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2013, 2013(special): 41-49. doi: 10.3934/proc.2013.2013.41

Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains

Sara Barile 1, and  Addolorata Salvatore 1,

1. 

Dipartimento di Matematica, Università degli Studi di Bari "Aldo Moro", Via E. Orabona 4, 70125 Bari, Italy, Italy

Received  September 2012 Revised  May 2013 Published  November 2013

We study a superlinear perturbed elliptic problem on $\mathbb R^N$ with rotational symmetry. Using variational and perturbative methods we find infinitely many radial solutions for any growth exponent $p$ of the nonlinearity greater than $2$ and less than $2^*$ if $N \geq 4$ and for any $p$ greater than $3$ and subcritical if $N =3$.
Keywords: variational and perturbative methods, Radial solutions, elliptic equations with broken symmetry., unbounded domains.
Mathematics Subject Classification: Primary: 35J20; Secondary: 35B38, 58E0.
Citation: Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41
References:
[1]

A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in cri\-ti\-cal point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  Google Scholar

[2]

A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc., 267 (1981), 1-32.  Google Scholar

[3]

A. Bahri and P.L. Lions, Morse index of some mini-max critical points, Comm. Pure Appl. Math., 41 (1988), 1027-1037.  Google Scholar

[4]

H. Berestycki and P.L. Lions, Nonlinear scalar field equations, I and II, Arch. Rat. Mech. Anal., 82 (1983), 313-376.  Google Scholar

[5]

F.A. Berezin and M.A. Shubin, The Schr\"odinger equation, Mathematics and its Applications (Soviet Series) 66, Kluwer Academic Publishers Group, Dordrecht, 1991.  Google Scholar

[6]

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

[7]

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

[8]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.  Google Scholar

[9]

A.M. 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

[10]

A.M. 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

[11]

A.M. Candela, A. Salvatore and M. Squassina, Semilinear elliptic systems with lack of symmetry, Dynam. Contin. Discrete Impuls. Systems Ser. A, 10 (2003), 181-192.  Google Scholar

[12]

M. Clapp, Y. Ding and S. Hern\`andez-Linares, Strongly indefinite functionals with perturbed symmetries and multiple solutions of non symmetric elliptic systems, Electron. J. Differential Equations, 100 (2004), 1-18.  Google Scholar

[13]

S.I. Pohozaev, On the global fibering method in nonlinear variational problems, Proc. Steklov Inst. Math., 219 (1997), 281-328.  Google Scholar

[14]

P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, 65 (1986).  Google Scholar

[15]

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

[16]

A. Salvatore, Multiple radial solutions for a superlinear elliptic problem in $\mathbb R^N$, Dynam. Systems and Applications, 4 (2004), 472-479.  Google Scholar

[17]

W.A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.  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

show all references

References:
[1]

A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in cri\-ti\-cal point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  Google Scholar

[2]

A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc., 267 (1981), 1-32.  Google Scholar

[3]

A. Bahri and P.L. Lions, Morse index of some mini-max critical points, Comm. Pure Appl. Math., 41 (1988), 1027-1037.  Google Scholar

[4]

H. Berestycki and P.L. Lions, Nonlinear scalar field equations, I and II, Arch. Rat. Mech. Anal., 82 (1983), 313-376.  Google Scholar

[5]

F.A. Berezin and M.A. Shubin, The Schr\"odinger equation, Mathematics and its Applications (Soviet Series) 66, Kluwer Academic Publishers Group, Dordrecht, 1991.  Google Scholar

[6]

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

[7]

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

[8]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.  Google Scholar

[9]

A.M. 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

[10]

A.M. 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

[11]

A.M. Candela, A. Salvatore and M. Squassina, Semilinear elliptic systems with lack of symmetry, Dynam. Contin. Discrete Impuls. Systems Ser. A, 10 (2003), 181-192.  Google Scholar

[12]

M. Clapp, Y. Ding and S. Hern\`andez-Linares, Strongly indefinite functionals with perturbed symmetries and multiple solutions of non symmetric elliptic systems, Electron. J. Differential Equations, 100 (2004), 1-18.  Google Scholar

[13]

S.I. Pohozaev, On the global fibering method in nonlinear variational problems, Proc. Steklov Inst. Math., 219 (1997), 281-328.  Google Scholar

[14]

P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, 65 (1986).  Google Scholar

[15]

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

[16]

A. Salvatore, Multiple radial solutions for a superlinear elliptic problem in $\mathbb R^N$, Dynam. Systems and Applications, 4 (2004), 472-479.  Google Scholar

[17]

W.A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.  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

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