Biharmonic systems involving multiple Rellich-type potentials and critical Rellich-Sobolev nonlinearities

In this paper, the minimizers of a Rellich-Sobolev constant are firstly investigated. Secondly, a system of biharmonic equations is investigated, which involves multiple Rellich-type terms and strongly coupled critical Rellich-Sobolev terms. The existence of nontrivial solutions to the system is established by variational arguments.

Furthermore, there exists a constant C(t) > 0 such that the following Rellich-Sobolev inequality holds ( [4,9,17]): In this paper, we use H := H 2 0 (Ω) to denote the completion of C ∞ 0 (Ω) with respect to the norm ( Ω |∆u| 2 dx) 1/2 . The functional of (1) is defined on H × H as follows Then J is said to satisfy the (P S) c condition if any sequence {(u n , v n )} ⊂ H ×H satisfying J(u n , v n ) → c, J (u n , v n ) → 0 in the dual space (H × H) −1 , has a subsequence converging strongly in H × H. For all λ, µ <μ, ν, σ > 0, 0 ≤ t < 4, 1 < q ≤ 2 * (t), a ∈ R N , by (2) and (3) the following best constants are well defined: |x − a| t dx paper). The minimizers of A(λ, µ, t, ν, σ, q) are related to the problem: By a ground state solution (u 0 , v 0 ) to (6) we mean that (u 0 , v 0 ) = (0, 0) is a solution to (6) and has the least energy among all solutions. That is, the solution (u 0 , v 0 ) is also a minimizer of A(λ, µ, t, ν, σ, q). For simplicity, we denote Second order singular elliptic equations related to (1) have been studied and the existence of solutions to the problems have been established (e.g. [6,10,11,13,14,19]). Biharmonic problems involving the Rellich inequality have been also studied (e.g. [2,4,5,8,9,12]). However, the system (1) has not been investigated and is attractive for its multiple singularity and multiple Rellich-Sobolev critical nonlinearities. In this paper, we will study the minimizers of A(λ, µ, t, ν, σ, q) and investigate the mountain-pass type solution to (1), which has the least energy among all solutions of (1).
Define the norm of H × H: For all N ≥ 5, 0 ≤ µ <μ, 0 ≤ t < 4, the following constants are well defined: The following conditions are also needed: (H 3 ) There exists l, 1 ≤ l ≤ k , such that 1 < q l < 2, 0 ≤ λ l = µ l < λ * and where λ * := 1 16 be the family of minimizers to S(µ, t) for all µ ∈ [0,μ) (e.g. [12], see also Lemma 2.1 of this paper). For all ν, σ > 0, define The main results of this paper are contained in the following theorems. We investigate the minimizers to A(λ, µ, t, ν, σ, q) and prove the existence of mountain-pass type solution to (1). The result is new to the best of our knowledge.
This paper is organized as follows: Some preliminary results are established in Section 2, Theorem 1.1 is proved in Section 3 and Theorem 1.2 is verified in Section 4. In the following argument, We always denote positive constants as C and omit dx in integrals for convenience.
2. Preliminary results. First we need to establish some preliminary results.
Then, the best constant S(µ, t) defined in (4) has the unique minimizers, up to multiplicative constants, and so, they satisfy Moreover, U µ (x) = U µ (|x|) > 0 is radially symmetric and decreasing. By setting ρ = |x| there holds that Then a standard argument shows that {(u n , v n )} is bounded in H × H (e.g. [7]). Up to a subsequence we have that Since 0 < t j < 4, 2 < p j < 2 * , 1 ≤ j ≤ k, by the concentration compactness principle ( [10,16,17]), there exists a subsequence (still denoted by {(u n , v n )}) and nonnegative real numbers µ aj , γ aj , ν aj , 1 ≤ j ≤ k, such that the following convergences hold in the sense of measures: where δ x is the Dirac mass at x. From (5) it follows that Now we consider the possibility of concentration at the points a j (1 ≤ j ≤ k). For all ε > 0 small enough, let ϕ j ε (x) ∈ C ∞ 0 (Ω) be a radial cut-off function centered at a j such that ϕ j ε (x) ≡ 1 in the ball B ε (a j ), ϕ j ε (x) = 0 in R N \ B 2ε (a j ) and 0 ≤ ϕ j ε (x) ≤ 1. Then by (7)- (9) we deduce that n→∞ Ω (u n ∆u n + v n ∆v n )∆ϕ j ε = 0. Consequently, From (10) and (11) we derive that which implies that Furthermore, by (9) we have that If there exists j ∈ {1, 2, · · · , k} such that ν aj = 0, by (12) we have that which contradicts the assumption c < c * . Then up to a subsequence, (u n , v n ) → (u, v) strongly in H × H.
Let y µ ε be the minimizers obtained as in Lemma 2.1. Take R > 0 small enough such that B 2R (a) ∈ Ω. Choose the radial cut-off function The following asymptotic properties hold.
Proof. For simplicity we set a = 0 and define A := A(λ, µ, t, ν, σ, q), By the Ekeland's variational principle, we can choose a minimizing sequence where f n → 0 and g n → 0 as n → ∞ in the dual space D −1 of D.