Weak solutions of semilinear elliptic equation involving Dirac mass

Chen, Huyuan; Felmer, Patricio; Yang, Jianfu

doi:10.1016/j.anihpc.2017.08.001

Chen, Huyuan ; Felmer, Patricio ; Yang, Jianfu

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 729-750

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Résumé

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

{\begin{matrix} - Δ u = V u^{p} + k δ_{0} in R^{N}, \\ \lim_{| x | \to + \infty} u (x) = 0, \end{matrix}

where

N > 2

,

p > 0

,

k > 0

,

δ_{0}

is the Dirac mass at the origin and the potential V is locally Lipchitz continuous in

R^{N} ∖ {0}

, with non-empty support and satisfying

0 \leq V (x) \leq \frac{σ_{1}}{| x |^{a_{0}} (1 + | x |^{a_{\infty} - a_{0}})},

with

a_{0} < N

,

a_{0} < a_{\infty}

and

σ_{1} > 0

. We obtain two positive solutions of (1) with additional conditions for parameters on

a_{\infty}, a_{0}

, p and k. The first solution is a minimal positive solution and the second solution is constructed via Mountain Pass Theorem.

MR Zbl

DOI : 10.1016/j.anihpc.2017.08.001

Classification : 35J60, 35J20
Keywords: 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},
     year = {2018},
     publisher = {Elsevier},
     volume = {35},
     number = {3},
     doi = {10.1016/j.anihpc.2017.08.001},
     mrnumber = {3778650},
     zbl = {1393.35057},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.08.001/}
}

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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

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