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Existence of solutions to a semilinear elliptic boundary value problem with augmented Morse index bigger than two. (English) Zbl 1370.35018

Summary: Building on the construction of least energy sign-changing solutions to variational semilinear elliptic boundary value problems introduced in [the first author et al., Rocky Mt. J. Math. 27, No. 4, 1041–1053 (1997; Zbl 0907.35050)], we prove the existence of a solution with augmented Morse index at least three when a sublevel of the corresponding action functional has nontrivial topology. We provide examples where the set of least energy sign changing solutions is disconnected, hence has nontrivial topology.

MSC:

35A16 Topological and monotonicity methods applied to PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J61 Semilinear elliptic equations

Citations:

Zbl 0907.35050
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