Abstract
In this paper, we investigate the following Kirchhoff-type equation
where \(N\ge 3\), \(1< p<N,\ p^*=\frac{Np}{N-p}\), \(0< h\in L^\frac{p^*}{p^*-q}(\mathbb {R}^{N})\) with \(q\in (1,p^*)\); M is a nonnegative continuous function with some growth conditions. We show that the above problem has infinitely many solutions by using variational methods.
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1 Introduction and main results
Kirchhoff equations of the type
are related to the stationary analogue of the Kirchhoff equation
which was proposed by Kirchhoff [1] in 1883 as an extension of the classical d’Alembert’s wave equation for free vibrations of elastic strings. After Lions [2] proposed an abstract framework to problem (1), various kinds of Kirchhoff-type equations have been widely concerned and studied by many scholars (see [3,4,5,6,7,8,9,10,11,12,13,14,15] and the references therein). Among them, the critical case has been studied in [7, 11, 13,14,15]. In particular, Faraci and Farkas [15] dealt with the following Kirchhoff-type problem involving a critical term
where \(\Omega \) is an open connected set of \(\mathbb {R}^{N}\) with smooth boundary, \(N\ge 3\), \(1<p<N\), \(M \in C([0,+\infty ),[0,+\infty ))\) and satisfies some of the following hypotheses.
- \((M_{1})\):
-
\(\hat{M}(t+s)\ge \hat{M}(t)+\hat{M}(s), \ \text {for every}\ t,s\in [0,+\infty )\);
- \((M_{2})\):
-
\(\displaystyle \inf _{t>0}\frac{\hat{M}(t)}{t^{\frac{p^{*}}{p}}}\ge c_{p}\);
- \((M_{3})\):
-
\(\displaystyle \inf _{t>0}\frac{{M}(t)}{t^{\frac{p^{*}}{p}-1}}> S_{N}^{-\frac{p^{*}}{p}}\),
where \(\hat{M} : [0,+\infty )\rightarrow [0,+\infty )\) is the primitive of the function M, defined by
\(c_{p}\) is a constant, defined by
\(S_{N}\) is the best Sobolev constant of \(W_{0}^{1,p}(\Omega )\hookrightarrow L^{p^{*}}(\Omega )\). If \((M_{1})\) and \((M_{2})\) hold, the authors proved that the energy functional associated with problem (2) is sequentially weakly lower semicontinuous in \(W_{0}^{1,p}(\Omega )\). When \((M_{3})\) holds, the property of Palais–Smale (for short (PS)) for the energy functional associated with problem (2) was got by using the second Concentration Compactness lemma of Lions [18] in \(W_{0}^{1,p}(\Omega )\). Moreover, the authors provided an application to a Kirchhoff-type problem on exterior domains
where \(\Omega =\mathbb {R}^{N}\setminus B_R(0)\), \(2\le p<r<p^*\). Under \((M_1)\), \((M_2)\) and
- \((M_{4})\):
-
\(\displaystyle \lim _{t\rightarrow 0}\frac{\hat{M}(t)}{t^{\frac{r}{p}}}=0\),
two nontrivial solutions of problem (3) were obtained for some \(\lambda \in (0,1)\) by employing an abstract well-posedness result for a class of constrained minimization problem.
Inspired by [15], we study the existence of infinitely many solutions for the following p-Laplacian equations of Kirchhoff type via variational methods
where \(N\ge 3\), \(1< p<N,\ p^*=\frac{Np}{N-p}\), \(q\in (1,p^*)\), \(D^{1,p}(\mathbb {R}^{N})\) is the classic Sobolev space (the definition is given in Sect. 2), \(M : [0,+\infty )\rightarrow [0,+\infty )\) is a continuous function, satisfies \((M_{3})\) with \(S_N\) is the best Sobolev constant of \(D^{1,p}(\mathbb {R}^{N})\hookrightarrow L^{p^{*}}(\mathbb {R}^{N})\) and
\((M_{5})\) \(\displaystyle \lim _{t\rightarrow 0}\frac{\hat{M}(t)}{t^{\frac{q}{p}}}=0\).
h satisfies the following assumptions.
- \((h_{1})\):
-
h is positive almost everywhere;
- \((h_{2})\):
-
\(h\in L^{\frac{p^{*}}{p^{*}-q}}(\mathbb {R}^{N})\).
The main result of this paper is the following theorem.
Theorem 1.1
Assume that assumptions \((M_{3}),(M_{5})\), \((h_{1})\) and \((h_{2})\) hold. Then, there are infinitely many solutions to problem (4).
Remark
Assumption \((M_3)\) indicates that the growth rate of \(\hat{M}\) at infinity is no less than \(\frac{p^*}{p}\); at the same time, the decay rate of \(\hat{M}\) at zero is no more than \(\frac{p^*}{p}\). It ensures that the functional associated with problem (4) is coercive in \(D^{1,p}(\mathbb {R}^{N})\). Assumption \((M_5)\) indicates that the decay rate of \(\hat{M}\) at zero is no less than \(\frac{q}{p}\). This assumption is mainly used to ensure the functional associated with problem (4) can take a value less than near zero. Assumption \((M_3)\) is a global constraint; we think it may be extended to some growth assumptions at zero and infinity. There are a lot of functions satisfy the assumptions \((M_3)\) and \((M_5)\). A class of example is
where \(a>S_N^{-\frac{p^{*}}{p}},\ \left\{ \begin{array}{ll} \frac{q}{p}-1<\alpha \le \frac{p^*}{p}-1, &{} q>p;\\ 0\le \alpha \le \frac{p^*}{p}-1,&{} 1<q<p, \end{array}\right. \ \beta \ge \frac{p^*}{p}-1.\) Here we can define \(M(0)=a\) if \(\alpha =0\).
The rest of the paper is organized as follows. In Sect. 2, we give the variational structure of problem (4). In Sect. 3, the main result is proved by using the second Concentration Compactness lemma of Loins and a variant of Clark’s theorem.
The following conventions and notations are used in this paper:
\(\bullet \) \(C, C_{1}, C_{2}, \ldots \) denote positive (possible different) constants.
\(\bullet \) We denote weak and strong convergence by \(u_{n}\rightharpoonup u\) and \(u_{n}\rightarrow u\), respectively.
\(\bullet \) \(o_n(1)\) is an infinitely small quantity of 1.
2 Preliminaries
In this section, we give some preliminary results which will be used to prove our main result.
As usual, the Sobolev space \(D^{1,p}(\mathbb {R}^{N})\) is defined by
equipped with the norm
The classical Lebesgue spaces \(L^{q}(\mathbb {R}^{N})(1\le q\le p^{*})\) are equipped with the norms
\(S_{N}\) is the best Sobolev constant of \(D^{1,p}(\mathbb {R}^{N})\hookrightarrow L^{p^{*}}(\mathbb {R}^{N})\), i.e.,
As it is well known, \(L^{p^{*}}(\mathbb {R}^{N})\) is a uniformly convex Banach space.
Under our assumptions, problem (4) has a variational structure. We denote by \(J: D^{1,p}(\mathbb {R}^{N})\rightarrow \mathbb {R}\), the energy functional associated with problem (4), which is given by
Obviously, \(J\in C^{1}(D^{1,p}(\mathbb {R}^{N}),\mathbb {R})\) with derivative at \(u\in D^{1,p}(\mathbb {R}^{N})\) given by
Consequently, the critical points of J are weak solutions for problem (4).
In order to prove our main result, we need the following lemma which is a variant of a result of Clark [16] and is given in [17].
Lemma 2.1
Assume X is a Banach space, \(I\in C^1(X, \mathbb {R})\) satisfying Palais–Smale condition is bounded from below and even, \(I(0)=0\). If for any \(k\in \mathbb {N}\), there exist k-dimensional subspaces \(X^k\) and \(\rho _k>0\) such that
where \( S_{\rho _k}=\{u\in X|\ \Vert u\Vert =\rho _k\}\); then, I has a sequence of critical values \(c_k<0\) satisfying \(c_k\rightarrow 0\) as \(k\rightarrow \infty .\)
3 Proof of Theorem 1.1
Under the assumptions of Theorem 1.1, we will show the existence of infinitely many solutions for problem (4).
Lemma 3.1
Under assumptions \((M_3)\), \((h_1)\) and \((h_2)\), the functional J is coercive bounded from below in \(D^{1,p}(\mathbb {R}^{N})\) and satisfies Palais–Smale condition.
Proof
From assumption \((M_{3})\), it follows that there exists a positive constant k such that \(k>S_{N}^{-\frac{p^{*}}{p}}\) and \(M(t)\ge kt^{\frac{p^{*}}{p}-1}\) for every \(t\ge 0\). Then, \(\hat{M}(t)\ge \frac{p}{p^{*}}kt^{\frac{p^{*}}{p}}\) for every \(t\ge 0\). Since \(h\in L^{\frac{p^{*}}{p^{*}-q}}(\mathbb {R}^{N})\), we have
Due to \(q<p^{*}\), we obtain that J is coercive and bounded from below in \(D^{1,p}(\mathbb {R}^{N})\).
Let \(\{u_{n}\}\) be a Palais–Smale sequence for J, that is,
Obviously, since J is coercive, \(\{u_{n}\}\) is bounded in \(D^{1,p}(\mathbb {R}^{N})\). Then, there exists \(u\in D^{1,p}(\mathbb {R}^{N})\) such that up to a subsequence
where \(\eta ,\ \nu \) are nonnegative and bounded measures on \(\mathbb {R}^{N}\). By the second Concentration Compactness lemma of Lions [18] and the Concentration compactness principle at infinity of Chabrowski [19], there exist an at most countable index set \(\Lambda \), a set of points \(\{x_{j}\}_{j\in \Lambda }\subset \mathbb {R}^{N}\) and two families of positive numbers \(\{\eta _{j}\}_{j\in \Lambda }\), \(\{\nu _{j}\}_{j\in \Lambda }\) such that
and
where \(\delta _{x_{j}}\) is the Dirac mass concentrated at \(x_{j}\);
where
satisfying \(S_{N}\nu _{\infty }^{\frac{p}{p^{*}}}\le \eta _{\infty }\).
Next, we will prove that the index set \(\Lambda \) is empty. Arguing by contradiction, we may assume that there exists a \(j_{0}\) such that \(\nu _{j_{0}}\ne 0\). Consider now, for \(\epsilon >0\) a nonnegative cut-off function \(\phi _{\epsilon }\) such that
It is easy to see that the sequence \(\{u_{n}\phi _{\epsilon }\}_{n}\) is bounded in \(D^{1,p}(\mathbb {R}^{N})\). Then,
That is to say,
Since \(\{u_n\}\) is bounded in \(D^{1,p}(\mathbb {R}^{N})\), by Hölder inequality, one has
Since
and the sequence \(\{M(\Vert u_{n}\Vert ^{p})\}\) is bounded in \(\mathbb {R}\), we can get that
Moreover, as \(0\le \phi _{\epsilon }\le 1\),
Together with \(\int \limits _{B(x_{j_{0}},2\epsilon )}|\nabla u|^{p}\phi _{\epsilon }{\mathrm{d}}x\rightarrow 0\) as \(\epsilon \rightarrow 0\), thus
In addition,
By assumptions \((h_1)\) and \((h_2)\),
and
Thus,
From (3.1),
Then,
Passing to the \(\liminf \) as \(\epsilon \rightarrow 0\) in both sides of the above inequality, it follows from (3.2)-(3.5) that
From \(S_{N}\nu _{j}^{\frac{p}{p^{*}}}\le \eta _{j}\) for every \(j\in \Lambda \), we obtain
This is a contradiction with the fact that \(k>S_{N}^{-\frac{p^{*}}{p}}\). Such a conclusion implies that \(\Lambda \) is empty.
Then, in order to get that
it suffices to show that \(\nu _\infty =0\). Indeed, let \(\psi _R\in C^\infty (\mathbb {R}^{N},[0,1])\) be a cut-off function such that
It is also easy to see that \(\{u_n\psi _R\}_n\) is bounded in \(D^{1,p}(\mathbb {R}^{N})\). Then,
By Hölder inequality, one has
Since
as \(R\rightarrow \infty \), and the sequence \(\{M(\Vert u_{n}\Vert ^{p})\}_{n}\) is bounded, we can get that
Moreover, by assumption \((M_3)\) again,
Then,
Thus,
In addition,
Since assumptions \((h_1)\) and \((h_2)\) imply that
and
then
Together with
we can get
From (3.6),
then,
Taking the \(\limsup \) as \(R\rightarrow \infty \) in both sides of the above inequality, it follows from (3.7)-(3.10) that
From \(S_{N}\nu _{\infty }^{\frac{p}{p^{*}}}\le \eta _{\infty }\), we obtain
If \(\nu _{\infty }\ne 0\), it also leads to a contradiction with the fact that \(k>S_{N}^{-\frac{p^{*}}{p}}\). Therefore, \(\nu _{\infty }=0\).
The uniform convexity of \(L^{p^{*}}(\mathbb {R}^{N})\) implies that
Since \(\{u_{n}\}\) is bounded in \(D^{1,p}(\mathbb {R}^{N})\),
By Hölder inequality,
We know from the definition of weak convergence and assumption \((h_{2})\) that
So we deduce that
We claim that
In fact, if \(\displaystyle \limsup _{n\rightarrow \infty }M(\Vert u_{n}\Vert ^{p})>0\), then, (3.11) follows at once( in the sense of subsequence). If \(\displaystyle \lim _{n\rightarrow \infty }M(\Vert u_{n}\Vert ^{p})=0\), then, by assumption \((M_{3})\) and Hölder inequality, we obtain that \(u_{n}\rightarrow 0\) in \(D^{1,p}(\mathbb {R}^{N})\) and (3.11) holds true also in this case.
It follows from the definition of weak convergence that
Then,
By the boundedness of \(\{u_n\}\) in \(D^{1,p}(\mathbb {R}^{N})\) and the well-known Simon inequalities
for all \(\xi , \eta \in \mathbb {R}^{N}\), where \(c_{p}\) and \(C_{p}\) are positive constants depending only on p, we can obtain
Therefore, \(u_{n}\rightarrow u\) in \(D^{1,p}(\mathbb {R}^{N})\). \(\square \)
Lemma 3.2
For any \(m\in \mathbb {N}\), there exists a m-dimensional subspace \(X^{m}\) of \(D^{1,p}(\mathbb {R}^{N})\) and \(\rho _{m}>0\) such that \(\displaystyle \sup _{u\in X^{m}\cap S_{\rho _{m}}}J(u)<0\), where \(S_{\rho _{m}}:=\{u\in D^{1,p}(\mathbb {R}^{N}): \Vert u\Vert =\rho _{m}\}\).
Proof
For any \(m\in \mathbb {N}\), we can find m functions \(e_{1}, e_{2}, \ldots , e_{m}\in C_0^{\infty }(\mathbb {R}^{N})\) of linearly independent. The m-dimensional subspace \(X^{m}\) is defined by \(X^{m}=\text {span}\{e_{1}, e_{2}, \ldots , e_{m}\}\) equipped with the norm of \(D^{1,p}(\mathbb {R}^{N})\). While \(\Vert u\Vert _{q,h}:=\left( \int \limits _{\mathbb {R}^{N}}h(x)|u|^{q}{\mathrm{d}}x\right) ^{\frac{1}{q}}\) is also a norm of \(X^{m}\). Because all norms are equivalent in \(X^{m}\), there exists \(C_m>0\) such that
We know from \((M_{5})\) that for some \(C_{0}\in (0,\frac{p}{qC_m^{q}})\), there exists \(\delta >0\) such that
Let \(\rho _{m}\in (0,\delta ^{\frac{1}{p}})\) be small sufficiently, we have that
when \(u\in X^{m}\cap S_{\rho _{m}}\). The proof of Lemma 3.2 is completed. \(\square \)
Proof of Theorem 1.1. Assumption \((M_5)\) implies that \(\hat{M}(0)=0\). By the definition of J, we can get that \(J(0)=0\) and \(J\in C^{1}(D^{1,p}(\mathbb {R}^{N}),\mathbb {R})\) is even. According to Lemma 3.1 and Lemma 3.2, J satisfies all the conditions of Lemma 2.1. Therefore, there are infinitely many solutions to problem (4).
References
Kirchhoff, G.: Vorlesungen über Mathematische Physik: Mechanik. Teubner, Leipzig (1883)
Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos., Inst. Mat. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1997, in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, pp. 284–346 (1978)
Alves, C.O., Corrêa, J.F.S.A., Ma, T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)
Perera, K., Zhang, Z.T.: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 221, 246–255 (2006)
Corrêa, J.F.S.A., Figueiredo, G.M.: On an elliptic equation of p-Kirchhoff type via variational methods. Bull. Aust. Math. Soc. 74, 263–277 (2006)
Mao, A.M., Zhang, Z.T.: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal. 70, 1275–1287 (2009)
Alves, C.O., Corrêa, F.J.S.A., Figueiredo, G.M.: On a class of nonlocal elliptic problems with critical growth. Differ. Equ. Appl. 2, 409–417 (2010)
Wu, X.: Existence of nontrivial solutions and high energy solutions for Schröinger-Kirchhoff-type equations in \(\mathbb{R}^N\). Nonlinear Anal. Real World Appl. 12, 1278–1287 (2011)
Li, Y.H., Li, F.Y., Shi, J.P.: Existence of a positive solution to Kirchhoff type problems without compactness conditions. J. Differ. Equ. 253, 2285–2294 (2012)
Li, G.B., Ye, H.Y.: Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in \(\mathbb{R}^{3}\). J. Differ. Equ. 257, 566–600 (2014)
Naimen, D.: The critical problem of Kirchhoff type elliptic equations in dimension four. J. Differ. Equ. 257, 1168–1193 (2014)
Li, A.R., Su, J.B.: Existence and multiplicity of solutions for Kirchhoff type equation with radial potentials in \(\mathbb{R}^{3}\). Z. Angew. Math. Phys. 66, 3147–3158 (2015)
Hebey, E.: Multiplicity of solutions for critical Kirchhoff type equations. Commun. Partial Differ. Equ. 41, 913–924 (2016)
Wang, L., Xie, K., Zhang, B.L.: Existence and multiplicity of solutions for critical Kirchhoff-type p-Laplacian problems. J. Math. Anal. Appl. 458, 361–378 (2018)
Faraci, F., Farkas, C.: On a critical Kirchhoff-type problem. Nonlinear Anal. 192, 111679 (2020)
Clark, D.C.: A variant of the Ljusternik–Schnirelmann theory. Indiana Univ. Math. J. 22, 65–74 (1972)
Heinz, H.P.: Free Ljusternik–Schnirelmann theory and the bifurcation diagrams of certain singular nonlinear problems. J. Differ. Equ. 66, 263–300 (1987)
Lions, P.L.: The concentration-compactness principle in the calculus of variations, The limit case. Part 1. Rev. Mat. Iberoamericana 1, 145–201 (1985)
Chabrowski, J.: Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents. Calc. Var. Partial Differ. Equ. 3, 493–512 (1995)
Acknowledgements
This work is supported by National Natural Science Foundation of China under Grant Numbers: 12071266, 11701346, 11801338, and Technological Innovation Projects of Colleges and Universities in Shanxi Province under Grant Number: 2019L0024.
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Li, A., Fan, D. & Wei, C. Infinitely many solutions for a class of critical Kirchhoff-type equations involving p-Laplacian operator. Z. Angew. Math. Phys. 73, 39 (2022). https://doi.org/10.1007/s00033-021-01674-9
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DOI: https://doi.org/10.1007/s00033-021-01674-9