Abstract
In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy-Sobolev-Maz’ya term:
where Ω is a bounded domain in ℝN (N ≥ 2), 0 ∈ Ω, x = (y, z) ∈ ℝk × ℝN-k and \(p_t = \frac{{N + 2 - 2t}} {{N - 2}}(0 \leqslant t \leqslant 2)\) For f(x) ∈ C 1(\(\bar \Omega \)){0}, we show that there exists a constant μ* > 0 such that the problem possesses at least two positive solutions if μ ∈ (0, μ*) and at least one positive solution if μ = μ*. Furthermore, there are no positive solutions if μ ∈ (μ*,+∞).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ambrosetti, A., Rabinowitz, P. H.: Dual variational methods in critical point throry and application. J. Funct. Anal., 14, 349–381 (1973)
Badiale, M., Tarantello, G.: A Sobolev–Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch. Ration. Mech. Anal., 163, 259–293 (2002)
Bahri, A., Coron, J.: On a nonlinear ellipitic equation involving the critical Sobolev exponent: The effect of the topology of the domain. Commun. Pure Appl. Math., 41, 253–294 (1988)
Bartsch, T., Peng, S., Zhang, Z.: Existence and non-existence of solutions to elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities. Calc. Var. Partial Differential Equations, 30, 113–136 (2007)
Batt, J., Faltenbacher, W., Horst, E.: Stationary spherically symmetric models in stellar dynamics. Arch. Ration. Mech. Anal., 93, 159–183 (1986)
Bhakta, M., Sandeep, K.: Hardy–Sobolev–Maz’ya type equations in bounded domains. J. Differential Equations., 247, 119–139 (2009)
Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical sobolev exponents. Commun. Pure Appl. Math., 36, 437–477 (1983)
Cao, D., Han, P.: Solutions for semilinear elliptic equations with critical exponents and Hardy potential. J. Differential Equations, 205, 521–537 (2004)
Cao, D., Peng, S.: A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms. J. Differential Equations., 193, 424–434 (2003)
Cao, D., Peng, S.: A global compactness result for singular elliptic problems involving critical Sobolev exponent. Proc. Amer. Math. Soc., 131, 1857–1866 (2003)
Capozzi, A., Fortunato, D., Palmieri, G.: An existence result for nonliear elliptic problems involving critical Sobolev exponents. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2, 463–470 (1985)
Cearmi, G., Solimini, S., Struwe, M.: Some existence results for superlinear elliptic boundary value problems involving critical exponents. J. Funct. Anal., 69, 289–306 (1986)
Chern, J., Lin, C.: Minimizer of Caffarelli–Kohn–Nirenberg inequlities with Singularity on the Boundary. Arch. Rational Mech. Anal., 197, 401–432 (2010)
Deng, Y.: Existence of multiple positive solutions for \( - \Delta u = \lambda u + u^{\frac{{N + 2}} {{N - 2}}} + uf(x)\). Acta Math. Sinica. (N.S.), 9, 311–320 (1993)
Deng, Y., Peng, S.: Existence of multiple positive solutions for inhomogeneous Neumann problem. J. Math. Anal. Appl., 271, 155–174 (2002)
Deng, Y., Yang, T.: Multiplicity of stationary solutions to the Euler–Poisson equations. J. Differential Equations., 231, 252–289 (2006)
Fabbri, I., Mancin, G., Sandeep, K.: Classification of solutions of a critical Hardy–Sobolev operator. J. Differential Equations, 224, 258–276 (2006)
Felli, V., Schneider, M.: A note on regularity of solutions to degenerate elliptic equations of Caffarelli–Kohn–Nirenberg type. Adv. Nolinear Stud., 3, 431–443 (2003)
Ghoussoub, N., Yuan, C.: Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans. Amer. Math. Soc., 352, 5703–5743 (2000)
Hsia, C. H., Lin, C., Hidemitsu. W: Revisiting an idea of Brézis and Nirenberg. J. Funct. Anal., 259, 1816–1849 (2010)
Jannelli, E.: The role played by space dimension in elliptic critical problems. J. Differential Equations, 156, 407–426 (1999)
Li, Y.: On the positive solutions of the Matukuma equation. Duke Math. J., 70, 575–589 (1993)
Li, Y., Ni, W. M.: On conformal scalar curvature equations in RN. Duke Math. J., 57, 895–924 (1988)
Li, Y., Ni, W. M.: On the existence and symmetry properties of finite total mass solutions of the Matukuma equation, the Eddington equation and their generalization. Arch. Ration. Mech. Anal., 108, 175–194 (1989)
Musina, R.: Ground state solutions of a critical problem involving cylindrical weights. Nonlinear Anal., 68, 3972–3986 (2008)
Struwe, M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z., 187, 511–517 (1984)
Tertikas, A., Tintarev, K.: On existence of minimizers for the Hardy–Sobolev–Maz’ya inequality. Ann. Mat. Pura Appl., 186, 645–662 (2007)
Wang, C., Wang, J.: Infinitely many solutions for Hardy–Sobolev–Maz’ya equation involving critical growth. Commun. Contemp. Math., 14(6), 1250044, 38pp (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by NSFC (Grant No. 11301204), the PhD specialized grant of the Ministry of Education of China (Grant No. 20110144110001), and the excellent doctorial dissertation cultivation grant from Central China Normal University (Grant No. 2013YBZD15)
Rights and permissions
About this article
Cite this article
Peng, S.J., Yang, J. Multiple positive solutions for a nonlinear elliptic equation involving Hardy-Sobolev-Maz’ya term. Acta. Math. Sin.-English Ser. 31, 893–912 (2015). https://doi.org/10.1007/s10114-015-4230-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-015-4230-8