Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ⁴

Abstract

In 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 to x and {f(x,t)=o(t)} as {t\rightarrow 0} .

Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc

“…They confirmed this conjecture indeed holds for any bounded and convex domain in R 2 in [25]. They proved Theorem C via an argument from local inequalities to global ones using the level sets of functions under consideration developed by Lam and Lu in [13,14] (see also [5,16,39]), together with the Riemann mapping theorem.…”

Existence and Non-existence of Ground states of bi-harmonic equations involving constant and degenerate Rabinowitz potentials

“…Combining the above estimate with the concentration compactness principle for the Adams inequality which was established in [8] in W 2,2 (R 4 ), one can derive that there Proof of Theorem 2.6. We will prove that if V ∞ < γ * , then m V is achieved by some u.…”

“…As we mentioned before, the loss of compactness for equations (1.1) may be produced not only by the concentration phenomena but also by the vanishing phenomena. In the literature, in order to exclude the vanishing phenomena, one can introduce the coercive potential (see [40,39,20]), or apply some symmetrization argument (see [9,8,6,2,15,34,28]). However, for our bi-harmonic equation (2.1), the symmetrization argument fails, since the nonlinearity f (t) in Theorem 2.2 needn't be an odd function.…”

“…In the case m = 2, the existence of a nontrivial solution to equation (1.1) was obtained in [8] under the assumption that (1.3) holds (see also [6]), and in [36] under the assumption that (1.4) holds. The existence of a nontrivial solution to (1.1) under the assumption weaker than both (1.3) and (1.4) was established in [9].…”

A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrödinger equations

“…It is well known that the imbedding H 1 (R 2 ) ↩→ L 2 (R 2 ) is continuous but not compact, even in the class of radial functions. For the existence of nontrivial solutions in this case, one can refer to [4,9,14,21,24,29,34] and the references therein.…”