Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ4
AbstractIn this paper, we establish a sharp concentration-compactness principle associated with the singular Adams inequality on the second-order Sobolev spaces in{\mathbb{R}^{4}}. We also give a new Sobolev compact embedding which states{W^{2,2}(\mathbb{R}^{4})}is compactly embedded into{L^{p}(\mathbb{R}^{4},|x|^{-\beta}\,dx)}for{p\geq 2}and{0<\beta<4}. As applications, we establish the existence of ground state solutions to the following bi-Laplacian equation with critical nonlinearity:\displaystyle\Delta^{2}u+V(x)u=\frac{f(x,u)}{|x|^{\beta}}\quad\mbox{in }% \mathbb{R}^{4},where{V(x)}has a positive lower bound and{f(x,t)}behaves like{\exp(\alpha|t|^{2})}as{t\to+\infty}. 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 toxand{f(x,t)=o(t)}as{t\rightarrow 0}.