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Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation

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  • Some critical Sobolev-Hardy inequalities with weight of distance function $d^{\frac{\alpha}{p}p^*}$ are established in a bounded domain $\Omega$, where $d$ is the distance to the boundary $\partial\Omega$. Using these inequalities we get the result that the embedding $\mathcal{D}^{1, 2}(\Omega, d^\alpha)\hookrightarrow L^q(\Omega, d^{\beta})$ is compact if $1\leq q<2^*$ and $\beta >\frac{\alpha}{2}q+\frac{q}{2^*}-1$. By the compactness result and critical-point theory about sign-changing solutions, we obtain infinitely many sign-changing solutions to a degenerate Dirichlet elliptic equation $-\hbox{div}(d^\alpha \nabla u)- \frac{(1-\alpha )^2}{4} d^{\alpha-2} u=f(x,u)$ provided that $f(x,u)$ satisfies suitable conditions.
    Mathematics Subject Classification: Primary: 35J57, 35J66; Secondary: 35J75.

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  • [1]

    G. H. Hardy, Note on a theorem of Hilbert, Mathematische Zeitschrift, 6 (1920), 314-317.

    [2]

    H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Revista Matemática de la Universidad Complutense de madrid, 10 (1997), 443-469.

    [3]

    F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Transactions of the American Mathematical Society, 356 (2004), 2149-2168.

    [4]

    Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality and its application, Proceedings of the American Mathematical Society, 130 (2002), 489-505.

    [5]

    Adimurthi and M. J. Esteban, An improved Hardy-Sobolev inequality in $W^{1,p}$ and its application to Schrödinger operators, Nonlinear Differential Equatons and Applications, 12 (2005), 243-263.

    [6]

    B. Abdellaoui, E. Colorado and I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities, Calculus of Variations and Partial Differential Equations, 23 (2005), 327-345.

    [7]

    Y. T. Shen, The Dirichlet problem for degenerate or singular elliptic equation of high order, Journal of China University of Science and Technology, 10 (1980), 1-11.

    [8]

    Y. T. Shen and X. K. Guo, Weighted Poincaré inequalities on unbounded domains and nonlinear elliptic boundary value problems, Acta Mathematica Scientia, 4 (1984), 277-286.

    [9]

    G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants, Trans. Amer. Math. Soc., 356 (2004), 2169-2196.

    [10]

    H. Brezis and M. Marcus, Hardy's inequalities revisited, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, Ser. IV 25 (1997), 217-237.

    [11]

    S. Filippas, V. G. Maz'ya and A. Tertikas, On a question of Brezis and marcus, Calc. of Variations and P.D.E., 25 (2006), 491-501.

    [12]

    S. Filippas, V. G. Maz'ya and A. Tertikas, Critical Hardy-Sobolev Inequalities, Journal de Mathématiques Pures et Appliquées, 87 (2007), 37-56.

    [13]

    J. Dávila and L. Dupaigne, Hardy-type inequalities, J. Eur. Math. Soc., 6 (2004), 335-365.

    [14]

    M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators, Annali Mat. Pura Appl., 80 (1968), 1-122.

    [15]

    A. Kristály and C. Varga, Multiple solutions for a degenerate elliptic equation involving sublinear terms at infinity, J. Math. Anal. Appl., 352 (2009), 139-148.

    [16]

    Y. M. Chen, Regularity of solutions to the Dirichlet problem for degenerate elliptic equation, Chin. Ann. Math., Ser. B, 24 (2003), 529-540.

    [17]

    Y. T. Shen and Y. X. Yao, Nonlinear elliptic equations with critical potential and critical parameter, Proceedings of the Royal Society of Edinburgh, Sect. A, 136 (2006), 1041-1051.

    [18]

    M. M. Zou, "Sign-Changing Critical Point Theory," Springer-Verlag, New York, 2008.

    [19]

    E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes in Mathematics, 5 (1999), A.M.S.

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