Abundance of entire solutions to nonlinear elliptic equations by the variational method

2018/11/07 by Lerman, L. M., Naryshkin, P. E., Nazarov, A. I.
#35J20 #35J91 #35J92 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1811.03143

Abstract

We study entire bounded solutions to the equation Δu - u + u³ = 0 in \mathbb R². Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in a unified way several types of solutions with various symmetries (radial, breather type, rectangular, triangular, hexagonal, etc.), both positive and sign-changing. The method is also applicable for more general equations in any dimension.

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