American Institute of Mathematical Sciences

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

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

Rossella Bartolo 1, , Anna Maria Candela 2, and  Addolorata Salvatore 2,

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.
Keywords: perturbative methods., non--null boundary data, $p$--Laplacian operator, variational approach, lack of symmetry, radial solution.
Mathematics Subject Classification: Primary: 35J92; Secondary: 35J60, 35J66, 58E0.
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.

[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.

[3]

R. Bartolo, Infinitely many solutions for quasilinear elliptic problems with broken symmetry, Adv. Nonlinear Stud., 13 (2013), 739-749.

[4]

P. Bolle, On the Bolza problem, J. Differential Equations, 152 (1999), 274-288.

[5]

P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in non-homogeneous boundary value problems, Manuscripta Math., 101 (2000), 325-350.

[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.

[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.

[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).

[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.

[10]

E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.

[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.

[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.

[13]

M. Guedda and L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13 (1989), 879-902.

[14]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.

[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.

[16]

P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc., 272 (1982), 753-769.

[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.

[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.

[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.

[20]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.

[21]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.

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.

[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.

[3]

R. Bartolo, Infinitely many solutions for quasilinear elliptic problems with broken symmetry, Adv. Nonlinear Stud., 13 (2013), 739-749.

[4]

P. Bolle, On the Bolza problem, J. Differential Equations, 152 (1999), 274-288.

[5]

P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in non-homogeneous boundary value problems, Manuscripta Math., 101 (2000), 325-350.

[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.

[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.

[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).

[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.

[10]

E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.

[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.

[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.

[13]

M. Guedda and L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13 (1989), 879-902.

[14]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.

[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.

[16]

P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc., 272 (1982), 753-769.

[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.

[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.

[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.

[20]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.

[21]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.

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