Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials
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Calculus of Variations
Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials Lu Chen1 · Guozhen Lu2 · Maochun Zhu3 Received: 11 September 2019 / Accepted: 31 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness of a class of nonlinear functionals in H 2 R4 which are of their independent interests. (See Theorems 2.1 and 2.2.) Using this result and the principle of symmetric criticality, we can present a relationship between the existence of the nontrivial solutions to the semilinear bi-harmonic equation of the form (−)2 u + γ u = f (u) in R4 and the range of γ ∈ R+ , where f (s) is the general nonlinear term having the critical exponential growth at infinity. (See Theorem 2.7.) Though the existence of the nontrivial solutions for the bi-harmonic equation with the critical exponential growth has been studied in the literature, it seems that nothing is known so far about the existence of the ground-state solutions for this class of equations involving the trapping potential introduced by Rabinowitz (Z Angew Math Phys 43:27–42, 1992). Since the trapping potential is not necessarily symmetric, classical radial method cannot be applied to solve this problem. In order to overcome this difficulty, we first establish the existence of the ground-state solutions for the equation (−)2 u + V (x)u = λs exp(2|s|2 )) in R4 ,
(0.1)
when V (x) is a positive constant using the Fourier rearrangement and the Pohozaev identity. Then we will explore the relationship between the Nehari manifold and the corresponding limiting Nehari manifold to derive the existence of the ground state solutions for the Eq. (2.5) when V (x) is the Rabinowitz type trapping potential, namely it satisfies 0 < inf V (x) < sup V (x) = x∈R4
x∈R4
lim
|x|→+∞
V (x).
Communicated by P. Rabinowitz. The Lu Chen was partly supported by the National Natural Science Foundation of China (No. 11901031) and a grant from Beijing Institute of Technology (No. 3170012221903). The Maochun Zhu was supported by Natural Science Foundation of China (11601190 and 12071185), Natural Science Foundation of Jiangsu Province (BK20160483) and Jiangsu University Foundation Grant (16JDG043). Extended author information available on the last page of the article 0123456789().: V,-vol
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(See Theorem 2.8.) The same result and proof applies to the harmonic equation with the critical exponential growth involving the Rabinowitz type trapping potential in R2 . (See Theorem 2.9.) Mathematics Subject Classification Primary 46E35; 35J91 · Secondary 26D10
1 Introduction Let be an open domain in Rn . We will consider the following nonlinear partial differential equation with critical growth (−)m u = f (u) in ⊂ Rn ,
(1.1)
where m = 1 or 2. Equation (1.1) have been extensively studied by many authors in bounded and unbounded domains. In the case n > 2m, t
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