Vol. 305, No. 1, 2020

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Ground state solutions of polyharmonic equations with potentials of positive low bound

Caifeng Zhang, Jungang Li and Lu Chen

Vol. 305 (2020), No. 1, 353–384

The purpose of this paper is threefold. First, we establish the critical Adams inequality on the whole space with restrictions on the norm

(mun m n m + τun m n m )m n

for any τ > 0. Second, we prove a sharp concentration-compactness principle for singular Adams inequalities and a new Sobolev compact embedding in Wm,2(2m). Third, based on the above results, we give sufficient conditions for the existence of ground state solutions to the following polyharmonic equation with singular exponential nonlinearity

(Δ)mu + V (x)u = f(x,u) |x|β  in 2m, (1)

where 0 < β < 2m, V (x) has a positive lower bound and f(x,t) behaves like exp(α|t|2) as t +. Furthermore, when β = 0, in light of the principle of the symmetric criticality and the radial lemma, we also derive the existence of nontrivial weak solutions by assuming f(x,t) and V (x) are radially symmetric with respect to x and f(x,t) = o(t) at origin. Thus our main theorems extend the recent results on bi-Laplacian in 4 by Chen, Li, Lu and Zhang (2018) to (Δ)m in m.

ground state solutions, Adams inequality, concentration-compactness principle, exponential growth
Mathematical Subject Classification 2010
Primary: 26D10, 35A23, 46E35
Received: 31 May 2018
Revised: 13 September 2018
Accepted: 9 October 2019
Published: 17 March 2020
Caifeng Zhang
Department of Applied Mathematics, School of Mathematics and Physics
University of Science and Technology Beijing
Jungang Li
Department of Mathematics
Brown University
Providence, RI
United States
Lu Chen
School of Mathematics and Statistics
Beijing Institute of Technology