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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

 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 $\lambda\in R$.

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), 489-505. 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), 443-469.  Google Scholar [3] M. Cuesta, Eigenvalue problems for the p-laplacian with indefinite weights, Electron. J. Differential Equations, 33 (2001), 1-9.  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), 967-979.  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), 286-304. 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), 907-919.  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), 139-153. 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), 139-174. 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), 1413-1420. 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), 446-476.  Google Scholar [11] N. Ghoussoub, "Duality and Perturbation Methods in Critical Point Theory," Cambridge Univ. Press, Cambridge, 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), 5703-5743. 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), 1077-1092. 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), 743-753. 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) 201-224. 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, Ser A mathematics, 47 (2004), 741-755.  Google Scholar [17] A. Szulkin and M. Willem, Eigenvalue problems with indefinite weights, Stud. Math., 135 (1999), 189-201.  Google Scholar [18] M. Willem, "Minimax Theorems," Birkhauser, Boston, 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), 929-936.  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), 489-505. 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), 443-469.  Google Scholar [3] M. Cuesta, Eigenvalue problems for the p-laplacian with indefinite weights, Electron. J. Differential Equations, 33 (2001), 1-9.  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), 967-979.  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), 286-304. 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), 907-919.  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), 139-153. 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), 139-174. 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), 1413-1420. 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), 446-476.  Google Scholar [11] N. Ghoussoub, "Duality and Perturbation Methods in Critical Point Theory," Cambridge Univ. Press, Cambridge, 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), 5703-5743. 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), 1077-1092. 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), 743-753. 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) 201-224. 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, Ser A mathematics, 47 (2004), 741-755.  Google Scholar [17] A. Szulkin and M. Willem, Eigenvalue problems with indefinite weights, Stud. Math., 135 (1999), 189-201.  Google Scholar [18] M. Willem, "Minimax Theorems," Birkhauser, Boston, 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), 929-936.  Google Scholar
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