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On the number of interior peak solutions for a singularly perturbed Neumann problem. (English) Zbl 1170.35424

Summary: We consider the following singularly perturbed Neumann problem:
\[ \varepsilon^2\Delta u-u+f(u)=0 \quad\text{in }\Omega, \qquad u>0\quad\text{in }\Omega, \qquad \frac{\partial u}{\partial\nu}=0\quad\text{on }\partial\Omega, \]
where \(\Delta= \sum_{i=1}^N \partial^2/\partial x_i^2\) is the Laplace operator, \(\varepsilon>0\) is a constant, \(\Omega\) is a bounded, smooth domain in \(\mathbb R^N\) with its unit outward normal \(\nu\), and \(f\) is superlinear and subcritical. A typical \(f\) is \(f(u)=u^p\) where \(1<p<+\infty\), when \(N=2\) and \(1<p<(N+2)/(N-2)\), when \(N\geq 3\).
We show that there exists an \(\varepsilon_0>0\) such that for \(0<\varepsilon< \varepsilon_0\) and for each integer \(K\) bounded by
\[ 1\leq K\leq \frac{\alpha_{N,\Omega,f}}{\varepsilon^N(|\ln \varepsilon|)^N}, \]
where \(\alpha_{N,\Omega,f}\) is a constant depending on \(N\), \(\Omega\), and \(f\) only, there exists a solution with \(K\) interior peaks. (An explicit formula for \(\alpha_{N,\Omega,f}\) is also given.) As a consequence, we obtain that for \(\varepsilon\) sufficiently small, there exists at least \([\alpha_{N,\Omega f}/\varepsilon^N(|\ln \varepsilon|)^N]\) number of solutions. Moreover, for each \(m\in (0,N)\) there exist solutions with energies in the order of \(\varepsilon^{N-m}\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B25 Singular perturbations in context of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
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