Existence of multiple positive solutions for a truncated Kirchhoff-type system involving weight functions and concave–convex nonlinearities

We consider the combined effect of concave–convex nonlinearities on the number of solutions for an indefinite truncated Kirchhoff-type system involving the weight functions. When α+β<4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha+ \beta<4$\end{document}, since the concave-convex nonlinearities do not satisfy the mountain pass geometry, it is difficult to obtain a bounded Palais–Smale sequence by the usual mountain pass theorem. To overcome the problem, we properly introduce a method of Nehari manifold and then establish the existence of multiple positive solutions when the pair of the parameters is under a certain range.

Nowadays scientists and researchers paid more attentions to problem (E 2 ) with sighchanging weight function. For instance, the case α + β = 2 * is considered in [13], whereas in [14,15] the case α + β < 2 * was studied, and the existence and multiplicity of positive solutions when (λ, μ) belongs to a certain subset of R 2 were obtained.
Meanwhile, the problem about Kirchhoff system has been studied. In [16,17] the Kirchhoff system with boundary value shows several physical and biological systems with u and v describing a process depending on the average of itself, such as population densities. Lv and Peng [18] established the existence of positive vector solutions and positive vector ground state solutions by using variational methods and also studied the asymptotic behavior of these solutions. In [19] the authors studied the nonlocal boundary value problem of Kirchhoff-type system, where Ω is a bounded domain in R N , N = 1, 2, 3, β ∈ R, a i , b i , λ i > 0 for i = 1, 2, and p and q are two positive numbers satisfying certain conditions. They obtained the existence of positive solutions by the Nehari manifold and mountain pass lemma and the multiplicity by using cohomological index of Fadell and Rabinowitz. Also, they considered the critical case and proved the existence of positive least energy solutions when β is sufficiently large.
Inspired by the works mentioned, in this paper, we mainly study the truncated Kirchhofftype system with concave-convex nonlinearities involving α + β < 4, since the case α + β ≥ 4 is trivial, which is easy to be proved by using the method in [20]. To the best of our knowledge, the usual mountain pass theorem cannot be directly applied because the concave-convex nonlinearities do not satisfy the mountain pass geometry, so it is difficult to obtain a bounded Palais-Smale sequence (see Theorem 1.15 in [21]). Hence, in this work, by using the method of Nehari manifold, we overcome this difficulty and obtain the existence of multiple positive solutions.
Let us state our knowledge framework and main result. For u ∈ H 1 0 (Ω), its usual norm is denoted by The energy functional associated with the equation (E λ,μ,M k ) is defined by

Preliminaries
Let us introduce the Nehari manifold We split N λ,μ,M k into three parts: The best Sobolev constant S r (1 < r < 2 * ) and S α,β are respectively defined by

Lemma 2.1 Assume that conditions (F) and (G) hold. Then the energy functional I λ,μ,M k is coercive and bounded below on N
α+β }, by the Sobolev and Hölder inequalities we obtain To finish this proof, we need the following claims.
First, by the Sobolev and Hölder inequalities we easily obtain Claim 3. Then, since the proof of Claim 2 is the same as that of Claim 1, here we only give the proof of Claim 1. If and if u 2 > k, then we conclude that which completes the proof of Claim 1. Thus, we could obtain that Since k < min{ a 1 (α+β-2) Proof We refer to Theorem 2.3 of [22]. Setting , M 2 (k)(α + β -2)}, we obtain the following result.
Proof For each (u, v) ∈ N 0 λ,μ,M k , in (2.1), we discuss the problem in four cases.
Similarly to the argument of [20], we conclude that for Λ ∈ (0,Λ), we have the following conclusion.
Case 1: If u 2 ≤ k and v 2 ≤ k, then we obtain that Case 2: If u 2 ≤ k and v 2 > k, then we get Case 3: If u 2 > k and v 2 ≤ k, then we have that Case 4: If u 2 > k and v 2 > k, then we deduce that .