New Classes of Solutions to Semilinear Equations in \({\mathbb{R}}^{n}\) with Fractional Laplacian

Bounded solutions to the fractional equation (–Δ)^su + u – |u|^q−2u = 0 in \({\mathbb{R}}^{n}\) ${\mathbb{R}}^n$ for n ≥ 2 and subcritical exponent q > 2 are studied. Applying the variational approach based on concentration arguments and symmetry considerations which was introduced by Lerman, Naryshkin, and Nazarov (2020), several types of solutions with various structures (radial, rectangular, triangular, hexagonal, breather type, etc.), both positive and sign-changing are constructed.