Conditions for two-peaked solutions of singularly perturbed elliptic equations
Let \(\Omega\) be a bounded smooth domain in \(\mathbb{R}^n\). This paper deals with spike-layer solutions \(u_\varepsilon\) of the perturbed problem \[ \varepsilon^2\Delta u-u+u^p=0, \quad u>0 \quad \text{in } \Omega, \quad u=0 \quad \text{on } \partial \Omega, \tag{1} \] with \(1<p< (n+2)/(n-2)\) if \(n\geq 3\), \(1<p< \infty\) if \(n=2\). In a previous paper [Commun. Pure Appl. Math. 48, 731-768 (1995; Zbl 0838.35009)], the author, in collaboration with \textit{W. M. Ni}, studied single-peaked solutions of (1). Here he investigates two-peaked solutions of (1). He obtains necessary geometric conditions for the existence and the locations of the two peaks. In particular, he shows that, if \(\Omega\) is strictly convex, then (1) does not admit two-peaked solutions. The proofs are based on a sharp analysis of the asymptotic behavior of \(u_\varepsilon\) around the peaks, up to the second order.
scientific article; zbMATH DE number 1160803