Existence And Properties Of Positive Solutions To Two Kinds Of Elliptic Equations

We consider the following equations and whereΔ2=Δ(Δ)is the bi-harmonic operator,a,b,λ>Oare constants,1<q<2<p<4 and V∈G(R3,R).Suppose potential V represents a potential well with the bottom V-1(0),we focus on the seldom-studied case 2<p<4 due to the presence of the nonlocal term(fR3 |▽u|2dx)Δu.Today we will focus on the above situation.For the first equation,by combining the truncation technique and the parameter-dependent compactness lemma,we prove the existence of positive solutions for b small and λ large in the case 2<p<4.Moreover,we also analyse asymptotic behavior of the positive solutions as b→ 0 and λ→∞as well as the decay rate as |x|→∞.For the second equation,it is seldom-studied case since the(PS)condition is still unsolved on H1(R3).By the Ekeland variational principle,one positive solution is proved to be in a ball Bρ0(0).The approach to get another solutions needs novel constraint approach.We will apply the filtration of the Nehari manifold on the outside of the ball Bρ0(0)as follows Mλ,c={u∈M|‖u‖λ>ρ0 and I(u)<c} for some c>0,one key point is to seek suitable numbers ρ0 and c.