# American Institute of Mathematical Sciences

2013, 2013(special): 51-59. doi: 10.3934/proc.2013.2013.51

## Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem

 1 Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via E. Orabona 4, 70125 Bari, Italy 2 Dipartimento di Matematica, Università degli Studi di Bari "Aldo Moro", Via E. Orabona 4, 70125 Bari, Italy

Received  September 2012 Published  November 2013

In this paper we investigate the existence of infinitely many radial solutions for the elliptic Dirichlet problem $\left\{ \begin{array}{ll} \displaystyle{-\Delta_p u\ =|u|^{q-2}u + f(x)} & \mbox{ in } B_R,\\ \displaystyle{u=\xi} & \mbox{ on } \partial B_R,\\ \end{array} \right.$ where $B_R$ is the open ball centered in $0$ with radius $R$ in $\mathbb{R}^N$ ($N \geq 3$), $2 < p < N$, $p< q < p^*$ (with $p^* = \frac{pN}{N-p}$), $\xi\in\mathbb{R}$ and $f$ is a continuous radial function in $\overline B_R$. The lack of even symmetry for the related functional is overcome by using some perturbative methods and the radial assumptions allow us to improve some previous results.
Citation: Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51
##### References:
 [1] A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc., 267 (1981), 1-32.  Google Scholar [2] A. Bahri and P. L. Lions, Morse index of some min-max critical points. I. Applications to multiplicity results, Comm. Pure Appl. Math., 41 (1988), 1027-1037.  Google Scholar [3] R. Bartolo, Infinitely many solutions for quasilinear elliptic problems with broken symmetry, Adv. Nonlinear Stud., 13 (2013), 739-749. Google Scholar [4] P. Bolle, On the Bolza problem, J. Differential Equations, 152 (1999), 274-288.  Google Scholar [5] P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in non-homogeneous boundary value problems, Manuscripta Math., 101 (2000), 325-350.  Google Scholar [6] A. Candela, G. Palmieri and A. Salvatore, Radial solutions of semilinear elliptic equations with broken symmetry, Topol. Methods Nonlinear Anal., 27 (2006), 117-132.  Google Scholar [7] A. Candela and A. Salvatore, Multiplicity results of an elliptic equation with non homogeneous boundary conditions, Topol. Methods Nonlinear Anal., 11 (1998), 1-18.  Google Scholar [8] A. Candela, A. Salvatore and M. Squassina, Semilinear elliptic systems with lack of symmetry, In: "Second International Conference on Dynamics of Continuous, Discrete and Impulsive Systems'' (London, ON, 2001).  Google Scholar [9] M. Clapp, Y. Ding and S. Hernández-Linares, Strongly indefinite functionals with perturbed symmetries and multiple solutions of nonsymmetric elliptic systems, Electron. J. Differential Equations, 100 (2004), 18 pp.  Google Scholar [10] E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  Google Scholar [11] G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with $p$-Laplacian, Port. Math. (N. S.), 58 (2001), 339-378.  Google Scholar [12] J. García Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895.  Google Scholar [13] M. Guedda and L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13 (1989), 879-902.  Google Scholar [14] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  Google Scholar [15] D. Liu and D. Geng, Infinitely many solutions for the $p$-Laplace equations with nonsymmetric perturbations, Electron. J. Differential Equations, 101 (2008), 7 pp.  Google Scholar [16] P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc., 272 (1982), 753-769.  Google Scholar [17] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986.  Google Scholar [18] M. Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta Math., 32 (1980), 335-364.  Google Scholar [19] K. Tanaka, Morse indices at critical points related to the symmetric mountain pass theorem and applications, Comm. Partial Differential Equations, 14 (1989), 99-128.  Google Scholar [20] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.  Google Scholar [21] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

show all references

##### References:
 [1] A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc., 267 (1981), 1-32.  Google Scholar [2] A. Bahri and P. L. Lions, Morse index of some min-max critical points. I. Applications to multiplicity results, Comm. Pure Appl. Math., 41 (1988), 1027-1037.  Google Scholar [3] R. Bartolo, Infinitely many solutions for quasilinear elliptic problems with broken symmetry, Adv. Nonlinear Stud., 13 (2013), 739-749. Google Scholar [4] P. Bolle, On the Bolza problem, J. Differential Equations, 152 (1999), 274-288.  Google Scholar [5] P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in non-homogeneous boundary value problems, Manuscripta Math., 101 (2000), 325-350.  Google Scholar [6] A. Candela, G. Palmieri and A. Salvatore, Radial solutions of semilinear elliptic equations with broken symmetry, Topol. Methods Nonlinear Anal., 27 (2006), 117-132.  Google Scholar [7] A. Candela and A. Salvatore, Multiplicity results of an elliptic equation with non homogeneous boundary conditions, Topol. Methods Nonlinear Anal., 11 (1998), 1-18.  Google Scholar [8] A. Candela, A. Salvatore and M. Squassina, Semilinear elliptic systems with lack of symmetry, In: "Second International Conference on Dynamics of Continuous, Discrete and Impulsive Systems'' (London, ON, 2001).  Google Scholar [9] M. Clapp, Y. Ding and S. Hernández-Linares, Strongly indefinite functionals with perturbed symmetries and multiple solutions of nonsymmetric elliptic systems, Electron. J. Differential Equations, 100 (2004), 18 pp.  Google Scholar [10] E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  Google Scholar [11] G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with $p$-Laplacian, Port. Math. (N. S.), 58 (2001), 339-378.  Google Scholar [12] J. García Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895.  Google Scholar [13] M. Guedda and L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13 (1989), 879-902.  Google Scholar [14] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  Google Scholar [15] D. Liu and D. Geng, Infinitely many solutions for the $p$-Laplace equations with nonsymmetric perturbations, Electron. J. Differential Equations, 101 (2008), 7 pp.  Google Scholar [16] P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc., 272 (1982), 753-769.  Google Scholar [17] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986.  Google Scholar [18] M. Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta Math., 32 (1980), 335-364.  Google Scholar [19] K. Tanaka, Morse indices at critical points related to the symmetric mountain pass theorem and applications, Comm. Partial Differential Equations, 14 (1989), 99-128.  Google Scholar [20] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.  Google Scholar [21] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar
 [1] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3851-3863. doi: 10.3934/dcdss.2020445 [2] Orlando Lopes. Uniqueness and radial symmetry of minimizers for a nonlocal variational problem. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2265-2282. doi: 10.3934/cpaa.2019102 [3] Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040 [4] Trad Alotaibi, D. D. Hai, R. Shivaji. Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4655-4666. doi: 10.3934/cpaa.2020131 [5] Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036 [6] Shixiu Zheng, Zhilei Xu, Huan Yang, Jintao Song, Zhenkuan Pan. Comparisons of different methods for balanced data classification under the discrete non-local total variational framework. Mathematical Foundations of Computing, 2019, 2 (1) : 11-28. doi: 10.3934/mfc.2019002 [7] Lorenzo Brasco, Enea Parini, Marco Squassina. Stability of variational eigenvalues for the fractional $p-$Laplacian. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1813-1845. doi: 10.3934/dcds.2016.36.1813 [8] Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Existence of radial solutions for the $p$-Laplacian elliptic equations with weights. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 447-479. doi: 10.3934/dcds.2006.15.447 [9] Antonio Greco, Vincenzino Mascia. Non-local sublinear problems: Existence, comparison, and radial symmetry. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 503-519. doi: 10.3934/dcds.2019021 [10] Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194 [11] Marco Degiovanni, Michele Scaglia. A variational approach to semilinear elliptic equations with measure data. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1233-1248. doi: 10.3934/dcds.2011.31.1233 [12] CÉSAR E. TORRES LEDESMA. Existence and symmetry result for fractional p-Laplacian in $\mathbb{R}^{n}$. Communications on Pure & Applied Analysis, 2017, 16 (1) : 99-114. doi: 10.3934/cpaa.2017004 [13] Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069 [14] Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125 [15] Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107 [16] Daniela Gurban, Petru Jebelean, Cǎlin Şerban. Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 133-151. doi: 10.3934/dcds.2020006 [17] Fioralba Cakoni, Houssem Haddar. A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media. Inverse Problems & Imaging, 2007, 1 (3) : 443-456. doi: 10.3934/ipi.2007.1.443 [18] Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez. Infinitely many radial solutions for a $p$-Laplacian problem with indefinite weight. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4805-4821. doi: 10.3934/dcds.2021058 [19] Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741 [20] Marc Briant. Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions. Kinetic & Related Models, 2017, 10 (2) : 329-371. doi: 10.3934/krm.2017014

Impact Factor: