Abstract.
We consider the problem
\[ \left \{ \begin{array}{rcl} \varepsilon^2 \Delta u - u + f(u) = 0 & \mbox{ in }& \Omega u > 0 \mbox{ in} \Omega, u = 0 & \mbox{ on }& \partial\Omega, \end{array} \right.\nonumber \]
where \(\Omega\) is a smooth domain in \(R^N\), not necessarily bounded, \(\varepsilon >0\) is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses a solution that concentrates, as \(\varepsilon\) approaches zero, at a maximum of the function \(d(x)=d(\cdot,\partial\Omega)\), the distance to the boundary. We obtain multi-peak solutions of the equation given above when the domain \(\Omega\) presents a distance function to its boundary d with multiple local maxima. We find solutions exhibiting concentration at any prescribed finite set of local maxima, possibly degenerate, of d. The proof relies on variational arguments, where a penalization-type method is used together with sharp estimates of the critical values of the appropriate functional. Our main theorem extends earlier results, including the single peak case. We allow a degenerate distance function and a more general nonlinearity.
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Received September 3, 1998 / Accepted February 29, 1999
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del Pino, M., Felmer, P. & Wei, J. Multi-peak solutions for some singular perturbation problems. Calc Var 10, 119–134 (2000). https://doi.org/10.1007/s005260050147
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DOI: https://doi.org/10.1007/s005260050147