Sharpened adams inequality and ground state solutions to the Bi-Laplacian equation in ℝ⁴

In this paper, we establish a sharp concentration-compactness principle associated with the singular Adams inequality on the second-order Sobolev spaces in ℝ⁴. We also give a new Sobolev compact embedding which states W2,2(ℝ⁴) is compactly embedded into L^p (ℝ⁴, x-βa dx) for p ≥ 2 and 0 < βa < 4. As applications,we establish the existence of ground state solutions to the following bi-Laplacian equation with critical nonlinearity: (Formula Presented) where V(x) has a positive lower bound and f(x, t) behaves like exp(αt2) as t →+∞. In the case β= 0, because of the loss of Sobolev compact embedding, we use the principle of symmetric criticality to obtain the existence of ground state solutions by assuming f(x, t) and V(x) are radial with respect to x and f(x, t) = o(t) as t → 0.