Geodesic
New classes of solutions to semilinear equations in $\mathbb R^n$ with fractional Laplacian
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 49, Tome 508 (2021), pp. 124-133

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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, breather type, etc.), both positive and sign-changing. 
@article{ZNSL_2021_508_a4,
     author = {A. I. Nazarov and A. P. Shcheglova},
     title = {New classes of solutions to semilinear equations in $\mathbb R^n$ with fractional {Laplacian}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {124--133},
     publisher = {mathdoc},
     volume = {508},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2021_508_a4/}
}
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A. I. Nazarov; A. P. Shcheglova. New classes of solutions to semilinear equations in $\mathbb R^n$ with fractional Laplacian. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 49, Tome 508 (2021), pp. 124-133. http://geodesic.mathdoc.fr/item/ZNSL_2021_508_a4/