Sign-changing solutions to Schrödinger-Kirchhoff-type equations with critical exponent

Xu and Chen Advances in Difference Equations (2016) 2016:121 DOI 10.1186/s13662-016-0828-0 RESEARCH Open Access Sign-changing solutions to Schrödinger-Kirchhoff-type equations with critical exponent Liping Xu1* and Haibo Chen2 * Correspondence: 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 1 Abstract In this paper, we study the following Schrödinger-Kirchhoff-type equations:   ∗ –(a + b R3 |∇u|2 dx)u + u = k(x)|u|2 –2 u + μh(x)u in R3 , u ∈ H1 (R3 ), where a, b, μ > 0 are constants, 2∗ = 6 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 1 Introduction In this paper, we investigate the following Schrödinger-Kirchhoff-type problem: ⎧ ⎨–(a + b  |∇u| dx)u + u = k(x)|u|∗ – u + μh(x)u in R , R ⎩u ∈ H  (R ), (.) where a, b >  are constants, ∗ =  is the critical Sobolev exponent in dimension three. We assume μ, functions k(x) and h(x) satisfy the following hypotheses: (μ )  < μ < μ̃, where μ̃ is defined by  μ̃ := 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) 2016:121 Page 2 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 Kirchhoff-type problem is related to the stationary analog of the equation  |∇u| dx u = f (x, u) in ,   utt – a + b  where  is a bounded domain in RN , u denotes the displacement, f (x, u) the external force and b the initial tension while 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 which 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; see, for example, [–] and the references therein. The solvability of the following Schrödinger-Kirchoff-type equation (.) has also been well studied in general dimensions 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 positive solution for (.) with critical exponents. More recently, Wang et al. [] and other author [] 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 above works, in this paper our aim is to study the existence of positive and sign-changing solutions for the problem (.). The method is inspired by Hirano and Shioji [] and Huang et al. []; however, the argument used by them cannot be directly applied here. To the best of our knowledge, there are very few works up to now studying sign-changing solutions for Schrödinger-Kirchhoff-type problem with critical exponent, i.e. the problem (.). Our main results are as follows. Theorem . Assume that (μ ), (k ), (k ), and (h )-(h ) hold, then for  < β < , the problem (.) possesses at least one positive solution. Xu and Chen Advances in Difference Equations (2016) 2016:121 Page 3 of 14 Theorem . Assume (μ ), (k ), (k ), and (h )-(h ) hold, then for  < β < , the problem (.) possesses at least one sign-changing solution. Notations  • H  (R ) is the Sobolev space equipped with the norm u H  (R ) = R (|∇u| + |u| ) dx.  • 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 ). • Let D, (R ) is 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 )\{} R |∇u| dx  ( R u dx)  • C >  denotes various positive constants. . The outline of the paper is given as follows: in Section , we present some preliminary results. In Sections  and , we give the proofs of Theorems . and ., respectively. 2 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) =   b  u +   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 (h (...truncated)