Polyharmonic equations with critical exponential growth in the whole space $ \mathbb{R}^{n}$

Abstract

In this paper, we apply the sharp Adams-type inequalities for the Sobolev space $W^{m,\frac{n}{m}}\left( \mathbb{R} ^{n}\right) $ for any positive real number $m$ less than $n$, established by Ruf and Sani [46] and Lam and Lu [30,31], to study polyharmonic equations in $\mathbb{R}^{2m}$. We will consider the polyharmonic equations in $\mathbb{R}^{2m}$ of the form \[ \left( I-\Delta\right) ^{m}u=f(x,u)\text{ in }% \mathbb{R} ^{2m}. \] We study the existence of the nontrivial solutions when the nonlinear terms have the critical exponential growth in the sense of Adams' inequalities on the entire Euclidean space. Our approach is variational methods such as the Mountain Pass Theorem ([5]) without Palais-Smale condition combining with a version of a result due to Lions ([39,40]) for the critical growth case. Moreover, using the regularity lifting by contracting operators and regularity lifting by combinations of contracting and shrinking operators developed in [14] and [11], we will prove that our solutions are uniformly bounded and Lipschitz continuous. Finally, using the moving plane method of Gidas, Ni and Nirenberg [22,23] in integral form developed by Chen, Li and Ou [12] together with the Hardy-Littlewood-Sobolev type inequality instead of the maximum principle, we prove our positive solutions are radially symmetric and monotone decreasing about some point. This appears to be the first work concerning existence of nontrivial nonnegative solutions of the Bessel type polyharmonic equation with exponential growth of the nonlinearity in the whole Euclidean space.