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On the second eigenvalue of combination between local and nonlocal \(p\)-Laplacian. (English) Zbl 1425.35131

The mountain pass characterization of the second eigenvalue of the perturbed \(p\)-Laplacian operator is studied. The variational characterization of the second eigenvalue and the sharp lower bounds on the first and second eigenvalues is an open question. The authors prove the variational characterization of the second eigenvalue of the operator associated to the considered problem. In particular, they prove the Faber-Krahn inequality and the nonlocal Hong-Krahn-Szegő inequality.

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
49Q10 Optimization of shapes other than minimal surfaces
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
35P15 Estimates of eigenvalues in context of PDEs
35R11 Fractional partial differential equations
35B40 Asymptotic behavior of solutions to PDEs
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