Singular solutions of elliptic equations on a perturbed cone. (English) Zbl 1412.35141
Authors’ abstract: In this work we obtain positive singular solutions of \[ \begin{cases} -\Delta u=u^p & \text{ in }\Omega_t,\\ u=0 & \text{ on }\partial\Omega_t, \end{cases} \] where \(\Omega_t\) is a sufficiently small \(C^{2,\alpha}\) perturbation of the cone \(\Omega:=\{x\in\mathbb{R}^N:\ x=r\theta,\ r>0,\ \theta\in S\}\) where \(S\in \mathbb{S}^{N-1}\) has a smooth nonempty boundary and where \(p>1\) satisfies suitable conditions. By singular solution we mean the solution is singular at the ‘vertex of the perturbed cone’. We also consider some other perturbations of the equation on the unperturbed cone \(\Omega\) and here we use a different class of function spaces.
Reviewer: Dian K. Palagachev (Bari)
MSC:
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35B09 | Positive solutions to PDEs |
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