Solutions with various structures for semilinear equations in ${\mathbb{R}}^n$ driven by fractional Laplacian

Abstract

We study bounded solutions to the fractional equation $(-\Delta {)}^{s}u+u-|u{|}^{q-2}u=0$ in ${\mathbb{R}}^n$ for $n\ge 2$ and subcritical exponent $q>2$. Applying the variational approach based on concentration arguments and symmetry considerations which was introduced by Lerman, Naryshkin and Nazarov (2020) we construct several types of solutions with various structures (radial, rectangular, triangular, hexagonal, quasi-periodic, breather type, etc.).

28 pages, 21 figures

References in corpus (3)

• Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian
• Multiplicity of 1D-concentrating positive solutions to the Dirichlet problem for equation with $p$-Laplacian
• Steklov-type 1D inequalities (a survey)