On existence, non-existence and blow-up results for a singular semilinear Laplacian problem
type: Item
instance of: scholarly article
title: On existence, non-existence and blow-up results for a singular semilinear Laplacian problem
zbMATH Open document ID: 1380.35108
author: Masoud Bayrami
author: Mahmoud Hesaaraki
published in: European Journal of Mathematics
publication date: 2017-07-28T00:00:00Z
review text: 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 \(\Omega\).
reviewed by: John Villavert
Mathematics Subject Classification ID: 35J91
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35B44
zbMATH DE Number: 6754823
zbMATH Keywords: semilinear Laplacian problems
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: positive solutions
MaRDI profile type: Publication
full work available at URL: s40879-017-0129-5
OpenAlex ID: W2585791134
cites work: Some remarks on elliptic problems with critical growth in the gradient
cites work: Elliptic problems with a Hardy potential and critical growth in the gradient: non-resonance and blow-up results
cites work: Semilinear elliptic equations for beginners. Existence results via the variational approach
cites work: Nonlinear elliptic and parabolic equations involving measure data
cites work: On a semilinear elliptic equation with inverse-square potential
cites work: Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
cites work: Approximated solutions of equations with \(L^ 1\) data. Application to the \(H\)-convergence of quasilinear parabolic equations
cites work: Q5189317
cites work: Elliptic partial differential equations
cites work: A simple approach to Hardy inequalities
cites work: Q3661978
cites work: Semilinear biharmonic problems with a singular term
cites work: A Nonlinear Dirichlet Problem on the Unit Ball
cites work: The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential
cites work: Semilinear elliptic equations with degenerate and singular weights related to Caffarelli-Kohn-Nirenberg inequalities
cites work: Applied functional analysis. Applications to mathematical physics. Vol. 1
DOI: 10.1007/S40879-017-0129-5
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description: scientific article; zbMATH DE number 6754823