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March  2011, 10(2): 571-581. doi: 10.3934/cpaa.2011.10.571

Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights

Guoqing Zhang 1, , Jia-yu Shao 2, and  Sanyang Liu 3,

1. 

College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China
2. 

Department of Mathematics, Tongji University, Shanghai, 200092, China
3. 

Department of Applied Mathematics, Xidian University, Xi'an, 710071, China

Received  April 2010 Revised  July 2010 Published  December 2010

In this paper, we investigate a class of N-Laplace elliptic equations with Hardy-Sobolev operator and indefinite weights

$ -\Delta_N u-\mu \frac{1}{(|x|\log(\frac{R}{|x|}))^N}|u|^{N-2}u= \lambda V(x)|u|^{N-2} u + f(x,u), u\in W_0^{1, N}(\Omega), $

where $\Omega$ be a bounded domain containing $0$ in $R^N$, $N \geq 2, 0 < \mu < (\frac{N-1}{N})^N$, and the weight function $V(x)$ may change sign and has nontrivial positive part. Using Moser-Trudinger inequality and nonstandard linking structure introduced by Degiovanni and Lancelotti [6], we prove the existence of a nontrivial solution for any $\lambda\in R$.

Keywords: Hardy-Sobolev operator, linking structure, N-Laplace elliptic equations, indefinite weights..
Mathematics Subject Classification: Primary: 35J20, 35J70; Secondary: 35J6.
Citation: Guoqing Zhang, Jia-yu Shao, Sanyang Liu. Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights. Communications on Pure & Applied Analysis, 2011, 10 (2) : 571-581. doi: 10.3934/cpaa.2011.10.571
References:
[1]

Adimurthi, M. Ramaswamy and N. Chaudhuri, Improved Hardy-Sobolev inequality and its application,, Proceeding of the American Mathematical Society, 130 (2002), 489. doi: doi:10.1090/S0002-9939-01-06132-9. Google Scholar

[2]

H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems,, Revista Mat. Univ. Complutense. Madrid, 10 (1997), 443. Google Scholar

[3]

M. Cuesta, Eigenvalue problems for the p-laplacian with indefinite weights,, Electron. J. Differential Equations, 33 (2001), 1. Google Scholar

[4]

J. M. do O, Semilinear Dirichlet problems for the N-Laplacian in $R^N$ with nonlinearities in critical growth range,, Differential Integral Equations, 9 (1996), 967. Google Scholar

[5]

J. M. do O, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimensional two,, J. Math. Anal. Appl., 345 (2008), 286. doi: doi:10.1016/j.jmaa.2008.03.074. Google Scholar

[6]

M. Degiovanni and S. Lancelotti, Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity,, Ann. I. H. Poincare-AN, 24 (2007), 907. Google Scholar

[7]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $R^2$ with nonlinearities in the critical growth range,, Calc. Var. Partial Differential Equations, 3 (1995), 139. doi: doi:10.1007/BF01205003. Google Scholar

[8]

E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems,, Invent. Math., 45 (1978), 139. doi: doi:10.1007/BF01390270. Google Scholar

[9]

X. Fan and Z. Li, Linking and existence results for perturbations of the p-Laplacian,, Nonlinear Anal, 42 (2000), 1413. doi: doi:10.1016/S0362-546X(99)00161-3. Google Scholar

[10]

J. P. Garcia and I. A. Peral, Hardy inequalities and some critical elliptic and parabolic problems,, J. Differential Equations, 144 (1998), 446. Google Scholar

[11]

N. Ghoussoub, "Duality and Perturbation Methods in Critical Point Theory,", Cambridge Univ. Press, (1993). doi: doi:10.1017/CBO9780511551703. Google Scholar

[12]

N. Ghoussoub and C. Yuan, Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponents,, Trans. Amer. Math. Soc., 352 (2000), 5703. doi: doi:10.1090/S0002-9947-00-02560-5. Google Scholar

[13]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (1970), 1077. doi: doi:10.1512/iumj.1971.20.20101. Google Scholar

[14]

K. Perera and A. Szulkin, p-Laplacian problems where the nonlinearity crosses an eigenvalue,, Discrete Contin. Dyn. Syst., 13 (2005), 743. doi: doi:10.3934/dcds.2005.13.743. Google Scholar

[15]

I. Peral and J. L. Vazquez, On the stability or instability of singular solutions with exponential reaction term,, Arch. Rational Mech. Anal., 129 (1995), 201. doi: doi:10.1007/BF00383673. Google Scholar

[16]

Y. T. Shen, Y. X. Yao and Z. H. Chen, On a nonlinear elliptic problem with critical potential in R,, Science in China, 47 (2004), 741. Google Scholar

[17]

A. Szulkin and M. Willem, Eigenvalue problems with indefinite weights,, Stud. Math., 135 (1999), 189. Google Scholar

[18]

M. Willem, "Minimax Theorems,", Birkhauser, (1996). Google Scholar

[19]

G. Zhang and S. Liu, On a class of elliptic equation with critical potential and indefinite weights in $R^2$,, Acta Mathematica Scientia, 28 (2008), 929. Google Scholar

show all references

References:
[1]

Adimurthi, M. Ramaswamy and N. Chaudhuri, Improved Hardy-Sobolev inequality and its application,, Proceeding of the American Mathematical Society, 130 (2002), 489. doi: doi:10.1090/S0002-9939-01-06132-9. Google Scholar

[2]

H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems,, Revista Mat. Univ. Complutense. Madrid, 10 (1997), 443. Google Scholar

[3]

M. Cuesta, Eigenvalue problems for the p-laplacian with indefinite weights,, Electron. J. Differential Equations, 33 (2001), 1. Google Scholar

[4]

J. M. do O, Semilinear Dirichlet problems for the N-Laplacian in $R^N$ with nonlinearities in critical growth range,, Differential Integral Equations, 9 (1996), 967. Google Scholar

[5]

J. M. do O, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimensional two,, J. Math. Anal. Appl., 345 (2008), 286. doi: doi:10.1016/j.jmaa.2008.03.074. Google Scholar

[6]

M. Degiovanni and S. Lancelotti, Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity,, Ann. I. H. Poincare-AN, 24 (2007), 907. Google Scholar

[7]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $R^2$ with nonlinearities in the critical growth range,, Calc. Var. Partial Differential Equations, 3 (1995), 139. doi: doi:10.1007/BF01205003. Google Scholar

[8]

E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems,, Invent. Math., 45 (1978), 139. doi: doi:10.1007/BF01390270. Google Scholar

[9]

X. Fan and Z. Li, Linking and existence results for perturbations of the p-Laplacian,, Nonlinear Anal, 42 (2000), 1413. doi: doi:10.1016/S0362-546X(99)00161-3. Google Scholar

[10]

J. P. Garcia and I. A. Peral, Hardy inequalities and some critical elliptic and parabolic problems,, J. Differential Equations, 144 (1998), 446. Google Scholar

[11]

N. Ghoussoub, "Duality and Perturbation Methods in Critical Point Theory,", Cambridge Univ. Press, (1993). doi: doi:10.1017/CBO9780511551703. Google Scholar

[12]

N. Ghoussoub and C. Yuan, Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponents,, Trans. Amer. Math. Soc., 352 (2000), 5703. doi: doi:10.1090/S0002-9947-00-02560-5. Google Scholar

[13]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (1970), 1077. doi: doi:10.1512/iumj.1971.20.20101. Google Scholar

[14]

K. Perera and A. Szulkin, p-Laplacian problems where the nonlinearity crosses an eigenvalue,, Discrete Contin. Dyn. Syst., 13 (2005), 743. doi: doi:10.3934/dcds.2005.13.743. Google Scholar

[15]

I. Peral and J. L. Vazquez, On the stability or instability of singular solutions with exponential reaction term,, Arch. Rational Mech. Anal., 129 (1995), 201. doi: doi:10.1007/BF00383673. Google Scholar

[16]

Y. T. Shen, Y. X. Yao and Z. H. Chen, On a nonlinear elliptic problem with critical potential in R,, Science in China, 47 (2004), 741. Google Scholar

[17]

A. Szulkin and M. Willem, Eigenvalue problems with indefinite weights,, Stud. Math., 135 (1999), 189. Google Scholar

[18]

M. Willem, "Minimax Theorems,", Birkhauser, (1996). Google Scholar

[19]

G. Zhang and S. Liu, On a class of elliptic equation with critical potential and indefinite weights in $R^2$,, Acta Mathematica Scientia, 28 (2008), 929. Google Scholar

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