# American Institute of Mathematical Sciences

October  2021, 41(10): 4805-4821. doi: 10.3934/dcds.2021058

## Infinitely many radial solutions for a $p$-Laplacian problem with indefinite weight

 1 Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA 2 Escuela de Matemáticas, Universidad Nacional de Colombia, Apartado Aéreo 3840, Medellín, Colombia

* Corresponding author: Alfonso Castro

Received  September 2020 Revised  February 2021 Published  October 2021 Early access  March 2021

Fund Project: The last two authors are supported by Facultad de Ciencias, Universidad Nacional de Colombia sede Medellín

We prove the existence of infinitely many sign changing radial solutions for a $p$-Laplacian Dirichlet problem in a ball. Our problem involves a weight function that is positive at the center of the unit ball and negative in its boundary. Standard initial value problems-phase plane analysis arguments do not apply here because solutions to the corresponding initial value problem may blow up near the boundary due to the fact that our weight function is negative at the boundary. We overcome this difficulty by connecting the solutions to a singular initial value problem with those of a regular initial value problem that vanishes at the boundary.

Citation: Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez. Infinitely many radial solutions for a $p$-Laplacian problem with indefinite weight. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4805-4821. doi: 10.3934/dcds.2021058
##### References:
 [1] K. Bal and P. Garain, Nonexistence results for weighted $p$-Laplace equations with singular nonlinearities, Electron. J. Differ. Equ., 95 (2019), 1-12. [2] H. Berestycki, P.-L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $\mathbb R^N$, Indiana Univ. Math. J., 30 (1981), 141-157.  doi: 10.1512/iumj.1981.30.30012. [3] A. Castro, J. Cossio, S. Herrón, R. Pardo and C. Vélez, Infinitely many radial solutions for a sub-super critical p-Laplacian problem in a ball, Annali di Matematica Pura ed Applicata, 199 (2020), 737-766.  doi: 10.1007/s10231-019-00898-x. [4] A. Castro and A. Kurepa, Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball, Proc. Am. Math. Soc., 101 (1987), 57-64.  doi: 10.1090/S0002-9939-1987-0897070-7. [5] T. Chen and R. Ma, Three positive solutions of $N$-dimensional $p$-Laplacian with indefinite weight, Electron. J. Qual. Theory Differ. Equ., (2019), Paper No. 19, 14 pp. doi: 10.14232/ejqtde.2019.1.19. [6] J. Cossio, S. Herrón and C. Vélez, Infinitely many radial solutions for a $p$-Laplacian problem p-superlinear at the origin, J. Math. Anal. Appl., 376 (2011), 741-749.  doi: 10.1016/j.jmaa.2010.10.075. [7] E. DiBenedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5. [8] A. El Hachimi and F. De Thelin, Infinitely many radially symmetric solutions for a quasilinear elliptic problem in a ball, J. Differ. Equ., 128 (1996), 78-102.  doi: 10.1006/jdeq.1996.0090. [9] W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233.  doi: 10.1007/BF00277151. [10] M. García-Huidobro, R. Manásevich and F. Zanolin, Infinitely many solutions for a Dirichlet problem with a nonhomogeneous $p$-Laplacian-like operator in a ball, Adv. Differential Equations, 2 (1997), 203-230. [11] S. Herrón and E. Lopera, Signed radial solutions for a weighted p-superlinear problem, Electron. J. Differ. Equ., 24 (2014), 1-13. [12] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3. [13] W. Reichel and W. Walter, Radial solutions of equations and inequalities involving the $p$-Laplacian, J. Inequal. Appl., (1997), 47–71. [14] A. S. Tersenov, On the existence of radially symmetric solutions of the inhomogeneous p-Laplace equation, Siberian Mathematical Journal, 57 (2016), 918-928.  doi: 10.1134/S0037446616050219.

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##### References:
 [1] K. Bal and P. Garain, Nonexistence results for weighted $p$-Laplace equations with singular nonlinearities, Electron. J. Differ. Equ., 95 (2019), 1-12. [2] H. Berestycki, P.-L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $\mathbb R^N$, Indiana Univ. Math. J., 30 (1981), 141-157.  doi: 10.1512/iumj.1981.30.30012. [3] A. Castro, J. Cossio, S. Herrón, R. Pardo and C. Vélez, Infinitely many radial solutions for a sub-super critical p-Laplacian problem in a ball, Annali di Matematica Pura ed Applicata, 199 (2020), 737-766.  doi: 10.1007/s10231-019-00898-x. [4] A. Castro and A. Kurepa, Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball, Proc. Am. Math. Soc., 101 (1987), 57-64.  doi: 10.1090/S0002-9939-1987-0897070-7. [5] T. Chen and R. Ma, Three positive solutions of $N$-dimensional $p$-Laplacian with indefinite weight, Electron. J. Qual. Theory Differ. Equ., (2019), Paper No. 19, 14 pp. doi: 10.14232/ejqtde.2019.1.19. [6] J. Cossio, S. Herrón and C. Vélez, Infinitely many radial solutions for a $p$-Laplacian problem p-superlinear at the origin, J. Math. Anal. Appl., 376 (2011), 741-749.  doi: 10.1016/j.jmaa.2010.10.075. [7] E. DiBenedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5. [8] A. El Hachimi and F. De Thelin, Infinitely many radially symmetric solutions for a quasilinear elliptic problem in a ball, J. Differ. Equ., 128 (1996), 78-102.  doi: 10.1006/jdeq.1996.0090. [9] W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233.  doi: 10.1007/BF00277151. [10] M. García-Huidobro, R. Manásevich and F. Zanolin, Infinitely many solutions for a Dirichlet problem with a nonhomogeneous $p$-Laplacian-like operator in a ball, Adv. Differential Equations, 2 (1997), 203-230. [11] S. Herrón and E. Lopera, Signed radial solutions for a weighted p-superlinear problem, Electron. J. Differ. Equ., 24 (2014), 1-13. [12] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3. [13] W. Reichel and W. Walter, Radial solutions of equations and inequalities involving the $p$-Laplacian, J. Inequal. Appl., (1997), 47–71. [14] A. S. Tersenov, On the existence of radially symmetric solutions of the inhomogeneous p-Laplace equation, Siberian Mathematical Journal, 57 (2016), 918-928.  doi: 10.1134/S0037446616050219.
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