Multi-peak solutions for some singular perturbation problems |
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Authors: | Manuel del Pino Patricio L Felmer Juncheng Wei |
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Institution: | (1) Departamento de Ingeniería Matemática F.C.F.M., Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile (e-mail: delpino@dim.uchile.cl / pfelmer@dim.uchile.cl) , CL;(2) Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong (e-mail: wei@math.cuhk.hk) , HK |
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Abstract: | We consider the problem
where is a smooth domain in , not necessarily bounded, is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses a solution that concentrates, as approaches zero, at a maximum of the function , the distance to the boundary. We obtain multi-peak solutions of the equation given above when the domain 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.
Received September 3, 1998 / Accepted February 29, 1999 |
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Keywords: | |
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