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

Previous Article
On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space
CPAA Home
This Issue
Next Article
The optimal weighted $W^{2, p}$ estimates of elliptic equation with non-compatible conditions

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
Related Papers

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 and 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.
[2]	H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Revista Mat. Univ. Complutense. Madrid, 10 (1997), 443-469.
[3]	M. Cuesta, Eigenvalue problems for the p-laplacian with indefinite weights, Electron. J. Differential Equations, 33 (2001), 1-9.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	J. P. Garcia and I. A. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 446-476.
[11]	N. Ghoussoub, "Duality and Perturbation Methods in Critical Point Theory," Cambridge Univ. Press, Cambridge, 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), 5703-5743. 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), 1077-1092. 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), 743-753. 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) 201-224. 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, Ser A mathematics, 47 (2004), 741-755.
[17]	A. Szulkin and M. Willem, Eigenvalue problems with indefinite weights, Stud. Math., 135 (1999), 189-201.
[18]	M. Willem, "Minimax Theorems," Birkhauser, Boston, 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), 929-936.

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.
[2]	H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Revista Mat. Univ. Complutense. Madrid, 10 (1997), 443-469.
[3]	M. Cuesta, Eigenvalue problems for the p-laplacian with indefinite weights, Electron. J. Differential Equations, 33 (2001), 1-9.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	J. P. Garcia and I. A. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 446-476.
[11]	N. Ghoussoub, "Duality and Perturbation Methods in Critical Point Theory," Cambridge Univ. Press, Cambridge, 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), 5703-5743. 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), 1077-1092. 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), 743-753. 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) 201-224. 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, Ser A mathematics, 47 (2004), 741-755.
[17]	A. Szulkin and M. Willem, Eigenvalue problems with indefinite weights, Stud. Math., 135 (1999), 189-201.
[18]	M. Willem, "Minimax Theorems," Birkhauser, Boston, 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), 929-936.

[1]	Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure and Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
[2]	Jinhui Chen, Haitao Yang. A result on Hardy-Sobolev critical elliptic equations with boundary singularities. Communications on Pure and Applied Analysis, 2007, 6 (1) : 191-201. doi: 10.3934/cpaa.2007.6.191
[3]	Yinbin Deng, Qi Gao, Dandan Zhang. Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$. Discrete and Continuous Dynamical Systems, 2007, 19 (1) : 211-233. doi: 10.3934/dcds.2007.19.211
[4]	J. Tyagi. Multiple solutions for singular N-Laplace equations with a sign changing nonlinearity. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2381-2391. doi: 10.3934/cpaa.2013.12.2381
[5]	Masato Hashizume, Chun-Hsiung Hsia, Gyeongha Hwang. On the Neumann problem of Hardy-Sobolev critical equations with the multiple singularities. Communications on Pure and Applied Analysis, 2019, 18 (1) : 301-322. doi: 10.3934/cpaa.2019016
[6]	M. Grossi, P. Magrone, M. Matzeu. Linking type solutions for elliptic equations with indefinite nonlinearities up to the critical growth. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 703-718. doi: 10.3934/dcds.2001.7.703
[7]	Jann-Long Chern, Yong-Li Tang, Chuan-Jen Chyan, Yi-Jung Chen. On the uniqueness of singular solutions for a Hardy-Sobolev equation. Conference Publications, 2013, 2013 (special) : 123-128. doi: 10.3934/proc.2013.2013.123
[8]	José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138
[9]	Huyuan Chen, Feng Zhou. Isolated singularities for elliptic equations with hardy operator and source nonlinearity. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2945-2964. doi: 10.3934/dcds.2018126
[10]	Patrizia Pucci, Raffaella Servadei. Nonexistence for $p$--Laplace equations with singular weights. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1421-1438. doi: 10.3934/cpaa.2010.9.1421
[11]	Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061
[12]	Edcarlos D. Silva, Jefferson S. Silva. Multiplicity of solutions for critical quasilinear Schrödinger equations using a linking structure. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5441-5470. doi: 10.3934/dcds.2020234
[13]	Melvin Faierman. Fredholm theory for an elliptic differential operator defined on $ \mathbb{R}^n $ and acting on generalized Sobolev spaces. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1463-1483. doi: 10.3934/cpaa.2020074
[14]	Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
[15]	Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164
[16]	Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1319-1345. doi: 10.3934/cpaa.2021022
[17]	Laura Gambera, Umberto Guarnotta. Strongly singular convective elliptic equations in $ \mathbb{R}^N $ driven by a non-homogeneous operator. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022088
[18]	Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Existence of radial solutions for the $p$-Laplacian elliptic equations with weights. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 447-479. doi: 10.3934/dcds.2006.15.447
[19]	Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Dead cores and bursts for p-Laplacian elliptic equations with weights. Conference Publications, 2007, 2007 (Special) : 191-200. doi: 10.3934/proc.2007.2007.191
[20]	Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Guoqing Zhang Jia-yu Shao Sanyang Liu
on Google Scholar
Guoqing Zhang Jia-yu Shao Sanyang Liu

[Back to Top]