Multiple solutions for a semilinear elliptic equation
Authors:
Manuel A. del Pino and Patricio L. Felmer
Journal:
Trans. Amer. Math. Soc. 347 (1995), 48394853
MSC:
Primary 35J65; Secondary 58E05
MathSciNet review:
1303117
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Abstract: Let be a bounded, smooth domain in , . We consider the problem of finding nontrivial solutions to the elliptic boundary value problem where , is Hölder continuous on and , are constants. Let denote the interior of the set where vanishes in . We assume a.e. on and consider the eigenvalues and of the Dirichlet problem in and respectively. We prove that no nontrivial solution of the equation exists if satisfies, for some , On the other hand, if, for some nonnegative integers , with , satisfies then the equation above possesses at least pairs of nontrivial solutions. For the proof of these results we use a variational approach. In particular, the existence result takes advantage of the even character of the associated functional.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199513031175
PII:
S 00029947(1995)13031175
Keywords:
Variational method,
multiplicity,
uniqueness,
blow up of solutions
Article copyright:
© Copyright 1995 American Mathematical Society
