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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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [5] G. Bonanno, Some remarks on a three critical points theorem,, Nonlinear Anal., 54 (2003), 651.  doi: 10.1016/S0362-546X(03)00092-0.  Google Scholar [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.  Google Scholar [7] J. M. do Ó, N-Laplacian equations in $\R^N$ with critical growth,, Abstr. Appl. Anal., 2 (1997), 301.  doi: 10.1155/S1085337597000419.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] S. Prashanth and K. Sreenadh, Multiplicity of positive solutions for N-Laplace equation in a ball,, Diff. Int. Equations, 17 (2004), 709.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar

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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [5] G. Bonanno, Some remarks on a three critical points theorem,, Nonlinear Anal., 54 (2003), 651.  doi: 10.1016/S0362-546X(03)00092-0.  Google Scholar [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.  Google Scholar [7] J. M. do Ó, N-Laplacian equations in $\R^N$ with critical growth,, Abstr. Appl. Anal., 2 (1997), 301.  doi: 10.1155/S1085337597000419.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] S. Prashanth and K. Sreenadh, Multiplicity of positive solutions for N-Laplace equation in a ball,, Diff. Int. Equations, 17 (2004), 709.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar
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