2018/01/19 by Jacques Giacomoni, Giacomoni, Jacques, Tuhina Mukherjee +3 · 1 citation
Mathematics · #Nonlinear Differential Equations Analysis #Differential Equations and Boundary Problems #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1801.06461
In this article, we prove the existence of at least three positive solutions\nfor the following nonlocal singular problem \(P_
la)
left
\n
beginsplit (-
De)su amp;=
la
fracf(u)uq,
;
; ugt;0
;
;
textin
;
;\n
Om,
u amp;= 0
;
;
textin
;
;
mb Rn
setminus
Om
endsplit
right.\n where (- De)s denotes the fractional Laplace operator for\ns\∈ (0,1), n>2s, q \∈ (0,1), la>0 and Om is smooth bounded domain\nin mb Rn. Here f :[0,\∞) \→ [0,\∞) is a continuous nondecreasing\nmap satisfying \lim\u\→ \∞ fracf(u)uq+1=0. We show that\nunder certain additional assumptions on f, (P_ la) possesses at least three\ndistinct solutions for a certain range of la. We use the method of\nsub-supersolutions and a critical point theorem by Amann citeamann to prove\nour results. Moreover, we prove a new existence result for a suitable infinite\nsemipositone nonlocal problem which played a crucial role to obtain our main\nresult and is of independent interest. medskip\n