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We study the semilinear elliptic equation with negative potential in the section 2:in BR = B（0, R） （?） RN, 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 BR = B（0, R） （?） RN, 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/rκ, 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
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