Abstract
In this paper, we study a class of Kirchhoff-type equation with asymptotically linear right-hand side and compute the critical groups at a point of mountain pass type under suitable Hilbert space. The existence results of three nontrivial solutions under the resonance and non-resonance conditions are established by using the minimax method and Morse theory.
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1 Introduction
In this article, we consider the following Kirchhoff-type problems with Dirichlet boundary conditions:
where \(\Omega \) is a smooth bounded domain in \({\mathbb {R}}^N\)\((N=1,2,3), a, b>0\), and \(f: {\bar{\Omega }}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is continuous and satisfies:
- \((f_1)\):
\(f\in C^1({\bar{\Omega }}\times {\mathbb {R}},{\mathbb {R}}),\ f(x,0)=0,\ f(x,t)t\ge 0\) for all \( x\in \Omega \), \(t\in {\mathbb {R}},\)
- \((f_2)\):
\(f'\) is subcritical in t, i.e., there is a constant \(p\in (2,2^*), 2^*=+\infty \) for \(N=1,2\) and \(2^*=6\) for \(N=3\) such that
$$\begin{aligned} \lim \limits _{t\rightarrow \infty }\frac{f_t(x,t)}{|t|^{p-1}}=0 \quad \text {uniformly for}\quad x\in {\bar{\Omega }}, \end{aligned}$$- \((f_3)\):
\(\lim \nolimits _{|t|\rightarrow 0}\frac{f(x,t)}{t}=f_0,\lim \nolimits _{|t|\rightarrow \infty }\frac{f(x,t)}{t^3} =l\) uniformly for \(x \in \Omega \), where \(f_0\) and l are constants;
- \((f_4)\):
\(\lim \nolimits _{|t|\rightarrow \infty }[f(x,t)t-4F(x,t)]=+\infty \), where \(F(x,t)=\int _0^tf(x,s) ds.\)
It is pointed out in [1] that the problem (1.1) models several physical and biological systems where u describes a process which depends on its average (for example, population density). Moreover, this problem is related to the stationary analogue of the Kirchhoff equation
which was proposed by Kirchhoff [16] as an extension of the classical D’Alembert’s wave equation for free vibration of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Some early studies of Kirchhoff equations may be seen [2, 4, 13]. Recently, by variational methods, Alves [1], Ma-Rivera [20] studied the existence of one positive solution, and He-Zou [15] studied the existence of infinitely many positive solutions for the problem (1.1), respectively; Perera-Zhang [22] studied the existence of nontrivial solutions for the problem (1.1) via the Yang index theory; Zhang-Perera [24] and Mao-Zhang [21] studied the existence of sign-changing solutions for problem (1.1) via invariant sets of descent flow. In [24], the authors considered the 4-superlinear case:
which implies that there exists a constant \(c>0\) such that
Note that condition (1.2) plays an important role for showing the boundedness of Palais–Smale sequences. Furthermore, by a simple calculation, it is easy to see that condition (1.2) implies that
Hence F(x, u) grows in a 4-superlinear rate as \(|t|\rightarrow +\infty .\) In the case of \(N>3\), some related work for problem (1.1), see [17, 18] and their references. In particular, in [11], Cheng-Wu studied the existence and non-existence of positive solutions for problem (1.1) with the asymptotic behavior assumption of f at zero and the more general asymptotically 4-linear than our condition \((f_3)\) of f at infinity. In the present paper, following the idea of [7, 9, 10, 12, 23] on the study of p-Laplacian problems, we can compute mountain pass-type critical groups under suitable Hilbert space and obtain the existence of multiple solutions of asymptotically 4-linear problem (1.1) by using Morse theory.
We need the following preliminaries. Let \(E:=H_0^1(\Omega )\) be the Sobolev space equipped with the inner product and the norm
and \(\lambda _1\) be the first eigenvalue of \((-\Delta ,H_0^1(\Omega ))\). We denote by \(|\cdot |_p\) the usual \(L^p\)-norm. Since \(\Omega \ (\Omega \subset {\mathbb {R}}^3)\) is a bounded domain, \(E\hookrightarrow L^p(\Omega )\) continuously for \(p\in [1,6],\) compactly for \(p\in [1,6),\) and there exists \( \gamma _p>0\) such that
Seeking a weak solution of problem (1.1) is equivalent to finding a critical point \(u^*\) of \(C^1\) functional
where \(F(x,u)=\int _0^uf(x,s)ds.\) Then
Definition 1.1
Let \((E,||\cdot ||_E)\) be a Hilbert space with its dual space \((E^*,||\cdot ||_{E^*})\) and \(I\in C^1(E,{\mathbb {R}})\). For \( c\in {\mathbb {R}},\) we say that I satisfies the \((PS)_c\) condition if for any sequence \(\{u_n\}\subset E\) with
there is a subsequence \(\{u_{n_k}\}\) such that \(\{u_{n_k}\}\) converges strongly in E. Also, we say that I satisfies \((C)_c\) condition (i.e., Cerami condition) if for any sequence \(\{u_n\}\subset E\) with
there is subsequence \(\{ u_{n_k}\}\) such that \(\{ u_{n_k}\}\) converges strongly in E.
Lastly, to state our results, we recall some basic facts on the eigenvalue problem:
\( \mu \) is an eigenvalue of problem (1.4) means that there is a non-zero \( u\in E\) such that
This u is called an eigenvector corresponding to eigenvalue \(\mu \). Set
Denote by \({\mathcal {A}}\) the class of closed symmetric subsets of S and denote by i(A) the yang index of A, let
and set
By Proposition 3.2 of Perera-Zhang [22], we know that \(\{\mu _m\}\) is an unbounded eigenvalues sequence of the nonlinear problem (1.4) and
Let \(\varphi _i\) be the normalized eigenfunction corresponding to the eigenvalue \(\mu _i\). Then the first eigenvalue \(\mu _1\) of problem (1.4) can be characterized as
and \(\mu _1\) can be achieved at some \(\varphi _1\in S\) and \(\varphi _1>0\) in \(\Omega \) (see [24]).
Now, we give our main results.
Theorem 1.1
Assume conditions \((f_1)\)–\((f_3)\) hold, \(f_0<a\lambda _1\) and \(l\in (b\mu _k,b\mu _{k+1})\) for some \(k\ge 2\), then problem (1.1) has at least three nontrivial solutions.
Theorem 1.2
Assume conditions \((f_1)\)–\((f_4)\) hold, \(f_0<a\lambda _1\) and \(l=b\mu _k\) for some \(k\ge 3\), then problem (1.1) has at least three nontrivial solutions.
Here, we also give an example for f(x, t). It satisfies all assumptions of our Theorem 1.2.
\(\textit{Example A}.\) Set
where \(0<\epsilon < \min \{\frac{6l}{a\lambda _1},1\}\) and \(l=b\mu _k\).
As previous introduction, assume condition \((f_3)\) holds, then problem (1.1) is called asymptotically 4 linear at infinity, which means that usual condition (1.2) is not satisfied. This will bring some difficulty if the mountain pass theorem is used to seek nontrivial solutions of problem (1.1). For standard Laplacian Dirichlet problem, Zhou [25] have overcome it by using some monotonicity condition. Novelties of our this paper are as following.
We consider multiple solutions of problem (1.1) in the cases of resonance and non-resonance by using the mountain pass theorem and Morse theory. At first, we use the truncated technique and mountain pass theorem to obtain a positive solution and a negative solution of problem (1.1) under our more general condition \((f_1)\), \((f_2)\) and \((f_3)\) with respect to the conditions \((H_1)\) and \((H_3)\) in [25]. In the course of proving the existence of positive solution and negative solution, the monotonicity condition \((H_2)\) of [25] on the nonlinear term f is not necessary, this point is very important because we can directly prove existence of positive solution and negative solution by using Rabinowitz’s mountain pass theorem. That is, the proof of our compact condition is more simple than that in [25]. Furthermore, we can obtain a nontrivial solution when the nonlinear term f is resonance or non-resonance at the infinity by computing mountain pass-type critical groups under suitable Hilbert space.
The paper is organized as follows. In Sect. 2, we prove some lemmas in order to prove our main results. In Sect. 3, we give the proofs for our main results.
2 Some Lemmas
Consider the following problem
where
Also we set \(F_+(x,t)=\int _0^tf_+(x,s)\mathrm{d}s\) and introduce the functional \(I_+:E\rightarrow {\mathbb {R}}\) defined by
Clearly \(I_+\in C^{2-0}(E,{\mathbb {R}}).\)
Lemma 2.1
\(I_+\) satisfies the (PS) condition.
Proof
Let \(\{u_n\}\subset E\) be a sequence such that \(|I_+(u_n)|\le c,\)\(\langle I_+'(u_n),\varphi \rangle \rightarrow 0\) as \(n\rightarrow \infty .\) Note that
for all \(\varphi \in E.\) Assume that \(|u_n|_4\) is bounded, taking \(\varphi =u_n\) in (2.1). By \((f_3)\), there exists \(c_1,c_2>0\) such that \(|f_+(x,u_n(x))|\le c_1|u_n(x)|+c_2|u_n(x)|^3,\) a.e. \(x\in \Omega .\) So \(u_n\) is bounded in E. If \(|u_n|_4\rightarrow +\infty ,\) as \(n\rightarrow \infty ,\) set \(v_n=\frac{u_n}{|u_n|_4}\), then \(|v_n|_4=1\). Taking \(\varphi =v_n\) in (2.1), it follows that \(\Vert v_n\Vert \) is bounded. Without loss of generality, we assume that \(v_n\rightharpoonup v\) in E, then \(v_n\rightarrow v\) in \(L^4(\Omega )\). Hence, \(v_n \rightarrow v\) a.e. in \(\Omega \). Dividing both sides of (2.1) by \(|u_n|_4^3\), we get
Then for a.e. \(x \in \Omega \), we deduce that \(\frac{f_+(x,u_n)}{|u_n|_4^3}\rightarrow lv_+^3\) as \(n\rightarrow \infty ,\) where \(v_+=\max \{v,0\}\). In fact, when \(v(x)>0,\) by \((f_3)\) we have
and
When \(v(x)=0\), we have
When \( v(x)<0\), we have
and
Since \(\frac{f_+(x,u_n)}{|u_n|_4^3}\le c_1|v_n||u_n|_4^{-2}+c_2|v_n^3|\), by (2.2) and the Lebesgue dominated convergence theorem, we arrive at
From the strong maximum principle, we deduce that \(v>0\). Choosing \(\varphi =\varphi _1\) in (2.3), we obtain
This is a contradiction. \(\square \)
Lemma 2.2
Let \(\varphi _1\) be the eigenfunction corresponding to \(\mu _1\) with \(\Vert \varphi _1\Vert =1\). If \(f_0<a\lambda _1\) and \(l>b\mu _1\), then
- (a)
There exist \(\rho ,\beta >0\) such that \(I_+(u)\ge \beta \) for all \(u\in E\) with \(\Vert u\Vert =\rho \);
- (b)
\(I_+(t\varphi _1)=-\infty \) as \(t\rightarrow +\infty \).
Proof
By \((f_1)\) and \((f_3)\), if \(l\in (b\mu _1,+\infty )\), for any \(\varepsilon >0\), there exist \(A=A(\varepsilon )\ge 0\) and \(B=B(\varepsilon )\) such that for all \( (x,s)\in \Omega \times {\mathbb {R}} \),
where \(p\in (1,5)\).
Choose \(\varepsilon >0\) such that \(f_0+\varepsilon <a\lambda _1.\) By (2.4) and the Sobolev inequality, we get
So, part (a) holds if we choose \(\Vert u\Vert =\rho >0\) small enough.
On the other hand, if \(l\in (b\mu _1,+\infty ),\) take \(\varepsilon >0\) such that \(l-\varepsilon >b\mu _1\). By (2.5), we have
Thus part (b) is proved. \(\square \)
Lemma 2.3
Let \(E=V \oplus W\), where \(V=span\{\varphi _1,\varphi _2,\cdot \cdot \cdot ,\varphi _k\}\) when \(l>b\mu _k\) or \(V=span\{\varphi _1,\varphi _2,\cdot \cdot \cdot ,\varphi _{k-1}\}\) when \(l=b\mu _k\) and \(W=V^{\perp }\). If f satisfies \((f_1), (f_3)\) and \((f_4)\) then
- (i)
the functional I is coercive on W, that is
$$\begin{aligned} I(u)\rightarrow +\infty \quad \text{ as }\ \Vert u\Vert \rightarrow +\infty , \ u\in W \end{aligned}$$and bounded from below on W,
- (ii)
the functional I is anti-coercive on V.
Proof
We firstly prove this conclusion for \(l>b\mu _k\).
For \(u\in W\), by \((f_1)\) and \((f_3)\), for any \(\varepsilon >0\), there exists \( B_1=B_1(\varepsilon )\) such that for all \( (x,s)\in \Omega \times {\mathbb {R}} \),
So we have
Choose \(\varepsilon >0\) such that \(l+\varepsilon <b\mu _{k+1}.\) This proves (i).
For \(u\in V\), again using \((f_1)\) and \((f_3)\), for any \(\varepsilon >0\), there exists \( B_2=B_2(\varepsilon )\) such that for all \( (x,s)\in \Omega \times {\mathbb {R}} \),
From (2.7), we have
Choose \(\varepsilon >0\) such that \(l-\varepsilon >b\mu _{k}.\) This proves (ii).
Now we consider the case \(l=b\mu _k\).
Write \(G(x,t)=F(x,t)-\frac{b}{4}\mu _kt^4,\)\(g(x,t)=f(x,t)-b\mu _kt^3\). Then \((f_3)\) and \((f_4)\) imply that
and
It follows from (2.8) that for every \(M>0\), there exists a constant \(T>0\) such that
For \(\tau >0,\) we have
Integrating (2.11) over \([t,s]\subset [T,+\infty )\), we deduce that
Letting \(s\rightarrow +\infty \) and using (2.9), we see that \(G(x,t)\le -\frac{M}{4},\) for \(t\in {\mathbb {R}} ,\ t\ge T,\) a.e. \(x\in \Omega .\) A similar argument shows that \(G(x,t)\le -\frac{M}{4},\) for \(t\in {\mathbb {R}},\ t\le -T,\) a.e. \(x\in \Omega \). Hence
For \(u\in W\), by (2.13), we get
for \(u\in W\) with \(\Vert u\Vert \rightarrow \infty .\)
The proof of conclusion (ii) is completely identical to the case \(l>b\mu _k\). Hence we omit it here. \(\square \)
Lemma 2.4
If \(b\mu _k<l<b\mu _{k+1}\), then I satisfies the (PS) condition.
Proof
Let \(\{u_n\}\subset E\) be a sequence such that \(|I(u_n)|\le c,\)\(\langle I'(u_n),\varphi \rangle \rightarrow 0\). Since
for all \(\varphi \in E.\) If \(|u_n|_4\) is bounded, we can take \(\varphi =u_n\). By \((f_3)\), there exists a constant \(c_1,c_2>0\) such that \(|f(x,u_n(x))|\le c_1|u_n(x)|+c_2|u_n(x)|^3,\) a.e. \( x\in \Omega .\) So \(u_n\) is bounded in E. If \(|u_n|_4\rightarrow +\infty \), as \(n\rightarrow \infty ,\) set \(v_n=\frac{u_n}{|u_n|_4}\), then \(|v_n|_4=1\). Taking \(\varphi =v_n\) in (2.14), it follows that \(\Vert v_n\Vert \) is bounded. Without loss of generality, we assume \(v_n\rightharpoonup v\) in E, then \(v_n\rightarrow v\) in \(L^4(\Omega )\). Hence, \(v_n\rightarrow v\) a.e. in \(\Omega \). Dividing both sides of (2.14) by \(|u_n|_4\), we get
Then for a.e. \(x \in \Omega \), we have \(\frac{f(x,u_n)}{|u_n|_4^3}\rightarrow lv^3\) as \(n\rightarrow \infty .\) In fact, if \(v(x)\ne 0,\) by \((f_3)\), we have
and
If \(v(x)=0\), we have
Since \(\frac{|f(x,u_n)|}{|u_n|_4^3}\le c_1|v_n||u_n|_4^{-2}+c_2|v_n|^3\), by (2.15) and the Lebesgue dominated convergence theorem, we arrive at
Obviously \(v\ne 0,\) hence, this contradicts our assumption. \(\square \)
Lemma 2.5
Suppose \(l=b\mu _k\) and I satisfies \((f_4)\). Then the functional I satisfies the (C) condition.
Proof
Suppose \({u_n}\in E\) satisfies
In view of \((f_3)\), it suffices to prove that \(u_n\) is bounded in E. Similar to the proof of Lemma 2.4, we have
Therefore, \(v\ne 0\) is an eigenfunction of \(\mu _k\), then \(|u_n(x)|\rightarrow \infty \) for a.e. \(x\in \Omega _0\) (\(\Omega _0\subset \Omega \)) with positive measure. It follows from \((f_4)\) that
holds uniformly in \(x\in \Omega _0\), which implies that
On the other hand, (2.16) implies that
Thus
which contradicts (2.18). Hence \(u_n\) is bounded. \(\square \)
It is well known that critical groups and Morse theory are the important tools in solving elliptic partial differential equation. Let us recall some results which will be used later. We refer the readers to the books [6] for more information on Morse theory.
Let E be a Hilbert space and \(I\in C^1(E,{\mathbb {R}})\) be a functional satisfying the (PS) condition or (C) condition, and \(H_q(X,Y)\) be the qth singular relative homology group with integer coefficients. Let \(u_0\) be an isolated critical point of I with \(I(u_0)=c,~c\in {\mathbb {R}},\) and U be a neighborhood of \(u_0\). The group
is said to be the qth critical group of I at \(u_0\), where \(I^c= \{ u\in E:I(u)\le c\}.\)
Let \(K:=\{u\in E:I'(u)=0\}\) be the set of critical points of I and \(a<\inf I(K)\), the critical groups of I at infinity are formally defined by (see [3])
The following result comes from [3, 6] and will be used to prove the results in this paper.
Proposition 2.6
[3] Assume that \(E=V\oplus W,\)I is bounded from below on W and \(I(u) \rightarrow -\infty \) as \(\Vert u\Vert \rightarrow \infty \) with \(u\in V\). Then
Next, we recall some similar results in [7, 8]. We assume that \((f_2)\) holds and \(u_0\) is an isolated critical point of the functional I. The second-order differential of I in \(u_0\) is given by
for any \(\varphi , w\in E\). Let \(H_{u_0}\) be the closure of \(C_0^\infty (\Omega )\) under the scalar product
then \(H_{u_0}\) is topological isomorphic to E. By \((f_2)\), we know that \(I''(u_0)\) is a Fredholm operator defined by setting
for any \(\varphi ,w\in H_{u_0}\). So we can consider splitting \(H_{u_0}=H^-\oplus H^0 \oplus H^+,\) where \(H^-, H^0, H^+\) are, respectively, the negative, null, and positive space, according to the spectral decomposition of \(I''(u_0)\) in \(L^2(\Omega ),\) and \(H^-, H^0\) have finite dimensions. If we set \(W=H^+\) and \(V=H^-\oplus H^0,\) then we get splitting
Now, by our assumptions \((f_1)\) and \((f_2)\), slightly modifying the proof of Lemmas 4.2–4.5 in [7], we will obtain four parallel results for Kirchhoff problem (1.1) as follows.
Lemma 2.7
If \(N=1\), then there exist \(r_0>0\) and \(C>0\) such that for any \(\eta \in E\), \(\Vert \eta -u_0\Vert <r_0,\) we have
for any \(\varphi \in W.\)
Lemma 2.8
Let \(\tau >0.\) If \(\eta \in B_{\tau }(u_0)\subset E\) is a solution of
for any \(w\in W,\) then \(\eta \in L^\infty (\Omega ).\) Moreover, there exists \(K^*>0\) such that \(\Vert \eta \Vert _{\infty }\le K^*\) with \(K^*\) depending on \(\tau \).
Lemma 2.9
If \(N=2,3\), for any \(M>0\), then there exist \(r_0>0\) and \(C>0\) such that for any \(\eta \in E\cap L^\infty (\Omega ) \), with \(\Vert \eta \Vert _\infty \le M,\)\(\Vert \eta -u_0\Vert <r_0,\) we have
for any \(\varphi \in W.\)
Lemma 2.10
There exists \(\delta >0\) such that for any \(w\in W\backslash \{0\},\) with \(\Vert w\Vert \le \delta \), we have
Next, we give three auxiliary results to prove our main results in this paper.
Lemma 2.11
There exist \(r\in (0,\delta )\) and \(\rho \in (0,r)\) such that for any \(v\in V \cap {\bar{B}}_\rho (0)\) there exists one and only one \({\bar{w}}\in W \cap B_r(0)\) such that for any \(z\in W\cap {\bar{B}}_r(0)\) we have
Moreover, \({\bar{w}}\) is the only element of \(W\cap {\bar{B}}_r(0)\) such that
Furthermore, \(u_0\) is the only critical point of \(B_r(u)\) and \(B_r(u) \subset I^{c+1},\) where \(c=I(u_0)\).
Proof
The proof of this result essentially derives from [7]. For convenience, we prove it. We first consider the case \(N=2,3.\) Since \(u_0\) is an isolated critical point of I and I is continuous, we can fix \(0<\tau <\delta \) such that \(u_0\) is the only critical point of I in \(B_\tau (u_0)\) and \(B_\tau (u_0)\subset I^{c+1}\). From Lemma 2.8, if \(\eta \in B_\tau (u_0)\) is a solution of \(\langle I'(\eta ),w\rangle =0\) for any \(w\in W\), then \(\Vert \eta \Vert _\infty \le M\), where \(M>0\) is a positive constant, depending on \(\tau \). Now, by Lemma 2.9, in correspondence of 2M, there exists \(r_0\in [0,\tau ]\) such that the conclusion of Lemma 2.9 holds.
Now let \(r\in [0,\frac{r_0}{3}].\) Since I is sequentially low semicontinuous with respect to the weakly topology of E. Therefore let us fix \(v\in B_r(0)\cap V \); there exists a minimum point \({\bar{w}}\in W\cap \bar{B_r}(0)\) of the function \( w\in W\cap \bar{B_r}(0)\mapsto I(u_0+v+w).\)
We shall prove that there exists \(\rho \in [0,r]\) such that for any \(v\in V \cap \bar{B_\rho }(0)\) we have
Arguing by contradiction, we assume that there exist a sequence \(\{w_n\}\) in \(W\cap \partial B_r(0)\) and a sequence \(\{v_n\}\) in V with \(\Vert v_n\Vert \rightarrow 0\) such that
Since \(\{w_n\}\) is bounded, there exists \({\tilde{w}}\in W\) such that \(\{w_n\} \) weakly converges to \({\tilde{w}}\) in E. From Lemma 2.10, 0 is unique minimum point of the function \(w\in W\cap \bar{B_r}(0)\mapsto I(u_0+w),\) therefore, we get
From (2.24) and (2.25), we can conclude that
Thus, we have \(w_n\rightarrow w\) in E. It follows that \(\Vert w\Vert =r\) which leads to a contradiction.
As a consequence, we infer that there exists \(\rho \in [0,r]\) such that for any \(v\in V\cap \bar{B_\rho }(0)\), (2.23) holds. Therefore, we have that for any \(v\in V \cap \bar{B_\rho }(0)\) the minimum point \({\bar{w}}\) belongs to \(W\cap B_r(0)\) and then \(\langle I'(u_0+v+{\bar{w}}),z\rangle =0\) for any \(z\in W\).
At last, by Lemmas 2.8, 2.9, similar to the last proof of Lemma 4.6 in [7], we also can prove that \({\bar{w}}\) is the only element of \(W\cap \bar{B_r}(0)\) such that
In the case \(N=1\) the proof is easier and the thesis immediately follows by Lemma 2.7, arguing as before. \(\square \)
Now we can introduce that map \(\psi : V\cap {\bar{B}}_\rho (0)\rightarrow W\cap {\bar{B}}_r(0)\) defined by \(\psi (v)={\bar{w}}\) and the function \(\varphi ^*: V \cap {\bar{B}}_\rho (0)\rightarrow {\mathbb {R}}\) defined by \(\varphi ^*(v)=I(u_0+v+\psi (v))\), which is a continuous map by [7]. Moreover, we have that
Lemma 2.12
For any \(v\in V \cap {\bar{B}}_\rho (0), z\in V, w\in V,\) we have
Proof
The proof of this lemma is essentially equal to the proof of Lemma 2.2 in [8]. We omit it here. \(\square \)
Lemma 2.13
If \((f_2)\) holds, then
Proof
By the crucial Lemma 2.11, we know that the proof of this lemma is essentially equal to the proof of two formulas (5.4) and (5.5) in [7]. We omit it here. \(\square \)
3 Proof of the Main Results
Proof
(Proof of Theorem 1.1.) By Lemmas 2.1, 2.2 and the mountain pass theorem, the functional \(I_+\) has a critical point \(u_1\) satisfying \(I_+(u_1)\ge \beta \). Since \(I_+(0)=0\), \(u_1\ne 0\) and by the maximum principle, we get \(u_1>0\). Hence \(u_1\) is a positive solution of the problem (1.1) and satisfies
By \((f_2)\), the functional \(I_+\) is \(C^{2-0}\). Now, we claim that
Using (2.21), for the isolated critical point \(u_1\) we can define \(V=H^-\oplus H^0\subset H_{u_1}, \) and it follows from Lemma 2.13 that there exists
such that
and
Set \(m=\dim H^-\) and \(n=\dim H^0,\) we know that \(m\le 1.\)
If \(n=0\), then 0 is a non-degenerate critical point of \(\varphi ^*\), and
which implies that (3.2) holds.
If \(n\ne 0\), then 0 is a degenerate critical point of \(\varphi ^*\), and from the Shifting theorem (see [5]), we have
where \( \tilde{\varphi ^*}(u)=\varphi ^* \mid _{H^0}.\)
Case 1. If \(m=1\), then \(C_{0}(\tilde{\varphi ^*},0)\ne 0\), which is equivalent to 0 being an isolated local minimum of \(\tilde{\varphi ^*}\), so
then (3.2) holds.
Case 2. If \(m=0\), then (3.5) implies that
Next, we show \(n=1\). For \(\ker \varphi ^*{''}(0)\) to be nontrivial it amounts to saying that 1 is the first eigenvalue of the following linear eigenvalue problem
From [11, Sect. 6.1], the first eigenvalue 1 is simple, then \(n=1\). Thus, by Theorem 2.7 in [19], we have
Now, we claim that \(C_1(I,u_1)=C_1(I_+,u_1)\). Set for all \((t,u)\in [0,1]\times E \),
Then, for all \(t\in [0,1], \, h_+(t,\cdot )\in C^1(E)\) and \(u_1\in K(h_+(t,\cdot )).\) We claim that \(u_1\) is an isolated critical point of \(h_+(t,\cdot )\), uniformly with respect to \(t\in [0,1]\). Arguing by contradiction, assume that there exist sequences \((t_n)\) in [0, 1] and \(\{u_n\}\) in \(E\setminus \{u_1\}\), respectively, such that \(u_n\in K(h_+(t_n,\cdot ))\) for all integer \(n\ge 1\) and \(u_n\rightarrow u_1\) in E. Thus, for all \(n\ge 1\), \(u_n\) solves the problem
By \((f_1)\) and the result of regularity in [1], the sequence \(\{u_n\}\) is bounded in \(C_0^1(\Omega )\) and \(u_n(x)>0.\) Thus (3.7) reduces to
i.e., \(u_n\in K(I_+)\). This leads to a contradiction. Thus, we have
Similarly, we can obtain another negative critical point \(u_2\) of I satisfying
Since \(f_0<a\lambda _1,\) the zero function is a local minimizer of I, then
On the other hand, by Lemmas 2.3, 2.4 and Proposition 2.6, we have
Hence I has a critical point \(u_3\) satisfying
Since \(k\ge 2\), it follows from (3.9)–(3.13) that \(u_1\), \(u_2\) and \(u_3\) are three different nontrivial solutions of the problem (1.1). \(\square \)
Proof
(Proof of Theorem 1.2.) By Lemmas 2.3, 2.5 and Proposition 2.6, we can prove the conclusion (3.12). The other proof is similar to that of Theorem 1.1. \(\square \)
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The authors would like to thank the referees for valuable comments and suggestions in improving this paper. This research was supported by the NSFC (Nos. 11661070 and 11571176).
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Pei, R., Ma, C. Multiple Solutions for a Kirchhoff-Type Equation. Mediterr. J. Math. 17, 78 (2020). https://doi.org/10.1007/s00009-020-01508-4
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DOI: https://doi.org/10.1007/s00009-020-01508-4