# American Institute of Mathematical Sciences

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

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

 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 $N =3$.
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

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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), 349-381.  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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