Elliptic equations with cylindrical potential and multiple critical exponents

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September 2013, 12(5): 1943-1957. doi: 10.3934/cpaa.2013.12.1943

Elliptic equations with cylindrical potential and multiple critical exponents

Xiaomei Sun ^1, and Yimin Zhang ^2,

1.	Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
2.	Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, PO Box 71010, Wuhan 430071E01103

Received November 2011 Revised August 2012 Published January 2013

Abstract
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In this paper, we deal with the following problem: \begin{eqnarray*} -\Delta u-\lambda |y|^{-2}u=|y|^{-s}u^{2^{*}(s)-1}+u^{2^{*}-1}\ \ \ in \ \ R^N , y\neq 0\\ u\geq 0 \end{eqnarray*} where $u(x)=u(y,z): R^m\times R^{N-m}\longrightarrow R$, $N\geq 4$, $2 < m < N$, $\lambda < (\frac{m-2}{2})^2$ and $0 < s < 2$, $2^*(s)=\frac{2(N-s)}{N-2}$, $2^*=\frac{2N}{N-2}$. Using the Variational method, we proved the existence of a ground state solution for the case $0 < \lambda < (\frac{m-2}{2})^2$ and the existence of a cylindrical weak solution under the case $\lambda<0$.

Keywords: Variational method, cylindrical weight, multiple critical exponents..

Mathematics Subject Classification: Primary: 35J15, 35J61; Secondary: 35J2.

Citation: Xiaomei Sun, Yimin Zhang. Elliptic equations with cylindrical potential and multiple critical exponents. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1943-1957. doi: 10.3934/cpaa.2013.12.1943

References:

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[2]	M. Badiale, V. Bergio and S. Rolando, A nonlinear elliptic equation with singular potential and applications to nonlinear field equations, J. Eur. Math. Soc., 9 (2007), 355-381.
[3]	M. Badiale, M. Guida and S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearities, Adv. Diff. Equ., 12 (2007), 1321-1362. doi: 10.1007/s00009-005-0055-5.
[4]	M. Badiale and S. Rolando, Nonlinear elliptic equations with subhomogeneous potentials, Nonlinear Analysis, 72 (2010), 602-617. doi: 10.1016/j.na.2009.06.111.
[5]	M. Badiale and G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal., 163 (2002), 252-293. doi: 10.1007/s002050200201.
[6]	M. Bhakta and K. Sandeep, Hardy-Sobolev-Maz'ya type equations in bounded domains, J. Differential Equations, 247 (2009), 119-139. doi: 10.1016/j.jde.2008.12.011.
[7]	H. Brezis and E. Lieb, A relation between pointwise convergence of functionals and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.
[8]	D. Castorina, I. Fabbri, G. Mancini and K. Sandeep, Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations, J. Differential Equations, 246 (2009), 1187-1206. doi: 10.1016/j.jde.2008.09.006.
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[12]	R. Filippucci, P. Pucci and F. Robert, On a p-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl., 91 (2009), 156-177. doi: 10.1016/j.matpur.2008.09.008.
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[15]	Y. Y. Li and C. S. Lin, A nonlinear Elliptic PDE with two Sobolev-Hardy critical exponents, Arch. Rational Mech. Anal., 203 (2012), 943-968. doi: 10.1007/s00205-011-0467-2.
[16]	G. Mancini, I. Fabbri and K. Sandeep, Classification of solutions of a critical Hardy-Sobolev operator, J. Differential Equations, 224 (2006), 258-276. doi: 10.1016/j.jde.2005.07.001.
[17]	G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\Bbb H^n$, Ann. Sc. Norm. Super. Pisa Cl. Sci., 5 (2008), 635-671.
[18]	R. Musina, Ground state solutions of a critical problem involving cylindrical weights, Nonlinear Anal., 68 (2008), 3972-3986. doi: 10.1016/j.na.2007.04.034.
[19]	R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322.
[20]	J. B. Su and Z. Q. Wang, Sobolev type embedding and quasilinear elliptic equations with radial potentials, J. Differential Equa., 250 (2011), 223-242. doi: 10.1016/j.jde.2010.08.025.
[21]	J. B. Su, Z. Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equa., 238 (2007), 201-219. doi: 10.1016/j.jde.2007.03.018.
[22]	G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.
[23]	S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 2 (1996), 241-264.
[24]	A. Tertikas and K. Tintarev, On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality, Ann. Mat. Pura e Appl., 186 (2007), 645-662. doi: 10.1007/s10231-006-0024-z.
[25]	S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space, Ann. Inst. Henry Poincaré-Analyse Nonlinéaire, 12 (1995), 319-337.
[26]	M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Third edition, Springer-Verlag, Berlin, 2000.

show all references

References:

[1]	J. Bellazzini and C. Bonanno, Nonlinear Schrödinger equations with strongly singular potentials, Proceedings of the Royal Society of Edinburgh, 140A (2010), 707-721. doi: 10.1017/S0308210509001401.
[2]	M. Badiale, V. Bergio and S. Rolando, A nonlinear elliptic equation with singular potential and applications to nonlinear field equations, J. Eur. Math. Soc., 9 (2007), 355-381.
[3]	M. Badiale, M. Guida and S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearities, Adv. Diff. Equ., 12 (2007), 1321-1362. doi: 10.1007/s00009-005-0055-5.
[4]	M. Badiale and S. Rolando, Nonlinear elliptic equations with subhomogeneous potentials, Nonlinear Analysis, 72 (2010), 602-617. doi: 10.1016/j.na.2009.06.111.
[5]	M. Badiale and G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal., 163 (2002), 252-293. doi: 10.1007/s002050200201.
[6]	M. Bhakta and K. Sandeep, Hardy-Sobolev-Maz'ya type equations in bounded domains, J. Differential Equations, 247 (2009), 119-139. doi: 10.1016/j.jde.2008.12.011.
[7]	H. Brezis and E. Lieb, A relation between pointwise convergence of functionals and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.
[8]	D. Castorina, I. Fabbri, G. Mancini and K. Sandeep, Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations, J. Differential Equations, 246 (2009), 1187-1206. doi: 10.1016/j.jde.2008.09.006.
[9]	D. M. Cao and P. G. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials, J. Differential Equations, 224 (2006), 332-372. doi: 10.1016/j.jde.2005.07.010.
[10]	D. M. Cao and Y. Y. Li, Results on positive solutions of elliptic equations with a critical Hardy-Sobolev operator, Methods and Applications of Analysis, 15 (2008), 081-096. doi: 10.1.1.140.417.
[11]	L. D'Ambrosio, Hardy type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci., 4 (2005), 451-486.
[12]	R. Filippucci, P. Pucci and F. Robert, On a p-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl., 91 (2009), 156-177. doi: 10.1016/j.matpur.2008.09.008.
[13]	F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc., 356 (2004), 2149-2168. doi: 10.1090/S0002-9947-03-03395-6.
[14]	N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire, 21 (2004), 767-793. doi: 10.1016/j.anihpc.2003.07.002.
[15]	Y. Y. Li and C. S. Lin, A nonlinear Elliptic PDE with two Sobolev-Hardy critical exponents, Arch. Rational Mech. Anal., 203 (2012), 943-968. doi: 10.1007/s00205-011-0467-2.
[16]	G. Mancini, I. Fabbri and K. Sandeep, Classification of solutions of a critical Hardy-Sobolev operator, J. Differential Equations, 224 (2006), 258-276. doi: 10.1016/j.jde.2005.07.001.
[17]	G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\Bbb H^n$, Ann. Sc. Norm. Super. Pisa Cl. Sci., 5 (2008), 635-671.
[18]	R. Musina, Ground state solutions of a critical problem involving cylindrical weights, Nonlinear Anal., 68 (2008), 3972-3986. doi: 10.1016/j.na.2007.04.034.
[19]	R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322.
[20]	J. B. Su and Z. Q. Wang, Sobolev type embedding and quasilinear elliptic equations with radial potentials, J. Differential Equa., 250 (2011), 223-242. doi: 10.1016/j.jde.2010.08.025.
[21]	J. B. Su, Z. Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equa., 238 (2007), 201-219. doi: 10.1016/j.jde.2007.03.018.
[22]	G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.
[23]	S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 2 (1996), 241-264.
[24]	A. Tertikas and K. Tintarev, On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality, Ann. Mat. Pura e Appl., 186 (2007), 645-662. doi: 10.1007/s10231-006-0024-z.
[25]	S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space, Ann. Inst. Henry Poincaré-Analyse Nonlinéaire, 12 (1995), 319-337.
[26]	M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Third edition, Springer-Verlag, Berlin, 2000.

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