Concentrating solutions for an exponential nonlinearity with Robin boundary condition. (English) Zbl 1329.35133
Summary: We study the following Robin boundary value problem
\[
\begin{cases} \Delta u+\epsilon^2u^{p-1}e^{u^p}=0\quad u>0,\quad & \text{in }\Omega\\ \frac{\partial u}{\partial n}+\lambda u=0\quad & \text{on }\partial\Omega,\end{cases}\tag{0.1}
\]
where \(\Omega\) is a bounded domain in \(\mathbb{R}^2\) with smooth boundary, as \(\epsilon \to 0\) and simultaneously \(\lambda \to + \infty\), \(0 < p < 2\). By using a Lyapunov-Schmidt reduction procedure, we extend for the whole range of exponents \(0 < p < 2\), the results of J. Dávila and E. Topp [J. Differ. Equations 252, No. 3, 2648–2697 (2012; Zbl 1236.35046)], who constructed bubbling solutions to problem for \(p = 1\).
MSC:
| 35J25 | Boundary value problems for second-order elliptic equations |
Citations:
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