Multi-layer radial solutions for a supercritical Neumann problem. (English) Zbl 1341.35061
The paper under review deals with the Neumann problem
\[ \begin{cases} -\Delta u+u=u^p & \text{in}\;B_1,\\ u>0, & \\ \partial_\nu u=0 & \text{on}\;\partial B_1, \end{cases} \]
where \(B_1\) is the unit ball in \(\mathbb{R}^N\) with \(N\geq3\) and \(p>1\).
The authors prove existence of multiple layer solutions as \(p\to\infty.\) These are radial solutions whose Laplacians weakly converge to measures concentrating at interior spheres, with a simple reflection rule.
The machinery employed rely on a gluing technique, using a variant of the Nehari method, adapted to deal with Neumann problems instead of the standard Dirichlet ones.
\[ \begin{cases} -\Delta u+u=u^p & \text{in}\;B_1,\\ u>0, & \\ \partial_\nu u=0 & \text{on}\;\partial B_1, \end{cases} \]
where \(B_1\) is the unit ball in \(\mathbb{R}^N\) with \(N\geq3\) and \(p>1\).
The authors prove existence of multiple layer solutions as \(p\to\infty.\) These are radial solutions whose Laplacians weakly converge to measures concentrating at interior spheres, with a simple reflection rule.
The machinery employed rely on a gluing technique, using a variant of the Nehari method, adapted to deal with Neumann problems instead of the standard Dirichlet ones.
Reviewer: Dian K. Palagachev (Bari)
MSC:
| 35J61 | Semilinear elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
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