How to break the uniqueness of W1,ploc(Ω)Wloc1,p(Ω) -solutions for very singular elliptic problems by non-local terms

Abstract

In this paper, we are going to show existence of branches of bifurcation of positive W1,ploc(Ω)Wloc1,p(Ω) -solutions for the very singular non-local λλ -problem −⎛⎝⎜∫Ωg(x,u)dx⎞⎠⎟rΔpu=λ(a(x)u−δ+b(x)uβ) in Ω,u>0 in Ω and u=0 on ∂Ω, −(∫Ωg(x,u)dx)rΔpu=λ(a(x)u−δ+b(x)uβ) in Ω,u>0 in Ω and u=0 on ∂Ω, where Ω⊂RNΩ⊂RN is a smooth bounded domain, δ>0δ>0 , 0<β<p−10<β<p−1 , a and b are nonnegative measurable functions and g is a positive continuous function. Our approach is based on sub- supersolutions techniques, fixed point theory, in the study of W1,ploc(Ω)Wloc1,p(Ω) -topology of a solution application and a new comparison principle for sub-supersolutions in W1,ploc(Ω)Wloc1,p(Ω) to a problem with p-Laplacian operator perturbed by a very singular term at zero and sublinear at infinity.