On the nonlinear Neumann problem with indefinite weight and Sobolev critical nonlinearity

• Autorzy: Chabrowski J.
• Języki publikacji: EN

Abstrakty

EN

We prove the existence of positive solutions of the Neumann problem with indefinite weight and critical Sobolev nonlinearity. Our approach is based on the concentration-compactness principle applied to a related variational problem.

Słowa kluczowe

EN

critical Sobolev exponent; Neumann problem

PL

krytyczna eksponenta Sobolewa; zagadnienie Neumanna

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2002
• Tom: Vol. 50, no 3
• Strony: 323--333
• Opis fizyczny: Bibliogr. 25 poz.

Twórcy

autor

Chabrowski J.

• The University of Queensland, Department of Mathematics, St. Lucia 4072, Qld, Australia

Bibliografia

• [1] Adimurthi, G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity, A tribute in honor of G. Prodi, Scuola Norm. Sup. Pisa (1991) 9-25.
• [2] Adimurthi, G. Mancini, Effect of geometry and topology of the boundary in critical Neumann problem, J. Reine Angew. Math., 456 (1994) 1-18.
• [3] Adimurthi, G. Mancini, S. L. Yadava, The role of the mean curvature in a semilinear Neumann problem involving critical exponent, Comm. Partial Differential Equations, 20 (3, 4) (1995) 591-631.
• [4] Adimurthi, F. Pacella, S. L. Yadava, Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal., 113 (1993) 318-350.
• [5] Adimurthi, F. Pacella, S. L. Yadava, Characterization of concentration points and L∞-estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent, Diff. Int. Eq., 8 (1995) 31-68.
• [6] Adimurthi, S. L. Yadava, Critical Sobolev exponent problem in Rn (N ≥ 4) with Neumann boundary condition, Proc. Indian Acad. Sci. Math. Sci., 100 (1990) 275-284.
• [7] C. Bandle, M. A. Pozio, A. Tesei, Existence and uniqueness of solutions of nonlinear problems, Math. Z., 199 (1988) 257-278.
• [8] H. Berestycki, I. Capuzzo-Dolcetta, L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl., 2 (1995) 553-572.
• [9] H. Brézis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983) 437-477.
• [10] J. Chabrowski, M. Willem, Least energy solutions of a critical Neumann problem with weight, Calc. Var. Partial Differential Equations, to be published.
• [11] J. F. Escobar, Positive solutions for some nonlinear elliptic equations with critical Sobolev exponents, Commun. Pure Appl. Math., 40 (1987) 623-657.
• [12] M. Grossi, F. Pacella, Positive solutions of nonlinear elliptic equations with critical Sobolev exponent and mixed boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990) 23-43.
• [13] C. Gui, N. Ghoussoub, Multi-peak solutions for semilinear Neumann problem involving the critical Sobolev exponent, Math. Z., (229) (1998) 443-474.
• [14] P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, Rev. Mat. Iberoamericana, 1 (1, 2) (1985) 145-201 and 45-120.
• [15] P. L. Lions, F. Pacella, M. Tricarico, Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions, Indiana Univ. Math. J., 37 (2) (1988) 301-324.
• [16] W. M. Ni, X. B. Pan, L. Takagi, Singular behavior of least energy solutions of a semilinear Neumann problem involving critical Sobolev exponent, Duke Math. J., 67 (1992) 1-20.
• [17] W. M. Ni, L. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991) 819-851.
• [18] H. T. Tehrani, On indefinite superlinear elliptic equations, Calc. Var. Partial Differential Equations, 4 (1996) 139-153.
• [19] X. J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents J. Differential Equations, 93 (1991) 283-310.
• [20] Z. Q. Wang, On the shape of solutions for a nonlinear Neumann problem in symmetric domains, Lect. in Appl. Math., 29 (1993) 433-442.
• [21] Z. Q. Wang, Remarks on a nonlinear Neumann problem with critical exponent, Houston J. Math., 20 (4) (1994) 671-694.
• [22] Z. Q. Wang, High-energy and multipeaked solutions for a nonlinear Neumann problem with critical exponents, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995) 1013-1029.
• [23] Z. Q. Wang, The effect of the domain geometry on number of positive solutions of Neumann problems with critical exponents, Differential Integral Equations, 8 (6) (1995) 1533-1554.
• [24] Z. Q. Wang, Construction of multi-peaked solutions for a nonlinear Neumann problem with critical exponent in symmetric domains, Nonlinear Anal., 27 (11) (1996) 1281-1306.
• [25] Z. Q. Wang, Existence and nonexistence of G-least energy solutions for a nonlinear Neumann problem with critical exponent in symmetric domains, Calc. Var. Partial Differential Equations, 8 (1999) 109-122.