Multiplicity results involving p-biharmonic Kirchhoff-type problems

This paper deals with the existence of multiple solutions for the following Kirchhoff type equations involving p-biharmonic operator: −M(∫Ω(|Δpu|2+|u|p)dx)(Δp2u−|u|p−2u)=λf(x)|u|q−2u+g(x)|u|m−2u,x∈Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned}& -M \biggl( \int_{\varOmega} \bigl( \vert \Delta_{p}u \vert ^{2}+ \vert u \vert ^{p} \bigr)\,dx \biggr) \bigl( \Delta _{p}^{2}u- \vert u \vert ^{p-2}u \bigr) =\lambda f(x) \vert u \vert ^{q-2}u+g(x) \vert u \vert ^{m-2}u,\quad x\in\varOmega, \end{aligned}$$ \end{document} where Ω is a bounded domain in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{N}$\end{document} (N>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N>1$\end{document}), λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda >0$\end{document}, p,q,m>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p, q, m>1$\end{document}, M is a continuous function, and the weight functions f and g are measurable. We obtain the existence results by combining the variational method with Nehari manifold and fibering maps.


Introduction
The theory of p-Laplacian and p-biharmonic operators has been developed very quickly.
The investigation of the existence and multiplicity of solutions has attracted a considerable attention of researchers (see, for instance, [1,3,15,18,22,24,[26][27][28] and the references therein). The motivation of this interest stems from the fact that these nonhomogeneous differential operators are a very productive and rich area of research in recent decades.
Kirchhof-type equations, known as nonlocal differential equations, have received specific attention in recent years. An important number of surveys dealing with this type of equations can model phenomenons arising from the study of elastic mechanics, in numerous physical phenomena such as systems of particles in thermodynamical equilibrium, dielectric breakdown, image restoration, biological phenomena, and so on (see turbed nonlinear p-biharmonic boundary value problem: 2 p u = |u| q-2 u + λh(x)|u| r-2 u, x ∈ Ω, u = ∇u = 0 on ∂Ω, h is a continuous function in Ω, which can change sign, and 2 p u := (| u| p-2 u) is the p-biharmonic operator.
Chen et al. [8] considered the following nonhomogeneous Kirchhof-type problem: , M(s) = a + bs, and a, b, λ, are positive real numbers. The weight functions f and g are continuous in Ω. Based on the Nehari manifold method and the fibering maps, the authors proved that problem (1.1) admits at least two nontrivial solutions.
Inspired by the works mentioned, we study the following Kirchhof-type system: where Ω ⊂ R N (N ≥ 2) is a bounded domain with smooth boundary ∂Ω, λ > 0, the functions f , g are measurable in Ω, and the function M is defined on [0, ∞) by M(s) = a + bs l for some a, b > 0 and 0 ≤ l < 2p N-2p . Before giving our main result, we assume the following hypotheses: (H 1 ) g is a measurable function such that g ∈ L p * p * -m (Ω) and g + := max(g, 0) = 0.
The rest of this paper is organized as follows. In Sect. 2, we give some definitions and basic results that will be used in this paper. Section 3 is devoted to the proof of Theorem 1.1.

Definitions and basic results
In this section, we collect some basic preliminary results that will be used in the proof of our main result. To state our main result, let us introduce some definitions and notations. First, we define the Sobolev space For 1 < s ≤ p * , we denote by C s the best Sobolev constant for the embedding operator W 2,p (Ω) → L s (Ω), which is given by In particular, we have that is, where · s is the usual norm in L s (Ω).

Definition 2.1
We say that a function u ∈ W 2,p (Ω) is a weak solution of (1.2) if for all v ∈ W 2,p (Ω), we have Associated with the problem (1.2), we define the functional energy J λ,M (u) : where M(t) = at + b l+1 t l+1 .
Proof From the hypotheses (H 1 )-(H 2 ) it is obvious that J λ,M ∈ C 1 (W 2,p (Ω), R) and its Gateaux derivative is given by This completes the proof of Lemma 2.1.
Since the energy functional is not bounded from bellow on W 2,p (Ω), we introduce the following subspace of W 2,p (Ω), which is called Nehari manifold: Note that the Nehari manifold N λ,M contains every nonzero solution of equation ( Since m < p(l + 1) < q, J λ,M is coercive and bounded below on N λ,M .
The Nehari manifold N λ,M is closely linked to the behavior of the function h u : t − → J λ,M (tu) for t > 0, defined as follows: Such maps, introduced by Drábek and Pohozaev [10], are known as fibering maps. A simple calculation shows that, for each u ∈ W 2,p (Ω), we have Clearly, Thus, for all u ∈ W 2,p (Ω) \ {0} and t > 0, we have In particular, h u (1) = 0 if and only if u ∈ N λ,M . Also, by equation (2.4) it is easy to see that for u ∈ N λ,M , In order to have multiplicity of solutions, we split N λ,M into three parts Furthermore, using arguments similar to those in of Theorem 2.3 in [6], we have the following lemma. Put Then we have the following lemma. Therefore and On the other hand, from (2.1) and the Hölder inequality we obtain (2.10) By combining (2.7) and (2.9) we get (2.11) Moreover, by combining (2.8) and (2.10) we get (2.12) Finally, by combining (2.11) and (2.12) we obtain λ 1 ≤ |λ|, which is a contradiction.
Then we have the following: (2.14) Proof (i) Let u ∈ N + λ,M . Then from (2.6) and the fact that h u (1) > 0 we obtain
(i) If λ Ω f (x)|u| q dx < 0, then there is a unique t + ∈ (0, T) such that Therefore h u (t + ) = 0 and h u (t + ) > 0, that is, h u has a global maximum at t + , and t + u ∈ N + λ,M . (ii) Assume that λ Ω f (x)|u| q dx > 0, and put Then we have Since ψ u is a decreasing function, we get T 0 > T. Moreover, since Ψ u is decreasing on (T, ∞), from (2.9) we have . Then by similar arguments as in the proof of Proposition 2.2 we can deduce the results of Proposition 2.3 Proof We omit the proof, which is almost the same as that in Wu ([29], Proposition 9).

Proof of our main result
In this section, we apply the method of Nehari manifold combined with the fibering maps to investigate the multiplicity of nontrivial solutions for problem (1.2). To this aim, we assume that |λ| ∈ (0, λ 0 ).
So, it is easy to see that From the Brezis-Lieb lemma [5] we obtain u + ku + λ,M p . Therefore u k − → u λ,M strongly in W 2,p (Ω). Now we will prove that u + λ,M ∈ N + λ,M . We proceed by contradiction assuming that u + λ,M ∈ Nλ,M . We have By letting k tend to infinity we obtain which is a contradiction. Therefore u + λ,M ∈ N + λ,M , Moreover, it is not difficult to see that (3.1) and (3.2) imply that u + λ,M is a weak solution of problem (1.2). The proof is now completed. a.e. in Ω. (3.4)