On the number of nodal solutions in singular perturbation problems

Abstract

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 $(P_1, P_2)$.