Xu and Chen Advances in Difference Equations (2016) 2016:176 DOI 10.1186/s13662-016-0864-9 RESEARCH Open Access Sign-changing solutions to Schrödinger-Kirchhoff-type equations with critical exponent Liping Xu1* and Haibo Chen2 * Correspondence: x.liping@126.com Department of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471003, P.R China Full list of author information is available at the end of the article Abstract In this paper, we study the following Schrödinger-Kirchhoff-type equation: ∗ –2 –(a + b R3 |∇u|2 dx) u + u = k(x)|u|2 u ∈ H1 (R3 ), u + μh(x)u in R3 , where a, b, μ > are constants, 2∗ = is the critical Sobolev exponent in three spatial dimensions Under appropriate assumptions on nonnegative functions k(x) and h(x), we establish the existence of positive and sign-changing solutions by variational methods MSC: 35J20; 35J65; 35J60 Keywords: Schrödinger-Kirchhoff-type equations; critical nonlinearity; positive solutions; sign-changing solutions; variational methods Introduction In this paper, we investigate the following Schrödinger-Kirchhoff-type problem: ⎧ ⎨–(a + b R ∗ – |∇u| dx) u + u = k(x)|u| u + μh(x)u in R , ⎩u ∈ H (R ), (.) where a, b > are constants, and ∗ = is the critical Sobolev exponent in dimension three We assume that μ and the functions k(x) and h(x) satisfy the following hypotheses: (μ ) < μ < μ, ˜ where μ˜ is defined by μ˜ := inf u∈H (R )\{} R a|∇u| + |u| dx : h(x)|u| dx = ; R (k ) k(x) ≥ , ∀x ∈ R ; (k ) there exist x ∈ R , σ > , ρ > , and ≤ α < such that k(x ) = maxx∈R k(x) and k(x) – k(x ) ≤ σ |x – x |α for |x – x | < ρ ; © 2016 Xu and Chen This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made Xu and Chen Advances in Difference Equations (2016) 2016:176 Page of 14 (h ) h(x) ≥ for any x ∈ R and h(x) ∈ L (R ); (h ) there exist σ > and ρ > such that h(x) ≥ σ |x – x |–β for |x – x | < ρ The Kirchhof-type problem is related to the stationary analogue of the equation utt – a + b |∇u| dx u = f (x, u) in , where is a bounded domain in RN , u denotes the displacement, f (x, u) is the external force, and b is the initial tension, whereas a is related to the intrinsic properties of the string (such as Young’s modulus) Equations of this type arise in the study of string or membrane vibration and were proposed by Kirchhoff in (see []) to describe the transversal oscillations of a stretched string, particularly, taking into account the subsequent change in string length caused by oscillations Kirchhoff-type problems are often referred to as being nonlocal because of the presence of the integral over the entire domain , which provokes some mathematical difficulties Similar nonlocal problems also model several physical and biological systems where u describes a process that depends on the average of itself, for example, the population density; see [, ] Kirchhoff-type problems have received much attention Some important and interesting results can be found in, for example, [–] and the references therein The solvability of the following Schrödinger-Kirchoff-type equation (.) has also been well studied in general dimension by various authors: – a+b RN |∇u| dx u + V (x)u = f (x, u) in RN (.) For example, Wu [] and many others [–], using variational methods, proved the existence of nontrivial solutions to (.) with subcritical nonlinearities Li and Ye [] obtained the existence of a positive solution for (.) with critical exponents More recently, Wang et al [] and Liang and Zhang [] proved the existence and multiplicity of positive solutions of (.) with critical growth and a small positive parameters The problem of finding sign-changing solutions is a very classical problem In general, this problem is much more difficult than finding a mere solution There were several abstract theories or methods to study sign-changing solutions; see, for example, [, ] and the references therein In recent years, Zhang and Perera [] obtained sign-changing solutions of (.) with superlinear or asymptotically linear terms More recently, Mao and Zhang [] use minimax methods and invariant sets of descent flow to prove the existence of nontrivial solutions and sign-changing solutions for (.) without the P.S condition Motivated by the works described, in this paper, our aim is to study the existence of positive and sign-changing solutions for problem (.) The method is inspired by Hirano and Shioji [] and Huang et al []; however, their arguments cannot be directly applied here To our best knowledge, there are very few works up to now studying sign-changing solutions for Schrödinger-Kirchhoff-type problem with critical exponent, that is, problem (.) Our main results are as follows Theorem . Assume that (μ ), (k ), (k ), and (h )-(h ) hold Then, for < β < , problem (.) possesses at least one positive solution Xu and Chen Advances in Difference Equations (2016) 2016:176 Page of 14 Theorem . Assume that (μ ), (k ), (k ), and (h )-(h ) hold Then, for (.) possesses at least one sign-changing solution < β < , problem Notation • H (R ) is the Sobolev space equipped with the norm u H (R ) = R (|∇u| + |u| ) dx • We define u := R (a|∇u| + |u| ) dx for u ∈ H (R ) Note that · is an equivalent norm on H (R ) • For any ≤ s ≤ ∞, u Ls := ( R |u|s dx) s denotes the usual norm of the Lebesgue space Ls (R ) • By D, (R ) we denote the completion of C∞ (R ) with respect to the norm u D, (R ) := R |∇u| dx • S denotes the best Sobolev constant defined by S = infu∈D, (R )\{} • C > denotes various positive constants R |∇u| dx ( R u dx) The outline of the paper is given as follows In Section , we present some preliminary results In Sections and , we give proofs of Theorems . and ., respectively The variational framework and preliminary In this section, we give some preliminary lemmas and the variational setting for (.) It is clear that system (.) is the Euler-Lagrange equations of the functional I : H (R ) → R defined by I(u) = u + b R |∇u| dx – k(x)|u| dx – R μ h(x)|u| dx (.) R Obviously, I is a well-defined C functional and satisfies I (u), v = (a∇u∇v + uv) dx + b R R |∇u| dx R ∇u∇v dx k(x)|u| uv + μh(x)uv dx – (.) R for v ∈ H (R ) It is well known that u ∈ H (R ) is a critical point of the functional I if and only if u is a weak solution of (.) Lemma . Assume that (h ) holds Then the function ψh : u ∈ H (R ) → R h(x)u dx is weakly continuous, and for each v ∈ H (R ), ϕh : u ∈ H (R ) → R h(x)uv dx is also weakly continuous The proof of Lemma . is a direct conclusion of [], Lemma . Lemma . Assume that (h ) holds Then the infimum μ˜ := inf u∈H (R )\{} is achieved R a|∇u| + |u| dx : h(x)|u| dx = R Xu and Chen Advances in Difference Equations (2016) 2016:176 Page of 14 Proof The proof of Lemma . is the same as that of [], Lemma . Here we omit it for simplicity Lemma . Assume that (k ), (h ), and (μ ) hold Then the functional I possesses the following properties () There exist ρ, γ > such that I(u) ≥ γ for u = ρ () There exists e ∈ H (R ) with e > ρ such that I(e) < Proof By Lemma . and the Sobolev inequality we obtain I(u) ≥ u –C u – μ u μ˜ = u μ – –C u μ˜ Set u = ρ small enough such that Cρ ≤ ( – μμ˜ ) Then we have I(u) ≥ μ – ρ μ˜ (.) Choosing γ = ( – μμ˜ )ρ , we complete the proof of () For t > and some u ∈ H (R ) with u = , it follows from (h ) and (μ ) that I(tu ) ≤ t u b + t R |∇u | dx – t R k(x)|u | dx, which implies that I(tu ) < for t > large enough Hence, we can take an e = t u for some t > large enough, and () follows Next, we define the Nehari manifold N associated with I by N := u ∈ H R \ {} : G(u) = , where G(u) = I (u), u Now we state some properties of N Lemma . Assume that (μ ) is satisfied Then the following conclusions hold () For all u ∈ H (R ) \ {}, there exists a unique t(u) > such that t(u)u ∈ N Moreover, I(t(u))u = maxt≥ I(tu) () < t(u) < in the case I (u), u < ; t(u) > in the case I (u), u > () t(u) is a continuous functional with respect to u in H (R ) () t(u) → +∞ as u → Proof The proof is similar to that of [], Lemma ., and is omitted here Positive solution In order to deduce Theorem ., the following lemmas are important Borrowing an idea from Lemma . in [], we obtain the first result Lemma . For s, t > , the system ⎧ ⎨f (t, s) = t – aS( s+t ) = , λ ⎩g(t, s) = s – bS ( s+t ) = , λ Xu and Chen Advances in Difference Equations (2016) 2016:176 Page of 14 has a unique solution (t , s ), where λ > is a constant Moreover, if ⎧ ⎨f (t, s) ≥ , ⎩g(t, s) ≥ , then t ≥ t and s ≥ s , where t = √ abS +a b S +λaS λ and s = √ bS +λabS +b S b S +λaS λ Lemma . Assume that (μ ), (k ), and (h ) hold Let a sequence {un } ⊂ N be such that un u in H (R ) and I(un ) → c, but any subsequence of {un } does not converge strongly to u Then one of the following results holds: () c > I(t(u)u) in the case u = and I (u), u < ; () c ≥ c∗ in the case u = ; () c > c∗ in the case u = and I (u), u ≥ ; where c∗ = abS k ∞ + b S k ∞ + (b S +a k ∞ S) k ∞ , and t(u) is defined as in Lemma . Proof Part of the proof is similar to that of [], Lemma . or [], Proposition . u in H (R ), we have For the reader’s convenience, we only sketch the proof Since un un – u Then by Lemma . we obtain that R h(x)|un – u| dx → (.) We obtain from the Brézis-Lieb lemma [], (.), and un ∈ N that c + o() = I(un ) = I(u) + – un – u R + b R ∇(un – u) dx k(x)|un – u| dx + o() (.) and = I (un ), un = I (u), u + un – u – R +b R ∇(un – u) dx k(x)|un – u| dx + o() (.) Up to a subsequence, we may assume that there exist li ≥ , i = , , , such that un – u → l , b R ∇(un – u) dx → l , (.) k(x)|un – u| dx → l R Since any subsequence of {un } does not converge strongly to u, we have l > Set γ (t) = l t + l t – l t and η(t) = g(t) + γ (t) By (.) and (.) we have η () = g () + γ () = , Xu and Chen Advances in Difference Equations (2016) 2016:176 Page of 14 and t = is the only critical point of η(t) in (, +∞), which implies that η() = max η(t) (.) t> We consider three situations: () u = and I (u), u < Then by (.) and (.) we have l + l – l > (.) Then, γ (t) = l t + l t – l t > l t + l t – (l + l )t = – t l t + (l + l )t ≥ (.) for any < t < , which implies that γ (t) > γ () = for any t ∈ (, ) (.) Since I (u), u < , by Lemma . there exists t(u) > such that < t(u) < Then it follows from (.) that γ (t(u)) > Therefore, we obtain from (.) and (.) that c = η() > η(t(u)) = g(t(u)) + γ (t(u)) > I(t(u)u), which implies that () holds () u = Then by (.), (.), and (.) we get ⎧ ⎨l + l – l = , ⎩ l + l – l = c By the definition of S we see that S |∇un | dx ≥ R k |∇un | dx b R k(x)|un | dx / ∞ R S ≥b k / ∞ , k(x)|un | dx R Then l ≥ aS l + l k ∞ and l + l k ∞ l ≥ bS Obviously, if l > , then l , l > It follows from Lemma . that c= l + l ≥ abS + a b S + k k ∞ = abS b S (b S + a k ∞ S) + + := c∗ k ∞ k ∞ k ∞ ∞ aS + bS + k ∞ abS + b S b S + k k ∞ ∞ aS (.) Xu and Chen Advances in Difference Equations (2016) 2016:176 Page of 14 () u = and I (u), u ≥ We prove this case in two steps Firstly, we consider u = and I (u), u = Then from Lemma . and Lemma . we get I(u) = max I(tu) > (.) t> Since u = and I (u), u = , as in (.), we obtain that c = η() = I(u) + l l + > c∗ (.) Secondly, we prove the case u = and I (u), u > Set t ∗∗ = ( attains its maximum at t ∗∗ , that is, √ l + l +l l ) l Then, γ (t) γ t ∗∗ = max γ (t) t> = l (l + l l ) l l + + l l l ≥ abS b S (b S + a k ∞ S) + + = c∗ k ∞ k ∞ k ∞ (.) It follows from Lemma . that < t ∗∗ < Then I(t ∗∗ u) ≥ Therefore, by (.), (.), and (.) we obtain c = η() > η t ∗∗ = I t ∗∗ u + γ t ∗∗ ≥ c∗ The proof of Lemma . is complete Lemma . If the hypotheses of Theorem . hold with < β < , then c < abS b S (b S + a k ∞ S) + + = c∗ , k ∞ k ∞ k ∞ where c is defined by infu∈N I(u) Proof To prove this lemma, we borrow an idea employed in [] For ε, r > , define wε (x) = Cϕ(x)ε (ε+|x–x | ) , where C is a normalizing constant, x is given in (k ), and ϕ ∈ C∞ (R ), ≤ ϕ ≤ , ϕ|Br () ≡ , and supp ϕ ⊂ Br () Using the method of [], we obtain R |∇wε | dx = K + O ε , R |wε | dx = K + O ε , (.) and |wε | dx = s R ⎧ s ⎪ ⎪ ⎨Kε , ⎪ ⎪ ⎩ s ∈ [, ), Kε | ln ε|, Kε –s , s = , s ∈ (, ), (.) Xu and Chen Advances in Difference Equations (2016) 2016:176 Page of 14 – where K , K , K are positive constants Moreover, the best Sobolev constant is S = K K By (.) we have R ( |∇wε | dx = S +O ε R wε dx) (.) By Lemma ., for this wε , there exists a unique t(wε ) > such that t(wε )wε ∈ N Thus, c < I(t(wε )wε ) Using (.), for t > , since I(twε ) → –∞ as t → ∞, we easily see that I(twε ) has a unique critical t(wε ) > that corresponds to its maximum, that is, I(tε wε ) = maxt> I(twε ) It follows from () of Lemma ., I(twε ) → –∞ as t → ∞, and the continuity of I that there exist two positive constants t and T such that t < tε < T Let I(tε wε ) = F(ε)+G(ε)+H(ε), where atε F(ε) = G(ε) = |∇wε | dx + R tε R btε k(x )|wε | dx – R tε R |∇wε | dx – tε R t R k(x )|wε | dx, k(x)|wε | dx, and H(ε) = tε R |wε | dx – μtε R h(x)|wε | dx Set (t) = Note that t∗ = at R |∇wε | dx + bt R |∇wε | dx – k(x )|wε | dx (t) attains its maximum at b( |∇wε | dx) + R b ( R |∇wε | dx) + a( R R |∇wε | dx) R k(x )|wε | dx k(x )|wε | dx Then t∗ = max (t) = t≥ abS b S (b S + a k ∞ S) + + +O ε k ∞ k ∞ k ∞ (.) for ε > small enough Then we have F(ε) ≤ c∗ + O ε (.) By (.) of [] we have G(ε) ≤ Cε (.) From (.) of [], (.), and the boundedness of tε we obtain H(ε) = tε R |wε | dx – β ≤ Cε – μCε– μtε R h(x)|wε | dx (.) Xu and Chen Advances in Difference Equations (2016) 2016:176 Page of 14 Since < β < , for fixed μ > , we obtain H(ε) → –∞ as ε → ε (.) It follows from (.), (.), and (.) that the proof of Lemma . is complete Proof of Theorem . By the definition of c there exists a sequence {un } ⊂ N such that I(un ) → c as n → ∞ Then we obtain that un +b R |∇un | dx – R μh(x)|un | dx = R k(x)|un | dx (.) It follows from (.) and Lemma . that c + o() = ≥ un –μ R μ – μ˜ b b – h(x)|un | dx + R |∇un | dx un , (.) which implies the boundedness of {un } in H (R ) since < μ < μ ˜ Then there exists a u in H (R ) By () of Lemma . subsequence of {un }, still denoted by {un }, such that un and Lemma . we have u = By the definition of t(u) we get t(u)u ∈ N So I(t(u)u) ≥ c We claim that un → u in H (R ) Otherwise, by () and () of Lemma ., we would get that c > I(t(u)u) or c > c∗ In any case, we get a contradiction since c < c∗ Therefore, {un } converges strongly to u Thus, u ∈ N and I(u) = c By the Lagrange multiplier rule there exists θ ∈ R such that I (u) = θ G (u) and thus = I (u), u = θ u + b R |∇u| dx k(x)|u| dx – μ – R h(x)|u| dx R Since u ∈ N , we get = θ – u h(x)|u| dx – b –μ R R |∇u| dx , which implies that θ = and u is a nontrivial critical point of the functional I in H (R ) Therefore, the nonzero function u can solve Eq (.), that is, – a+b R |∇u| dx ∗ – u + u = k(x)|u| u + μh(x)u (.) In (.), using u– = max{–u, } as a test function and integrating by parts, by (k ), (h ), and (μ ) we obtain = R a ∇u– dx + k(x) u– + R ∗ – u– dx + b R R |∇u| dx u– dx + R R ∇u– dx μh(x) u– dx ≥ Xu and Chen Advances in Difference Equations (2016) 2016:176 Page 10 of 14 Then u– = and u ≥ From Harnack’s inequality [] we can infer that u > for all x ∈ R Therefore, u is a positive solution of (.) The proof is complete by choosing ω = u Sign-changing solution This subsection is devoted to proving the existence of sign-changing solution of Eq (.) Let N = {u = u+ – u– ∈ H (R ) : u+ ∈ N, u– ∈ N}, where u± = max{±u, } If u+ = and u– = , then u is called a sign-changing function We define c = infu∈N I(u) Lemma . Assume that (μ ), (k )-(k ), and (h )-(h ) hold Then for < β < , c < c + c∗ Proof By Lemma ., using first the same argument as in [] or [], we have that there are s > and s ∈ R such that s ω + s ωε ∈ N (.) Next, we prove that there exists ε > small enough such that sup I(s ω + s ωε ) < c + c∗ (.) s >,s ∈R Obviously, it follows from () of Lemma . that, for any s > and s ∈ R satisfying s ψ + s ωε > ρ, I(s ω + s ωε ) < We only estimate I(s ω + s ωε ) for all s ω + s ωε ≤ ρ By calculation we see that I(s ω + s ωε ) = I(s ω ) + + + + + + , (.) where = = = = = as s R R R s b |∇wε | dx + bs k(x )|wε | dx – R s R |∇wε | dx s – R k(x )|wε | dx, k(x)|wε | dx, k(x) |s ω | + |s wε | – |s ω + s wε | dx, R |wε | dx – μs R h(x)|wε | dx, R ∇(s ω + s ωε ) dx – R ∇(s ωε ) dx – R ∇(s ω ) dx , and = R a∇(s ω )∇(s ωε ) + (s ω )(s ωε ) – μh(x)(s ω )(s ωε ) dx By (.) we obtain that sup s ∈R = abS b S (b S + a k ∞ S) + + +O ε k ∞ k ∞ k ∞ (.) Xu and Chen Advances in Difference Equations (2016) 2016:176 Page 11 of 14 It follows from (.) that ≤ Cε (.) From the elementary inequality |s + t|q ≥ |s|q + |t|q – C |s|q– t + |t|q– s for any q ≥ , the fact that ω ∈ H (R ) ∩ L∞ (R ), and from (.) we have ≤C k(x) |ω | ωε + ω |wε | dx R ≤ k ω ∞ ∞ R |wε | dx + k ω ∞ ∞ R wε dx ≤ Cε (.) By (.) we have β ≤ Cε – Cε– , (.) and using (.), we have ≤ b R R b R R ∇(s ωε ) dx R ∇(s ω ) dx + – + ∇(s ω ) dx – = ∇(s ω ) dx b ∇(s ωε ) dx R ∇(s ωε ) dx ≤ C + Cε (.) Since ω is a positive solution of (.), by the Sobolev inequality we obtain = s s ≤ k R k(x)|ω | ωε dx – b ∞ ω ∞ R wε dx + b R ∇(s ω ) dx R ∇(s ω ) dx R ∇(s ω )∇(s ωε ) dx R ∇(s ωε ) dx ≤ Cε (.) It follows from (.)-(.) that, for < β < , β I(s ω + s ωε ) ≤ I(s ω ) + c∗ + C + Cε + Cε – Cε– < I(s ω ) + c∗ = c + c∗ as ε → , which implies that (.) holds This finishes the proof of Lemma . Xu and Chen Advances in Difference Equations (2016) 2016:176 Page 12 of 14 Lemma . Suppose that (μ ), (k )-(k ), and (h )-(h ) hold Then, for < β < , there exists ω ∈ N such that I(ω ) = c Proof Let {un } ⊂ N be such that I(un ) → c Since un ∈ N, we may assume that there exist constants d and d such that I(u+n ) → d and I(u–n ) → d and d + d = c Then d ≥ c , d ≥ c (.) Just as the proof of (.), we can prove the boundedness of {u+n } and {u–n } Going, if necu± in H (R ) as n → ∞ essary, to a subsequence, we may assume that u± n + – We claim u = and u = Arguing by contradiction, if u+ = or u– = , then by (.) and Lemma ., c + c∗ ≤ d + d = c , ± which contradicts Lemma . Hence, u+ = and u– = We claim that u± n → u strongly in H (R ) Indeed, according to Lemma ., we get one of the following: (i) {u+n } converges strongly to u+ ; (ii) d > I(t(u+ )u+ ); (iii) d > c∗ ; and we also have one of the following: (iv) {u–n } converges strongly to u– ; (v) d > I(t(u– )u– ); (vi) d > c∗ We will prove that only cases (i) and (iv) hold For example, in cases (i) and (v) or (ii) and (v), from u+ – t(u– )u– ∈ N or t(u+ )u+ – t(u– )u– ∈ N we have c ≤ I u+ – t u– u– = I u+ + I –t u– u– < d + d = c or c ≤ I t u+ u+ – t u– u– = I t u+ u+ + I –t u– u– < d + d = c Any one of the two inequalities is impossible In cases (i) and (vi) or (ii) and (vi) or (iii) and (vi), we have c + c∗ ≤ I u+ + c∗ < d + d = c , c + c∗ ≤ I t u+ u+ + c∗ < d + d = c , c + c∗ ≤ c∗ + c∗ < d + d = c , and any one of the three inequalities is a contradiction Therefore, we prove that only (i) and (iv) hold Hence, we obtain that {u+n } and {u–n } converge strongly to u+ and u– , respectively, and we obtain u+ , u– ∈ N Denote ω = u+ – u– Then ω ∈ N and I(ω ) = d + d = c Proof of Theorem . Now we show that ω is a critical point of I in H (R ) Arguing by contradiction, assume that I (ω ) = For any u ∈ N , we claim that G (u) H – = Xu and Chen Advances in Difference Equations (2016) 2016:176 sup v have = | Page 13 of 14 G (u), v | = In fact, by the definition of N and Lemma ., for any u ∈ N , we G (u), u = u h(x)|u| dx + b –μ R R |∇u| dx + b R |∇u| dx k(x)|u| dx – R = u h(x)|u| dx + b –μ R R |∇u| dx + b R |∇u| dx – u h(x)|u| dx + b –μ R R |∇u| dx = – u h(x)|u| dx + b –μ R ≤ – – μ μ˜ R |∇u| dx + b R u +b R |∇u| dx |∇u| dx + b R |∇u| dx < Then we define (u) = I (u) – I (u), G (u) G (u) G (u) , G (u) u ∈ N Choose λ ∈ (, min{ u+ , u– }/) such that (v) – (u) ≤ (ω ) for any v ∈ N with v – ω ≤ λ Let χ : N → [, ] be a Lipschitz mapping such that ⎧ ⎨, v ∈ N with v – ω ≥ λ, χ(v) = ⎩, v ∈ N with v – ω ≤ λ, and for positive constant s , let η : [, s ] × N → N be the solution of the differential equation η(, v) = , dη(s, v) = –χ η(s, v) ds η(s, v) for (s, v) ∈ [, s ] × N We set ψ(τ ) = t ( – τ )ω+ + τ ω– ( – τ )ω+ + τ ω– , ξ (τ ) = η s , ψ(τ ) for ≤ τ ≤ We now give the proof of the fact that I(ξ (τ )) < I(u) for some τ ∈ (, ) Obviously, if τ ∈ (, ) ∪ ( , ), then we have I(ξ ( )) < I(ψ( )) < I(ω ) and I(ξ (τ )) ≤ I(ψ(τ )) < I(ω ) Since t(ξ + (τ )) – t(ξ – (τ )) → –∞ as τ → + and t(ξ + (τ )) – t(ξ – (τ )) → +∞ as τ → – , there exists τ ∈ (, ) such that t(ξ + (τ )) = t(ξ – (τ )) Thus, ξ (τ ) ∈ N and I(ξ (τ )) < I(ω ), which contradicts to the definition of c Hence, we get that I (ω ) = and ω is a signchanging solution of problem (.) The proof of Theorem . is complete Competing interests The authors declare that they have no competing interests Xu and Chen Advances in Difference Equations (2016) 2016:176 Page 14 of 14 Authors’ contributions Both authors contributed to each part of this work equally and read and approved the final version of the manuscript Author details Department of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471003, P.R China School of Mathematics and Statistics, Central South University, Changsha, 410075, P.R China Acknowledgements The authors would like to thank the referees for their valuable suggestions and comments, which led to improvement of the manuscript Research was supported by Natural Science Foundation of China 11271372 and by Hunan Provincial Natural Science Foundation of China 12JJ2004 Received: 28 March 2016 Accepted: 12 May 2016 References Kirchhoff, G: Mechanik Teubner, Leipzig (1883) Chipot, M, Lovat, B: Some remarks on non local elliptic and parabolic problems Nonlinear Anal 30, 4619-4627 (1997) Corrêa, FJSA: On positive solutions of nonlocal and nonvariational elliptic problems Nonlinear Anal 59, 1147-1155 (2004) He, X, Zou, W: Infinitely many positive solutions for Kirchhoff-type problems Nonlinear Anal 70(3), 1407-1414 (2009) Cheng, B, Wu, X: Existence results of positive solutions of Kirchhoff type problems Nonlinear Anal 71, 4883-4892 (2009) Alves, CO, Corrêa, FJSA, Ma, TF: Positive solutions for a quasilinear elliptic equation of Kirchhoff type Comput Math Appl 49, 85-93 (2005) Wu, X: Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in RN Nonlinear Anal., Real World Appl 12, 1278-1287 (2011) He, X, Zou, W: Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3 J Differ Equ 252, 1813-1834 (2012) Nie, J, Wu, X: Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential Nonlinear Anal 75, 3470-3479 (2012) 10 Liu, Z, Guo, S: Positive solutions for asymptotically linear Schrödinger-Kirchhoff-type equations Math Methods Appl Sci (2013) doi:10.1002/mma.2815 11 Sun, J, Wu, TF: Ground state solutions for an indefinite Kirchhoff type problem with steep potential well J Differ Equ 256, 1771-1792 (2014) 12 Liu, H, Chen, H, Yuan, Y: Multiplicity of nontrivial solutions for a class of nonlinear Kirchhoff-type equations Bound Value Probl 2015, 187 (2015) 13 Xu, L, Chen, H: Nontrivial solutions for Kirchhoff-type problems with a parameter J Math Anal Appl 433, 455-472 (2016) 14 Li, G, Ye, H: Existence of positive solutions for nonlinear Kirchhoff type problems in R3 with critical Sobolev exponent and sign-changing nonlinearities (2013) arXiv:1305.6777v1 [math.AP] 15 Wang, J, Tian, L, Xu, J, Zhang, F: Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth J Differ Equ 253, 2314-2351 (2012) 16 Liang, S, Zhang, J: Existence of solutions for Kirchhoff type problems with critical nonlinearity in R3 Nonlinear Anal., Real World Appl 17, 126-136 (2014) 17 Ambrosetti, A, Malchiodi, A: Perturbation Methods and Semilinear Elliptic Problems on RN Birkhäuser, Basel (2005) 18 Bartsch, T: Critical point theory on partially ordered Hilbert spaces J Funct Anal 186, 117-152 (2001) 19 Zhang, Z, Perera, K: Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow J Math Anal Appl 317(2), 456-463 (2006) 20 Mao, A, Zhang, Z: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S condition Nonlinear Anal 70(3), 1275-1287 (2009) 21 Hirano, N, Shioji, N: A multiplicity result including a sign-changing solution for an inhomogeneous Neumann problem with critical exponent Proc R Soc Edinb., Sect A 137, 333-347 (2007) 22 Huang, L, Rocha, EM, Chen, J: Positive and sign-changing solutions of a Schrödinger-Poisson system involving a critical nonlinearity J Math Anal Appl 408, 55-69 (2013) 23 Willem, M: Minimax Theorems Birkhäuser, Boston (1996) 24 Huang, L, Rocha, EM: A positive solution of a Schrödinger-Poisson system with critical exponent Commun Math Anal 15, 29-43 (2013) 25 Chen, J, Rocha, EM: Four solutions of an inhomogeneous elliptic equation with critical exponent and singular term Nonlinear Anal 71, 4739-4750 (2009) 26 Brézis, H, Lieb, EH: A relation between pointwise convergence of functions and convergence of functionals Proc Am Math Soc 88, 486-490 (1983) 27 Gilbarg, D, Trudinger, N: Elliptic Partial Differential Equations of Second Order, 2nd edn Grundlehren Math Wiss., vol 224 Springer, Berlin (1983) 28 Tarantello, G: Multiplicity results for an inhomogeneous Neumann problem with critical exponent Manuscr Math 81, 51-78 (1993) 29 Brézis, H, Nirenberg, L: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents Commun Pure Appl Math 36, 437-477 (1983)