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

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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

Abstract
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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.
[2]	H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems,, Revista Mat. Univ. Complutense. Madrid, 10 (1997), 443.
[3]	M. Cuesta, Eigenvalue problems for the p-laplacian with indefinite weights,, Electron. J. Differential Equations, 33 (2001), 1.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	J. P. Garcia and I. A. Peral, Hardy inequalities and some critical elliptic and parabolic problems,, J. Differential Equations, 144 (1998), 446.
[11]	N. Ghoussoub, "Duality and Perturbation Methods in Critical Point Theory,", Cambridge Univ. Press, (1993). doi: doi:10.1017/CBO9780511551703.
[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.
[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.
[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.
[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.
[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.
[17]	A. Szulkin and M. Willem, Eigenvalue problems with indefinite weights,, Stud. Math., 135 (1999), 189.
[18]	M. Willem, "Minimax Theorems,", Birkhauser, (1996).
[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.

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.
[2]	H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems,, Revista Mat. Univ. Complutense. Madrid, 10 (1997), 443.
[3]	M. Cuesta, Eigenvalue problems for the p-laplacian with indefinite weights,, Electron. J. Differential Equations, 33 (2001), 1.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	J. P. Garcia and I. A. Peral, Hardy inequalities and some critical elliptic and parabolic problems,, J. Differential Equations, 144 (1998), 446.
[11]	N. Ghoussoub, "Duality and Perturbation Methods in Critical Point Theory,", Cambridge Univ. Press, (1993). doi: doi:10.1017/CBO9780511551703.
[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.
[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.
[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.
[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.
[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.
[17]	A. Szulkin and M. Willem, Eigenvalue problems with indefinite weights,, Stud. Math., 135 (1999), 189.
[18]	M. Willem, "Minimax Theorems,", Birkhauser, (1996).
[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.

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