# American Institute of Mathematical Sciences

• Previous Article
Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance
• CPAA Home
• This Issue
• Next Article
Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition
November  2013, 12(6): 2381-2391. doi: 10.3934/cpaa.2013.12.2381

## Multiple solutions for singular N-Laplace equations with a sign changing nonlinearity

 1 Indian Institute of Technology Gandhinagar, Vishwakarma Government Engineering College Complex, Chandkheda, Visat-Gandhinagar Highway, Ahmedabad, Gujarat, 382424, India

Received  November 2011 Revised  September 2012 Published  May 2013

In this article, we prove the existence of multiple weak solutions to N-Laplace equation \begin{eqnarray} -\Delta_N u-\mu \frac{g(x)}{(|x| \log\frac{R}{|x|})^N }|u|^{N-2}u=\lambda f(x,u), \ in\ \Omega.\\ u =0, \ on\ \partial \Omega, \end{eqnarray} using Bonanno's three critical point theorem.
Citation: J. Tyagi. Multiple solutions for singular N-Laplace equations with a sign changing nonlinearity. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2381-2391. doi: 10.3934/cpaa.2013.12.2381
##### References:
 [1] Adimurthi, M. Ramaswamy and N. Chaudhuri, Improved Hardy-Sobolev inequality and its applications,, Proc. Amer. Math. Soc., 130 (2002), 489. doi: 10.1090/S0002-9939-01-06132-9. [2] Adimurthi and K. Sandeep, Existence and non-existence of first eigenvalue of perturbed Hardy-Sobolev operator,, Proc. Royal. Soc. Edinburg, 132 (2002), 1021. doi: 10.1017/S0308210500001992. [3] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the N-Laplacian,, Ann. Sc. Norm. Super. Pisa, 17 (1990), 393. [4] R. P. Agarwal, D. Cao. H. Lü and Donal O'Regan, Existence and multiplicity of positive solutions for singular semipositone p-Laplacian,, Canad. J. Math., 58 (2006), 449. doi: 10.4153/CJM-2006-019-2. [5] G. Bonanno, Some remarks on a three critical points theorem,, Nonlinear Anal., 54 (2003), 651. doi: 10.1016/S0362-546X(03)00092-0. [6] H. Brezis and E. Lieb, A relation between point convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486. doi: 10.1090/S0002-9939-1983-0699419-3. [7] J. M. do Ó, N-Laplacian equations in $\R^N$ with critical growth,, Abstr. Appl. Anal., 2 (1997), 301. doi: 10.1155/S1085337597000419. [8] J. M. do Ó, E. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in $R^n$,, J. Diff. Equ., 246 (2009), 1363. doi: 10.1016/j.jde.2008.11.020. [9] P. Drábek, A. Kufner and F. Nicolosi, "Quasilinear Elliptic Equations with Degenerations and Singularities,'', De Gruyter Series in Nonlinear Analysis and Applications, (1997). doi: 10.1515/9783110804775. [10] J. P. Garcia Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems,, J. Diff. Equations, 144 (1998), 441. doi: 10.1006/jdeq.1997.3375. [11] J. Giacomoni, S. Prashanth and K. Sreenadh, A global multiplicity result for N-Laplacian with critical nonlinearity of concave-convex type,, J. Diff. Equations, 232 (2007), 544. doi: 10.1016/j.jde.2006.09.012. [12] D. D. Hai, On a class of sublinear quasilinear elliptic problems,, Proc. Amer. Math. Soc., 131 (2003), 2409. doi: 10.1090/S0002-9939-03-06874-6. [13] D. Jiang, Donal O'Regan and R. P. Agarwal, Existence theory for single and multiple solutions to singular boundary value problems for the one-dimensional p-Laplacian,, Adv. Math. Sci. Appl., 13 (2003), 179. doi: ~aiki/AMSA/vol13.html. [14] A. Kristály and C. Varga, Multiple solutions for elliptic problems with singular and sublinear potentials,, Proc. Amer. Math. Soc., 135 (2007), 2121. doi: 10.1090/S0002-9939-07-08715-1. [15] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1,, Rev. Mat. Iberoamericana, 1 (1985), 145. doi: 10.4171/RMI/6. [16] E. Montefusco, Lower semicontinuity of functionals via the concentration-compactness principle,, J. Math. Anal. Appl., 263 (2001), 264. doi: 10.1006/jmaa.2001.7631. [17] J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Uni. Math. J., 20 (1970), 1077. doi: 10.1512/iumj.1971.20.20101. [18] 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: 10.1007/BF00383673. [19] K. Perera, R. P. Agarwal and Donal O'Regan, Multiplicity results for p-sublinear p-Laplacian problems involving indefinite eigenvalue problems via Morse theory,, Electronic J. Diff. Equations, 41 (2010), 1. doi: ISSN: 1072-6691. [20] S. Prashanth and K. Sreenadh, Multiplicity of positive solutions for N-Laplace equation in a ball,, Diff. Int. Equations, 17 (2004), 709. [21] J. Saint Raymond, On the multiplicity of solutions of the equations $-\Delta u = \lambda. f(u)$,, J. Diff. Equations, 180 (2002), 65. doi: 10.1006/jdeq.2001.4057. [22] Y. T. Shen, Y. X. Yao and Z. H. Chen, On a nonlinear elliptic problem with critical potential in $\R^2$,, Science in China, 47 (2004), 741. doi: 10.1360/03ys0194. [23] M. Souza and J. M. do Ó, On a singular and nonhomogeneous N-Laplacian equation involving critical growth,, J. Math. Anal. Appl., 380 (2011), 241. doi: 10.1016/j.jmaa.2011.03.028. [24] J. Tyagi, Existence of nontrivial solutions for singular quasilinear equations with sign changing nonlinearity,, Electronic J. Diff. Equations, 117 (2010), 1. doi: ISSN: 1072-6691. [25] Z. Yang, D. Geng and H. Yan, Three solutions for singular p-Laplacian type equations,, Electronic J. Diff. Equations, 61 (2008), 1. doi: ISSN: 1072-6691. [26] G. Zhang, J. Shao and S. Liu, Linking solutions for N-Laplace elliptic equations with Hardy-Sobolev operator and indefinite weights,, Comm. Pure. Appl. Anal., 10 (2011), 571.

show all references

##### References:
 [1] Adimurthi, M. Ramaswamy and N. Chaudhuri, Improved Hardy-Sobolev inequality and its applications,, Proc. Amer. Math. Soc., 130 (2002), 489. doi: 10.1090/S0002-9939-01-06132-9. [2] Adimurthi and K. Sandeep, Existence and non-existence of first eigenvalue of perturbed Hardy-Sobolev operator,, Proc. Royal. Soc. Edinburg, 132 (2002), 1021. doi: 10.1017/S0308210500001992. [3] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the N-Laplacian,, Ann. Sc. Norm. Super. Pisa, 17 (1990), 393. [4] R. P. Agarwal, D. Cao. H. Lü and Donal O'Regan, Existence and multiplicity of positive solutions for singular semipositone p-Laplacian,, Canad. J. Math., 58 (2006), 449. doi: 10.4153/CJM-2006-019-2. [5] G. Bonanno, Some remarks on a three critical points theorem,, Nonlinear Anal., 54 (2003), 651. doi: 10.1016/S0362-546X(03)00092-0. [6] H. Brezis and E. Lieb, A relation between point convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486. doi: 10.1090/S0002-9939-1983-0699419-3. [7] J. M. do Ó, N-Laplacian equations in $\R^N$ with critical growth,, Abstr. Appl. Anal., 2 (1997), 301. doi: 10.1155/S1085337597000419. [8] J. M. do Ó, E. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in $R^n$,, J. Diff. Equ., 246 (2009), 1363. doi: 10.1016/j.jde.2008.11.020. [9] P. Drábek, A. Kufner and F. Nicolosi, "Quasilinear Elliptic Equations with Degenerations and Singularities,'', De Gruyter Series in Nonlinear Analysis and Applications, (1997). doi: 10.1515/9783110804775. [10] J. P. Garcia Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems,, J. Diff. Equations, 144 (1998), 441. doi: 10.1006/jdeq.1997.3375. [11] J. Giacomoni, S. Prashanth and K. Sreenadh, A global multiplicity result for N-Laplacian with critical nonlinearity of concave-convex type,, J. Diff. Equations, 232 (2007), 544. doi: 10.1016/j.jde.2006.09.012. [12] D. D. Hai, On a class of sublinear quasilinear elliptic problems,, Proc. Amer. Math. Soc., 131 (2003), 2409. doi: 10.1090/S0002-9939-03-06874-6. [13] D. Jiang, Donal O'Regan and R. P. Agarwal, Existence theory for single and multiple solutions to singular boundary value problems for the one-dimensional p-Laplacian,, Adv. Math. Sci. Appl., 13 (2003), 179. doi: ~aiki/AMSA/vol13.html. [14] A. Kristály and C. Varga, Multiple solutions for elliptic problems with singular and sublinear potentials,, Proc. Amer. Math. Soc., 135 (2007), 2121. doi: 10.1090/S0002-9939-07-08715-1. [15] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1,, Rev. Mat. Iberoamericana, 1 (1985), 145. doi: 10.4171/RMI/6. [16] E. Montefusco, Lower semicontinuity of functionals via the concentration-compactness principle,, J. Math. Anal. Appl., 263 (2001), 264. doi: 10.1006/jmaa.2001.7631. [17] J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Uni. Math. J., 20 (1970), 1077. doi: 10.1512/iumj.1971.20.20101. [18] 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: 10.1007/BF00383673. [19] K. Perera, R. P. Agarwal and Donal O'Regan, Multiplicity results for p-sublinear p-Laplacian problems involving indefinite eigenvalue problems via Morse theory,, Electronic J. Diff. Equations, 41 (2010), 1. doi: ISSN: 1072-6691. [20] S. Prashanth and K. Sreenadh, Multiplicity of positive solutions for N-Laplace equation in a ball,, Diff. Int. Equations, 17 (2004), 709. [21] J. Saint Raymond, On the multiplicity of solutions of the equations $-\Delta u = \lambda. f(u)$,, J. Diff. Equations, 180 (2002), 65. doi: 10.1006/jdeq.2001.4057. [22] Y. T. Shen, Y. X. Yao and Z. H. Chen, On a nonlinear elliptic problem with critical potential in $\R^2$,, Science in China, 47 (2004), 741. doi: 10.1360/03ys0194. [23] M. Souza and J. M. do Ó, On a singular and nonhomogeneous N-Laplacian equation involving critical growth,, J. Math. Anal. Appl., 380 (2011), 241. doi: 10.1016/j.jmaa.2011.03.028. [24] J. Tyagi, Existence of nontrivial solutions for singular quasilinear equations with sign changing nonlinearity,, Electronic J. Diff. Equations, 117 (2010), 1. doi: ISSN: 1072-6691. [25] Z. Yang, D. Geng and H. Yan, Three solutions for singular p-Laplacian type equations,, Electronic J. Diff. Equations, 61 (2008), 1. doi: ISSN: 1072-6691. [26] G. Zhang, J. Shao and S. Liu, Linking solutions for N-Laplace elliptic equations with Hardy-Sobolev operator and indefinite weights,, Comm. Pure. Appl. Anal., 10 (2011), 571.
 [1] Rong Xiao, Yuying Zhou. Multiple solutions for a class of semilinear elliptic variational inclusion problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 991-1002. doi: 10.3934/jimo.2011.7.991 [2] Tomas Godoy, Alfredo Guerin. Existence of nonnegative solutions to singular elliptic problems, a variational approach. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1505-1525. doi: 10.3934/dcds.2018062 [3] L. Ke. Boundary behaviors for solutions of singular elliptic equations. Conference Publications, 1998, 1998 (Special) : 388-396. doi: 10.3934/proc.1998.1998.388 [4] Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2469-2494. doi: 10.3934/dcds.2013.33.2469 [5] Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527 [6] J. Chen, K. Murillo, E. M. Rocha. Two nontrivial solutions of a class of elliptic equations with singular term. Conference Publications, 2011, 2011 (Special) : 272-281. doi: 10.3934/proc.2011.2011.272 [7] Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133 [8] Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105 [9] Qingfang Wang. Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1671-1688. doi: 10.3934/cpaa.2016008 [10] Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249 [11] Monica Musso, Donato Passaseo. Multiple solutions of Neumann elliptic problems with critical nonlinearity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 301-320. doi: 10.3934/dcds.1999.5.301 [12] Weichung Wang, Tsung-Fang Wu, Chien-Hsiang Liu. On the multiple spike solutions for singularly perturbed elliptic systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 237-258. doi: 10.3934/dcdsb.2013.18.237 [13] Emmanuel Hebey, Jérôme Vétois. Multiple solutions for critical elliptic systems in potential form. Communications on Pure & Applied Analysis, 2008, 7 (3) : 715-741. doi: 10.3934/cpaa.2008.7.715 [14] David L. Finn. Noncompact manifolds with constant negative scalar curvature and singular solutions to semihnear elliptic equations. Conference Publications, 1998, 1998 (Special) : 262-275. doi: 10.3934/proc.1998.1998.262 [15] Lei Wei, Xiyou Cheng, Zhaosheng Feng. Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7169-7189. doi: 10.3934/dcds.2016112 [16] Xiaomei Sun, Yimin Zhang. Elliptic equations with cylindrical potential and multiple critical exponents. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1943-1957. doi: 10.3934/cpaa.2013.12.1943 [17] Marco Degiovanni, Michele Scaglia. A variational approach to semilinear elliptic equations with measure data. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1233-1248. doi: 10.3934/dcds.2011.31.1233 [18] Andrzej Szulkin, Shoyeb Waliullah. Infinitely many solutions for some singular elliptic problems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 321-333. doi: 10.3934/dcds.2013.33.321 [19] Patrizia Pucci, Marco Rigoli. Entire solutions of singular elliptic inequalities on complete manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 115-137. doi: 10.3934/dcds.2008.20.115 [20] Shuangjie Peng. Remarks on singular critical growth elliptic equations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 707-719. doi: 10.3934/dcds.2006.14.707

2017 Impact Factor: 0.884