On the Elliptic Problems Involving Multisingular Inverse Square Potentials and Concave-Convex Nonlinearities

and Applied Analysis 3 conditions on masses and location of singularities for the minimum to be achieved. In 9 , both the case of the whole R and bounded domains are taken into account. To proceed, wemake somemotivations of the present paper. In 6 , the authors studied more general problem than problem 1.5 with μ ∈ 0, μ , s 0, and they proved that there exists Λ > 0 such that problem 1.5 has at least two positive solutions for all λ ∈ 0,Λ . A natural question is whether the above results remain true for problem Eλ withmultisingular inverse square potentials. In recent work 10 , the author studied problem 1.1 withQ x ≡ 1 on Ω and showed that there exists Λ > 0 such that problem 1.1 has at least two positive solutions for all λ ∈ 0,Λ . In this paper, we continue the study of 10 by considering the more general function Q x instead of Q x ≡ 1 and extend the results of 10 to the more general function Q x . For 0 ≤ μi < μ and ai ∈ Ω, i 1, 2, . . . , k, we can define the best constant Sμi inf u∈H\{0} ∫ Ω ( |∇u| − μi ( u2/|x − ai| )) dx (∫ Ω |u| ∗ dx )2/2∗ , 1.7 and from 11 , we get that Sμi is independent of Ω. For 0 ≤ μ < μ, 0 ≤ μi < μ, setting β √ μ − μ, γ √ μ β, γ ′ √ μ − β, βi √ μ − μi, γi √ μ βi, γ ′ i √ μ − βi, 1.8 the authors in 1, 2 proved that Sμi is attained in R N by the function Uμi x − ai ( 22∗β2 i )1/ 2∗−2 |x − ai| ′ i ( 1 |x − ai| 2∗−2 βi )2/ 2∗−2 , 1.9 and, moreover, for all ε > 0, V ai μi,ε x ε 2−N Uμi x − ai /ε solve the problem −Δu − μi |x − ai| u |u|2−2u in R \ {ai} 1.10 and satisfy ∫ RN ⎛ ⎝∣∇V ai μi,ε ∣∣2 − μi ∣V ai μi,ε ∣∣2 |x − ai| ⎞ ⎠dx ∫ RN ∣V ai μi,ε ∣2dx SN/2 μi . 1.11 4 Abstract and Applied Analysis Note that Sμ is a decreasing function of μ for μ ∈ 0, μ and Ui μi x 1 ( |x − ai| √ μ |x − ai| ′ k / √ μ )√μ 1.12 also attains Sμi for i 1, 2, . . . , k. Now we recall the following standard definition. Assume that X is a Banach space and X−1 is the dual space of X. The functional I ∈ C1 X,R is said to satisfy the Palais-Smale condition at level c PS c in short , if every sequence {un} ⊂ X satisfying I un → c and I ′ un → 0 in X−1 has a convergent subsequence. In this paper, we will take I Jλ and X H. To proceed, we need the following assumptions: H1 there exists an l ∈ {1, 2, . . . , k} such that SN/2 μl Q al 2−N /2 min { SN/2 μi Q ai 2−N , i 1, 2, . . . , k } , 1.13 H2 Q x is a positive bounded function on Ω, and there exists an x0 ∈ Ω such that Q x0 is a strict local maximum. Furthermore, there exists τ > √ μ − μlN / √ μ such that Q x0 QM max Ω Q x , Q x −Q x0 o |x − x0| ) as x −→ x0, Q x −Q al o |x − al| ) as x −→ al, 1.14 H3 0 ≤ μi < μ for every i 1, 2, . . . , k and ∑k i 1 μi < μ. We define the following constants: S inf u∈H1 0 Ω \{0} ∫ Ω ( |∇u| −ki 1 μi ( u2/|x − ai| )) dx (∫ Ω |u| ∗ dx )2/2∗ , 1.15 Λ0 ( 2 − q ( 2∗ − qQM ) 2−q / 2∗−2 ( 2∗ − 2 2∗ − q ) |Ω|− 2∗−q /2∗ S 2∗ 2−q / 2 2∗−2 q/2. 1.16 The main result of this paper is the following theorem. Abstract and Applied Analysis 5 Theorem 1.1. Assume that conditions H1 – H3 hold; then one has the following. i If λ ∈ 0,Λ0 , then problem Eλ has at least one positive solution. ii If λ ∈ 0, q/2 Λ0 , then problem Eλ has at least two positive solutions. This paper is organized as follows. In Section 2, we give some properties of Nehari manifold. In Sections 3 and 4, we complete proofs of Theorem 1.1. At the end of this section, we explain some notations employed in this paper. L Ω, |x−ai| denotes the usual weighted L Ω space with the weight |x−ai|. |Ω| is the Lebesguemeasure ofΩ. Br x is a ball centered at x with radius r. O ε denotes |O ε |/εt ≤ C, and on 1 denotes on 1 → 0 as n → ∞. C, Ci will denote various positive constants and omit dx in the integration for convenience. 2. Nehari Manifold In this section, we will give some properties of Nehari manifold. As the energy functional Jλ is not bounded below on H, it is useful to consider the functional on the Nehari manifold Mλ { u ∈ H \ {0} : J ′ λ u , u 〉 0 } . 2.1and Applied Analysis 5 Theorem 1.1. Assume that conditions H1 – H3 hold; then one has the following. i If λ ∈ 0,Λ0 , then problem Eλ has at least one positive solution. ii If λ ∈ 0, q/2 Λ0 , then problem Eλ has at least two positive solutions. This paper is organized as follows. In Section 2, we give some properties of Nehari manifold. In Sections 3 and 4, we complete proofs of Theorem 1.1. At the end of this section, we explain some notations employed in this paper. L Ω, |x−ai| denotes the usual weighted L Ω space with the weight |x−ai|. |Ω| is the Lebesguemeasure ofΩ. Br x is a ball centered at x with radius r. O ε denotes |O ε |/εt ≤ C, and on 1 denotes on 1 → 0 as n → ∞. C, Ci will denote various positive constants and omit dx in the integration for convenience. 2. Nehari Manifold In this section, we will give some properties of Nehari manifold. As the energy functional Jλ is not bounded below on H, it is useful to consider the functional on the Nehari manifold Mλ { u ∈ H \ {0} : J ′ λ u , u 〉 0 } . 2.1 Thus, u ∈ Mλ if and only if 〈 J ′ λ u , u 〉 ‖u‖ − ∫