Multiple interior peak solutions for some singularly perturbed Neumann problems. (English) Zbl 1061.35502
Summary: We consider the problem \(\epsilon^2\Delta u-u+f(u)=0,\) \(u>0\), in \(\Omega, \partial u/\partial\nu=0\) on \(\partial\), where \(\Omega\) is a bounded smooth domain in \(\mathbb R^N, \epsilon>0\) is a small parameter, and \(f\) is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions that concentrate, as \(\epsilon\) approaches zero, at a critical point of the mean curvature function \(H(P),\;P\in\partial\Omega\). It is also proved that this equation has single interior spike solutions at a local maximum point of the distance function \(d(P,\partial\Omega), P\in\Omega\). In this paper, we prove the existence of interior \(K\)-peak \((K\geq2)\) solutions at the local maximum points of the following function: \(\phi(P_1,P_2,\cdots,P_K)=\min_{i,k,l=1, \cdots,K;k\not=l}(d(P_i,\partial\Omega),{1\over2}|P_k-P_l|)\). We first use the Lyapunov-Schmidt reduction method to reduce the problem to a finite-dimensional problem. Then we use a maximizing procedure to obtain multiple interior spikes. The function \(\phi(P_1,\cdots,P_K)\) appears naturally in the asymptotic expansion of the energy functional.
MSC:
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35B25 | Singular perturbations in context of PDEs |
| 47J30 | Variational methods involving nonlinear operators |
| 47N20 | Applications of operator theory to differential and integral equations |
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