Differential and Integral Equations

On the number of nodal solutions in singular perturbation problems

Ezzat S. Noussair

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We establish the existence of nodal solutions for the problem \begin{eqnarray*} -{\varepsilon}^2 \Delta u + u = |u|^{p-2} u \quad \hbox{in}\, \Omega ,\ \ \ u \in H_0^1 (\Omega ), \end{eqnarray*} where $\Omega $ is a bounded domain, $2 <p <2N/(N-2)$ for $N\geq 3,$ and $\ 2 <p <\infty$ for $N=2$. It is shown that, corresponding to each pair of points $P_1, P_2 \in (\Omega \times \Omega )$, whose location depends on the geometry of $\Omega $, there is a nodal solution, which has, for small ${\varepsilon}$, exactly one positive and one negative peak, and that the peak points converge, as ${\varepsilon} \to 0$, to $(P_1, P_2)$.

Article information

Differential Integral Equations, Volume 14, Number 9 (2001), 1093-1109.

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B25: Singular perturbations
Secondary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations


Noussair, Ezzat S. On the number of nodal solutions in singular perturbation problems. Differential Integral Equations 14 (2001), no. 9, 1093--1109. https://projecteuclid.org/euclid.die/1356124309

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