Applications of Natural Constraints in Critical Point Theory to Boundary Value Problems on Domains with Rotation Symmetry.
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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In this paper a nonlinear Dirichlet problem for the Laplace operator is considered on a disc in R sub 2. It is shown that if the nonlinearity, which may explicitly depend on the radial variable, is odd and superlinear at infinity, there exist infinitely many non-radial solutions. If the nonlinearity is odd and sublinear at infinity, and satisfies certain conditions at zero, a finite number of radial and non-radial solutions will be found. This number is given by the number of radial, respectively non-radial, eigenvalues that are crossed by the nonlinearity. In any case, as a consequence of the oddness of the nonlinearity, these solutions inherit the nodal line structure of the eigenfunctions corresponding to the eigenvalues that are crossed. The results are obtained by using natural constraints in a variational approach of the problem. Author
- Theoretical Mathematics