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Multiple radial solutions for a semilinear Dirichlet problem in a ball. (English) Zbl 0799.35024

The authors prove that the semilinear elliptic boundary value problem \(\Delta u+ f(u)=0\) in \(\Omega\), \(u=0\) on \(\partial\Omega\) has at least \(4j-1\) radially symmetric solutions if the nonlinearity has a positive zero and satisfies \(f(0)=0\), \(f'(0)= f'(\infty) >\lambda_ j\), where \(\lambda_ 1< \lambda_ 2<\dots\) are the eigenvalues of \(-\Delta\) in the space of radial functions of \(H_ 0^ 1(\Omega)\). Extensive use is made of the global bifurcation theorem, bifurcation from infinity and bifurcation from simple eigenvalues.

MSC:

35B32 Bifurcations in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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