# American Institute of Mathematical Sciences

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 and 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), 427-466. 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), 297-329. 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), 947-985. 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), 109-138. 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), 221-236. 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), 266-289. 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), 322-326. 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), 347-368. [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), 1-38. [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), 1765-1790. 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), 227-248. [12] T. D'Aprile, On a class of solutions with non-vanishing angular momentum for nonlinear Schrödinger equations, Differential Integral Equations, 16 (2003), 349-384. [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), 155-177. doi: 10.1080/03605300008821511. [14] O. Druet, F. Robert and J. Wei, The Lin-Ni's problem for mean convex domains, preprint, arXiv:1103.3811. [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), 1-15. 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), 1654-1682. 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), 537-561. doi: 10.1016/j.matpur.2007.03.001. [18] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer-Verlag, Berlin, 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), 1397-1420. [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), 243-266. 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), 1445-1490. [22] F. Mahmoudi and A. Malchiodi, Concentration on minimal submanifolds for a singularly perturbed Neumann problem, Adv. Math., 209 (2007), 460-525. 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), 1162-1222. 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), 1507-1568. 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), 105-143. 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. Poincaré Anal. Non Linéaire, 22 (2005), 143-163. 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. Poincaré Anal. Non Linéaire, 23 (2006), 63-84. 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), 9-18. [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), 819-851. 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), 247-281. 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), 731-768. doi: 10.1002/cpa.3160480704. [32] Maristella Petralla, "Asymptotic Analysis for a Singularly Perturbed Dirichlet Problem," Ph.D thesis, University of Rome III, 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), 301-319. [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), 413-427. 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), 315-333. 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), 849-874. doi: 10.1017/S030821050002182X. [38] J. Wei, On the effect of the domain geometry in singular perturbation problems, Differential Integral Equations, 13 (2000), 15-45.

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), 427-466. 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), 297-329. 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), 947-985. 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), 109-138. 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), 221-236. 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), 266-289. 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), 322-326. 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), 347-368. [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), 1-38. [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), 1765-1790. 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), 227-248. [12] T. D'Aprile, On a class of solutions with non-vanishing angular momentum for nonlinear Schrödinger equations, Differential Integral Equations, 16 (2003), 349-384. [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), 155-177. doi: 10.1080/03605300008821511. [14] O. Druet, F. Robert and J. Wei, The Lin-Ni's problem for mean convex domains, preprint, arXiv:1103.3811. [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), 1-15. 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), 1654-1682. 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), 537-561. doi: 10.1016/j.matpur.2007.03.001. [18] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer-Verlag, Berlin, 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), 1397-1420. [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), 243-266. 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), 1445-1490. [22] F. Mahmoudi and A. Malchiodi, Concentration on minimal submanifolds for a singularly perturbed Neumann problem, Adv. Math., 209 (2007), 460-525. 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), 1162-1222. 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), 1507-1568. 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), 105-143. 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. Poincaré Anal. Non Linéaire, 22 (2005), 143-163. 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. Poincaré Anal. Non Linéaire, 23 (2006), 63-84. 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), 9-18. [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), 819-851. 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), 247-281. 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), 731-768. doi: 10.1002/cpa.3160480704. [32] Maristella Petralla, "Asymptotic Analysis for a Singularly Perturbed Dirichlet Problem," Ph.D thesis, University of Rome III, 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), 301-319. [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), 413-427. 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), 315-333. 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), 849-874. doi: 10.1017/S030821050002182X. [38] J. Wei, On the effect of the domain geometry in singular perturbation problems, Differential Integral Equations, 13 (2000), 15-45.
 [1] Futoshi Takahashi. Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data. Conference Publications, 2013, 2013 (special) : 729-736. doi: 10.3934/proc.2013.2013.729 [2] Zongming Guo, Zhongyuan Liu, Juncheng Wei, Feng Zhou. Bifurcations of some elliptic problems with a singular nonlinearity via Morse index. Communications on Pure and Applied Analysis, 2011, 10 (2) : 507-525. doi: 10.3934/cpaa.2011.10.507 [3] Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure and Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521 [4] Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661 [5] Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042 [6] Philippe Souplet, Juan-Luis Vázquez. Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 221-234. doi: 10.3934/dcds.2006.14.221 [7] Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure and Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697 [8] C. Y. Chan. Recent advances in quenching and blow-up of solutions. Conference Publications, 2001, 2001 (Special) : 88-95. doi: 10.3934/proc.2001.2001.88 [9] Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020 [10] Marina Chugunova, Chiu-Yen Kao, Sarun Seepun. On the Benilov-Vynnycky blow-up problem. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1443-1460. doi: 10.3934/dcdsb.2015.20.1443 [11] Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1 [12] Marek Fila, Hiroshi Matano. Connecting equilibria by blow-up solutions. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 155-164. doi: 10.3934/dcds.2000.6.155 [13] Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399 [14] W. Edward Olmstead, Colleen M. Kirk, Catherine A. Roberts. Blow-up in a subdiffusive medium with advection. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1655-1667. doi: 10.3934/dcds.2010.28.1655 [15] Yukihiro Seki. A remark on blow-up at space infinity. Conference Publications, 2009, 2009 (Special) : 691-696. doi: 10.3934/proc.2009.2009.691 [16] Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267 [17] Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086 [18] José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43 [19] Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315 [20] Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881

2021 Impact Factor: 1.273