Unusual existence theorems for nonlocal inhomogeneous elliptic equations

In this paper, we prove two unusual existence theorems for nonlocal inhomogeneous elliptic equations. A very particular case of one of them reads as follows: Let k:[0,1]×R→R be a continuous function such that k(t,⋅) is nondecreasing for all x∈[0,1] and k(x0,⋅) is not constant for some x0∈[0,1]. Then, for every a>infξ∈R⁡∫01K(x,ξ)dx (where K(x,ξ)=∫0ξk(x,t)dt) and for every convex set S⊆C0([0,1]) dense in L2([0,1]), there exists δ˜∈S having the following property: for every continuous function f:[0,1]×R→R and for every nonincreasing nonpositive function γ:R→R, there exists ϵ>0 such that, for each λ∈[−ϵ,ϵ], the problem−u″=λf(x,u)+γ(∫01K(x,u(x))dx)k(x,u)+δ˜(x)in[0,1],u(0)=u(1)=0,∫01K(x,u(x))dx<a has at least one classical solution. Besides the presence of the convex dense set S, the most important novelty is that δ˜ is fully independent of f and γ. Moreover, we have the localization of the found solution expressed by the inequality ∫ΩK(x,u(x))dx<a.