September  2012, 11(5): 1935-1957. doi: 10.3934/cpaa.2012.11.1935

Symmetries and blow-up phenomena for a Dirichlet problem with a large parameter

1. 

Dipartimento di Matematica, Universitá di Roma Tre, Largo San Leonardo Murialdo, 1, I-00146 Roma

2. 

Dipartimento di Matematica, Università degli Studi "Roma Tre", Largo S. Leonardo Murialdo 1, Rome, 00146, Italy

Received  May 2011 Revised  September 2011 Published  March 2012

\noindent For the Dirichlet problem $-\Delta u+\lambda V(x) u=u^p$ in $\Omega \subset \mathbb R^N$, $N\geq 3$, in the regime $\lambda \to +\infty$ we aim to give a description of the blow-up mechanism. For solutions with symmetries an uniform bound on the ``invariant" Morse index provides a localization of the blow-up orbits in terms of c.p.'s of a suitable modified potential. The main difficulty here is related to the presence of fixed points for the underlying group action.
Citation: Pierpaolo Esposito, Maristella Petralla. Symmetries and blow-up phenomena for a Dirichlet problem with a large parameter. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1935-1957. doi: 10.3934/cpaa.2012.11.1935
References:
[1]

A. Ambrosetti, A. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I,, Comm. Math. Phys., 235 (2003), 427. doi: 10.1007/s00220-003-0811-y.

[2]

A. Ambrosetti, A. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. II,, Indiana Univ. Math. J., 53 (2004), 297. doi: 10.1512/iumj.2004.53.2400.

[3]

M. Badiale and T. D'Aprile, Concentration around a sphere for a singularly perturbed Schrödinger equation,, Nonlinear Anal., 49 (2002), 947. doi: 10.1016/S0362-546X(01)00717-9.

[4]

V. Benci and T. D'Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential,, J. Differential Equations, 184 (2002), 109. doi: 10.1006/jdeq.2001.4138.

[5]

D. Cao, E. N. Dancer, E. Noussair and S. Yan, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems,, Discrete Contin. Dyn. Syst., 2 (1996), 221. doi: 10.3934/dcds.1996.2.221.

[6]

D. Cao and E. Noussair, Multi-peak solutions for a singularly perturbed semilinear elliptic problem,, J. Differential Equations, 166 (2000), 266. doi: 10.1006/jdeq.2000.3795.

[7]

E. N. Dancer, Some singularly perturbed problems on annuli and a counterexample to a problem of Gidas, Ni and Nirenberg,, Bull. London Math. Soc., 29 (1997), 322. doi: 10.1112/S0024609396002391.

[8]

E. N. Dancer and S. Yan, A singularly perturbed elliptic problem in bounded domains with nontrivial topology,, Adv. Differential Equations, 4 (1999), 347.

[9]

E. N. Dancer and S. Yan, Effect of the domain geometry on the existence of multipeak solution for an elliptic problem,, Topol. Methods Nonlinear Anal., 14 (1999), 1.

[10]

E. N. Dancer and S. Yan, A new type of concentration solutions for a singularly perturbed elliptic problem,, Trans. Amer. Math. Soc., 359 (2007), 1765. doi: 10.1090/S0002-9947-06-04386-8.

[11]

E. N. Dancer and J. Wei, On the effect of domain topology in a singular perturbation problem,, Topol. Methods Nonlinear Anal., 11 (1998), 227.

[12]

T. D'Aprile, On a class of solutions with non-vanishing angular momentum for nonlinear Schrödinger equations,, Differential Integral Equations, 16 (2003), 349.

[13]

M. del Pino, P. Felmer and J. Wei, On the role of distance function in some singular perturbation problems,, Comm. Partial Differential Equations, 25 (2000), 155. doi: 10.1080/03605300008821511.

[14]

O. Druet, F. Robert and J. Wei, The Lin-Ni's problem for mean convex domains,, preprint, ().

[15]

P. Esposito, G. Mancini, Sanjiban Santra and P. N. Srikanth, Asymptotic behavior of radial solutions for a semilinear elliptic problem on an annulus through Morse index,, J. Differential Equations, 239 (2007), 1. doi: 10.1016/j.jde.2007.04.008.

[16]

P. Esposito and M. Petralla, Pointwise blow-up phenomena for a Dirichlet problem,, Comm. Partial Differential Equations, 36 (2011), 1654. doi: 10.1080/03605302.2011.574304.

[17]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb R^N$,, J. Math. Pures Appl., 87 (2007), 537. doi: 10.1016/j.matpur.2007.03.001.

[18]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2$^{nd}$ edition, (1983).

[19]

M. Grossi and A. Pistoia, On the effect of critical points of distance function in superlinear elliptic problems,, Adv. Differential Equations, 5 (2000), 1397.

[20]

M. K. Kwong, Uniqueness of positive solutions of positive solutions of $\Delta u - u + u^p =0$ in $\mathbb R^N$,, Arch. Rat. Mech. Anal., 105 (1989), 243. doi: 10.1007/BF00251502.

[21]

Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations,, Comm. Pure Appl. Math., 51 (1998), 1445.

[22]

F. Mahmoudi and A. Malchiodi, Concentration on minimal submanifolds for a singularly perturbed Neumann problem,, Adv. Math., 209 (2007), 460. doi: 10.1016/j.aim.2006.05.014.

[23]

A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains,, Geom. Funct. Anal., 15 (2005), 1162. doi: 10.1007/s00039-005-0542-7.

[24]

A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem,, Comm. Pure Appl. Math., 55 (2002), 1507. doi: 10.1002/cpa.10049.

[25]

A. Malchiodi and M. Montenegro, Multidimensional boundary-layers for a singularly perturbed Neumann problem,, Duke Math. J., 124 (2004), 105. doi: 10.1215/S0012-7094-04-12414-5.

[26]

A. Malchiodi, W. M. Ni and J. Wei, Multiple clustered layer solutions for semilinear Neumann problems on a ball,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 22 (2005), 143. doi: 10.1016/j.anihpc.2004.05.003.

[27]

R. Molle and D. Passaseo, Concentration phenomena for solutions of superlinear elliptic problems,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 23 (2006), 63. doi: 10.1016/j.anihpc.2005.02.002.

[28]

W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 45 (1998), 9.

[29]

W. M. Ni and I. Takagi, On the shape of least-energy solution to a semilinear Neumann problem,, Comm. Pure Appl. Math., 44 (1991), 819. doi: 10.1002/cpa.3160440705.

[30]

W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem,, Duke Math. J., 70 (1993), 247. doi: 10.1215/S0012-7094-93-07004-4.

[31]

W. M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems,, Comm. Pure Math. Appl., 48 (1995), 731. doi: 10.1002/cpa.3160480704.

[32]

Maristella Petralla, "Asymptotic Analysis for a Singularly Perturbed Dirichlet Problem,", Ph.D thesis, (2010).

[33]

M. Petralla, Non existence of bounded Morse index solutions for a super-critical Dirichlet problem with a large parameter,, in preparation., ().

[34]

A. Pistoia, The role of the distance function in some singular perturbation problem,, Methods Appl. Anal., 8 (2001), 301.

[35]

B. Ruf and P. N. Srikanth, Singularly perturbed elliptic equations with solutions concentrating on a $1-$dimensional orbit,, J. Eur. Math. Soc., 12 (2010), 413. doi: 10.4171/JEMS/203.

[36]

J. Wei, On the construction of single-peaked solutions of a singularly perturbed semilinear Dirichlet problem,, J. Differential Equations, 129 (1996), 315. doi: 10.1006/jdeq.1996.0120.

[37]

J. Wei, On the interior spike solutions for some singular perturbation problems,, Proc. Royal Soc. of Edinburgh Sect. A, 128 (1998), 849. doi: 10.1017/S030821050002182X.

[38]

J. Wei, On the effect of the domain geometry in singular perturbation problems,, Differential Integral Equations, 13 (2000), 15.

show all references

References:
[1]

A. Ambrosetti, A. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I,, Comm. Math. Phys., 235 (2003), 427. doi: 10.1007/s00220-003-0811-y.

[2]

A. Ambrosetti, A. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. II,, Indiana Univ. Math. J., 53 (2004), 297. doi: 10.1512/iumj.2004.53.2400.

[3]

M. Badiale and T. D'Aprile, Concentration around a sphere for a singularly perturbed Schrödinger equation,, Nonlinear Anal., 49 (2002), 947. doi: 10.1016/S0362-546X(01)00717-9.

[4]

V. Benci and T. D'Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential,, J. Differential Equations, 184 (2002), 109. doi: 10.1006/jdeq.2001.4138.

[5]

D. Cao, E. N. Dancer, E. Noussair and S. Yan, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems,, Discrete Contin. Dyn. Syst., 2 (1996), 221. doi: 10.3934/dcds.1996.2.221.

[6]

D. Cao and E. Noussair, Multi-peak solutions for a singularly perturbed semilinear elliptic problem,, J. Differential Equations, 166 (2000), 266. doi: 10.1006/jdeq.2000.3795.

[7]

E. N. Dancer, Some singularly perturbed problems on annuli and a counterexample to a problem of Gidas, Ni and Nirenberg,, Bull. London Math. Soc., 29 (1997), 322. doi: 10.1112/S0024609396002391.

[8]

E. N. Dancer and S. Yan, A singularly perturbed elliptic problem in bounded domains with nontrivial topology,, Adv. Differential Equations, 4 (1999), 347.

[9]

E. N. Dancer and S. Yan, Effect of the domain geometry on the existence of multipeak solution for an elliptic problem,, Topol. Methods Nonlinear Anal., 14 (1999), 1.

[10]

E. N. Dancer and S. Yan, A new type of concentration solutions for a singularly perturbed elliptic problem,, Trans. Amer. Math. Soc., 359 (2007), 1765. doi: 10.1090/S0002-9947-06-04386-8.

[11]

E. N. Dancer and J. Wei, On the effect of domain topology in a singular perturbation problem,, Topol. Methods Nonlinear Anal., 11 (1998), 227.

[12]

T. D'Aprile, On a class of solutions with non-vanishing angular momentum for nonlinear Schrödinger equations,, Differential Integral Equations, 16 (2003), 349.

[13]

M. del Pino, P. Felmer and J. Wei, On the role of distance function in some singular perturbation problems,, Comm. Partial Differential Equations, 25 (2000), 155. doi: 10.1080/03605300008821511.

[14]

O. Druet, F. Robert and J. Wei, The Lin-Ni's problem for mean convex domains,, preprint, ().

[15]

P. Esposito, G. Mancini, Sanjiban Santra and P. N. Srikanth, Asymptotic behavior of radial solutions for a semilinear elliptic problem on an annulus through Morse index,, J. Differential Equations, 239 (2007), 1. doi: 10.1016/j.jde.2007.04.008.

[16]

P. Esposito and M. Petralla, Pointwise blow-up phenomena for a Dirichlet problem,, Comm. Partial Differential Equations, 36 (2011), 1654. doi: 10.1080/03605302.2011.574304.

[17]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb R^N$,, J. Math. Pures Appl., 87 (2007), 537. doi: 10.1016/j.matpur.2007.03.001.

[18]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2$^{nd}$ edition, (1983).

[19]

M. Grossi and A. Pistoia, On the effect of critical points of distance function in superlinear elliptic problems,, Adv. Differential Equations, 5 (2000), 1397.

[20]

M. K. Kwong, Uniqueness of positive solutions of positive solutions of $\Delta u - u + u^p =0$ in $\mathbb R^N$,, Arch. Rat. Mech. Anal., 105 (1989), 243. doi: 10.1007/BF00251502.

[21]

Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations,, Comm. Pure Appl. Math., 51 (1998), 1445.

[22]

F. Mahmoudi and A. Malchiodi, Concentration on minimal submanifolds for a singularly perturbed Neumann problem,, Adv. Math., 209 (2007), 460. doi: 10.1016/j.aim.2006.05.014.

[23]

A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains,, Geom. Funct. Anal., 15 (2005), 1162. doi: 10.1007/s00039-005-0542-7.

[24]

A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem,, Comm. Pure Appl. Math., 55 (2002), 1507. doi: 10.1002/cpa.10049.

[25]

A. Malchiodi and M. Montenegro, Multidimensional boundary-layers for a singularly perturbed Neumann problem,, Duke Math. J., 124 (2004), 105. doi: 10.1215/S0012-7094-04-12414-5.

[26]

A. Malchiodi, W. M. Ni and J. Wei, Multiple clustered layer solutions for semilinear Neumann problems on a ball,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 22 (2005), 143. doi: 10.1016/j.anihpc.2004.05.003.

[27]

R. Molle and D. Passaseo, Concentration phenomena for solutions of superlinear elliptic problems,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 23 (2006), 63. doi: 10.1016/j.anihpc.2005.02.002.

[28]

W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 45 (1998), 9.

[29]

W. M. Ni and I. Takagi, On the shape of least-energy solution to a semilinear Neumann problem,, Comm. Pure Appl. Math., 44 (1991), 819. doi: 10.1002/cpa.3160440705.

[30]

W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem,, Duke Math. J., 70 (1993), 247. doi: 10.1215/S0012-7094-93-07004-4.

[31]

W. M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems,, Comm. Pure Math. Appl., 48 (1995), 731. doi: 10.1002/cpa.3160480704.

[32]

Maristella Petralla, "Asymptotic Analysis for a Singularly Perturbed Dirichlet Problem,", Ph.D thesis, (2010).

[33]

M. Petralla, Non existence of bounded Morse index solutions for a super-critical Dirichlet problem with a large parameter,, in preparation., ().

[34]

A. Pistoia, The role of the distance function in some singular perturbation problem,, Methods Appl. Anal., 8 (2001), 301.

[35]

B. Ruf and P. N. Srikanth, Singularly perturbed elliptic equations with solutions concentrating on a $1-$dimensional orbit,, J. Eur. Math. Soc., 12 (2010), 413. doi: 10.4171/JEMS/203.

[36]

J. Wei, On the construction of single-peaked solutions of a singularly perturbed semilinear Dirichlet problem,, J. Differential Equations, 129 (1996), 315. doi: 10.1006/jdeq.1996.0120.

[37]

J. Wei, On the interior spike solutions for some singular perturbation problems,, Proc. Royal Soc. of Edinburgh Sect. A, 128 (1998), 849. doi: 10.1017/S030821050002182X.

[38]

J. Wei, On the effect of the domain geometry in singular perturbation problems,, Differential Integral Equations, 13 (2000), 15.

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