to the Dirichlet problem (1). More precisely, they show that if \(\lambda \leq \Lambda_N := (N-2)^2/4\), \(p < p_{+}(\lambda,\alpha)\) and small \(f \lesssim |x|^{-2}\), then (1) admits a positive distribution solution in \(\Omega\). The existence of this solution is obtained as the limit of approximate solutions coming from closely related Dirichlet problems on ball domains. Conversely, the authors show that their existence result is in some sense optimal by establishing Liouville-type non-existence results (for super-solutions) to (1). Namely, they show that, if \(\lambda > \Lambda_N\) and \(p > 1\), or if \(\lambda \leq \Lambda_N\) and \(p \geq p_{+}(\lambda,\alpha)\), then problem (1) does not have any distributional super-solution in \(\Omega\). In fact, they show that the ``strong'' blow-up phenomenon occurs in the cases in that their associated approximate (super-)solutions blow up in \(\Omega\).
scientific article; zbMATH DE number 6754823
scientific article