Concentration of solutions for some singularly perturbed mixed problems: Asymptotics of minimal energy solutions
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1 , p. 37-56
doi : 10.1016/j.anihpc.2009.06.005
URL stable : http://www.numdam.org/item?id=AIHPC_2010__27_1_37_0

Classification:  35B25,  35B34,  35J20,  35J60
In this paper we carry on the study of asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions, started in the first paper [J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems: Existence results, Arch. Ration. Mech. Anal., in press]. Here we are mainly interested in the analysis of the location and shape of least energy solutions when the singular perturbation parameter tends to zero. We show that in many cases they coincide with the new solutions produced in [J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems: Existence results, Arch. Ration. Mech. Anal., in press].

Bibliographie

[1] A. Ambrosetti, A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on ${ℝ}^{n}$, Progr. Math. vol. 240, Birkhäuser (2005)

[2] A. Ambrosetti, A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Stud. Adv. Math. vol. 104, Cambridge Univ. Press, Cambridge (2007) Zbl 1125.47052

[3] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381 Zbl 0273.49063

[4] H. Berestycki, L. Caffarelli, L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 no. 1–2 (1997), 69-94 Zbl 1079.35513 |

[5] E. Colorado, I. Peral, Eigenvalues and bifurcation for elliptic equations with mixed Dirichlet–Neumann boundary conditions related to Caffarelli–Kohn–Nirenberg inequalities, Topol. Methods Nonlinear Anal. 23 no. 2 (2004), 239-273 Zbl 1075.35014

[6] L. Damascelli, F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoamericana 20 no. 1 (2004), 67-86 Zbl 1330.35146 |

[7] E.N. Dancer, S. Yan, Interior and boundary peak solutions for a mixed boundary value problem, Indiana Univ. Math. J. 48 no. 4 (1999), 1177-1212 Zbl 0948.35055

[8] M. Del Pino, P. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J. 48 no. 3 (1999), 883-898 Zbl 0932.35080

[9] J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems: Existence results, Arch. Ration. Mech. Anal., in press

[10] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in ${ℝ}^{n}$, Adv. Math. (Suppl. Stud. A) 7 (1981), 369-402

[11] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin/Heidelberg/New York/Tokyo (1983) Zbl 0691.35001

[12] M.K. Kwong, Uniqueness of positive solutions of $-\Delta u+u-{u}^{p}=0$ in ${ℝ}^{n}$, Arch. Ration. Mech. Anal. 105 (1989), 243-266 Zbl 0676.35032

[13] C.S. Lin, W.M. Ni, I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations 72 (1988), 1-27 Zbl 0676.35030

[14] W.M. Ni, J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math. 48 (1995), 731-768 Zbl 0838.35009

[15] W.M. Ni, I. Takagi, On the shape of least-energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math. 41 (1991), 819-851 Zbl 0754.35042

[16] W.M. Ni, I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), 247-281 Zbl 0796.35056

[17] G. Stampacchia, Problemi al contorno ellitici, con dati discontinui, dotati di soluzionie hölderiane, Ann. Mat. Pura Appl. (4) 51 (1960), 1-37 Zbl 0204.42001