Geodesic
The multiplicity of positive solutions to the quasilinear equation generated by the Il'in--Caffarelli--Kohn--Nirenberg inequality
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 45, Tome 444 (2016), pp. 98-109

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We consider the Euler–Lagrange equation for the functional related to the V. P. Il'in inequality also known as the Caffarelli–Kohn–Nirenberg inequality. We prove that if the space dimension is even then, changing some parameters, we can obtain arbitrary many different positive solutions for this equation. 
@article{ZNSL_2016_444_a4,
     author = {A. I. Nazarov and B. O. Neterebskii},
     title = {The multiplicity of positive solutions to the quasilinear equation generated by the {Il'in--Caffarelli--Kohn--Nirenberg} inequality},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {98--109},
     publisher = {mathdoc},
     volume = {444},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_444_a4/}
}
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A. I. Nazarov; B. O. Neterebskii. The multiplicity of positive solutions to the quasilinear equation generated by the Il'in--Caffarelli--Kohn--Nirenberg inequality. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 45, Tome 444 (2016), pp. 98-109. http://geodesic.mathdoc.fr/item/ZNSL_2016_444_a4/