Elliptic equations with cylindrical potential and multiple critical exponents

Xiaomei Sun; Yimin Zhang

doi:10.3934/cpaa.2013.12.1943

Article Contents

2013, Volume 12, Issue 5: 1943-1957. Doi: 10.3934/cpaa.2013.12.1943

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

Received: October 31, 2011

Revised: July 31, 2012

Published: August 2013

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Abstract

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 $\lambda<0$.

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Mathematics Subject Classification: Primary: 35J15, 35J61; Secondary: 35J20.

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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.