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

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

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

Abstract
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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.
[2]	A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications,, Trans. Amer. Math. Soc., 267 (1981), 1.
[3]	A. Bahri and P.L. Lions, Morse index of some mini-max critical points,, Comm. Pure Appl. Math., 41 (1988), 1027.
[4]	H. Berestycki and P.L. Lions, Nonlinear scalar field equations, I and II,, Arch. Rat. Mech. Anal., 82 (1983), 313.
[5]	F.A. Berezin and M.A. Shubin, The Schr\"odinger equation,, Mathematics and its Applications (Soviet Series) 66, (1991).
[6]	P. Bolle, On the Bolza Problem,, J. Differential Equations, 152 (1999), 274.
[7]	P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in non-homogeneous boundary value problems,, Manuscripta Math., 101 (2000), 325.
[8]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer, (2011).
[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.
[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.
[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.
[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.
[13]	S.I. Pohozaev, On the global fibering method in nonlinear variational problems,, Proc. Steklov Inst. Math., 219 (1997), 281.
[14]	P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations,, CBMS Regional Conference Series in Mathematics, 65 (1986).
[15]	P.H. Rabinowitz, Multiple critical points of perturbed symmetric functionals,, Trans. Amer. Math. Soc., 272 (1982), 753.
[16]	A. Salvatore, Multiple radial solutions for a superlinear elliptic problem in $\mathbb R^N$,, Dynam. Systems and Applications, 4 (2004), 472.
[17]	W.A. Strauss, Existence of solitary waves in higher dimensions,, Commun. Math. Phys., 55 (1977), 149.
[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.
[19]	K. Tanaka, Morse indices at critical points related to the Symmetric Mountain Pass Theorem and applications,, Comm. Partial Differential Equations, 14 (1989), 99.

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References:

[1]	A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in cri\-ti\-cal point theory and applications,, J. Funct. Anal., 14 (1973), 349.
[2]	A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications,, Trans. Amer. Math. Soc., 267 (1981), 1.
[3]	A. Bahri and P.L. Lions, Morse index of some mini-max critical points,, Comm. Pure Appl. Math., 41 (1988), 1027.
[4]	H. Berestycki and P.L. Lions, Nonlinear scalar field equations, I and II,, Arch. Rat. Mech. Anal., 82 (1983), 313.
[5]	F.A. Berezin and M.A. Shubin, The Schr\"odinger equation,, Mathematics and its Applications (Soviet Series) 66, (1991).
[6]	P. Bolle, On the Bolza Problem,, J. Differential Equations, 152 (1999), 274.
[7]	P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in non-homogeneous boundary value problems,, Manuscripta Math., 101 (2000), 325.
[8]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer, (2011).
[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.
[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.
[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.
[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.
[13]	S.I. Pohozaev, On the global fibering method in nonlinear variational problems,, Proc. Steklov Inst. Math., 219 (1997), 281.
[14]	P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations,, CBMS Regional Conference Series in Mathematics, 65 (1986).
[15]	P.H. Rabinowitz, Multiple critical points of perturbed symmetric functionals,, Trans. Amer. Math. Soc., 272 (1982), 753.
[16]	A. Salvatore, Multiple radial solutions for a superlinear elliptic problem in $\mathbb R^N$,, Dynam. Systems and Applications, 4 (2004), 472.
[17]	W.A. Strauss, Existence of solitary waves in higher dimensions,, Commun. Math. Phys., 55 (1977), 149.
[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.
[19]	K. Tanaka, Morse indices at critical points related to the Symmetric Mountain Pass Theorem and applications,, Comm. Partial Differential Equations, 14 (1989), 99.

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