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On the number of interior peaks of solutions to a non-autonomous singularly perturbed Neumann problem. (English) Zbl 1162.35009

Let \(N\geq2\) and suppose that \(\Omega\subseteq\mathbb{R}^N\) is a bounded smooth domain, \(a\in C^\infty(\overline{\Omega})\), and \(1<p<2^*-1\), where \(2^*\) is the usual critical Sobolev exponent that corresponds to this problem. It is never stated but implicitly assumed that \(a\) is positive. For a small positive parameter \(\varepsilon\) consider the elliptic problem \[ \begin{cases} -\varepsilon^2\Delta u+u=a(x)u^p&\text{in }\Omega\\ u>0&\text{in }\Omega\\ \frac{\partial u}{\partial\nu}=0&\text{on }\partial\Omega. \end{cases}\tag{1} \] Here \(\partial/\partial\nu\) denotes the partial derivative in the normal direction on \(\partial\Omega\).
If \(Q_0\in\Omega\) is a strict local minimum of \(a\) then it is shown that there are \(\varepsilon_0>0\) and \(K_0(\varepsilon)>0\), \(K_0(\varepsilon)\to\infty\) as \(\varepsilon\to0\), such that for \(\varepsilon\in(0,\varepsilon_0]\) and \(K\leq K_0(\varepsilon)\) there is a multipeak solution of \((1)\) with exactly \(K\) local maximum points. These maxima tend to \(Q_0\) as \(\varepsilon\to0\).
In the result of [F.-H. Lin, W.-M. Ni and J.-C. Wei, Commun. Pure Appl. Math. 60, No. 2, 252–281 (2007; Zbl 1170.35424)], a similar statement is made for the case \(a\equiv1\), with asymptotics \(K_0(\varepsilon)\sim\varepsilon^{-N}|\log\varepsilon|^{-N}\). In contrast, in the paper under review the asymptotics are \(K_0(\varepsilon) \sim\varepsilon^{-M}|\log\varepsilon|^{-N}\) for some \(M\in(0,1)\) that depends on the order of the zero of \(\nabla a\) in \(Q_0\).

MSC:

35B25 Singular perturbations in context of PDEs
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations

Citations:

Zbl 1170.35424
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