Some Semilinear Elliptic Equations With Singular Potential

We study the semilinear elliptic equation with negative potential in the section 2:in B_R = B(0, R) (?) R^N, N≥3, whereβ> 0, q > 1, 0≤c≤c₀ and c₀ is the best constant in the Hardy inequality. We find that ifβ> 2 , it has no positive solutions. But for 0 <β< 2 , if and only if q∈(1, q⁺) it has positive solutions.Then we research the existence of solution of semilinear equation with negative potential of four orderin B_R = B(0, R) (?) R^N, N≥5 ,where q > 1, 0 < c < c₁ and c₁ is the best constant in the improved Hardy inequality. If 1 < q < (?), the equation has a nontrivial solution. And if q^- < q < q⁺, we find a solution u = A/r^κ, whereκ= (?), q⁺ and q^- are all critical points, A is related with N,κ.At last, we examine the singularly perturbed variational problemin two dimensional bounded domain, whereψ∈H²(Ω) .Our main result is a compactness theorem: if E_?(ψ_?) is uniformly bounded then (?) is compact in...