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 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

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},
     year = {2021},
     volume = {508},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2021_508_a4/}
}
TY  - JOUR
AU  - A. I. Nazarov
AU  - A. P. Shcheglova
TI  - New classes of solutions to semilinear equations in $\mathbb R^n$ with fractional Laplacian
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2021
SP  - 124
EP  - 133
VL  - 508
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2021_508_a4/
LA  - ru
ID  - ZNSL_2021_508_a4
ER  - 
%0 Journal Article
%A A. I. Nazarov
%A A. P. Shcheglova
%T New classes of solutions to semilinear equations in $\mathbb R^n$ with fractional Laplacian
%J Zapiski Nauchnykh Seminarov POMI
%D 2021
%P 124-133
%V 508
%U http://geodesic.mathdoc.fr/item/ZNSL_2021_508_a4/
%G ru
%F ZNSL_2021_508_a4
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/

[1] M. Sh. Birman, M. Z. Solomyak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Lan, SPb., 2010

[2] J. F. Bonder, N. Saintier, A. Silva, “The concentration-compactness principle for fractional order Sobolev spaces in unbounded domains and applications to the generalized fractional Brezis-Nirenberg problem”, Nonlin. Diff. Eq. Appl., 25 (2018), 52 | DOI | Zbl

[3] S. Dipierro, G. Palatucci, E. Valdinoci, “Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian”, Matematiche, 68 (2013), 201–216 | Zbl

[4] L. M. Lerman, P. E. Naryshkin, A. I. Nazarov, “Abundance of entire solutions to nonlinear elliptic equations by the variational method”, Nonlin. Analysis, 190 (2020), 111590 | DOI | Zbl

[5] S. Mosconi, K.Perera, M. Squassina, Y. Yang, “The Brezis-Nirenberg problem for the fractional $p$-Laplacian”, Calc. Var. Part. Differ. Eqs., 55:4 (2016), 105 | DOI | Zbl

[6] R. Musina, A. I. Nazarov, “On the Sobolev and Hardy constants for the fractional Navier Laplacian”, Nonlin. Analysis, 121 (2015), 123–129 | DOI | Zbl

[7] V. G. Osmolovskii, Nelineinaya zadacha Shturma-Liuvillya, Izd-vo SPb. un-ta, SPb., 2003, 230 pp.

[8] G. Palatucci, A. Pisante, “Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces”, Calc. Var. Part. Diff. Eqs., 50:3–4 (2014), 799–829 | DOI | Zbl

[9] P. R. Stinga, J. L. Torrea, “Extension problem and Harnack's inequality for some fractional operators”, Comm. Part. Diff. Eqs., 35:11 (2010), 2092–2122 | DOI | Zbl

[10] Kh. Tribel, Teoriya interpolyatsii, funktsionalnye prostranstva, differentsialnye operatory, Mir, M., 1980

[11] N. S. Ustinov, “O postoyanstve ekstremali v teoreme vlozheniya drobnogo poryadka”, Funkts. analiz i ego prilozh., 54:4 (2020), 85–97 | MR | Zbl

[12] N. S. Ustinov, “O razreshimosti polulineinoi zadachi so spektralnym drobnym laplasianom Neimana i kriticheskoi pravoi chastyu”, Algebra i analiz, 33:1 (2021), 194–212 | MR