Multipeak solutions for a semilinear Neumann problem
The paper is concerned with the semilinear Neumann problem: \[ \varepsilon^2 \Delta u-u+f(u) =0,\;u>0, \text{ in } \Omega,\quad {\partial u\over \partial\nu} =0\text{ on } \partial \Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(\nu\) is the outer normal to \(\partial\Omega\) and \(\varepsilon\) is a positive constant. In addition to suitable conditions on \(f(t)\), typically satisfied by the function \(f(t)= t^p-at^q\) if \(a\geq 0\) and \(1<q<p <(N+2)/(N-2)\), the domain \(\Omega\) is assumed to satisfy the condition that there exist \(k\) disjoint patches \(\Lambda_1\), \(\Lambda_2, \dots, \Lambda_k\) on \(\partial \Omega\) such that \(\max_{P\in \Lambda_i} H(P)> \max_{P\in\partial \Lambda_i} H(P)\), where \(H(P)\) denotes the mean curvature of \(\partial \Omega\) at \(P\). Under these conditions, the author proves the existence of a classical solution \(u_\varepsilon\) which has exactly \(k\) local maxima, precisely one on each \(\Lambda_i (i=1,2, \dots,k)\), and then analyses the asymptotic behavior as \(\varepsilon \downarrow 0\).
scientific article; zbMATH DE number 943724