The solvability of quasilinear Brezis-Nirenberg-type problems with singular weights. (English) Zbl 1130.35061

Summary: We consider the existence and non-existence of non-trivial solutions to quasilinear Brezis-Nirenberg-type problems with singular weights. First, we shall obtain a compact imbedding theorem which is an extension of the classical Rellich-Kondrachov compact imbedding theorem, and consider the corresponding eigenvalue problem. Secondly, we deduce a Pokhozhaev-type identity and obtain a non-existence result. Thirdly, thanks to the generalized concentration compactness principle, we will give some abstract conditions when the functional satisfies the (PS)\(_c\) condition. Finally, basing on the explicit form of the extremal function, we will obtain some existence results.


35J60 Nonlinear elliptic equations
Full Text: DOI arXiv


[1] Ambrosetti, A.; Rabinowitz, P.H., Dual variational methods in critical point theory and applications, J. funct. anal., 14, 349-381, (1973) · Zbl 0273.49063
[2] Brezis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. amer. math. soc., 88, 486-490, (1983) · Zbl 0526.46037
[3] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical exponents, Comm. pure appl. math., 36, 437-477, (1983) · Zbl 0541.35029
[4] J. Byeon, Z.Q. Wang, Symmetry breaking of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities, June 27, 2002, preprint. · Zbl 1010.35029
[5] Caffarrelli, L.; Kohn, R.; Nirenberg, L., First order interpolation inequalities with weights, Compositio Mathematica, 53, 259-275, (1984) · Zbl 0563.46024
[6] Catrina, F.; Wang, Z.Q., On the caffarelli – kohn – nirenberg inequalities: sharp constants, existence (and nonexistence) and symmetry of extremal functions, Comm. pure appl. math., LIV, 229-258, (2001) · Zbl 1072.35506
[7] Chou, K.S.; Chu, C.W., On the best constant for a weighted sobolev – hardy inequality, J. London math. soc., 2, 137-151, (1993) · Zbl 0739.26013
[8] Chou, K.-S.; Geng, D., On the critical dimension of a semilinear degenerate elliptic equation involving critical sobolev – hardy exponent, Nonlinear anal. theory methods appl., 26, 1965-1984, (1996) · Zbl 0855.35042
[9] Egnell, H., Semilinear elliptic equations involving critical Sobolev exponents, Arch. rational mech. anal., 104, 27-56, (1988) · Zbl 0674.35033
[10] Egnell, H., Existence and nonexistence results for m-Laplace equations involving critical Sobolev exponents, Arch. rational mech. anal., 104, 57-77, (1988) · Zbl 0675.35036
[11] Guedda, M.; Veron, L., Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear anal. theory methods appl., 13, 879-902, (1989) · Zbl 0714.35032
[12] Horiuchi, T., Best constant in weighted Sobolev inequality with weights being powers of distance from the origin, J. inequal. appl., 1, 275-292, (1997) · Zbl 0899.35034
[13] E. Jannelli, S. Solomini, Critical behaviour of some elliptic equations with singular potentials, Rapport No. 41/96, Dipartimento di Mathematica Universita degi Studi di Bari, 70125 Bari, Italia.
[14] Lions, P.L., The concentration-compactness principle in the calculus of variations, The locally compact case, ann. inst. H. poincare anal. nonlineaire, 1, part 1, 109-145, (1984), (part 2) (1984) 223-283 · Zbl 0541.49009
[15] Lions, P.L., The concentration-compactness principle in the calculus of variations, the limit case, Rev. mat. ibero americana, 1, part 1, 145-201, (1985), 2 (part 2) (1985) 45-121 · Zbl 0704.49005
[16] Nicolaescu, L., A weighted semilinear elliptic equation involving critical Sobolev exponents, Differential integral equations, 3, 653-671, (1991) · Zbl 0736.35049
[17] Pucci, P.; Serrin, J., A general variational identity, Indiana univ. math. J., 35, 681-703, (1986) · Zbl 0625.35027
[18] Pucci, P.; Serrin, J., Critical exponents and critical dimensions for polyharmonic operators, J. math. pures appl., 69, 55-83, (1990) · Zbl 0717.35032
[19] Struwe, M., Variational methods, applications to nonlinear partial differential equations and Hamiltonian systems, (1996), Springer Berlin · Zbl 0864.49001
[20] Tolksdorff, P., Regularity for a more general class of quasilinear elliptic equations, J. differential equations, 51, 126-150, (1984)
[21] Xuan, B.-J.; Chen, Z.-C., Existence, multiplicity and bifurcation for critical polyharmonic equations, System. sci. math. sci., 12, 59-69, (1999) · Zbl 0984.35067
[22] Zhu, X.-P., Nontrivial solution of quasilinear elliptic involving critical Sobolev exponent, Sci. sinica, ser. A, 31, 1166-1181, (1988) · Zbl 0677.35039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.