## 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.

### MSC:

 35J60 Nonlinear elliptic equations
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### References:

 [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
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