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 R-4. We also give a new Sobolev compact embedding which states W-2,W-2(R-4) is compactly embedded into Lp(R-4, vertical bar x vertical bar(-beta) dx) for p >= 2 and 0 < beta < 4. As applications, we establish the existence of ground state solutions to the following bi-Laplacian equation with critical nonlinearity: Delta(2)u + V(x)u = f(x, u/vertical bar x vertical bar)(beta) in R-4, where V(x) has a positive lower bound and f(x, t) behaves like exp(alpha vertical bar t vertical bar(2)) as t -> +infinity. In the case beta = 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.