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

TL;DR

This paper constructs various structured solutions to a fractional semilinear equation in multi-dimensional space using variational and symmetry methods, expanding understanding of solution types for fractional Laplacian equations.

Contribution

It introduces a variational approach combined with symmetry considerations to generate multiple solution structures for fractional semilinear equations.

Findings

Existence of radial, rectangular, triangular, hexagonal, quasi-periodic, and breather solutions.

Application of concentration arguments and symmetry in the variational framework.

Extension of solution types for fractional Laplacian equations in n".

Abstract

We study bounded solutions to the fractional equation $(-\Delta)^s u + u - |u|^{q-2}u = 0$ in $\mathbb R^n$ for $n\ge2$ 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.).