Abstract
In this present paper, we consider the following critical Kirchhoff problem
where \(a,\lambda \in {\mathbb {R}}\) and \(m,\mu \in {\mathbb {R}}^+\cup \{0\}\), \(N\ge 3\) and \(2<p<2^*\). In this first part, a pure critical Kirchhoff problem (\(m=\mu =0\)) has been considered for both \(a>0\) and \(a\le 0\). We obtain a series of fairly complete existence and multiplicity results and have a clear understand the solutions of this pure critical Kirchhoff problem. In particular, if \(N\ge 5\), \(a>0\) and \(\lambda >0\) is suitable small, we obtain two positive solutions, in which one is a mountain pass solution and another one is a global (local) minimum solution. In the second part, the original perturbation problem with \(m,\mu >0\) has been considered and two positive solutions also have been obtained for \(N\ge 5\), which is rather different compared with the case that \(\lambda =0\).
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1 Introduction and main results
In this present paper, we investigate the following critical Kirchhoff problem
where a, m are two constants, \(\lambda ,\mu \) are two parameters, \(2<p<2^*\) and \(2^*=\frac{2N}{N-2}\) for \(N\ge 3\) is the critical exponent. The original model comes from the following equation
presented in Kirchhoff [23]. This kind of equations is an extension of the classical d’Alembert’s wave equations because of taking into account the effects of the changes in the length of a string during vibrations. Kirchhoff type problem is concerning not only the effects of the changes in length of a string, but also the non-Newton mechanics, the physical laws of the universe, the problem of plasma, elastic theory, population dynamics models and so on.
The stationary general problem
has been studied by many authors under variant conditions on V(x) and g(x, u). Li and Ye [25] proved that problem (1.2) has a positive ground state solution for \(g(x,u)=u^{p-1}\), \(3<p<6\) when V satisfies some suitable conditions. For more related results, we refer the readers to the bounded domain in [31, 36, 39], the ground state solutions in [15, 41], the nodal solutions [10, 37, 40, 47], the periodic potential cases [3, 27, 48], the semi-classical and multi-peak solutions [13, 21, 26].
In what follows, we briefly recall some known results about the critical Kirchhoff problem. A large number of papers have been published in three-dimensional space. In Alves et al. [2], the authors studied the following Kirchhoff problem
where the parameter \(\nu >0\), M(t) and f(x, u) are continuous functions, and \(\Omega \) is a bounded domain in \({\mathbb {R}}^3\). They obtained a positive solution if \(\nu >0\) is large enough, among other things, f(x, u) satisfies the well-known Ambrosetti-Rabinowitz sup-linear condition, i.e.,
After that, the authors in He and Zou [18] and Wang et al. [42] studied the following critical Kirchhoff problem
where \(\varepsilon >0\) is a small parameter, \(a,b>0\) are two constants and V(x) is a suitable potential. By recovering a local \((PS)_c\) condition for \(c<c'_3\) or \(c''_3\), where
and S is the best Sobolev constant from \(D^{1,2}({\mathbb {R}}^N)\) to \(L^{2^*}({\mathbb {R}}^N)\), they obtained a semi-classical solution if \(\nu >0\) large enough and f(u) satisfies the sup-4-linear condition.
In Li and Ye [24] and Xie et al. [44], the authors obtained a positive solution of the critical Kirchhoff problem either in \({\mathbb {R}}^3\) or in a bounded domain when the perturbation term f satisfies the sup-4-linear condition by improving a local \((PS)_c\) condition for \(c<c_3\), where
For more related results concerning the perturbation term f with the sup-4-linear condition, we refer the readers to Figueiredo et al. [12], He and Li [17], Liu and Guo [28, 29], Naimen [33] and references therein.
In He and Li [16], the authors considered the following critical Kirchhoff problem with a sub-4-linear perturbation term
They obtained some existence results of the semi-classical solutions for \(2<p<4\) and \(\nu >0\) large enough. Sun and Liu [38] obtained a positive solution for \(1<p<2\) and \(\nu >0\) small enough. Some results concerning the critical Kirchhoff problem with a nonhomogeneous term can be found in [8, 22].
Little results have been obtained for the high-dimensional cases \(N\ge 4\). The authors in Xie et al. [45, 46] studied the following critical Kirchhoff problem in \(D^{1,2}({\mathbb {R}}^N)\),
With some suitable assumptions on b and the potential V, they obtained a bound stated solution for \(N=3,4\) and two solutions for \(N\ge 5\) by recovering a local \((PS)_c\) condition with
where \(c_{N,\pm }\) are two fixed levels depending on N, a, b for \(N\ge 5\) and \(c_{N}\) is defined in (1.5) for \(N=3\) and
In Naimen [32], the author considered the following problem in high-dimensional space
where \(p\in [2,2^*)\) and \(\Omega \) is a bounded domain in \({\mathbb {R}}^N\), \(N\ge 4\). With some suitable assumptions on the positive constants a, b and the parameters \(\nu \), \(\mu \), the author obtained a positive solution in four-dimensional case by recovering a local \((PS)_c\) condition for \(c<c_{4,\mu }\), where \(c_{4,\mu }=\frac{(aS)^{2}}{4(\mu -bS^2)}.\) After that, Naimen and Shibata [34] obtained two positive solutions for problem (1.9) in high-dimensional case \(N\ge 5\) by using the critical levels \(c_{N,\pm }\), which coincides with (1.7). A related result can be found in Naimen and Shibata [35] for four-dimensional case. We also refer the readers to Hebey [19, 20] for the critical Kirchhoff problem in manifolds.
The rescaling argument is an effective method dealing with a kind of autonomous Kirchhoff problem. In Azzollini [6], the author considered the following Kirchhoff problem by a rescaling argument
where g satisfies some suitable assumptions. The author proved that there exist some positive constants t such that \(u(t\cdot )\) solving the Kirchhoff problem, where u is a solution of \(-\Delta u=g(u)\) in \({\mathbb {R}}^N\). More related results by using this similar method can be found in Azzollini [5], Lu [30] and Wu et al. [43].
From the above-mentioned works, we can find that it is necessary to make a clear and complete study on the solutions and corresponding critical levels of the pure critical Kirchhoff problem, which are interesting and basic works and heavily affect the compactness results or the local \((PS)_c\) condition. The levels \(c_N\), which are defined by (1.5) and (1.8) for \(N=3\) and 4, respectively, are the first and optimal threshold for the lack of compactness result. It should be mentioned that these are not clear for the high-dimensional cases. From this point, we try to give a complete research on the pure critical Kirchhoff problem, i.e., problem (1.1) with \(m=\mu =0\), especially for the high-dimensional cases \(N\ge 5\).
In the first part of this paper, we investigate the following pure critical Kirchhoff problem
where a is a constant, \(\lambda \) is a parameter and \(N\ge 3\). Roughly speaking, we try to find solutions of \(K_{\lambda }^*\) from the solutions of the following classical critical elliptic equation by a multiplying argument
It should be mentioned that this method has no difference comparing with the rescaling argument used by Azzollini [6].
To state our results, we need some basic and suitable variational setting. It is well-known that the solutions of \(K_{\lambda }^*\) correspond to the critical points of \(C^2\) functional I: \(D^{1,2}({\mathbb {R}}^N)\rightarrow {\mathbb {R}}\) given by
For each \(u\in D^{1,2}({\mathbb {R}}^N)\), we denote the fibering map \(Q_{u}(t)\) : \([0,+\infty )\rightarrow {\mathbb {R}}\),
A good candidate for appropriate subset of \(D^{1,2}({\mathbb {R}}^N)\) is so-called Nehari manifold
With the help of the fibering map, it is natural to divide \({\mathcal {N}}^{N}\) into three subsets \({\mathcal {N}}^{N,-}\), \({\mathcal {N}}^{N,+}\) and \({\mathcal {N}}^{N,0}\) corresponding to the local maxima, local minima and points of inflexion of fibering map, respectively,
We denote the unique positive solution of \(S^*\) by U, which achieves the best Sobolev constant S and \(S^{N/2}=|U|_{2^*}^{2^*}=|\nabla U|_2^{2}\), where \(|\cdot |_q\) is the standard norm in \(L^q\) for \(q\ge 1\).
The results for the three- or four-dimensional cases can be stated as follows:
Theorem 1.1
Assume \(a>0\) and \(\lambda >0\). Then, the following results hold:
- (i):
-
For \(N=3\), problem \(K_{\lambda }^*\) has infinitely many distinct solutions \(\{\varphi _i\}_{i=1}^{\infty }\). Moreover, \({\varphi _1}\) is a positive solution,
$$\begin{aligned} \inf _{u\in {\mathcal {N}}^{3}}I(u)=c_{3}=I(\varphi _1)>0,\ \ I(\varphi _{i})\rightarrow +\infty \ \ as\ i\rightarrow +\infty . \end{aligned}$$ - (ii):
-
For \(N=4\), problem \(K_{\lambda }^*\) has a positive solution \(\varphi _1\) if and only if \(\lambda <S^{-2}\). Moreover, for any \(n\in {\mathbb {N}}\), there exists \(\lambda _n>0\) such that problem \(K_{\lambda }^*\) with \(\lambda \in (0,\lambda _n)\) has n solutions \(\{\varphi _i\}_{i=1}^{n}\) satisfying
$$\begin{aligned} \inf _{u\in {\mathcal {N}}^{4}}I(u)=c_4=I(\varphi _1)>0, \\ I(\varphi _{1})<I(\varphi _{2})<\cdots <I(\varphi _{n}) =\frac{a^2}{4(\lambda _n-\lambda )}. \end{aligned}$$
If \(N\ge 5\), the solutions of problem \(K_{\lambda }^*\) exist in pairs and the main results can be stated as follows:
Theorem 1.2
Assume \(N\ge 5\), \(a>0\) and \(\lambda >0\). Then, the following results hold:
- (i):
-
Problem \(K_{\lambda }^*\) has two positive solutions \(\varphi _{1,\pm }\) if and only if \(\lambda \in (0, \Lambda _0)\). Both of them are two local minimum points on the Nehari manifolds
$$\begin{aligned} \inf _{u\in {\mathcal {N}}^{N,-}}I(u)=I(\varphi _{1,-})=c_{N,-}>0, \ \ \inf _{u\in {\mathcal {N}}^{N,+}}I(u)=I(\varphi _{1,+})=c_{N,+},\ \ \end{aligned}$$and \(c_{N,-}>c_{N,+}\). Moreover, \(c_{N,+}<0\) for \(\lambda \in (0,\Lambda _1)\) and \(c_{N,+}>0\) for \(\lambda \in (\Lambda _1,\Lambda _0)\), where \(\Lambda _0\) and \(\Lambda _1\) are defined as follows, respectively,
$$\begin{aligned} \ \Lambda _0:=\frac{2}{N-2} \left( \frac{N-4}{a(N-2)}\right) ^{\frac{N-4}{2}}S^{-\frac{N}{2}}\ { and}\ \ \Lambda _1:=\frac{4}{N}\left( \frac{N-4}{aN}\right) ^{\frac{N-4}{2}} S^{-\frac{N}{2}}. \end{aligned}$$(1.12) - (ii):
-
For any \(n\in {\mathbb {N}}\), there exists \(\lambda _n>0\) such that problem \(K_{\lambda }^*\) with \(\lambda \in (0, \lambda _n)\) has 2n solutions \(\varphi _{i,\pm }\in {\mathcal {N}}^{N,\pm }\), \(i\in \{1,2,\ldots , n\}\) and
$$\begin{aligned} I(\varphi _{i,-})>I(\varphi _{i,+}), \\ I(\varphi _{1,\pm })<I(\varphi _{2,\pm })<\cdots<I(\varphi _{n,\pm }) <\frac{a^2}{N(N-4)\lambda } . \end{aligned}$$
Remark 1.3
More details can be stated as follows for \(N\ge 5\).
(i). It should be mentioned that 0 and \(\varphi _{1,+}\) are two local minimum points of the energy functional I and one of them is the global minimum point in \(D^{1,2}({\mathbb {R}}^N)\). In other words, there hold,
where \(B_{r_\lambda }=\{u\in D^{1,2}({\mathbb {R}}^N): |\nabla u|_2 \le r_\lambda \}\) and \(r_\lambda :=\xi _{1,-}\) is a constant defined by (1.21). Moreover,
Actually, by the Sobolev inequalities and some basic computations, one obtains
where \(\theta (t)=\frac{a}{2}t^2+ \frac{\lambda }{4}t^4-\frac{1}{2^*}S^{-\frac{2^*}{2}}t^{2^*}\). Here \(I(0)=\inf _{B_{r_\lambda }}I(u)\) holds.
On the one hand, for \(u\in {\mathcal {N}}^{N,+}\), we have that \(|\nabla u|_2>\eta _\lambda >r_\lambda \) (see (1.23)), which implies \({\mathcal {N}}^{N,+}\subset B^c_{r_\lambda }\). Thus, \(\inf _{u\in {\mathcal {N}}^{N,+}}I(u)\ge \inf _{u\in B^c_{r_\lambda }}I(u)\). On the other hand, it follows that
where \(\xi _{1,+}\) is given in (1.21). Thus, \(I(\varphi _{1,+})=\inf _{u\in B^c_{r_\lambda }}I(u)\) holds. Moreover, by Theorem 1.2 (i), we have
(ii) Moreover, \(\varphi _{1,-}\) is a mountain pass solution. Actually, by (1.13), Theorem 1.2 (i) and \(|\nabla \varphi _{1,+}|_2>r_\lambda =|\nabla \varphi _{1,-}|_2\), we can check that
Thus, we can define a mountain pass level as following:
where \(\Gamma =\{\gamma (t)\in C([0,1]): \gamma (0)=0\ \mathrm{and}\ \gamma (1)=\varphi _{1,+}\}\). We claim that
In fact, since \(|\nabla \gamma (0)|_2<r_\lambda <|\nabla \gamma (1)|_2\) for any \(\gamma \in \Gamma \), there exists \(t_\gamma \in (0,1)\) such that \(\gamma (t_\gamma )\in \partial B_{r_\lambda }\). Thus, it follows that
which implies \(c_{N,-}\le m_-\). Let \(\gamma _1(t):=t\varphi _{1,+}\) and \(\gamma _1(t)\in \Gamma \). Then, we have
which implies \(c_{N,-}\ge m_-\). Then, \(c_{N,-}= m_-\) holds.
Theorem 1.4
Assume that \(N\ge 3\), \(a>0\) and \(\lambda <0\), problem \(K_{\lambda }^*\) has infinitely many distinct solutions \(\{\varphi _i\}_{i=1}^{\infty }\). Moreover, \(\varphi _1\) is the positive ground state solution with \(I(\varphi _1)=c_{N}>0\) and \(I(\varphi _{i})<-\frac{a^2}{4\lambda }.\)
Remark 1.5
(i). Similar to Remark 1.3, the critical level \(c_N\) obtained in Theorem 1.1 (i)-(ii) and Theorem 1.4 are the mountain pass levels. In fact, for \(a>0\), if \(N=3\) or \(N=4\) and \(\lambda <S^{-2}\) or \(N\ge 5\) and \(\lambda <0\), we take \(T_0>0\) large such that \(\Vert T_0U\Vert >\xi _1\) and \(I(T_0U)<0\), where \(\xi _1\) comes from (1.19), (1.20) and (1.28),
Thus, we can define a mountain pass level as following:
where \(\Gamma :=\{\gamma (t)\in C([0,1]): \gamma (0)=0\ \mathrm{and}\ \gamma (1)=T_0U\}\). It is evident to prove that
(ii). We have a clear description of the above critical levels \(c_N\) and \(c_{N,\pm }\) as follows. Set
If \(N=3\) or \(N=4\) and \(\lambda <S^{-2}\) or \(N\ge 5\) and \(\lambda <0\), the critical levels \(c_N\) defined in Theorem 1.1 (i)-(ii) and Theorem 1.4 can be rewritten as
If \(N\ge 5\) and \(0<\lambda <\Lambda _0\), the critical levels \(c_{N,\pm }\) defined in Theorem 1.2 (i) can be rewritten as
where \(\eta _\lambda =\left( \frac{2a}{(N-4)\lambda }\right) ^{1/2}\). It is easy to check that (1.17) coincide with (1.5) for \(N=3\) and (1.8) for \(N=4\) and \(0<\lambda <S^{-2}\), (replace b by \(\lambda \)).
Here, we begin to give the proof of the above Theorem 1.1–1.4 by the multiplying argument.
The proof of Theorem 1.1 (i).
Equation \(S^*\) admits infinitely many distinct solutions \(\{u_i\}_{i=1}^{\infty }\) satisfying \(|\nabla u_i|_2\rightarrow +\infty \) as \(i\rightarrow \infty \) (see Ding [11]). Without loss of generality, we assume \(|\nabla u_1|_2<|\nabla u_2|_2<\cdots <|\nabla u_i|_2\rightarrow +\infty \), where \(u_1=U\) is the positive solution. It is evident to check that the existence of a positive root \(K_{i}\) for equation \(g_{i}(t)=0\) defined by
where \(i\in {\mathbb {N}}\), \(a>0\) and \(\lambda >0\). Let \(\varphi _{i}:=K_{i}u_i\), then
Thus, \(\{\varphi _{i}\}_{i=1}^{\infty }\) is a sequence of solutions for problem \(K_{\lambda }^*\) and \(\varphi _i\in {\mathcal {N}}^{3,-}\) because of
Set \(\xi _{i}:=|\nabla \varphi _i|_2\), then \(\xi _{i}\) is the positive root of equation \(f_i(t)=0\) defined by
where \(i\in {\mathbb {N}}\), \(a>0\) and \(\lambda >0\). By the fact that \(|\nabla u_i|_2<|\nabla u_{i+1}|_2\) and \(|\nabla u_i|_2\rightarrow +\infty \) as \(i\rightarrow \infty \) and some calculations, one has \(f_i(t)<f_{i+1}(t)\) and
Therefore, \(I(\varphi _{i})=\frac{1}{3}a|\nabla \varphi _i|_2^2+\frac{1}{12}\lambda |\nabla \varphi _i|_2^4\). By (1.19), one obtains
Moreover, it follows from \(u\in {\mathcal {N}}^{3}\) and the Sobolev inequalities that
which implies \(|\nabla u|_2\ge \xi _{1}\). Combing with \(\xi _1=|\nabla \varphi _1|_2\), it follows that
The proof is completed. \(\square \)
The Proof of Theorem 1.1 (ii)
On the one hand, if \(0<\lambda <S^{-2}\), let \(K_1=\sqrt{\frac{a}{1-\lambda S^{2}}}\) and U be the positive solution of problem \(S^*\). Then, it is evident to check that \(\varphi _{1}:=K_{1}U\) is a positive solution of problem \(K_{\lambda }^*\). On the other hand, if \(\varphi _{1}\) is a positive solution of problem \(K_{\lambda }^*\), then \(U=\left( a+\lambda |\nabla \varphi _{1}|_2^2\right) ^{-1/2}\varphi _{1}\) is a positive solution of problem \(S^*\). Remind that \(|\nabla U|_2^2=S^{N/2}\), then \(\xi _{1}:=|\nabla \varphi _{1}|_2\) is the root of equation \(f(t)=0\) defined by
where \(a>0\) and \(\lambda >0\). The existence of a positive root for \(f(t)=0\) implies that \(\lambda <S^{-2}\).
For any \(n\in {\mathbb {N}}\), let \(\{u_i\}_{i=1}^{n}\) be n solutions of equation \(S^*\) satisfying \(|\nabla U|_2=|\nabla u_1|_2<|\nabla u_2|_2<\cdots <|\nabla u_n|_2\). Setting \(\lambda _n=|\nabla u_n|_2^{-2}\), it is evident to check that the existence of a positive root \(K_{i}\) for equation \(g_i(t)=0\) defined by
where \(i\in \{1,2,\ldots , n\}\), \(a>0\) and \(\lambda \in (0,\lambda _n)\). Let \(\varphi _{i}:=K_{i}u_i\), then \(\{\varphi _{i}\}_{i=1}^{n}\) is a sequence of solutions for problem \(K_{\lambda }^*\) and \(\varphi _i\in {\mathcal {N}}^{4,-}\) because of \(Q_{\varphi _i}''(1)=-2a|\nabla \varphi _i|_2^2<0\). Set \(\xi _{i}:=|\nabla \varphi _i|_2\) for any \(i\in \{1,2,\ldots , n\}\), then \(\xi _{i}\) is the positive root of equation \(f_i(t)=0\) defined by
where \(i\in \{1,2,\ldots , n\}\), \(a>0\) and \(\lambda \in (0,\lambda _n)\). By some calculations and \(|\nabla u_i|_2<|\nabla u_{i+1}|_2\) for \(i\in \{1,2\ldots ,n-1\}\), one has \(f_i(t)<f_{i+1}(t)\) and
It follows from \(I(\varphi _{i})=\frac{1}{4}a|\nabla \varphi _{i}|_2^2\) and (1.20) that
By the Sobolev inequalities, for \(u\in {\mathcal {N}}^{4}\), we obtain
which implies that \(|\nabla u|_2\ge \xi _{1}\). Combing with \(|\nabla \varphi _1|_2=\xi _1\), it follows that
The proof is completed. \(\square \)
The proof of Theorem 1.2 (i).
On the one hand, by the assumption \(\lambda \in (0, \Lambda _0)\), there exist two positive roots for equation \(g(t)=0\) defined by
where \(a>0\). Without loss of generality, we denote the roots by \(K_{1,\pm }\) and \(K_{1,-}<K_{1,+}\). Let U be the positive solution of problem \(S^*\), then it is evident to check that \(\varphi _{1,\pm }:=K_{1,\pm }U\) are two positive solutions of problem \(K_{\lambda }^*\). On the other hand, if \(\varphi _{1,\pm }\) are two positive solutions of problem \(K_{\lambda }^*\), then \(U=\left( a+\lambda |\nabla \varphi _{1,\pm }|_2^2\right) ^{-(N-2)/4}\varphi _{1,\pm }\) are the positive solutions of problem \(S^*\). Remind that \(|\nabla U|_2^2=S^{N/2}\). Set
then \(\xi _{1,\pm }\) are the roots of equation \(f(t)=0\) defined by
where \(a>0\) and \(\lambda >0\). The existence of two positive roots for \(f(t)=0\) implies that \(\lambda \in (0, \Lambda _0)\).
Secondly, it follows from \(u\in {\mathcal {N}}^{N}\) and the Sobolev inequalities that
which implies that \(\xi _{1,-}\le |\nabla u|_2\le \xi _{1,+}\). Therefore, for any \({\mathcal {N}}^{N}\), one obtains
where \(\eta _\lambda :=(\frac{2a}{(N-4)\lambda })^{1/2}\). Combining with \(\xi _{1,-}\le |\nabla u|_2\le \xi _{1,+}\) for any \(u\in {\mathcal {N}}^{N}\) and \(\xi _{1,-}<\eta _{\lambda }<\xi _{1,+}\) (see Lemma 3.3), we can rewrite the Nehari manifolds as follows:
Directly, \(\varphi _{1,\pm }\in {\mathcal {N}}^{N,\pm }\). For any \(u\in {\mathcal {N}}^{N}\), there holds,
where
It is easy to check that \({\mathcal {I}}(t)\) is increasing in \((0, \eta _\lambda ^2)\) and decreasing in \((\eta _\lambda ^2,+\infty )\). \({\mathcal {I}}\) achieves its maximum point at \(\eta ^2_\lambda \). It follows from \(\varphi _{1,\pm }\in {\mathcal {N}}^{N,\pm }\) and \( 0<\xi _{1,-}<\eta _\lambda <\xi _{1,+}\) that
which imply that \(\inf _{u\in {\mathcal {N}}^{N,\pm }}I(u)=I(\varphi _{1,\pm })\) and \(I(\varphi _{1,-})>0\). Moreover, by the fact that
in Lemma 3.3, one obtains \(I(\varphi _{1,+})<I(\varphi _{1,-})\).
Lastly, it remains to prove that \(I(\varphi _{1,+})<0\) for \(\lambda \in (0,\Lambda _1)\) and \(I(\varphi _{1,+})>0\) for \(\lambda \in (\Lambda _1,\Lambda _0)\). Actually, it is evident to check that \({\mathcal {I}}(t)>0\) for \(t\in (0,2\eta ^2_\lambda )\), \({\mathcal {I}}(2\eta ^2_\lambda )=0\) and \({\mathcal {I}}(t)<0\) for \(t\in (2\eta ^2_\lambda , +\infty )\). Then, it is sufficient to prove that \(\xi _{1,+}>\sqrt{2}\eta _\lambda \) for \(\lambda \in (0,\Lambda _1)\) and \(\xi _{1,+}<\sqrt{2}\eta _\lambda \) for \(\lambda \in (\Lambda _1,\Lambda _0)\). It follows that
Here, \(f(\sqrt{2}\eta _\lambda )<0\) for \(\lambda \in (0,\Lambda _1)\) and \(f(\sqrt{2}\eta _\lambda )>0\) for \(\lambda \in (\Lambda _1,\Lambda _0)\). Combining this and \(f(\xi _{1,-})=f(\xi _{1,+})=0\), one has that \(\xi _{1,+}>\sqrt{2}\eta _\lambda \) for \(\lambda \in (0,\Lambda _1)\) and \(\xi _{1,+}<\sqrt{2}\eta _\lambda \) for \(\lambda \in (\Lambda _1,\Lambda _0)\). The proof is completed. \(\square \)
The Proof of Theorem 1.2 (ii)
For any \(n\in {\mathbb {N}}\), let \(\{u_i\}_{i=1}^{n}\) be n solutions of equation \(S^*\) satisfying \(|\nabla U|_2=|\nabla u_1|_2<|\nabla u_2|_2<\cdots <|\nabla u_n|_2\). Let
it is evident to check that the existence of two positive roots \(K_{i, \pm }\) (\(K_{i, -}<K_{i, +}\)) for equation \(g_i(t)=0\) defined by
where \(i\in \{1,2,\ldots , n\}\), \(a>0\) and \(\lambda \in (0,\lambda _n)\). Let \(\varphi _{i,\pm }:=K_{i, \pm }u_i\), then \(\{\varphi _{i,\pm }\}_{i=1}^{n}\) are the solutions of equation \(K_{\lambda }^*\). Set \(\xi _{i,\pm }:=|\nabla \varphi _{i,\pm }|_2\) for \(i\in \{1,2,\ldots , n\}\), then \(\xi _{i,\pm }\) are the roots of equation \(f_i(t)=0\) defined by
where \(i\in \{1,2,\ldots , n\}\), \(a>0\) and \(\lambda \in (0,\lambda _n)\). With a similar argument of Lemma 3.3, we obtain
which implies \(\varphi _{i,\pm }\in {\mathcal {N}}^{N,\pm }\) and \(I(\varphi _{i,-})>I(\varphi _{i,+}).\) From the assumption \(|\nabla u_i|_2<|\nabla u_{i+1}|_2\) for \(i\in \{1,2,\ldots ,n-1\}\), it follows that \(f_i(t)<f_{i+1}(t)\) and
Combining with the monotonicity of \({\mathcal {I}}(t)\), the desired result holds,
The proof is completed. \(\square \)
The proof of Theorem 1.4
Let \(\{u_i\}_{i=1}^{\infty }\) be a sequence of solutions for equation \(S^*\) and satisfy \(|\nabla U|_2=|\nabla u_1|_2<|\nabla u_2|_2<\cdots <|\nabla u_i|_2\rightarrow +\infty \). It is evident to check that the existence of a positive root \(K_{i}\) for equation \(g_i(t)=0\) defined by
where \(i\in {\mathbb {N}}\), \(a>0\) and \(\lambda <0\). Let \(\varphi _{i}:=K_{i}u_i\), then \(\{\varphi _i\}_{i=1}^{\infty }\) is a sequence of solutions of equation \(K_{\lambda }^*\). Set
then \(\xi _{i}\) is a positive root of \(f_i(t)=0\) defined by
where \(i\in {\mathbb {N}}\), \(a>0\) and \(\lambda <0\). From the assumption \(|\nabla u_i|_2<|\nabla u_{i+1}|_2\), it follows that \(f_i(t)<f_{i+1}(t)<\lambda t^2+a\) and
By (1.29), \(a>0\) and \(\lambda <0\), one has
which implies that \(\varphi _{i}\in {\mathcal {N}}^{N,-}\). By (1.29) and the monotonicity of \({\mathcal {I}}(t)\), the desired result \(I(\varphi _{i}) <-\frac{a^2}{4\lambda }\) holds. It follows from \(u\in {\mathcal {N}}^{N}\) that
which implies that \(|\nabla u|_2\ge \xi _{1}\). Combing with \(|\nabla \varphi _{1}|_2=\xi _1\), it follows that
The proof is completed. \(\square \)
Remark 1.6
For \(N\ge 5\), let \(\{\lambda _n\}\) be a positive and decreasing sequence satisfying that \(\lambda _n<\Lambda _0\) and \(\lambda _n\rightarrow 0^+\) as \(n\rightarrow \infty \). It follows from Theorem 1.2 (i) that \(\varphi ^{n}_{1,\pm }:=K^{n}_{1,\pm }U\) being two positive solutions of problem \(K_{\lambda }^*\), where \(K^{n}_{1,\pm }\) are the positive roots of the equation \(\lambda _n S^{\frac{N}{2}}t^2-t^{\frac{4}{N-2}}+a=0.\) It is evident to check that \(a^{\frac{N-2}{4}}<K^{n}_{1,-}<\left( \frac{2}{(N-2)\lambda S^{N/2}}\right) ^{\frac{N-2}{2(N-4)}}<K^{n}_{1,+}\) and
Moreover, let \(\{\lambda _n\}\) be a negative and increasing sequence satisfying that \(\lambda _n\rightarrow 0^-\). It follows from Theorem 1.4 that \(\varphi ^{n}_{1,\pm }:=K^{n}_{1}U\) being a positive solutions of problem \(K_{\lambda }^*\), where \(K^{n}_{1}<a^{\frac{N-2}{4}}\) and \(\lim _{n\rightarrow \infty }K^{n}_{1}=a^{\frac{N-2}{4}}\). It is easy to check that \(\varphi ^{0}_{1}:=a^{\frac{N-2}{4}}U\) solves \(K_{\lambda }^*\) with \(\lambda =0\). Thus, from this aspect, the existence results are continuous in \(\lambda =0\).
From all the above analysis, we can give the following bifurcation and the red one is the positive solution.
\(a>0\) and \(N=3\)
\(a>0\) and \(N=4\)
\(a>0\) and \(N\ge 5\)
In the last part of this section, we consider the case that \(a\le 0\).
Theorem 1.7
Assume \(a=0\) and \(\lambda >0\). Then, the following results hold:
- (i):
-
If \(N=3\), problem \(K_{\lambda }^*\) has infinitely many distinct solutions \(\{\varphi _i\}_{i=1}^{\infty }\). Moreover, \({\varphi _1}\) is a positive ground state solution
$$\begin{aligned} 0<\inf _{u\in {\mathcal {N}}^{3}}I(u)=\inf _{u\in {\mathcal {N}}^{3,-}}I(u)=I(\varphi _1) , \ \ I(\varphi _{i})\rightarrow +\infty \ \ as\ i\rightarrow +\infty . \end{aligned}$$ - (ii):
-
If \(N=4\), problem \(K_{\lambda }^*\) admits a nontrivial solution \(\varphi \) if and only if \(\lambda =\Vert w\Vert ^{-2}\), where w is a nontrivial solution of problem \(S^*\). Moreover, \(I(\varphi )=0\).
- (iii):
-
If \(N\ge 5\), problem \(K_{\lambda }^*\) has infinitely many distinct solutions \(\{\varphi _i\}_{i=1}^{\infty }\). Moreover, \({\varphi _1}\) is a positive ground state solution
$$\begin{aligned} \inf _{u\in {\mathcal {N}}^{N}}I(u)=\inf _{u\in {\mathcal {N}}^{N,+}}I(u)=I(\varphi _1), \ \ I(\varphi _{i})<0,\ I(\varphi _{i})\rightarrow 0\ \ as\ i\rightarrow +\infty . \end{aligned}$$
Proof
Since the proof is similar, we omit it here. \(\square \)
Theorem 1.8
Assume \(a<0\) and \(\lambda >0\). Then, the following results hold:
- (i):
-
For \(N=3\), problem \(K_{\lambda }^*\) has two sequences of distinct solutions \(\{\varphi _{i,\pm }\}_{i=1}^{\infty }\).
$$\begin{aligned} I(\varphi _{i,-})<I(\varphi _{i,+}), \\ I(\varphi _{1,-})<I(\varphi _{2,-})<\cdots<I(\varphi _{i,-})<\cdots<0, \\ I(\varphi _{1,+})<I(\varphi _{2,+})<\cdots <I(\varphi _{i,+})\rightarrow +\infty . \end{aligned}$$Moreover, if \(\lambda >\lambda _1=-4aS^{-3}\), then \(\varphi _{1,\pm }\) are two positive solution. Both of them are local minimum points on the Nehari manifolds,
$$\begin{aligned} \inf _{u\in {\mathcal {N}}^{N,-}}I(u)=I(\varphi _{1,-})<0, \ \ \inf _{u\in {\mathcal {N}}^{N,+}}I(u)=I(\varphi _{1,+}).\ \ \end{aligned}$$ - (ii):
-
For \(N\ge 4\), problem \(K_{\lambda }^*\) has infinitely many distinct solutions \(\{\varphi _{i}\}_{i=1}^{\infty }\) satisfying
$$\begin{aligned} I(\varphi _{1})<I(\varphi _{2})<\cdots<I(\varphi _{i})<\cdots <-\frac{a^2}{4\lambda }. \end{aligned}$$For \(N=4\) and \(\lambda >\lambda _1=S^{-2}\) or \(N\ge 5\) and \(\lambda >0\), then \(\varphi _{1}\) is a positive solution,
$$\begin{aligned} \inf _{u\in {\mathcal {N}}^{N}}I(u)=\inf _{u\in {\mathcal {N}}^{N,+}}I(u)=I(\varphi _1). \end{aligned}$$
The proof Theorem 1.8 (i)
Equation \(S^*\) admits infinitely many distinct solutions \(\{u_i\}_{i=1}^{\infty }\) satisfying \(|\nabla U|_2=|\nabla u_1|_2<|\nabla u_2|_2<\cdots <|\nabla u_i|_2\rightarrow +\infty \). Set
Then, we can check that the existence of two positive roots \(K_{i, \pm }\) (\(K_{i, -}<K_{i, +}\)) for equation \(g_i(t)=0\) defined by
where \(i\in {\mathbb {N}}\), \(a<0\) and \(\lambda >0\). Let \(\varphi _{i,\pm }:=K_{i, \pm }u_{i+i_0-1}\), \(i\in {\mathbb {N}}\), then \(\{\varphi _{i,\pm }\}_{i=1}^{\infty }\) are two sequences of solutions of equation \(K_{\lambda }^*\). Set \(\xi _{i,\pm }:=|\nabla \varphi _{i,\pm }|_2\) for any \(i\in {\mathbb {N}}\), then \(\xi _{i,\pm }\) admit the following forms:
where \(A_i:=|\nabla u_{i+i_0-1}|_2^4\). Then by some calculations and \(A_{i}<A_{i+1}\), one has \(f_{i}(t)<f_{i+1}(t)<\lambda t^2+a\) and
Then, \(\varphi _{i,\mp }\in {\mathcal {N}}^{3,\pm }\) because of \( \pm Q_{\varphi _{i,\mp }}''(1)=\pm 2\lambda |\nabla \varphi _{i,\mp }|_2^2\left( |\nabla \varphi _{i,\mp }|_2^2+\frac{2a}{\lambda }\right) <0\). Therefore, \(I(\varphi _{i,\pm })=\frac{1}{3}a|\nabla \varphi _{i,\pm }|_2^2+\frac{1}{12}\lambda |\nabla \varphi _{i,\pm }|_2^4.\) The desired results follow from (1.31) and (1.32).
Moreover, if \(\lambda >-4aS^{-3}\), then \(i_0=1\), that is, \(u_1=U\). It means that \(\varphi _{1,\pm }\) are positive solutions. It follows from \(u\in {\mathcal {N}}^{4}\) that
which implies that \(|\nabla u|_2\le \xi _{1,-}\) for \(u\in {\mathcal {N}}^{3,+}\) and \(|\nabla u|_2\ge \xi _{1,+}\) for \(u\in {\mathcal {N}}^{3,-}\). Combing with \(\Vert \varphi _{1,\pm }\Vert =\xi _{1,\pm }\), it follows that
The proof is completed. \(\square \)
The Proof of Theorem 1.8 (ii)
Equation \(S^*\) admits infinitely many distinct solutions \(\{u_i\}_{i=1}^{+\infty }\) satisfying \(|\nabla U|_2=|\nabla u_1|_2<|\nabla u_2|_2<\cdots <|\nabla u_i|_2\rightarrow +\infty \).
Then, we can check that the existence of a positive root \(K_{i}\) for equation \(g_i(t)=0\) defined by
where \(\lambda >0\) and \(i\in {\mathbb {N}}\). Let \(\varphi _{i}:=K_{i}u_{i+i_0-1}\), \(i\in {\mathbb {N}}\), then \(\{\varphi _{i}\}_{i=1}^{\infty }\) a sequence of solutions of equation \(K_{\lambda }^*\) . Set \(\xi _{i}:=|\nabla \varphi _{i}|_2\) for any \(i\in {\mathbb {N}}\), then \(\xi _{i}\) is the positive solution of \(f_i(t)=0\), \(i\in {\mathbb {N}}\),
Then by some calculations and \(|\nabla u_{i}|_2<|\nabla u_{i+1}|_2\), one has \(f_{i}(t)<f_{i+1}(t)<\lambda t^2+a\) and
Then, \(\varphi _i\in {\mathcal {N}}^{4,+}\) for \(Q_{\varphi _{i}}''(1)=-2a|\nabla \varphi _{i}|_2^2>0\) and \(\varphi _i\in {\mathcal {N}}^{N,+}\) for
By (1.33), the following desired results hold,
Similarly, for \(N=4\) and \(\lambda >S^{-2}\) or \(N\ge 5\) and \(\lambda >0\), we can obtain that \(\varphi _{1}\) is a positive solution satisfying
The proof is completed. \(\square \)
We also have the following bifurcation and the red one is the positive solution.
\(a<0\) and \(N=3\)
\(a<0\) and \(N=4\)
\(a<0\) and \(N\ge 5\)
2 A perturbed problem
In this present paper, we consider the following critical Kirchhoff problem
where \(a>0,m>0\) are two constants, \(\lambda>0,\mu >0\) are two parameters, \(2<p<2^*\) and \(2^*=\frac{2N}{N-2}\) for \(N\ge 3\). It can be easily checked that problem \(K_{\lambda ,\mu }^*\) has no nontrivial solutions for any \(m>0\) if \(\mu =0\) by the Pohoz̆aev identity.
Now, we state our existence results as follows.
Theorem 2.1
Assume \(a>0\), \(m>0\) and \(\lambda >0\). Then, the following results hold.
- (i):
-
If \(N=3\), problem \(K_{\lambda ,\mu }^*\) has a positive ground solution for one of the following cases:
$$\begin{aligned} p\in (2,4)\ and\ \mu>0\ large\ enough\ or \ p\in (4,6)\ and\ for\ any\ \mu >0. \end{aligned}$$ - (ii):
-
If \(N=4\), there exists \(0<\Lambda _2<S^{-2}\) such that problem \(K_{\lambda ,\mu }^*\) has a positive ground solution for
$$\begin{aligned} p\in (2,4),\ \ 0<\lambda <\Lambda _2\ \ and \ \mu >0. \end{aligned}$$
The critical Kirchhoff problem \(K_{\lambda ,\mu }^*\) with \(p\in (2,4)\) or \(p\in (4,6)\) in three-dimensional case is considered in Proposition 3.8 of He and Li [16] and in Theorem 1.3 of Li and Ye [24], respectively. We list it in Theorem 2.1(i) for the sake of the completeness.
Theorem 2.2
Assume \(a>0\), \(m>0\), \(\mu >0\), \(p\in (2,2^*)\) and \(N\ge 5\).
- (i):
-
If \(\lambda \in (0,\Lambda _1)\), problem \(K_{\lambda ,\mu }^*\) has a positive ground state solution with negative energy, which is the global minima point of the corresponding functional, where
$$\begin{aligned} \Lambda _1=\frac{4}{N}\left( \frac{N-4}{aN}\right) ^{\frac{N-4}{2}} S^{-\frac{N}{2}}. \end{aligned}$$ - (ii):
-
There exists \(0<\Lambda _2<\Lambda _1\), problem \(K_{\lambda ,\mu }^*\) with \(\lambda \in (0,\Lambda _2)\) has two positive solutions.
Remark 2.3
If \(a=1\) and \(\lambda =0\), the problem reduces to a semi-linear elliptic equation
The authors in Zhang and Zou [49] and Alves et al. [4] obtained a positive ground state solution for problem \(S_{m}^*\) either \(N=3\) and \(p\in (4,6)\) or \(N\ge 4\) and \(p\in (2,2^*)\). A simple proof also is found in Proposition 1.1 of Akahori et al. [1], in which the authors proved that problem \(S_{m}^*\) admits a unique positive ground state solution in \(H^{1}_{rad}({\mathbb {R}}^N)\) if \(N\ge 5\), \(p\in (2,2^*)\) and \(m>0\). However, from this point, we found a different result that problem \(K_{\lambda ,\mu }^*\) admits two positive solutions if \(\lambda >0\) suitable small.
Since problem \(K_{\lambda ,\mu }^*\) is an autonomous problem, we consider this problem on the space \(H:=H_{rad}^{1}({\mathbb {R}}^3)\), the subspace formed by radially symmetric functions, with the norm
To get the positive solution, the functional is defined by
for any \(u\in H\), where \(u_+=\max \{u,0\}\). Similar to the above section, we set that \(Q_{u,m}(t):=J(tu)\) for any \(u\in H\) and \(t\ge 0\). Recall that \(U_{\varepsilon }\) is the unique positive solution of \(S^*\),
and \(|\nabla U|_2^2=|U|_{2^*}^{2^*}=S^{N/2}\). Similar to Brezis and Nirenberg [9], by some directly computations, it is evident to check that
and
Lemma 2.4
Assume that \(N=4\), \(a,m,\mu >0\) and \(0<\lambda <S^{-2}\). Let \(Q_{U_\varepsilon ,m}(t):=J(tU_\varepsilon )\) for \(t\ge 0\). There exists \(t_{\varepsilon }>0\) such that \(Q_{U_\varepsilon ,m}'(t_{\varepsilon })=0\) and \(Q_{U_\varepsilon ,m}''(t_{\varepsilon })<0\) for small \(\varepsilon >0\). Moreover, \(I(t_{\varepsilon }U_\varepsilon )<c_{4}\), where \(c_{4}\) is defined in Theorem 1.1(ii).
Since the proof of Lemma 2.4 is similar, we omit it here and only prove the following lemma.
Lemma 2.5
Assume that \(N\ge 5\), \(a,m,\mu >0\) and \(0<\lambda <\Lambda _0\). Let \(Q_{U_\varepsilon ,m}(t):=J(tU_\varepsilon )\) for \(t\ge 0\). There exist \(0<t_{\varepsilon ,-}<t_{\varepsilon ,+}\) such that \(Q_{U_\varepsilon ,m}'(t_{\varepsilon ,\pm })=0\) and \(\pm Q_{U_\varepsilon ,m}''(t_{\varepsilon ,\pm })>0\) for small \(\varepsilon >0\). Moreover, \(I(t_{\varepsilon ,\pm }U_\varepsilon )<c_{N,\pm }\), where \(c_{N,\pm }\) is defined in Theorem 1.2 (i).
Proof
By \(\lambda \in (0,\Lambda _0)\), the following equation admits two positive roots \(0<K_{-}<K_+\),
It is evident to check that \(\pm (2\lambda S^{N/2}K_\pm -(2^*-2)K_\pm ^{2^*-3})>0\). Moreover,
It is evident to check that there exists \(\varepsilon _0\) such that \(Q_{U_\varepsilon ,m}'(t)=0\) admits two positive roots \(t_{\varepsilon ,\pm }\) and \(\pm Q_{U_\varepsilon ,m}''(t_{\varepsilon ,\pm })>0\) for \(\varepsilon <\varepsilon _0\). Set \(t_{\varepsilon ,\pm }=K_{\pm }+\delta _{\varepsilon ,\pm }\), then \(\delta _\varepsilon \rightarrow 0\) as \(\varepsilon \rightarrow 0\). We claim that
In fact, it follows from \(Q_{U_\varepsilon ,m}'(t_{\varepsilon ,\pm })=0\) that
where \(C_{\lambda ,\pm }=2\lambda S^{N/2}K_{\pm }-(2^*-2)K_{\pm }^{2^*-3}\) and \(\pm C_{\lambda ,\pm }>0\). So our claim holds. It follows from (2.3) and (2.4) that
This completes the proof. \(\square \)
Thanks to the radially symmetric functions space H and the compact embedding from H to \(L^{q}({\mathbb {R}}^N)\) for \(q\in (2,2^*)\), we avoid to deal with the lack of the compactness caused by the translation. Thus, we have the following analysis on the Palais–Smale sequence by a standard argument (see Naimen [32] or Xie et al. [45]).
Proposition 2.6
Let \(\{u_n\}\) be a bounded Palais–Smale sequence for J in H. Then, \(\{u_n\}\) has a subsequence which strongly converges in H. Otherwise, replacing \(\{u_n\}\) if necessary by a subsequence, there exist a function \(u_0\in H\), a number \(A\in {\mathbb {R}}\), a number \(l\in {\mathbb {N}}\), l sequences of number \(\{\sigma _n^i\}\subset {\mathbb {R}}^+\), points \(\{y_n^i\}\subset {\mathbb {R}}^N\) and l functions \(u^i\in D^{1,2}({\mathbb {R}}^N)\), \(i\in \{1,2,\ldots ,l\}\), which satisfy
and
such that, up to subsequences, there hold
and
as \(n\rightarrow \infty \), where \(J_1(u)\) and \(J_2(u)\) are defined by
Remark 2.7
(i). From the fact that equation \(S^*\) admits a unique positive solution and (2.6), it is evident to check that \(u^i\) can be rewritten as
which implies that \(|u^i|_{2^*}^{2^*}=S^{-2^*/2}|\nabla u^i|^{2^*}_2\) and \(|\nabla u^i|^2_2=|\nabla u^j|^2_2\) for any \(i\ne j\). Moreover, the constant A can be rewritten as
(ii). Set \(j_1(t)\) and \(j_2(t)\), respectively,
for \(t\ge 0\). It follows from (2.5), (2.6) and (2.7) that
By a standard argument (see [7]), we get a Pohoz̆aev identity from (2.5) as follows:
Thus, combining with \(P(u_0)=j_1'(1)=0\), one has
Proposition 2.8
For \(N\ge 5\), then J(u) is coercive, i.e., \(J(u)\rightarrow +\infty \) as \(\Vert u\Vert \rightarrow \infty \).
Proof
By the Gagliardo–Nirenberg inequality in [14], we have that for all \(u\in H\),
where \(\alpha =N(\frac{1}{2}-\frac{1}{p})\). Let \(\delta _1=(p-2)(N-4)/4\). Then, for \(0<\delta <\delta _1\), we set
It is evident to check that \(q'_1>1\), \(q'_2>1\) and \(1/q'_1+1/q'_2=1\). Here by the Gagliardo–Nirenberg inequality and Young’s inequality, one obtains
where \(q_1:=q'_1p\alpha \) and \(q_2:=q'_2p(1-\alpha )\). It follows from \(\delta \in (0, \delta _1)\) that \(q_1<4\) and \(q_2<2\). By the above fact and the Sobolev inequalities, we obtain
which implies that \(J(u)\rightarrow +\infty \) as \(\Vert u\Vert \rightarrow \infty \) because of \(2^*<4\), \(q_1<4\) and \(q_2<2\). \(\square \)
The proof of Theorem 2.2 (i)
Firstly, we define that
Combining with Proposition 2.8, Lemma 2.5 and Theorem 1.2(i), one has
It is evident to obtain a (PS) sequence \(\{u_n\}\) of J at \(c_+\) by Ekeland variational principle. By the coerciveness of J, \(\{u_n\}\) is bounded in H. Assume that \(u_n\rightharpoonup u_0\) weakly in H. We claim that \(u_0\ne 0\). Otherwise, if \(u_0=0\), it follows from (2.11) and (2.12) that \(A=l |\nabla u^i|_{2}^2\),
which imply that \(j''_2(1)>0\). Thus, it obtains
which implies \(l|\nabla u^i|_2^2>\eta ^2_\lambda \). Moreover, by \(j'_2(1)=0\), one has
which implies that \(\xi ^2_{1,-}\le l|\nabla u|_2^2\le \xi ^2_{1,+}\). Thus, we obtain
It follows that
which is impossible for \(c_+<c_{N,+}\). Thus, \(u_0\ne 0\).
It remains to prove that \(u_n\rightarrow u_0\) strongly in H. Arguing indirectly, let \({\overline{u}}_n=u_n-u_0 \in H\). Thus, \({\overline{u}}_n\rightharpoonup 0\) weakly in H and \(\lim _{n\rightarrow \infty }|\nabla {\overline{u}}_n|_2^2=l |\nabla u^i|_{2}^2\). Set that
where \(e=|\nabla u_0|_{2}^2/(l |\nabla u^i|_{2}^2)\) and \(s\in (-1,1/e)\) is a parameter. Then,
for \(s\in (-1,1/e)\). Set \(h(s):=\lim _{n\rightarrow \infty }J(u_{n,s})\). Thus, we know that \(h(0)=c_+\) and
By some direct calculations, it obtains
and
That is \(h(0)=c_+\), \(h'(0)=0\) and \(h''(0)<0\). Thus, there exists \(s_0\) small, such that \(h(s_0)<h(0)\). Combining with \(h(s_0)=\lim _{n\rightarrow \infty }J(u_{n,s_0})\), we have \(J(u_{n,s_0})<c_+\) for n large enough, which is impossible because of the definition of \(c_+\). The proof is completed. \(\square \)
In what follows, we will prove the existence of the solution for \(N=4\) and the second solution for \(N\ge 5\). With the help of the fibering map, it is useful to understand the structure of \({\mathcal {N}}_{\mu ,m}^{N}\) for \(N\ge 4\), where
We define that for \(N\ge 4\),
It follows from the Sobolev inequalities that for any \(u\in {\mathcal {N}}_{\mu ,m}^{N}\),
By \(2<p<2^*\), there exists \(\rho _0>0\) such that \(\Vert u\Vert \ge \rho _0\) for any \(u\in {\mathcal {N}}_{\mu ,m}^{N}\).
For \(N\ge 4\), set
It follows from Lemma 2.4 and Lemma 2.5 that \({\mathcal {N}}_{\mu ,m}^{N,-}\ne \emptyset \). By \(u\in {\mathcal {N}}_{\mu ,m}^{N,-}\), we obtain
It follows that
which implies that \(c_->0\). To get a PS sequence, we set
where \(\widetilde{{\mathcal {N}}}^{N,0}: =\left\{ \{u_n\}\subset {\mathcal {N}}_{\mu ,m}^{N}: \lim _{n\rightarrow \infty } Q''_{u_n,m}(1)=0\right\} \). If \(\widetilde{{\mathcal {N}}}^{N,0}=\emptyset \), we set \(c_0=\infty \).
Lemma 2.9
There exists \(0<\Lambda _2<S^{-2}\) for \(N=4\) or \(0<\Lambda _2<\Lambda _1\) for \(N\ge 5\) such that
for \(0<\lambda <\Lambda _2\).
Proof
Firstly, we prove the second inequality. Let \(\{u_n\}\in \widetilde{{\mathcal {N}}}^{N,0}\), then
which implies that
Then, we have
and
By (2.24), we have
With the help of (2.25), it obtains
By the arbitrariness of \(\{u_n\}\in \widetilde{{\mathcal {N}}}^{N,0}\) and the definition of \(c_0\), we have
For \(N=4\), we have \(c^{\lambda _1}_{N}\le c^{\lambda _2}_{N}\) for \(0<\lambda _1\le \lambda _2<S^{-2}\) directly from
Similarly, for \(N\ge 5\), we have \(c^{\lambda _1}_{N,-}\le c^{\lambda _2}_{N,-}\) for \(0<\lambda _1\le \lambda _2<\Lambda _0\) from
Moreover, by Remark 1.6, we can easily obtain that
Thus, \(c^{\lambda }_{4}\) and \(c^{\lambda }_{N,-}\) are nondecreasing with respect to \(\lambda \in (0,S^{-2})\) and \(\lambda \in (0,\Lambda _1)\), respectively. Combining with the above inequalities, there exists \(\Lambda _2\) such that
for \(0<\lambda <\Lambda _2\). The proof is completed. \(\square \)
The proof of Theorem 2.1 (i) and Theorem 2.2 (ii)
We divide the proof into two steps.
Step 1. Construct a PS sequence for J at level \(c_-\) in \({\mathcal {N}}_{\mu ,m}^{N,-}\).
From the boundedness of \({\mathcal {N}}_{\mu ,m}^{N}\), there exists a minimizing sequence \(\{u_n\}\subset {\mathcal {N}}_{\mu ,m}^{N,-}\cup {\mathcal {N}}_{\mu ,m}^{N,0}\) satisfying that
for any \(w\in {\mathcal {N}}_{\mu ,m}^{N,-}\cup {\mathcal {N}}_{\mu ,m}^{N,0},\) by Ekeland variational principle. With the help of Lemma 2.9, we know that
Then, it is evident to check that \(J'(u_n)\rightarrow 0\) as \(n\rightarrow \infty \) since \(\lim _{n\rightarrow \infty }|Q''_{u_n,m}(1)|>0\). Thus \(\{u_n\}\) is a PS sequence for J at level \(c_-\) in \({\mathcal {N}}_{\mu ,m}^{N,-}\).
Step 2. Claim \(u_n\rightarrow u_0\) strongly in H.
If \(N=4\), then we have
implies the boundedness of \(\{u_n\}\). Without loss of generality, we assume that \(u_n\rightharpoonup u_0\) weakly in H as \(n\rightarrow \infty \). It remains to prove that \(u_n\rightarrow u_0\) strongly in H. Arguing indirectly, by Proposition 2.6, Lemma 2.4, and (2.12), we get
Firstly, we have \(j_1(1)= j_1(1)-\frac{1}{4}j'_1(1)=\frac{1}{4}a|\nabla u_0|_2^2\ge 0.\) Let
It follows from \(j'_1(1)=0\) that
which implies that \(\theta '(|\nabla u^i|_{2})\le 0\). It follows \(j_1'(1)=0\) and \(j''_2(1)<0\) that
which is impossible for \(j_2(1)\le j_1(1)+lj_2(1)=c_-<c_{4}\). Thus, \(u_n\rightarrow u_0\) strongly in H.
If \(N\ge 5\), the sequence \(\{u_n\}\) is bounded in H from the coerciveness of J(u) in H. We also assume that \(u_n\rightharpoonup u_0\) weakly in H as \(n\rightarrow \infty \). It remains to prove that \(u_n\rightarrow u_0\) strongly in H. Arguing indirectly, by Proposition 2.6 and (2.12), we get
which implies that \(0< A<2\eta ^2_\lambda =\frac{4a}{(N-4)\lambda }\). Then, we have
We claim that
In fact, it obviously holds if \(u_0=0\) because of
If \(u_0\ne 0\), arguing indirectly, we assume \(j''_2(1)>0\), that is,
Then,
which is impossible. Thus, our claim \(j''_2(1)\le 0\) holds. Recall that
It follows from \(j'_1(1)=0\) that
which implies that \(\theta '(|\nabla u^i|_{2})\le 0\). It follows from \(j_1'(1)=0\) and \(j''_2(1)\le 0\) that
which is impossible for \(j_2(1)<j_1(1)+j_2(1)\le c_-<c_{N,-}\). \(\square \)
Remark 2.10
The existence and multiplicity results also can be obtained by the rescaling argument as follows. Problem \(K_{\lambda ,\mu }^*\) has a positive solution for any \(\mu >0\) and \(m>0\) if one of the following cases holds,
-
\(N=3\), \(p\in (4,6)\) and \(\lambda >0\);
-
\(N=4\), \(p\in (2,4)\) and \(0<\lambda <S^{-2}\).
Problem \(K_{\lambda ,\mu }^*\) has two positive solutions for any \(\mu >0\) and \(m>0\) if
-
\(N\ge 5\), \(p\in (2,2^*)\) and \(0<\lambda <\Lambda _{0}\), where
$$\begin{aligned} \Lambda _{0}=\frac{2}{N-2} \left( \frac{N-4}{a(N-2)}\right) ^{(N-4)/2}S^{-\frac{N}{2}}. \end{aligned}$$
Actually, the authors in Akahori et al. [1] obtained a positive ground state solution \(u_1\) of the following problem
where \(m>0\), \(\mu >0\), \(p\in (4,6)\) for \(N=3\) and \(p\in (2,2^*)\) for \(N\ge 4\). It follows from that Lemma A.2 in Akahori et al. [1] that the ground state solution \(u_1\) satisfies
where \(J_{0,m}(u)=\frac{1}{2}(|\nabla u|_2+m|u|_2^2)-\frac{1}{p}\mu |u|_p^p-\frac{1}{2^*}|u|_{2^*}^{2^*}\). Since \(u_1\) is a solution, we have
and \( P_2(u_1)=\frac{1}{2^*}|\nabla u_1|_2+\frac{1}{2}m|u_1|_2^2-\frac{1}{p}\mu |u_1|_p^p -\frac{1}{2^*}|u_1|_{2^*}^{2^*}=0, \) which implies that
By (2.27), (2.28) and (2.29), we obtain that
which implies that
Now, we try to prove the main results. Firstly, we consider the existence of positive roots of equation \(G_1(t)=0\) defined by
Combining with (2.30), if \(N=3\) and \(\lambda >0\) or \(N=4\) and \(0<\lambda <S^{-2},\) then \(G_1(t)=0\) has a positive root. If \(N=5\) and
then \(G_1(t)=0\) has two positive roots.
Let \(\varphi _1(x):=u_1(\delta x)\), where \(\delta >0\) such that \(G_1(\delta )=0\) and \(u_1\) is a solution of \(S_{m}^*\). Then, \(\varphi _1\) is a solution of \(K_{\lambda ,\mu }^*\). In fact, \(G_1(\delta )=0\) implies that \(\Big (a+\lambda \delta ^{2-N}| \nabla u_1|_2^2\Big )\delta ^2 =1\). Then, we obtain
which implies that \(\varphi _1\) is a solution of \(K_{\lambda ,\mu }^*\). By the above argument, if \(N=3\) and \(\lambda >0\) or \(N=4\) and \(0<\lambda <S^{-2}\), then problem \(K_{\lambda ,\mu }^*\) has a positive solution. If \(N=5\) and \(0<\lambda <\Lambda _{0}\), then problem \(K_{\lambda ,\mu }^*\) has two positive solutions.
Compared with these results, we have more information about the energies of the solutions obtained in Theorem 2.1 and 2.2. Though the exact intervals of \(\lambda \) for the existence of solutions seem different, we tend to believe that the solutions obtained in the rescaling argument and in Theorems 2.1 or 2.2 are the same.
Moreover, some existence results also can be obtained if \(\lambda <0\) by the same method and we do not state it here.
3 Some general results and an auxiliary lemma
In this section, we consider a general nonlocal equation by the multiplying argument
where \(M_i\): \([0, +\infty )\rightarrow (0,\infty )\), are two continuous functions, \(i=1,2\). Actually, the case that \(M_1(t)=a+\lambda t\) and \(M_2(t)=1\) has been considered in Section 1. As to the general cases, we have the following results:
Theorem 3.1
If there exists a positive root of the following equation,
where \(A=S^{N/2}\), problem (3.1) has a positive solution. Moreover, if there exists a positive root of equation (3.2) for any positive parameter A, problem (3.1) has infinitely many distinct solutions.
Proof
Firstly, we denote the positive root of the following equation by \(t_1\),
Thus, let \(\varphi _1=t_1^{(N-2)/4}U\), we can easily check that
and \(\varphi _1\) is a positive solution of problem (3.1).
Similarly, let \(\{u_i\}_{i=1}^{\infty }\) be a sequence of solutions for equation \(S^*\) satisfying \(|\nabla U|_2=|\nabla u_1|_2<|\nabla u_2|_2<\cdots <|\nabla u_i|_2\rightarrow +\infty \). By the assumption, there exists a positive root \(t_{i}\) for the following equation:
where \(A_i=|\nabla u_i|^2_2=|u_i|_{2^*}^{2^*}\). Thus, \(\varphi _i=t_iu_i\) solves problem (3.1). Then, problem (3.1) admits a sequence of solutions \(\{\varphi _i\}_{i=1}^{\infty }\), and the proof is completed. \(\square \)
In the last part of this section, we investigate the existence of positive solutions for two linear growth terms case
where \(a>0\), \(\lambda >0\) and \(\mu \in {\mathbb {R}}\).
Theorem 3.2
Assume that \(a>0\) and \(\lambda >0,\mu \in {\mathbb {R}}\). The following statements hold.
- (i):
-
If \(N=3\) or 4, problem (3.3) has a positive solution for \(\mu \in (0,+\infty )\).
- (ii):
-
If \(N\ge 5\) and \(\lambda \in (0,\Lambda _0)\), there exist \(\mu _1<0<\mu _2\) such that problem (3.3) has two positive solutions for \(\mu \in (\mu _1, 0)\), three positive solutions for \(\mu \in (0, \mu _2)\) and a positive solution for \(\mu \in (\mu _2,+\infty )\).
- (iii):
-
If \(N\ge 5\) and \(\lambda \in (\Lambda _0,{\overline{\Lambda }})\), there exist \(\mu _4>\mu _3>0\) such that problem (3.3) has three positive solutions for \(\mu \in (\mu _3, \mu _4)\) and a positive solution for \(\mu \in (0,\mu _3)\cup (\mu _4,+\infty )\), where \({\overline{\Lambda }}\) is defined by a exact form,
$$\begin{aligned} {\overline{\Lambda }}=\frac{N}{4}\left( \frac{N}{N+2}\right) ^{\frac{N-4}{2}}\Lambda _0. \end{aligned}$$(3.4) - (iv):
-
If \(N\ge 5\) and \(\lambda \in ({\overline{\Lambda }},+\infty )\), problem (3.3) has a positive solution for \(\mu \in (0,+\infty )\).
The proof of Theorem 3.2
Setting that \(G(t):=G_1(t)-\mu S^{\frac{N}{2}}\), where
It is obvious that \(G_1(0)=0\) and \(G_1(t)\rightarrow +\infty \) as \(t\rightarrow +\infty \). If \(N=3\) or 4, then \(G(t)=0\) admits a positive root denoted by \(t_1\). Let \(K=t_1^{-(N-2)/2}\) and U be the positive solution of problem \(S^*\), then it is evident to check that \(\varphi =KU\) solves problem (3.3). Thus, (i) holds.
It remains to prove that \(N\ge 5\). By some calculations, for \(\lambda \in (0,{\overline{\Lambda }})\), \(G_1(t)\) admits a local maximum point \(t_{\max }\) and a local minimum point \(t_{\min }\) satisfying \(0<t_{\max }<t_{\min }\) and
-
(I).
For \(\lambda \in (0,\Lambda _0)\), set that \(\mu _1=-G_1(t_{\min })S^{-N/2}\) and \(\mu _2=G_1(t_{\max })S^{-N/2}\). Then, G(t) admits two roots for \(\mu \in (\mu _1, 0)\), three roots for \(\mu \in (0, \mu _2)\) and a root for \(\mu \in (\mu _2,+\infty )\).
-
(II).
For \(\lambda \in (\Lambda _0,{\overline{\Lambda }})\), set that \(\mu _3=G_1(t_{\min })S^{-N/2}\) and \(\mu _4=G_1(t_{\max })S^{-N/2}\). Then, G(t) admits three roots for \(\mu \in (\mu _3, \mu _4)\) and a root for \(\mu \in (0,\mu _3)\cup (\mu _4,+\infty )\).
-
(III).
For \(\lambda \in ({\overline{\Lambda }},+\infty )\), then G(t) admits a root for \(\mu \in (0,+\infty )\).
Lastly, assume that \(t_i\) is a root of \(G(t)=0\). Let \(K_i:=t_i^{-(N-2)/2}\) and U be the positive solution of problem \(S^*\), then it is evident to check that \(\varphi _i=K_iU\) solves problem (3.3). The desired results from (I)–(III). \(\square \)
By some careful analysis on the roots of \(G(t)=0\), we can give the following bifurcation of the positive solutions.
\(N\le 5\) and \(\lambda \epsilon (0\Lambda _0)\)
\(N\le 5\) and \(\lambda \epsilon (\Lambda _0 \underline{\Lambda })\)
\(N\le 5\) and \(\lambda \epsilon (\underline{\Lambda }+\infty )\)
We complete this paper by proving an auxiliary lemma.
Lemma 3.3
Let \(N\ge 5\), \(a>0\) and
For \(\lambda \in (0, \Lambda _0)\), we denote the roots of \(f(t)=0\) by \(\xi _{1,-}\) and \(\xi _{1,+}\) (\(0<\xi _{1,-}<\xi _{1,+}\)). Let \(\eta _\lambda =(\frac{2a}{(N-4)\lambda })^{\frac{1}{2}}\), then \(\xi _{1,-}<\eta _{\lambda }<\xi _{1,+}\) and
Proof
For \(\lambda \in (0, \Lambda _0)\), there exists \(\zeta _\lambda \) such that
Therefore, \(\xi _{1,-}<\zeta _\lambda <\xi _{1,+}\) and \(\eta ^2_\lambda <\zeta ^2_\lambda \) because of \(0<\lambda <\Lambda _0\). By \(\lambda \in (0,\Lambda _0)\), then \(f(\eta _\lambda )<0\) implies \(\xi _{1,-}<\eta _{\lambda }<\xi _{1,+}\).
If we set that \({\widetilde{\zeta }}_{\lambda }:=\zeta ^2_{\lambda }\), \({\widetilde{\xi }}_\pm :=\xi ^2_{1,\pm }\) and
then \(f'(\zeta _{\lambda })=k'({\widetilde{\zeta }}_{\lambda })= f(\xi _{1,\pm })=k({\widetilde{\xi }}_\pm )=0\). By the above setting, to prove (3.5), it is sufficient to prove that
We claim that
The desired result (3.6) follows from the claim (3.7). If (3.6) does not hold, that is, \({\widetilde{\xi }}_+-{\widetilde{\zeta }}_{\lambda }\le {\widetilde{\zeta }}_{\lambda }-{\widetilde{\xi }}_-\). Setting \(\delta :={\widetilde{\xi }}_+-{\widetilde{\zeta }}_{\lambda }\), then \(\delta \in (0, {\widetilde{\zeta }}_{\lambda })\). It follows from \({\widetilde{\xi }}_-\le 2{\widetilde{\zeta }}_{\lambda }-{\widetilde{\xi }}_+< {\widetilde{\xi }}_+\) and (3.7) that
which is a contradiction.
Lastly, it remains to prove our claim (3.7). In fact, by some basic calculations, it follows that
Since \(N\ge 5\), \(k'''(t)<0\) for any \(t>0\). Then, one obtains that for any \(0< t<{\widetilde{\zeta }}_{\lambda }\),
For \(0<s<{\widetilde{\zeta }}_{\lambda }\), integrating the above inequality from 0 to s gives us
Moreover, by \(k'({\widetilde{\zeta }}_{\lambda })=0\) and (3.8), then one obtains
Similarly, integrating the inequality (3.9) again from 0 to t, it follows that \(k({\widetilde{\zeta }}_{\lambda }-t)>k({\widetilde{\zeta }}_{\lambda }+t)\) for \(0<t<{\widetilde{\zeta }}_{\lambda }\). The proof is completed. \(\square \)
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Acknowledgements
This work was partially done while Qilin Xie was visiting the Stochastic Analysis and Application Research Center of the Korea Advanced Institute of Science and Technology (KAIST). The first author would like to express his gratitude to Prof. Jaeyoung Byeon for his warm hospitality and support. The first author is deeply grateful to Dr. Sangdon Jin for his kindly help during the visit. This research is supported by the National Natural Science Foundation of China (No. 11701113), the Natural Science Foundation of Guangdong Province (No. 2021A1515010383), the Project of Science and Technology of Guangzhou (No. 202102020730) and the grant from Guangdong University of Technology (No. 220413278), the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University (No. 2021023).
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Xie, Q., Zhou, BX. A study on the critical Kirchhoff problem in high-dimensional space. Z. Angew. Math. Phys. 73, 4 (2022). https://doi.org/10.1007/s00033-021-01626-3
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DOI: https://doi.org/10.1007/s00033-021-01626-3