Weak solutions of semilinear elliptic equation involving Dirac mass
Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 729-750.

In this paper, we study the elliptic problem with Dirac mass

{Δu=Vup+kδ0inRN,lim|x|+u(x)=0,
where N>2, p>0, k>0, δ0 is the Dirac mass at the origin and the potential V is locally Lipchitz continuous in RN{0}, with non-empty support and satisfying
0V(x)σ1|x|a0(1+|x|aa0),
with a0<N, a0<a and σ1>0. We obtain two positive solutions of (1) with additional conditions for parameters on a,a0, p and k. The first solution is a minimal positive solution and the second solution is constructed via Mountain Pass Theorem.

DOI : 10.1016/j.anihpc.2017.08.001
Classification : 35J60, 35J20
Mots clés : Weak solution, Mountain Pass theorem, Dirac mass
@article{AIHPC_2018__35_3_729_0,
     author = {Chen, Huyuan and Felmer, Patricio and Yang, Jianfu},
     title = {Weak solutions of semilinear elliptic equation involving {Dirac} mass},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {729--750},
     publisher = {Elsevier},
     volume = {35},
     number = {3},
     year = {2018},
     doi = {10.1016/j.anihpc.2017.08.001},
     mrnumber = {3778650},
     zbl = {1393.35057},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2017.08.001/}
}
TY  - JOUR
AU  - Chen, Huyuan
AU  - Felmer, Patricio
AU  - Yang, Jianfu
TI  - Weak solutions of semilinear elliptic equation involving Dirac mass
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2018
SP  - 729
EP  - 750
VL  - 35
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2017.08.001/
DO  - 10.1016/j.anihpc.2017.08.001
LA  - en
ID  - AIHPC_2018__35_3_729_0
ER  - 
%0 Journal Article
%A Chen, Huyuan
%A Felmer, Patricio
%A Yang, Jianfu
%T Weak solutions of semilinear elliptic equation involving Dirac mass
%J Annales de l'I.H.P. Analyse non linéaire
%D 2018
%P 729-750
%V 35
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2017.08.001/
%R 10.1016/j.anihpc.2017.08.001
%G en
%F AIHPC_2018__35_3_729_0
Chen, Huyuan; Felmer, Patricio; Yang, Jianfu. Weak solutions of semilinear elliptic equation involving Dirac mass. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 729-750. doi : 10.1016/j.anihpc.2017.08.001. http://www.numdam.org/articles/10.1016/j.anihpc.2017.08.001/

[1] Aviles, P. Local behaviour of the solutions of some elliptic equations, Commun. Math. Phys., Volume 108 (1987), pp. 177–192 | DOI | MR | Zbl

[2] Ambrosetti, A.; Rabinowitz, P. Dual variational methods in critical point theory and applications, J. Funct. Anal., Volume 14 (1973), pp. 349–381 | DOI | MR | Zbl

[3] Baras, P.; Pierre, M. Singularités éliminables pour des équations semi linéaires, Ann. Inst. Fourier Grenoble, Volume 34 (1984), pp. 185–206 | DOI | Numdam | MR | Zbl

[4] Baras, P.; Pierre, M. Critere d'existence de soltuions positive pour des equations semi-lineaires non monotones, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 2 (1985), pp. 185–212 | DOI | Numdam | MR | Zbl

[5] Bénilan, Ph.; Brezis, H. Nonlinear problems related to the Thomas–Fermi equation, J. Evol. Equ., Volume 3 (2003), pp. 673–770 | DOI | MR | Zbl

[6] Bidaut-Véron, M.F.; Hung, N.; Véron, L. Quasilinear Lane–Emden equations with absorption and measure data, J. Math. Pures Appl., Volume 102 (2014) no. 2, pp. 315–337 | MR | Zbl

[7] Bidaut-Véron, M.F.; Vivier, L. An elliptic semilinear equation with source term involving boundary measures: the subcritical case, Rev. Mat. Iberoam., Volume 16 (2000), pp. 477–513 | MR | Zbl

[8] Brezis, H. Variational Inequalities and Complementarity Problems, Wiley, Chichester (1980), pp. 53–73 (Proc. Internat. School, Erice) | MR | Zbl

[9] Caffarelli, L.; Gidas, B.; Spruck, J. Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math., Volume 42 (1989), pp. 271–297 | DOI | MR | Zbl

[10] Egnell, H. Asymptotic results for finite energy solutions of semilinear elliptic equations, J. Differ. Equ., Volume 98 (1992), pp. 34–56 | DOI | MR | Zbl

[11] Gidas, B.; Spruck, J. Global and local behaviour of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., Volume 34 (1981), pp. 525–598 | DOI | MR | Zbl

[12] Gmira, A.; Véron, L. Boundary singularities of solutions of some nonlinear elliptic equations, Duke Math. J., Volume 64 (1991), pp. 271–324 | DOI | MR | Zbl

[13] Han, Q.; Lin, F. Elliptic partial differential equations, Am. Math. Soc., Volume 1 (2011) | Zbl

[14] Lions, P. Isolated singularities in semilinear problems, J. Differ. Equ., Volume 38 (1980) no. 3, pp. 441–450 | DOI | MR | Zbl

[15] Marcus, M.; Véron, L. The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case, Arch. Ration. Mech. Anal., Volume 144 (1998), pp. 201–231 | DOI | MR | Zbl

[16] Marcus, M.; Véron, L. The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case, J. Math. Pures Appl., Volume 77 (1998), pp. 481–524 | DOI | MR | Zbl

[17] Marcus, M.; Véron, L. Removable singularities and boundary traces, J. Math. Pures Appl., Volume 80 (2001), pp. 879–900 | DOI | MR | Zbl

[18] Marcus, M.; Véron, L. The boundary trace and generalized B.V.P. for semilinear elliptic equations with coercive absorption, Commun. Pure Appl. Math., Volume 56 (2003), pp. 689–731 | DOI | MR | Zbl

[19] Mazzeo, R.; Pacard, F. A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis, J. Differ. Geom., Volume 44 (1996), pp. 331–370 | DOI | MR | Zbl

[20] Naito, Y.; Sato, T. Positive solutions for semilinear elliptic equations with singular forcing terms, J. Differ. Equ., Volume 235 (2007) no. 2, pp. 439–483 | DOI | MR | Zbl

[21] Pacard, F. Existence and convergence of positive weak solutions of Δu=uNN2 in bounded domains of RN , Calc. Var. Partial Differ. Equ., Volume 1 (1993), pp. 243–265 | DOI | MR | Zbl

[22] Véron, L. Elliptic equations involving measures, Stationary Partial Differential Equations, Handb. Differ. Equ., vol. I, North-Holland, Amsterdam, 2004, pp. 593–712 | MR | Zbl

Cité par Sources :