# On an elliptic equation with singular cylindrical growth

Keywords:
Singular cylindrical grouwth, concave term, Nehari manifold, mountain pass theorem

### Abstract

In the present paper, an elliptic equation with singular cylindrical grouwth, is considered. By using the Nehari manifold and mountain pass theorem, the existence of at least four distinct solutions is obtained. The result depends crucially on the parameters k, λ, g and u.

### References

[1] M. Badiale, M. Guida, S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearities, Adv. Differential Equations, 12 (2007) 1321-1362.

[1] M. Badiale, M. Guida, S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearities, Adv. Differential Equations, 12 (2007) 1321-1362.

[2] M. Bouchekif, A. Matallah, On singular nonhomogeneous elliptic equations involving critical Caffarelli-Kohn-Nirenberg exponent, Ric. Mat., 58 (2009) 207-218.

[3] M. Bouchekif, M. E. O. El Mokhtar, On nonhomogeneous singular elliptic equations with cylindrical weight, Ric. Mat. 61 (2012) 147-156.

[4] K. J. Brown, Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign changing weight function, J. Differential Equations, 2 (2003) 481-499 .

[5] D. Cao, S. Peng, A note on the sign-changing solutions to elliptic problems with critical Sobolev exponent and Hardy terms, J. Differential Equations 193 (2003) 424-434.

[6] J. Chen, Existence of solutions for a nonlinear PDE with an inverse square potential, J. Differential Equations 195 (2003) 497-519.

[7] P. Drabek, A. Kufner, and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter Series in Nonlinear Analysis and Applications Vol. 5 (New York), 1997.

[8] I. Ekeland, N. Ghoussoub, Selected new aspects of the calculus of variations in the large, Bull. Amer. Math. Soc. 39 (2002) 207-265.

[9] M. E. O. El Mokhtar, Five nontrivial solutions of p-Laplacian problems involving critical exposants and singular cylindrical potential, J. of Physical Science and Application 5(2) (2015) 163-172.

[10] M. E. O. El Mokhtar, On Elliptic Problem with Singular Cylindrical Potential, a Concave Term, and Critical Caffarelli-Kohn-Nirenberg Exponent, Applied Mathematics, 2015, 6, 1891-1901

[11] M. E. O. El Mokhtar, Existence for Elliptic Equation Involving Decaying Cylindrical Potentials with Subcritical and Critical Exponent, International Journal of Differential Equations., 9 (2015) 1-5.

[12] M. E. O. El Mokhtar, Existence of Multiple Solutions for P-Laplacian Problems Involving Critical Exponents and Singular Cylindrical Potential, J. Appl. Computat. Math., 2015, 4:4.

[13] M. Gazzini, R. Musina, On the Hardy-Sobolev-Maz'ja inequalities: symmetry and breaking symmetry of extremal functions, Commun. Contemp. Math., 11 (2009) 993-1007.

[14] D. Kang, On the elliptic problems with critical weighted Sobolev-Hardy exponents, Nonlinear Anal., 66 (2007) 1037-1050.

[15] D. Kang, S. Peng, Positive solutions for singular elliptic problems, Appl. Math. Lett., 17 (2004) 411-416.

[16] Z. Liu, P. Han, Existence of solutions for singular elliptic systems with critical exponents, Nonlinear Anal., 69 (2008) 2968-2983.

[17] R. Musina, Ground state solutions of a critical problem involving cylindrical weights, Nonlinear Anal., 68 (2008) 3972-3986.

[18] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non. Linéaire, 9 (1992) 281-304.

[19] S. Terracini, On positive entire solutions to a class of equations with singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996) 241-264.

[20] Z. Wang, H. Zhou, Solutions for a nonhomogeneous elliptic problem involving critical Sobolev-Hardy exponent in ℝ^{N}. Acta Math. Sci., 26 (2006) 525-536.

[21] T.-F. Wu, The Nehari manifold for a semilinear system involving sign-changing weight functions, Nonlinear Anal., 68 (2008) 1733-1745.

[1] M. Badiale, M. Guida, S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearities, Adv. Differential Equations, 12 (2007) 1321-1362.

[2] M. Bouchekif, A. Matallah, On singular nonhomogeneous elliptic equations involving critical Caffarelli-Kohn-Nirenberg exponent, Ric. Mat., 58 (2009) 207-218.

[3] M. Bouchekif, M. E. O. El Mokhtar, On nonhomogeneous singular elliptic equations with cylindrical weight, Ric. Mat. 61 (2012) 147-156.

[4] K. J. Brown, Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign changing weight function, J. Differential Equations, 2 (2003) 481-499 .

[5] D. Cao, S. Peng, A note on the sign-changing solutions to elliptic problems with critical Sobolev exponent and Hardy terms, J. Differential Equations 193 (2003) 424-434.

[6] J. Chen, Existence of solutions for a nonlinear PDE with an inverse square potential, J. Differential Equations 195 (2003) 497-519.

[7] P. Drabek, A. Kufner, and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter Series in Nonlinear Analysis and Applications Vol. 5 (New York), 1997.

[8] I. Ekeland, N. Ghoussoub, Selected new aspects of the calculus of variations in the large, Bull. Amer. Math. Soc. 39 (2002) 207-265.

[9] M. E. O. El Mokhtar, Five nontrivial solutions of p-Laplacian problems involving critical exposants and singular cylindrical potential, J. of Physical Science and Application 5(2) (2015) 163-172.

[10] M. E. O. El Mokhtar, On Elliptic Problem with Singular Cylindrical Potential, a Concave Term, and Critical Caffarelli-Kohn-Nirenberg Exponent, Applied Mathematics, 2015, 6, 1891-1901

[11] M. E. O. El Mokhtar, Existence for Elliptic Equation Involving Decaying Cylindrical Potentials with Subcritical and Critical Exponent, International Journal of Differential Equations., 9 (2015) 1-5.

[12] M. E. O. El Mokhtar, Existence of Multiple Solutions for P-Laplacian Problems Involving Critical Exponents and Singular Cylindrical Potential, J. Appl. Computat. Math., 2015, 4:4.

[13] M. Gazzini, R. Musina, On the Hardy-Sobolev-Maz'ja inequalities: symmetry and breaking symmetry of extremal functions, Commun. Contemp. Math., 11 (2009) 993-1007.

[14] D. Kang, On the elliptic problems with critical weighted Sobolev-Hardy exponents, Nonlinear Anal., 66 (2007) 1037-1050.

[15] D. Kang, S. Peng, Positive solutions for singular elliptic problems, Appl. Math. Lett., 17 (2004) 411-416.

[16] Z. Liu, P. Han, Existence of solutions for singular elliptic systems with critical exponents, Nonlinear Anal., 69 (2008) 2968-2983.

[17] R. Musina, Ground state solutions of a critical problem involving cylindrical weights, Nonlinear Anal., 68 (2008) 3972-3986.

[18] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non. Linéaire, 9 (1992) 281-304.

[19] S. Terracini, On positive entire solutions to a class of equations with singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996) 241-264.

[20] Z. Wang, H. Zhou, Solutions for a nonhomogeneous elliptic problem involving critical Sobolev-Hardy exponent in ℝ^{N}. Acta Math. Sci., 26 (2006) 525-536.

[21] T.-F. Wu, The Nehari manifold for a semilinear system involving sign-changing weight functions, Nonlinear Anal., 68 (2008) 1733-1745.

Published

2020-02-03

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*Journal of Progressive Research in Mathematics*,

*16*(1), 2824-2836. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/1794

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