Sign changing solutions for elliptic equations with critical growth in cylinder type domains
ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002) , pp. 407-419.

We prove the existence of positive and of nodal solutions for -Δu=|u| p-2 u+μ|u| q-2 u, uH 0 1 (Ω), where μ>0 and 2<q<p=2N(N-2), for a class of open subsets Ω of N lying between two infinite cylinders.

DOI : https://doi.org/10.1051/cocv:2002061
Classification : 35J20,  35J25,  35J65,  35B05
Mots clés : nodal solutions, cylindrical domains, semilinear elliptic equation, critical Sobolev exponent, concentration-compactness
@article{COCV_2002__7__407_0,
     author = {Gir\~ao, Pedro and Ramos, Miguel},
     title = {Sign changing solutions for elliptic equations with critical growth in cylinder type domains},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {407--419},
     publisher = {EDP-Sciences},
     volume = {7},
     year = {2002},
     doi = {10.1051/cocv:2002061},
     mrnumber = {1925035},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2002__7__407_0/}
}
Girão, Pedro; Ramos, Miguel. Sign changing solutions for elliptic equations with critical growth in cylinder type domains. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002) , pp. 407-419. doi : 10.1051/cocv:2002061. http://www.numdam.org/item/COCV_2002__7__407_0/

[1] A.K. Ben-Naoum, C. Troestler and M. Willem, Extrema problems with critical Sobolev exponents on unbounded domains. Nonlinear Anal. TMA 26 (1996) 823-833. | Zbl 0851.49004

[2] G. Bianchi, J. Chabrowski and A. Szulkin, On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent. Nonlinear Anal. TMA 25 (1995) 41-59. | MR 1331987 | Zbl 0823.35051

[3] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36 (1983) 437-476. | MR 709644 | Zbl 0541.35029

[4] G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents. J. Funct. Anal. 69 (1986) 289-306. | MR 867663 | Zbl 0614.35035

[5] M. Del Pino and P. Felmer, Least energy solutions for elliptic equations in unbounded domains. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996) 195-208. | MR 1378841 | Zbl 0849.35037

[6] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Second Edition. Springer, New York, Grundlehren Math. Wiss. 224 (1983). | MR 737190 | Zbl 0562.35001

[7] P.-L. Lions, The concentration-compactness principle in the Calculus of Variations. The limit case, Part 2. Rev. Mat. Iberoamericana 1 (1985) 45-121. | MR 850686 | Zbl 0704.49006

[8] M. Ramos, Z.-Q. Wang and M. Willem, Positive solutions for elliptic equations with critical growth in unbounded domains, in Calculus of Variations and Differential Equations, edited by A. Ioffe, S. Reich and I. Shafrir. Chapman & Hall/CRC, Boca Raton, FL, Res. Notes in Math. Ser. 140 (2000) 192-199. | MR 1713847 | Zbl 0968.35050

[9] I. Schindler and K. Tintarev, Abstract concentration compactness and elliptic equations on unbounded domains, in Prog. Nonlinear Differential Equations Appl., Vol. 43, edited by M.R. Grossinho, M. Ramos, C. Rebelo and L. Sanchez. Birkhäuser, Boston (2001) 369-380. | MR 1800637 | Zbl 1030.35080

[10] G. Tarantello, Nodal solutions of semilinear elliptic equations with critical exponent. Differential Integral Equations 5 (1992) 25-42. | MR 1141725 | Zbl 0758.35035

[11] M. Willem, Minimax theorems, in Prog. Nonlinear Differential Equations Appl., Vol. 24. Birkhäuser, Boston (1996). | MR 1400007 | Zbl 0856.49001

[12] X.-P. Zhu, Multiple entire solutions of a semilinear elliptic equations. Nonlinear Anal. TMA 12 (1998) 1297-1316. | MR 969507 | Zbl 0671.35023