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Multiple boundary peak solutions for some singularly perturbed Neumann problems. (English) Zbl 0944.35020

Summary: We consider the problem \[ \begin{cases} \varepsilon^2\Delta u-u+f(u)=0\quad & \text{on }\Omega,\\ u>0\text{ in }\Omega,\;\partial u/\partial \nu=0 \quad & \text{on }\partial\Omega, \end{cases} \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\), \(\varepsilon>0\) is a small parameter and \(f\) is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions such that the spike concentrates, as \(\varepsilon\) approaches zero, at a critical point of the mean curvature function \(H(P)\), \(P \in \partial\Omega\). It is also known that this equation has multiple boundary spike solutions at multiple nondegenerate critical points of \(H(P)\) or multiple local maximum points of \(H(P)\). In this paper, we prove that for any fixed positive integer \(K\) there exist boundary \(K\)-peak solutions at a local minimum point of \(H(P)\). This implies that for any smooth and bounded domain there always exist boundary \(K\)-peak solutions. We first use the Lyapunov-Schmidt method to reduce the problem to finite dimensions. Then we use a maximizing procedure to obtain multiple boundary spikes.

MSC:

35J40 Boundary value problems for higher-order elliptic equations
35B25 Singular perturbations in context of PDEs
35B45 A priori estimates in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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