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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[14]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

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

[33]

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

[34]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[14]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

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

[33]

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

[34]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

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

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