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Solution set splitting at low energy levels in Schrödinger equations with periodic and symmetric potential. (English) Zbl 1156.35024

Summary: The time-independent superlinear Schrödinger equation with spatially periodic and positive potential admits sign-changing two-bump solutions if the set of positive solutions at the minimal nontrivial energy level is the disjoint union of period translates of a compact set. Assuming a reflection symmetric potential we give a condition on the equation that ensures this splitting property for the solution set. Moreover, we provide a recipe to verify explicitly the condition, and we carry out the calculation in dimension one for a specific class of potentials.

MSC:

35J60 Nonlinear elliptic equations
35J10 Schrödinger operator, Schrödinger equation
35J20 Variational methods for second-order elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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