Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 34, pp. 1–18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTION TO CRITICAL KIRCHHOFF-TYPE EQUATION WITH DIPOLE-TYPE POTENTIAL SAINAN WANG, YU SU Abstract. Dipole-type potential arises in the area of nonrelativistic molecular physics. In this paper, we establish the existence and nonexistence of solution to critical Kirchhoff-type equation with dipole-type potential. 1. Introduction We consider the Kirchhoff-type equation − ( 1 + b ∫ RN |∇u|2dx ) ∆u− µΦ(x/|x|) |x|2 u = |u|2 ∗−2u, x ∈ RN , (1.1) where N > 3, b > 0 and 2∗ = 2N N−2 is the Sobolev critical exponent. The function Φ and the parameter µ satisfy the following condition: (A1) 0 ≤ Φ ∈ Lp(SN−1), p ≥ (N−2)2 2(N−1) + 1, and µ ∈ (0,ΛΦ), where ΛΦ := (N − 2)2 4 |SN−1|1/p‖Φ‖−1 Lp(SN−1) . On the other hand the Laplace operator with dipole-type potential is LΦ := −∆− µΦ(x/|x|) |x|2 , x ∈ RN , where N ≥ 3. This kind of operator arises in the area of nonrelativistic molecular physics. Specifically, the Schrödinger equation for the wave function of an electron interacting with a polar molecule can be written as H = − ~ 2m ∆ + e x ·D |x|3 − E, where D is the dipole moment of the molecule, e and m denote the charge and the mass of the electron, see [19]. The operator with different kinds of singular potentials have been largely studied, see [7, 8, 9, 10, 23, 26, 28] and references therein. On the other hand, equation (1.1) is related to the stationary analogue of equa- tion ρ ∂2u ∂t2 − (P0 h + E 2L ∫ L 0 |∂u ∂x |dx )∂2u ∂x2 = 0, 2020 Mathematics Subject Classification. 35A15, 35J20. Key words and phrases. Kirchhoff-type equation; dipole-type potential; critical exponent. ©2022. This work is licensed under a CC BY 4.0 license. Submitted June 4, 2021. Published April 22, 2022. 1 2 S. WANG, Y. SU EJDE-2022/34 which was proposed by Kirchhoff in [18] as an extension of the classical D’Alembert wave equation for free vibrations of elastic strings. The existence of solution of Kirchhoff-type equation with Laplacian was explored in [3, 25], and with fractional Laplacian was investigated in [21]. Liu-Liao-Tang [20] studied equation (1.1) with Φ = 0: − ( a+ b ∫ RN |∇u|2dx ) ∆u = |u|2 ∗−2u, x ∈ RN . (1.2) By using the minimizing of best constant S := inf u∈D1,2(RN )\{0} ‖u‖2D1,2(RN ) ( ∫ RN |u|2 ∗dx)2/2∗ as follows Uε,y = [N(N − 2)] N−2 4 ε N−2 2 (ε2 + |x− y|2) , they established the existence and nonexistence of solutions for equation (1.2) with respect to parameters N , a and b. The existence of solution of equation (1.2) with p-Laplacian was presented in [17, 22]. For Φ =Constant, Fiscella-Pucci [11] established the Concentration Compact- ness Principle with Hardy potential, and then they established the existence of solutions for Kirchhoff-type equations involving Hardy potential and different crit- ical nonlinearities. For more recent work, we refer to [1, 12, 13]. The case where the potential Φ is a constant was discussed in [11, 17, 20, 22]. Therefore, it is natural to ask whether equation (1.1) admits a solution for Φ non- constant. To the best of our knowledge, there is no result on this problem. If b = 0, equation (1.1) becomes −∆u− µΦ(x/|x|) |x|2 u = |u|2 ∗−2u, x ∈ RN . (1.3) We study the following minimizing problem: SΦ := inf u∈D1,2 rad(RN )\{0} ‖u‖2Φ( ∫ RN |u|2 ∗dx )2/2∗ . Extremals for SΦ are solutions of the Euler-Lagrange equation (1.3). The following is our first result. Theorem 1.1. Assume that N ≥ 3 and (A1) hold. Then equation (1.3) has a radially symmetric solution v̄ ∈ D1,2 rad(RN ), and infinitely many nonradial solutions v̄k such that ∫ RN |v̄k| 2∗ dx→∞ as k →∞. Remark 1.2. Note that the Sobolev embedding D1,2(RN ) ↪→ L2∗ (RN ) is not compact. Hence, it is hard to show that the minimizing sequence of SΦ has a convergence subsequence. We investigate this problem by two different methods. In the first method, we obtain a radially symmetric solution. In the second method, we obtain infinitely many nonradial solutions. For b > 0 and N = 3⇔ 2∗ > 4, we have Theorem 1.3. Assume that N = 3, b > 0 and condition (A1) holds. Then (1.1) has a radially symmetric ground state solution v ∈ D1,2 rad(RN ). Moreover, if µ ∈ (0, 4ΛΦ/(2 ∗)2), then v ∈ L2∗· 2∗2 (RN ). EJDE-2022/34 CRITICAL KIRCHHOFF-TYPE EQUATION 3 When N ≥ 4⇔ 2∗ ≤ 4, equation (1.1) is more complicated. Theorem 1.4. Assume that N ≥ 4, b > 0 and condition (A1) holds. Then the following statements are true. (1) For N = 4 and b ≥ S−2, equation (1.1) has no nontrivial solution. (2) For N > 4 and b > 2∗−2 2 ( ΛΦ−µ ΛΦ ) 4−2∗ 2∗−2 ( 4−2∗ 2 ) 4−2∗ 2∗−2S− 2∗ 2∗−2 , equation (1.1) has no nontrivial solution, where ΛΦ and µ are defined in condition (A1). (3) For N ≥ 4, there exists b0 > 0 small enough such that for all b ∈ (0, b0), equation (1.1) has a radially symmetric. We summarize of Theorems 1.1–1.4 as follows: b = 0, N ≥ 3 { a radially symmetric solution, infinitely many nonradial solutions, b > 0  N = 3, a radially symmetric ground state solution, N = 4, b ≥ S−2, no nontrivial solution, N ≥ 5 { b > 2∗−2 2 ( ΛΦ−µ ΛΦ ) 4−2∗ 2∗−2 ( 4−2∗ 2 ) 4−2∗ 2∗−2S− 2∗ 2∗−2 , no nontrivial solution, b ∈ (0, b0), a radially symmetric solution. This article is organized as follows. In Section 2, we present notation. In Sections 3-5, we give the proofs of Theorems 1.1–1.4, respectively. 2. Preliminaries The space D1,2(RN ) is the completion of C∞0 (RN ) with respect to the semi-norm ‖u‖2D1,2(RN ) := ∫ RN |∇u|2dx. We denote by D1,2 rad(RN ) the space of radial functions in D1,2(RN ). We define the best constant S := inf u∈D1,2(RN )\{0} ‖u‖2D1,2(RN ) ( ∫ RN |u|2 ∗dx)2/2∗ . We know that S can be attained in RN , see [5]. For all u ∈ D1,2(RN ), we have the Hardy inequality, see [14], (N − 2)2 4 ∫ RN |u|2 |x|2 dx ≤ ∫ RN |∇u|2dx. We introduce the measure dϑ induced by Lebesgues measure on the unit sphere SN−1 ⊂ RN . We denote by ‖ · ‖Lq(SN−1) the quantity ‖Φ‖q Lq(SN−1) = ∫ SN−1 |Φ(ϑ)|qdϑ. Lemma 2.1 ([15]). Let N ≥ 3, 0 ≤ Φ ∈ Lp(SN−1) and p ≥ (N−2)2 2(N−1) + 1. Then∫ RN |∇u|2dx ≥ ΛΦ ∫ RN Φ(x/|x|)|u|2 |x|2 dx, where u ∈ D1,2(RN ) and ΛΦ := (N−2)2 4 |SN−1|1/p‖Φ‖−1 Lp(SN−1) . 4 S. WANG, Y. SU EJDE-2022/34 By using Lemma 2.1 and µ ∈ (0,ΛΦ), ‖u‖2Φ =: ∫ RN |∇u|2dx− µ ∫ RN Φ(x/|x|)|u|2 |x|2 dx is an equivalent norm in D1,2(RN ). A measurable function u : RN → R belongs to the Morrey space ‖u‖Lq,$(RN ) with q ∈ [1,∞) and $ ∈ (0, N ] if and only if ‖u‖qLq,$(RN ) = sup R>0,x∈RN R$−3 ∫ B(x,R) |u(y)|qdy <∞. Lemma 2.2 ([24]). For N ≥ 3, there exists C > 0 such that for ι and ϑ satisfying 2 2∗ ≤ ι < 1, 1 ≤ ϑ < 2∗, we have(∫ RN |u|2 ∗ dx )1/2∗ ≤ C‖u‖ιD1,2(RN )‖u‖ 1−ι Lϑ, ϑ(N−2) 2 (RN ) , for any u ∈ D1,2(RN ). Equation (1.1) is variational and its solutions are the critical points of the func- tional defined in D1,2(RN ) by Ib(u) = 1 2 ‖u‖2D1,2(RN ) − µ 2 ∫ RN Φ(x/|x|)|u|2 |x|2 dx+ b 4 ‖u‖4D1,2(RN ) − 1 2∗ ∫ RN |u|2 ∗ dx. It is easy to see that the functional Ib ∈ C1(D1,2(RN ),R). It is easy to see that if u ∈ D1,2(RN ) is a critical point of Ib, i.e., 0 = 〈I ′b(u), ϕ〉 = ( 1 + b‖u‖2D1,2(RN ) )∫ RN ∇u∇ϕdx − µ ∫ RN Φ ( x |x| ) uϕ |x|2 dx− ∫ RN |u|2 ∗−2uϕdx, for all ϕ ∈ D1,2(RN ). 3. Proof of Theorem 1.1 We separate the proof of Theorem 1.1 into two parts: (i) radially symmetric solution; (ii) nonradial solution. Proof of Theorem 1.1. (radially symmetric solution). Step 1. Note that µ ∈ (0,ΛΦ). Applying Lemma 2.2 with ϑ = 2, we obtain(∫ RN |u|2 ∗ dx )1/2∗ ≤ C‖u‖2ιΦ ‖u‖ 2(1−ι) L2,N−2(RN ) , (3.1) for u ∈ D1,2(RN ). Let {un} ⊂ D1,2 rad(RN ) be a minimizing sequence of SΦ, that is ‖un‖2Φ → SΦ as n→∞, and ∫ RN |un|2 ∗ dx = 1. According to (3.1), there exists C > 0 such that for any n it holds ‖un‖L2,N−2(RN ) ≥ C > 0. EJDE-2022/34 CRITICAL KIRCHHOFF-TYPE EQUATION 5 On the other hand, we note that {un} is bounded in D1,2 rad(RN ) and D1,2 rad(RN ) ↪→ L2∗ (RN ) ↪→ L2,N−2(RN ). Then ‖un‖L2,N−2(RN ) ≤ C, Hence, there exists C0 > 0 such that for any n it holds C0 ≤ ‖un‖L2,N−2(RN ) ≤ C−1 0 . From above inequality, we deduce that for any n ∈ N there exist σn > 0 and xn ∈ RN such that 1 σ2 n ∫ B(xn,σn) |un(y)|2dy ≥ ‖un‖2L2,N−2(RN ) − C 2n ≥ C1 > 0. Let vn(x) = σ N−2 2 n un(σnx). By scaling invariance, we have ‖vn‖2Φ → SΦ, as n→∞,∫ RN |vn|2 ∗ dx = 1, and ∫ B( xnσn ,1) |vn(y)|2dy = 1 σ2 n ∫ B(xn,σn) |un(y)|2dy ≥ C1 > 0. (3.2) Hence, we assume that vn ⇀ v in D1,2 rad(RN ), vn → v a.e. in RN , vn → vin Lqloc(RN ) for all q ∈ [2, 2∗). Step 2. We show that {xnσn } is bounded. Suppose on the contrary that xn σn → ∞ as n → ∞. By the boundedness of {un} in D1,2 rad(RN ), we have ‖vn‖D1,2(RN ) = ‖un‖D1,2(RN ) ≤ C. It follows from the uniform decay estimates of radial functions that |vn(x)| ≤ C |x|N−2 2 ‖vn‖D1,2(RN ) ≤ C |x|N−2 2 , a.e. RN . For √ C1 |B(0,1)| > ε > 0, there exists M > 0 for any n > M it holds |vn(x)| ≤ C3 |xnσn − 1|N−2 2 ≤ ε, x ∈ Bc(0, |xn σn − 1|). Note that B(xnσn , 1) ⊂ Bc(0, |xnσn − 1|). Then∫ B( xnσn ,1) |vn(y)|2dy ≤ ε2 ∫ B( xnσn ,1) dy = ε2|B( xn σn , 1)| = ε2|B(0, 1)| < C1. This contradicts (3.2). Hence, {xnσn } is bounded. There exists R > 0 such that∫ B(0,R) |vn(y)|2dy ≥ ∫ B( xnσn ,1) |vn(y)|2dy ≥ C1 > 0. Since the embedding D1,2 rad(RN ) ↪→ Lrloc(RN ), r ∈ [2, 2∗) is compact, we deduce that v 6≡ 0. Step 3. Set h(t) = t2 ∗ , t ≥ 0. 6 S. WANG, Y. SU EJDE-2022/34 It is easy to see that h(t) is a convex function. By h(0) = 0 and l ∈ [0, 1], we know h(lt) = h(lt+ (1− l) · 0) ≤ lh(t) + (1− l)h(0) = lh(t). For t1, t2 ∈ [0,∞), applying last inequality, we obtain h(t1) + h(t2) = h ( (t1 + t2) t1 t1 + t2 ) + h ( (t1 + t2) t2 t1 + t2 ) ≤ t1 t1 + t2 h ( t1 + t2 ) + t2 t1 + t2 h(t1 + t2) = h(t1 + t2). Step 4. We claim that vn → v strongly in D1,2(RN ). It follows from Brézis-Lieb type lemma [2] that ‖v‖2Φ + lim n→∞ ‖vn − v‖2Φ = lim n→∞ ‖vn‖2Φ = SΦ,α, lim n→∞ ∫ RN |vn|2 ∗ dx = lim n→∞ ∫ RN |vn − v|2 ∗ dx+ ∫ RN |v|2 ∗ dx. Therefore, 1 = lim n→∞ ∫ RN |vn|2 ∗ dx = lim n→∞ ∫ RN |vn − v|2 ∗ dx+ ∫ RN |v|2 ∗ dx ≤ S− 2∗ 2 Φ lim n→∞ ‖vn − v‖2 ∗ Φ + S − 2∗ 2 Φ ‖v‖2 ∗ Φ ≤ S− 2∗ 2 Φ ( lim n→∞ ‖vn − v‖Φ + ‖v‖Φ )2∗ = 1. Therefore, all the inequalities above have to be equalities. We know that lim n→∞ ‖vn − v‖2 ∗ Φ + ‖v‖2 ∗ Φ = ( lim n→∞ ‖vn − v‖Φ + ‖v‖Φ )2∗ . This further gives: either limn→∞ ‖vn − v‖Φ = 0 or ‖v‖Φ = 0. From v 6≡ 0, so we have ‖v‖Φ 6= 0. Then lim n→∞ ‖vn − v‖Φ = 0. We can choose v ≥ 0. There exists C > 0 such that v̄ = Cv satisfies −∆v̄ − µΦ(x/|x|) |x|2 v̄ = |v̄|2 ∗−1, x ∈ RN . The proof is complete. � To study the nonradial solution of equation (1.1), we need the following result. Lemma 3.1 ([6]). Let X be a closed subspace of H1(SN−1). Suppose that the embedding X ⊂ Lq(SN−1) is compact. Then the restriction of function K on X, K|X satisfies the Palais-Smale condition. Furthermore, if X is infinite dimensional, then K|X has a sequence of critical points φk in X, such that ∫ SN−1 |φk|qdϑ → ∞ as k →∞. EJDE-2022/34 CRITICAL KIRCHHOFF-TYPE EQUATION 7 Proof of Theorem 1.1. (nonradial solutions). It is easy to see that u(x) = |x| 2−N 2 φ ( x |x| ) (3.3) solves equation (1.3), if and only if φ is a solution of the equation −∆ϑφ+ (N − 2)2 4 φ− µΦφ = |φ|2 ∗−2φ, in SN−1. (3.4) The energy functional of equation (3.4) is K(φ) = 1 2 ∫ SN−1 |∇φ|2dϑ+ (N − 2)2 8 ∫ SN−1 |φ|2dϑ− µ 2 ∫ SN−1 Φ|φ|2dϑ − 1 2∗ ∫ SN−1 |φ|2 ∗ dϑ and 〈K ′(φ), ϕ〉 = ∫ SN−1 ∇φ∇ϕdϑ+ (N − 2)2 4 ∫ SN−1 φϕdϑ− µ ∫ SN−1 Φφϕdϑ − ∫ SN−1 |φ|2 ∗−2φϕdϑ. Suppose that G = O(k) × O(m) ⊂ O(N), where k + m = N , then H1 G(SN−1) is an infinite dimensional closed subspace of H1(SN−1), and H1 G(SN−1) is compactly embedded in Lq(SN−1) for every q ∈ [1, 2(N−1) N−3 ), see [6]. Since 2∗ ∈ [1, 2(N−1) N−3 ), so we have that H1 G(SN−1) is compactly embedded in L2∗ (SN−1). Applying Lemma 3.1 with X = H1 G(SN−1) and q = 2∗, then we have that K|H1 G(SN−1) has a sequence of critical points φk in H1 G(SN−1), such that∫ SN−1 |φk|2 ∗ dϑ→∞ as k →∞. According to (3.3), we know that v̄k(x) = |x| 2−N2 φk( x |x| ) are solutions of equation (1.3), and ∫ RN |v̄k| 2∗ dx = ∫ SN−1 |φk|2 ∗ dϑ→∞ as k →∞. � 4. Proof of Theorem 1.3 Define Jb = Ib|D1,2 rad(RN ), c = inf Υ∈Γ max t∈[0,1] Jb(Υ(t)), where Γ = {Υ ∈ C([0, 1], D1,2 rad(RN ))|Υ(0) = 0, Jb(Υ(1)) < 0}. It is easy to see that Jb possesses the mountain pass geometry, there exists {un} ⊂ D1,2 rad(RN ) such that Jb(un)→ c > 0 and J ′b(un)→ 0 as n→∞. And {un} is uniformly bounded in D1,2 rad(RN ). The Nehari manifold on D1,2 rad(RN ) is defined by Nb = {u ∈ D1,2 rad(RN )|〈J ′b(u), u〉 = 0, u 6= 0}, and ¯̄c = inf u∈Nb Jb(u) and c̄ = inf u∈D1,2 rad(RN ) max t≥0 Jb(tu). 8 S. WANG, Y. SU EJDE-2022/34 With minor change the proof of [27, Theorem 4.2], we can show that ¯̄c = c̄ = c. Lemma 4.1. Assume the assumptions in Theorem 1.3 hold. Then for each u ∈ D1,2 rad(RN ) \ {0}, there exists a unique tu > 0 such that tuu ∈ Nb. Moreover, Jb(tuu) = maxt≥0 Jb(tu). Proof. For each u ∈ D1,2 rad(RN ) \ {0}, and t ∈ (0,∞), we set f1(t) = Jb(tu) = t2 2 ‖u‖2Φ + bt4 4 ‖u‖4D1,2(RN ) − t2 ∗ 2∗ ∫ RN |u|2 ∗ dx, f ′1(t) = t‖u‖2Φ + bt3‖u‖4D1,2(RN ) − t 2∗−1 ∫ RN |u|2 ∗ dx. This implies that f ′1(·) = 0 if and only if t2−2∗ ‖u‖2Φ + bt4−2∗ ‖u‖4D1,2(RN ) = ∫ RN |u|2 ∗ dx. Set f2(t) = t2−2∗ ‖u‖2Φ + bt4−2∗ ‖u‖4D1,2(RN ). We know that limt→0 f2(t) = ∞, limt→∞ f2(t) = 0 and f2(·) is strictly decreasing on (0,∞). Then there exists a unique 0 < tu <∞ such that f2(t)  < ∫ RN |u| 2∗ dx, tu < t <∞, = ∫ RN |u| 2∗ dx, t = tu, > ∫ RN |u| 2∗ dx, 0 < t < tu. This is showing that tuu ∈ Nb. Moreover, f ′1(t)  < 0, tu < t <∞, = 0, t = tu, > 0, 0 < t < tu. This shows that f1(·) admits a unique critical point tu on (0,∞) such that f1(·) takes the maximum at tu. To prove the uniqueness of tu, let us assume that 0 < t̄ < ¯̄t satisfy f ′1(t̄) = f ′1(¯̄t) = 0. We obtain ∫ RN |u|2 ∗ dx = f2(t̄) = f2(¯̄t). Since 0 < t̄ < ¯̄t, the above equality leads to the contradiction: u = 0. Hence, for each u ∈ D1,2 rad(RN ) \ {0}, there exists a unique tu > 0 such that tuu ∈ Nb. � Lemma 4.2. Assume that the assumptions in Theorem 1.3 hold. Let {un} be a (PS)c sequence of Jb at c > 0. Then up to a subsequence, un ⇀ u in D1,2 rad(RN ) with u 6≡ 0 being a weak solution of equation (1.1). Proof. It is easy to see that {un} is uniformly bounded in D1,2 rad(RN ). In order to see that u is a weak solution of Jb, we recall un ⇀ u in D1,2 rad(RN ), un → u a.e. in RN , un → u in Lrloc(RN ) for all r ∈ [2, 2∗). Moreover, there exists A ∈ R, such that lim n→∞ ‖un‖2D1,2(RN ) = A. (4.1) EJDE-2022/34 CRITICAL KIRCHHOFF-TYPE EQUATION 9 Then by Fatou’s lemma, ‖u‖2D1,2(RN ) ≤ A. We claim that ‖u‖2D1,2(RN ) = A. To obtain a contradiction, we assume that ‖u‖2D1,2(RN ) < A. Since un ⇀ u weakly in D1,2 rad(RN ), we know that for each ϕ ∈ D1,2 rad(RN ) lim n→∞ ∫ RN ∇un∇ϕdx− µ ∫ RN Φ ( x |x| )unϕ |x|2 dx = ∫ RN ∇u∇ϕdx− µ ∫ RN Φ ( x |x| ) uϕ |x|2 dx (4.2) and lim n→∞ ∫ RN |un|2 ∗−2unϕdx = ∫ RN |u|2 ∗−2uϕdx. (4.3) From limn→∞〈J ′b(un), ϕ〉 = 0, we have 0 = lim n→∞ (1 + b‖un‖2D1,2(RN )) ∫ RN ∇un∇ϕdx− µ ∫ RN Φ ( x |x| )unϕ |x|2 dx − ∫ RN |un|2 ∗−2unϕdx. Applying (4.1), we obtain 0 = (1 + bA) ∫ RN ∇un∇ϕdx− µ ∫ RN Φ ( x |x| )unϕ |x|2 dx− ∫ RN |un|2 ∗−2unϕdx. By using (4.2), (4.3) and ‖u‖2D1,2(RN ) < A, we know that 〈J ′b(u), u〉 < 0. (4.4) On the other hand, we have 〈J ′b(tu), tu〉 = f ′1(t)t = t2‖u‖2Φ + bt4‖u‖4D1,2(RN ) − t 2∗ ∫ RN |u|2 ∗ dx, (4.5) Applying Lemma 4.1, there exists a unique t0 > 0 satisfying f ′1(t0) = 0, which implies that 〈J ′b(t0u), t0u〉 = f ′1(t0)t0 = 0 (4.6) Now, we show that t0 < 1. Combining (4.4) and (4.5), we know that f ′1(1) < 0. Taking tε > 0 small enough in (4.5), we know f ′1(tε)tε > 0, which implies f ′1(tε) > 0. According to Intermediate value theorem, there exists t1 ∈ (tε, 1) such that f ′1(t1) = 0. By using the uniqueness of t0, we have t0 = t1 ∈ (tε, 1) (4.7) From (4.5)-(4.7), we obtain c = Jb(t0u) = Jb(t0u)− 1 4 〈J ′b(t0u), t0u〉 = t20 4 ‖u‖2Φ + (1 4 − 1 2∗ ) t2 ∗ 0 ∫ RN |u|2 ∗ dx < 1 4 ‖u‖2Φ + (1 4 − 1 2∗ ) ∫ RN |u|2 ∗ dx 10 S. WANG, Y. SU EJDE-2022/34 ≤ 1 4 lim n→∞ ‖un‖2Φ + (1 4 − 1 2∗ ) lim n→∞ ∫ RN |un|2 ∗ dx = lim n→∞ Jb(un)− 1 4 lim n→∞ 〈J ′b(un), un〉 = c which is a contradiction. Then lim n→∞ ‖un‖2D1,2(RN ) = A = ‖u‖2D1,2(RN ). Thus for any ϕ ∈ D1,2(RN ), we obtain lim n→∞ 〈J ′b(un), ϕ〉 = 0 = 〈J ′b(u), ϕ〉. The proof is complete. � The following result implies the non-vanishing of (PS)c sequence. Lemma 4.3. Assume that all the assumptions descripted in Theorem 1.3 hold. Let {un} be a (PS)c sequence of Jb at c > 0. Then lim n→∞ ∫ RN |un|2 ∗ dx > 0. Proof. It is easy to see that {un} is uniformly bounded in D1,2 rad(RN ). Then there exists a constant 0 < C <∞ such that ‖un‖Φ ≤ C. Suppose on the contrary that lim n→∞ ∫ RN |un|2 ∗ dx = 0. (4.8) According to (4.8) and the definition of (PS)c sequence, we obtain c+ o(1) = 1 2 ‖un‖2Φ + b 4 ‖un‖4D1,2(RN ) and o(1) = ‖un‖2Φ + b‖un‖4D1,2(RN ). This implies c+ o(1) = − 1 4‖un‖ 2 Φ, which contradicts 0 < c. � Proof of Theorem 1.3. (i) Note that {un} is a bounded sequence of Jb at level c in D1,2 rad(RN ). Up to a subsequence, we assume un ⇀ u in D1,2 rad(RN ), un → ua.e. in RN , un → u in Lrloc(RN ) for all r ∈ [2, 2∗). Let vn(x) = σ N−2 2 n un(σnx). We assume that vn ⇀ v in D1,2 rad(RN ), vn → v a.e. in RN , vn → v in Lqloc(RN ) for all q ∈ [2, 2∗). From Lemma 4.3, we have lim n→∞ ∫ RN |un|2 ∗ dx > 0. Similar to the proof of Theorem 1.1 Steps 1 and 2, we deduce that v 6≡ 0. From Lemma 4.2, we know v ∈ Nb. We show that vn → v strongly inD1,2 rad(RN ). Applying EJDE-2022/34 CRITICAL KIRCHHOFF-TYPE EQUATION 11 Brézis-Lieb lemma [2], we obtain c = lim n→∞ Jb(vn)− lim n→∞ 1 2∗ 〈J ′b(vn), vn〉 = lim n→∞ (1 2 − 1 2∗ ) ‖vn‖2Φ + lim n→∞ (1 4 − 1 2∗ ) ‖vn‖4D1,2(RN ) ≥ (1 2 − 1 2∗ ) ‖v‖2Φ + (1 4 − 1 2∗ ) ‖v‖4D1,2(RN ) = Jb(v) ≥ c. (4.9) Thus, the inequalities above have to be equalities. We know that lim n→∞ ‖vn‖2Φ = ‖v‖2Φ. By Brézis-Lieb lemma again, we have lim n→∞ ‖vn‖2Φ − lim n→∞ ‖vn − v‖2Φ = ‖v‖2Φ, which implies lim n→∞ ‖vn − v‖2Φ = 0. Using (4.9) again, we know that Jb(v) = c. This implies that v attains the minimum of Jb at c. Moreover, we can choose v ≥ 0. The principle of symmetric criticality implies that the critical point of Jb is also a critical point of Ib. (ii) For each L > 1, define vL(x) = { v(x) if v(x) ≤ L, L if v(x) > L. For β = 2∗/2 > 1. Set φ = vv 2(β−1) L . It is easy to see that φ ∈ D1,2 rad(RN ). We know that v is a nonnegative solution of equation (1.1). Then( 1 + b‖v‖2D1,2(RN ) )∫ RN ∇v∇ϕdx− µ ∫ RN Φ ( x |x| ) vϕ |x|2 dx = ∫ RN |v|2 ∗−2vϕdx. Plugging φ into above equation, we obtain( 1 + b‖v‖2D1,2(RN ) )∫ RN ∇v∇φdx− µ ∫ RN Φ ( x |x| ) vφ |x|2 dx = ∫ RN |v|2 ∗−2vφdx. A direct calculation yields∫ RN ∇v∇φdx ≥ ∫ RN v 2(β−1) L |∇v|2dx. (4.10) Notice that |∇(vvβ−1 L )|2 = v 2(β−1) L |∇v|2 + (β − 1)2v2v 2(β−2) L |∇vL|2 + 2(β − 1)vv2β−3 L ∇v∇vL. Then one has ∫ RN v2v 2(µ−2) L |∇vL|2dx ≤ ∫ RN v 2(µ−1) L |∇v|2dx,∫ RN vv2µ−3 L ∇v∇vLdx ≤ ∫ RN v 2(µ−1) L |∇v|2dx. Therefore, ∫ RN |∇(vvβ−1 L )|2dx ≤ β2 ∫ RN v 2(β−1) L |∇v|2dx. (4.11) 12 S. WANG, Y. SU EJDE-2022/34 It follows from (4.10) and (4.11) that 1 β2 ∫ RN |∇(vvβ−1 L )|2dx ≤ ∫ RN ∇v∇φdx. Hence,∫ RN |v|2 ∗−2|vvβ−1 L |2dx = ( 1 + b‖v‖2D1,2(RN ) )∫ RN ∇v∇φdx− µ ∫ RN Φ ( x |x| ) vφ |x|2 dx ≥ ∫ RN ∇v∇φdx− µ ∫ RN Φ ( x |x| ) vφ |x|2 dx ≥ 1 β2 ∫ RN |∇(vvβ−1 L )|2dx− µ ∫ RN Φ ( x |x| ) |vvβ−1 L |2 |x|2 dx ≥ ( 1 β2 − µ ΛΦ ) ‖vvβ−1 L ‖2Φ. Then, combining above inequality and Moser iteration technique, we deduce that v ∈ L2∗· 2∗2 (RN ). � 5. Proof of Theorem 1.4 5.1. Perturbation equation. In this subsection, we look equation (1.1) as a per- turbation of (1.3). The energy functional of equation (1.3) is I0(u) = 1 2 ‖u‖2Φ − 1 2∗ ∫ RN |u|2 ∗ dx. Set J0 = I0|D1,2 rad(RN ), and define c0 = inf Υ∈Γ0 max t∈[0,1] J0(Υ(t)), where Γ0 = {Υ ∈ C([0, 1], D1,2 rad(RN ))|Υ(0) = 0, J0(Υ(1)) < 0}. The Nehari mani- fold is N0 = {u ∈ D1,2 rad(RN )|〈J ′0(u), u〉 = 0, u 6= 0}, and c̄0 = inf u∈D1,2 rad(RN ) max t≥0 J0(tu) and ¯̄c0 = inf u∈N0 J0(u). We can show that c0 = c̄0 = ¯̄c0. Lemma 5.1. Assume that the assumptions in Theorem 1.4 hold. Then the energy functional J0 satisfies the following properties (M1) There exist ρ, ι > 0 such that if ‖u‖D1,2(RN ) = ρ, then J0(u) ≥ ι, and e0 ∈ D1,2 rad(RN ) exists such that ‖e0‖D1,2(RN ) > ρ and J0(e0) < 0. (M2) There exists v0 6≡ 0 such that J0(v0) = c0 := minΥ∈Γ0 maxt∈[0,1] J0(Υ(t)), where Γ0 = {Υ ∈ C([0, 1], D1,2 rad(RN ))|Υ(0) = 0, J0(Υ(1)) < 0}. (M3) c0 = inf{J0(u)|‖J ′0(u)‖D−1,2(RN ) = 0, u ∈ D1,2 rad(RN ) \ {0}}. (M4) There exists a path Υ0(t) ∈ Γ0 passing through v0 at t = t0 and satisfying J0(v0) > J0(Υ0(t)) for all t 6= t0. (M5) The set S := {u ∈ D1,2 rad(RN )|‖J ′0(u)‖D−1,2(RN ) = 0, J0(u) = c0} is compact in D1,2 rad(RN ) with the strong topology up to dilations in RN . EJDE-2022/34 CRITICAL KIRCHHOFF-TYPE EQUATION 13 Proof. As in Theorem 1.3, we have (M1)–(M4). (M5) Note that J0 is invariant by dilations. It follows from Theorem 1.3 that the weak convergence of the dilated subsequence can be upgraded into strong con- vergence. This further implies that the set S is compact in D1,2 rad(RN ) with the topology up to dilations in RN . � 5.2. Perturbation method. We define a modified mountain pass level of Jb cb := min Υ∈ΓM max t∈[0,1] Jb(Υ(t)), where ΓM = {Υ ∈ Γ0 : sup t∈[0,1] ‖Υ(t)‖D1,2(RN ) ≤M} with M = 2{sup u∈S ‖u‖D1,2(RN ), sup t∈[0,1] ‖Υ(t)‖D1,2(RN )} fixed. By the choice of M , Υ0 ∈ ΓM , we have c0 = minΥ∈ΓM maxt∈[0,1] J0(Υ(t)). becasue ΓM ( Γ0, the standard mountain pass theorem becomes unavailable. Lemma 5.2. Let b > 0. Then limb→0 cb = c0. Proof. For b > 0, it is easy to obtain cb ≥ c0. We take e0 = Tv0 in (M1), where T > (2∗/2) 1 2∗−1 . Then Υ0(t) ∈ C([0, 1], D1,2 rad(RN )) defined as Υ0(t) = te0 = tTv0, and t0 = 1 T in (M4). We know that lim b→0 cb = lim b→0 Jb(Υ0(t)) ≤ J0(Υ0(t)) + lim λ→0 b 4 ‖Υ0(t)‖4D1,2(RN ) = J0(v0) = c0. � For any d > 0, and any subset A of D1,2 rad(RN ), we set Ad := ⋃ u∈A Bd(u), where Bd(u) := {v ∈ D1,2 rad(RN )|‖u− v‖D1,2(RN ) ≤ d}. Lemma 5.3. Let d > 0 and {uj} ⊂ Sd. Then there exists {σj} such that ‖ūj‖D1,2(RN ) = ‖uj‖D1,2(RN ) where ūj(x) = σ N−2 2 j uj(σjx). Up to a subsequence, ūj ⇀ ū ∈ S2d. Proof. Let {uj} ⊂ Sd. From Sd and Lemma 5.1 (M5), there exists wj ∈ S such that ‖uj − wj‖D1,2(RN ) ≤ d. From (M5), there exists {σj} such that w̄j ∈ S,where w̄j(x) = σ N−2 2 j wj(σjx). It is easy to prove that w̄j → w̄ ∈ S. And ‖ūj‖D1,2(RN ) = ‖uj‖D1,2(RN ), ‖ūj − w̄j‖D1,2(RN ) = ‖uj − wj‖D1,2(RN ) ≤ d. For j large enough, we have ‖ūj − w̄‖D1,2(RN ) = ‖ūj − w̄j + w̄j − w̄‖D1,2(RN ) ≤ ‖ūj − w̄j‖D1,2(RN ) + ‖w̄j − w̄‖D1,2(RN ) ≤ 2d. 14 S. WANG, Y. SU EJDE-2022/34 This shows that {ūj} is bounded. Up to a subsequence, we assume that ūj ⇀ ū in D1,2 rad(RN ). Note that B2d(w̄) is weakly closed in D1,2 rad(RN ). We obtain ū ∈ B2d(w̄) ⊂ S2d. � Lemma 5.4. Let d1 := 1 2 √ 2·2∗ 2∗−2c0 and d ∈ (0, d1). Suppose that there exist se- quences bj > 0, bj → 0, and {uj} ⊂ Sd satisfying lim j→∞ Jbj (uj) ≤ c0 and lim j→∞ ‖J ′bj (uj)‖D−1,2(RN ) = 0. Then there exists a sequence {σj} such that ‖ūj‖D1,2(RN ) = ‖uj‖D1,2(RN ), where ūj(x) = σ N−2 2 j uj(σjx). Up to a subsequence, {ūj} converges to ū ∈ S. Proof. Let limj→∞ ‖J ′bj (uj)‖D−1,2(RN ) = 0 and {uj} be bounded. From Lemma 5.3, up to a subsequence, ūj ⇀ ū ∈ S2d. From d1, we know that ū 6≡ 0. Let ūj(x) = σ N−2 2 j uj(σjx). We have lim j→∞ Jbj (ūj) = lim j→∞ Jbj (uj) ≤ c0. For all ϕ ∈ D1,2 rad(RN ), we obtain |〈J ′bj (ūj), ϕ〉| = |〈J ′bj (uj), ϕ̄〉| ≤ ‖J ′bj (uj)‖D−1,2(RN )‖ϕ̄‖D1,2(RN ) = o(1)‖ϕ̄‖D1,2(RN ), where ϕ̄ = σ −N−2 2 j ϕ(x/σj). Note that ‖ϕ̄‖D1,2(RN ) = ‖ϕ‖D1,2(RN ). We know that ‖J ′bj (ūj)‖D−1,2(RN ) → 0 as j →∞, which further implies 〈J ′0(ū), ϕ〉 = lim j→∞ 〈J ′bj (ūj), ϕ〉 − bj 4 ‖ūj‖4D1,2(RN ) = 0. This shows that ‖J ′0(ū)‖D−1,2(RN ) = 0. It follows from ūj ∈ S2d that lim j→∞ 〈J ′0(ūj), ϕ〉 = lim j→∞ 〈J ′bj (ūj), ϕ〉 − lim j→∞ bj‖ūj‖2D1,2(RN ) ∫ RN ∇ūj(x)∇ϕ(x)dx = o(1)‖ϕ‖D1,2(RN ). On the other hand, c0 ≥ lim j→∞ Jbj (ūj) = lim j→∞ J0(ūj) + lim j→∞ bj 4 ‖ūj‖4D1,2(RN ) = lim j→∞ J0(ūj). (5.1) So {ūj} is a (PS)m sequence for J0 with m := limj→∞ J0(ūj). Up to a subsequence, ūj ⇀ ū and J0(ū) = 1 2 ‖ū‖2Φ − 1 2∗ ∫ RN |ū|2 ∗ dx EJDE-2022/34 CRITICAL KIRCHHOFF-TYPE EQUATION 15 = (1 2 − 1 2∗ ) ‖ū‖2Φ ≤ (1 2 − 1 2∗ ) lim inf j→∞ ‖ūj‖2Φ = lim inf j→∞ ( J0(ūj)− 1 2∗ 〈J ′0(ūj), ūj〉 ) = m. It follows from (M3) that m ≥ J0(ū) ≥ c0. From (5.1), one has m = J0(ū) = c0, which implies ū ∈ S. � Set mb := max t∈[0,1] Jb(Υ0(t)). (5.2) Then cb ≤ mb. It is easy to see that limb→0mb ≤ c0. From this inequality and Lemmas 5.2 and 5.4 , one has lim b→0 cb = lim b→0 mb = c0. We define Jmbb = {u ∈ D1,2 rad(RN )|Jb(u) ≤ mb}. Proposition 5.5. Let d2, d3 > 0 satisfying d3 < d2 < d1. Then there exist ι > 0 and b̃ > 0 depending on d2, d3 such that for b ∈ (0, b̃), it holds ‖J ′b(u)‖D−1,2(RN ) ≥ ι, u ∈ Jmbb ∩ (Sd2\Sd3). Proof. Suppose on the contrary that d2, d3 > 0 satisfying d3 < d2 < d1, there exist sequences {bj} with limj→∞ bj = 0, and {uj} ∈ J mbj bj ∩ (Sd2 \ Sd3) such that lim j→∞ Jbj (uj) ≤ c0 and lim j→∞ ‖J ′bj (uj)‖D−1,2(RN ) = 0. From (M5), there exists sequence {σj} such that {ūj} ∈ J mbj bj ∩ (Sd2 \ Sd3), lim j→∞ Jbj (ūj) ≤ c0, lim j→∞ ‖J ′bj (ūj)‖D−1,2(RN ) = 0, where ūj(x) = σ N−2 2 j uj(σjx). Hence, we can apply Lemma 5.4 and the existence of ū ∈ S such that ūj → ū in D1,2 rad(RN ). As a consequence, dist(ūj ,S)→ 0 as j →∞. This is a contradiction with ūj 6∈ Sd3 . � Proposition 5.6. For any d > 0, there exists δ > 0 such that if b > 0 small enough, then Jb(Υ0(t)) ≥ cb − δ implies Υ0(t) ∈ Sd, t ∈ [0, 1]. The proof of the above proposition follows by repeating the proof of [16, Propo- sitions 4]. Proposition 5.7. For any d ∈ (0, d1), there exist b0 > 0 and a sequence {uj} ⊂ Jmbb ∩ Sd such that ‖J ′b(uj)‖D−1,2(RN ) → 0 as j →∞, for all b ∈ (0, b0). The proof of the above proposition follows from a discussion in [4, Propositions 5.3], by Propositions 5.5 and 5.6. 16 S. WANG, Y. SU EJDE-2022/34 Proof of Theorem 1.4. (i) Suppose on the contrary that u ∈ D1,2(RN )\{0} is a solution of (1.1). It follows from 2∗ = 4 and b ≥ S−2 that 〈I ′b(u), u〉 = ‖u‖2D1,2(RN ) − µ ∫ RN Φ(x/|x|)|u|2 |x|2 dx+ b‖u‖4D1,2(RN ) − ∫ RN |u|2 ∗ dx ≥ ‖u‖2D1,2(RN ) − µ ∫ RN Φ(x/|x|)|u|2 |x|2 dx+ b‖u‖4D1,2(RN ) − S −2‖u‖4D1,2(RN ) ≥ ‖u‖2D1,2(RN ) − µ ∫ RN Φ(x/|x|)|u|2 |x|2 dx > 0. This is a contradiction. (ii) Suppose on the contrary that u ∈ D1,2(RN )\{0} is a solution of (1.1). Ap- plying Young’s inequality and b > 2∗ − 2 2 (ΛΦ − µ ΛΦ ) 4−2∗ 2∗−2 (4− 2∗ 2 ) 4−2∗ 2∗−2 S− 2∗ 2∗−2 , we have ( 1− µ ΛΦ ) ‖u‖2D1,2(RN ) + b‖u‖4D1,2(RN ) ≤ ‖u‖2D1,2(RN ) − µ ∫ RN Φ(x/|x|)|u|2 |x|2 dx+ b‖u‖4D1,2(RN ) = ∫ RN |u|2 ∗ dx ≤ S− 2∗ 2 ‖u‖2 ∗ D1,2(RN ) = [ S− 2∗ 2 ( 2b 2∗ − 2 ) 2−2∗ 2 ‖u‖4−2∗ D1,2(RN ) ][( 2b 2∗ − 2 ) 2∗−2 2 ‖u‖2(2∗−2) D1,2(RN ) ] ≤ 4− 2∗ 2 [ S− 2∗ 2 ( 2b 2∗ − 2 ) 2−2∗ 2 ‖u‖4−2∗ D1,2(RN ) ] 2 4−2∗ + 2∗ − 2 2 [( 2b 2∗ − 2 ) 2∗−2 2 ‖u‖2(2∗−2) D1,2(RN ) ] 2 2∗−2 = 4− 2∗ 2 S− 2∗ 4−2∗ (2∗ − 2 2b ) 2∗−2 4−2∗ ‖u‖2D1,2(RN ) + b‖u‖4D1,2(RN ). which is a contradiction. (iii) Taking d ∈ (0, d1), by Proposition 5.7, there exists b0 > 0 such that for all λ ∈ (0, b0), there exists a Palais-Smale sequence {uj} ⊂ Sd/2. By applying (M5), there exists sequence {σj} such that {ūj} ⊂ Sd/2 where ūj(x) = σ 3−2s 2 j uj(σjx). Clearly, {ūj} is bounded in D1,2 rad(RN ). Then by Lemma 5.4, up to a subsequence, there exists ū ∈ S d2 ·2 = Sd such that ūj ⇀ ū. Then we obtain ‖J ′b(ū)‖D−1,2(RN ) = 0. It follows from d ∈ (0, d1) that ū 6≡ 0. Hence ū is a nontrivial critical point of Jb. The principle of symmetric criticality implies that the critical point of Jb is also a critical point of Ib. � Acknowledgments This research is supported by the University-level key projects of Anhui Univer- sity of Science and Technology (xjzd2020-23), and by the Key Program of University Natural Science Research Fund of Anhui Province (Grant No. KJ2021A0452). EJDE-2022/34 CRITICAL KIRCHHOFF-TYPE EQUATION 17 References [1] V. Ambrosio, A. Fiscella, T. 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Proof of Theorem 1.1 4. Proof of Theorem 1.3 5. Proof of Theorem 1.4 5.1. Perturbation equation 5.2. Perturbation method Acknowledgments References