Abstract
We are interested in the existence of sign-changing solutions for the following Kirchhoff-type equation
where \(a,b>0\), \(\Omega \subset {\mathbb {R}}^3\) is a bounded domain with smooth boundary, the potential \(h:\overline{\Omega }\rightarrow {\mathbb {R}}\) is a sign-changing continuous function, and \(\lambda >0\) is a parameter. If \(p\in (4,6)\), we prove the existence of least energy sign-changing solution \(u_{b,\lambda }\), the asymptotic behavior of \(u_{b,\lambda }\) as \(b\rightarrow 0^+\) or \(\lambda \rightarrow +\infty \) are also analyzed. Moreover, if the set \(\{x\in \Omega :\ h(x)>0\}\) possesses several disjoint components, we also prove the existence of multi-bump sign-changing solutions.
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1 Introduction
In the past decades, the following Kirchhoff-type equations
has been investigated by many authors, where \(V:{\mathbb {R}}^3\rightarrow {\mathbb {R}}\), \(f\in \mathcal {C}({\mathbb {R}}^3 \times {\mathbb {R}}, {\mathbb {R}})\) and \(a,b>0\) are constants. If \(V(x)\equiv 0\) and replace \({\mathbb {R}}^3\) by a bounded domain \(\Omega \subset {\mathbb {R}}^3\) in (1.1), we then obtain the following Kirchhoff Dirichlet problem
Equation (1.2) is related to the stationary analogue of the following equation
which is proposed by Kirchhoff in [19] as an extension of the classical D’Alembert’s wave equations for free vibration of elastic strings. After the pioneer work of Lions [20], where a functional analysis approach was proposed to the equation
equation (1.3) began to call attention of several researchers, see [2, 5, 8] and the references therein.
Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. In (1.2), u denotes the displacement, f(x, u) is the external force, b is the initial tension and a is related to the intrinsic properties of the string. We point out that such nonlocal problems also appear in other fields as biological systems, where u describes a process which depends on the average of itself, for example, population density. For more mathematical and physical background of (1.2), we refer the reader to the papers [1, 2, 5, 15, 16, 19, 21] and the references therein.
Mathematically, Eq. (1.1) is a nonlocal problem as the appearance of the nonlocal term \(\int \limits _{{\mathbb {R}}^3}|\nabla u|^2\mathrm{{d}}x \Delta u\), which implies that (1.1) is not a pointwise identity. This causes some mathematical difficulties which make the study of (1.1) particularly interesting. A lot of interesting results on the existence of positive solutions, multiple solutions, semiclassical state solutions and sign-changing solutions for (1.1) are obtained in last decade, see for examples, [6, 9, 11, 12, 15,16,17,18, 21, 22, 24, 26,27,28] and the references therein.
In particular, Chen, Kuo and Wu [6] studied the following nonlinear Kirchhoff-type equation with indefinite nonlinearity
where \(a,b>0\), \(\Omega \) is a smooth bounded domain in \({\mathbb {R}}^N\) with \(1<q<2<p<2^*\) (\(2^*=\frac{2N}{N-2}\) if \(N\ge 3\), \(2^*=+\infty \) if \(N= 1,2\)), \(\lambda >0\) is a parameter, the weight functions \(f, g\in \mathcal {C}(\Omega )\) satisfy \(f^+(x):=\max \{f(x),0\}\not \equiv 0\) and \(g^+(x):=\max \{g(x),0\}\not \equiv 0\). By using Nehari manifold and fibering map, the authors proved the existence of multiple positive solutions for Eq. (1.4). We point out that Kirchoff-type equations with potential well and indefinite nonlinearities were also investigated in [26, 30].
Recently, Figueiredo et al [13] investigated ground states of elliptic problems over cones. As an application, the authors [13] proved the following Kirchhoff-type equation
has a positive ground state solution provided \(b^+(x):=\max \{b(x),0\}\not \equiv 0\) and \(r\in (4,6)\), where \(\Omega \subset {\mathbb {R}}^3\) is a bounded domain with smooth boundary, \(M:[0,+\infty )\rightarrow [0,+\infty )\) is a monotone increasing \(\mathcal {C}^1\) function such that \(M(0):=m_0>0\) and \(t\mapsto \frac{M(t)}{t}\) is increasing on \((0,+\infty )\).
Based on the above results, a natural question is whether Eq. (1.5) has sign-changing solutions with b(x) is a sign-changing function. The present paper is devoted to this aspect and partially answers this question. More precisely, we devoted to study the existence of sign-changing solutions for the following Kirchhoff-type equation
where a, \(b>0\), \(\Omega \subset {\mathbb {R}}^3\) is a bounded domain with smooth boundary, the potential \(h:\overline{\Omega }\rightarrow {\mathbb {R}}\) is a sign-changing continuous function, \(\lambda >0\) is a parameter, and
Throughout this paper, we denote \(H^1_0(\Omega )\) the usual Sobolev space equipped with the inner product and norm
Define the energy functional \(I_{b,\lambda }: H^1_0(\Omega )\rightarrow {\mathbb {R}}\) by
Obviously, the functional \(I_{b,\lambda }\) is well-defined and belongs to \(\mathcal {C}^1(H^1_0(\Omega ),{\mathbb {R}})\). Moreover, for any \(u, \varphi \in H^1_0(\Omega )\), we have
In the case \(h(x)\equiv 1\), by constrained minimization method, Figueiredo and Nascimento [12] and Shuai [25] proved the existence of least energy sign-changing solution for Eq. (1.6). The authors first proved the following set
is nonempty, which is a crucial step. Then, the authors sought a minimizer of the energy functional \(I_{b,\lambda }\) restricted on \(\mathcal {M}_{b,\lambda }\) and proved the minimizer is a sign-changing solution of (1.6) by quantitative deformation lemma. In the first step, the authors proved that, for each \(u\in H^1_0(\Omega )\) with \(u^\pm \ne 0\), there exists a unique pair \((s,t)\in {\mathbb {R}}_+\times {\mathbb {R}}_+\) such that \(su^+ +tu^- \in \mathcal {M}_{b,\lambda }\), see Lemma 2.3 in [12] and Lemma 2.1 in [25]. However, if h(x) is a sign-changing continuous function, this fact does not hold for all \(u\in H^1_0(\Omega )\) with \(u^\pm \ne 0\), but rather in some part of it. A direct observation is that, a necessary condition for \(u\in \mathcal {M}_{b,\lambda }\) is \(u^+, u^-\in \mathcal {A}\), where
Thus, the method that used in [12, 25] cannot be applied to Eq. (1.6), we need some crucial modifications.
Our first main result can be stated as follows.
Theorem 1.1
Assume \(h:\overline{\Omega }\rightarrow {\mathbb {R}}\) is a sign-changing continuous function, \(p\in (4,6)\) and \(\lambda >0\), then Eq. (1.6) possesses one least energy sign-changing solution \(u_{b,\lambda }\), which has precisely two nodal domains. Moreover, \(I_{b,\lambda }(u_{b,\lambda })>2c_{b,\lambda }\), where
and
Theorem 1.1 implies that, the energy of any sign-changing solution of Eq. (1.6) is larger than two times the least energy, this property is called energy doubling by Weth in [29]. It is obvious that the least energy of the sign-changing solution \(u_{b,\lambda }\) obtained in Theorem 1.1 depends on b and \(\lambda \). We next focus on the convergence property of \(u_{b,\lambda }\) as \(b\rightarrow 0^+\) or \(\lambda \rightarrow +\infty \). Our main results in this direction can be stated as follows.
Theorem 1.2
If the assumptions of Theorem 1.1 hold, for any sequence \(\{b_n\}\) with \(b_n \rightarrow 0^+\) as \(n\rightarrow \infty \), there exists a subsequence, still denoted by \(\{b_n\}\), such that \(u_{b_n,\lambda }\rightarrow u_{0,\lambda }\) strongly in \(H^1_0(\Omega )\) as \(n\rightarrow \infty \), where \(u_{b_n,\lambda }\) denote the least energy sign-changing solution of Eq. (1.6) with \(b=b_n\) obtained by Theorem 1.1, and \(u_{0,\lambda }\) is a least energy sign-changing solution of the following equation
which changes sign only once.
The proof of Theorem 1.2 includes three steps, we first prove \(\{u_{b_n,\lambda }\}\) is bounded in \(H^1_0(\Omega )\), then we prove \(u_{b_n,\lambda }\rightarrow u_{0,\lambda }\) strongly in \(H^1_0(\Omega )\), and we finally prove that \(u_{0,\lambda }\) is just a least energy sign-changing solution of (1.13).
Theorem 1.3
If the assumptions of Theorem 1.1 hold, for any sequence \(\{\lambda _n\}\) with \(\lambda _n \rightarrow +\infty \) as \(n\rightarrow \infty \), there exists a subsequence, still denoted by \(\{\lambda _n\}\), such that \(u_{b,\lambda _n}\rightarrow \bar{u}\) strongly in \(H^1_0(\Omega )\) as \(n\rightarrow \infty \), where \(u_{b,\lambda _n}\) denote the least energy sign-changing solution of Eq. (1.6) with \(\lambda =\lambda _n\) obtained by Theorem 1.1, and \(\bar{u}\) is a least energy sign-changing solution of following equation
which changes sign only once, here \(\Omega ^-:=\{x\in \Omega \ |\ h(x)<0\}\).
Next, we study the existence of multi-bump sign-changing solutions for Eq. (1.6). We now assume \(h:\overline{\Omega }\rightarrow {\mathbb {R}}\) is a sign-changing continuous function satisfying
(\(h_1\)) \(\Omega ^+:=\{x\in \Omega \ |\ h(x)>0\}=\Omega {\setminus } \overline{\Omega ^-}\);
(\(h_2\)) the set \(\Omega ^+\) is the union of k (\(k\ge 2\)) open connected and disjoint Lipschitz components, that is
Theorem 1.4
Assume \(h:\overline{\Omega }\rightarrow {\mathbb {R}}\) is a sign-changing continuous function and \((h_1)\)–\((h_2)\) hold. If \(p\in (4,6)\), then, for any non-empty subset \(\Gamma \subset \{1,2,\ldots ,k\}\) with
there exists a constant \(\Lambda _\Gamma >0\) such that for \(\lambda \ge \Lambda _\Gamma \), Eq. (1.6) has a sign-changing multi-bump solution \(u_{b,\lambda }\), which possesses the following property: for any sequence \(\{\lambda _n\}\) with \(\lambda _n \rightarrow +\infty \) as \(n\rightarrow \infty \), there exists a subsequence, still denoted by \(\{\lambda _n\}\), such that \(u_{b,\lambda _n}\rightarrow u\) strongly in \(H^1_0(\Omega )\) as \(n\rightarrow \infty \), where u solves the following equation
Moreover, \(u|_{\Omega _i}\) is positive for \(i\in \Gamma _1\), \(u|_{\Omega _i}\) is negative for \(i\in \Gamma _2\), and \(u|_{\Omega _i}\) changes sign exactly once for \(i\in \Gamma _3\).
If \(b=0\), Eq. (1.6) does not depend on the nonlocal term \(\int \limits _{\Omega }|\nabla u|^2\mathrm{{d}}x\Delta u\) any more. In this case, Eq. (1.6) becomes to the following semilinear elliptic equation
Under the conditions (\(h_1\))–(\(h_2\)), separate the components of \(\Omega ^+\) arbitrarily into three families, i.e.,
by using constrained minimization method, Gir\(\tilde{a}\)o and Gomes [14] proved the existence of multi-bump nodal solution \(u_\lambda \) for Eq. (1.18) if \(\lambda >0\) large enough. Moreover, for any sequence \(\{\lambda _n\}\) with \(\lambda _n \rightarrow +\infty \) as \(n\rightarrow \infty \), there exists a subsequence, still denoted by \(\{\lambda _n\}\), such that \(u_{\lambda _n} \rightarrow u\) strongly in \(H^1_0(\Omega )\) as \(n\rightarrow \infty \), where u solves the following equation
here \(u|_{\widetilde{\omega }_i}\) changes sign exactly once for \(i=1,2,\ldots ,I\), \(u|_{\widehat{\omega }_j}\) is positive for \(j=1,2,\ldots ,J\), \(u|_{\overline{\omega }_k}\equiv 0\) for \(k=1,2,\ldots ,K\). We refer the reader to [4] for multiple positive solutions for Eq. (1.18).
However, we cannot apply the same method that used in [14] to Eq. (1.6), because Kirchhoff-type equation depends on the global information of its solution. Different from the method used in [14], we first construct a special minimax value of the energy functional; Then, by careful analysis of the deformation flow to the energy functional, we prove the existence of multi-bump sign-changing solutions for Eq. (1.6); Finally, we show that the multi-bump sign-changing solutions are localized near the components of \(\Omega ^+\) and converge to the solutions (1.17) with prescribed sign properties. We remark that our method also can be used to study the existence of multi-bump sign-changing solutions for Eq. (1.18).
The paper is organized as follows. In Sect. 2, we give some primarily results. In Sect. 3, we prove Theorems 1.1–1.3. In Sect. 4 and 5, we devote to proving Theorem 1.4.
2 Some preliminary results
In this section, we give some preliminary results.
Lemma 2.1
Assume \(h:\overline{\Omega }\rightarrow {\mathbb {R}}\) is a sign-changing continuous function and \(p\in (4,6)\). If \(u\in \mathcal {A}\), then there exists a unique \(t>0\) such that \(tu\in \mathcal {N}_{b,\lambda }\), where \(\mathcal {A}\) is defined by (1.10), \(\mathcal {N}_{b,\lambda }\) is defined by (1.12).
Proof
For \(u\in \mathcal {A}\), we define
Since \(u\in \mathcal {A}\), then
Therefore, \(V_u(t)>0\) for \(t>0\) small enough and \(V_u(t)<0\) for \(t<0\) large enough, since \(p\in (4,6)\). Thus, there exists \(t_0>0\) such that \(I'_{b,\lambda }(t_0u)=0\), that is \(t_0u\in \mathcal {N}_{b,\lambda }\).
Assume \(t_1\), \(t_2>0\) such that \(t_1u,~t_2u\in \mathcal {N}_{b,\lambda }\), that is
and
It follows from (2.1) and (2.2) that
which implies \(t_1=t_2\). \(\square \)
Lemma 2.2
Assume \(h:\overline{\Omega }\rightarrow {\mathbb {R}}\) is a sign-changing continuous function and \(p\in (4,6)\), if \(u\in H^1_0(\Omega )\) with \(u^{\pm }\in \mathcal {A}\), then there is a unique pair \((s_u,t_u)\) of positive numbers such that \(s_uu^+ +t_uu^-\in \mathcal {M}_{b,\lambda }\).
Proof
We prove the lemma by two steps.
Step 1: Define \(\overrightarrow{F}(s,t):=\left( f_1(s,t),f_2(s,t)\right) \), where
Since \(u^{\pm }\in \mathcal {A}\), then
We deduce that there exist \(0<r<R\) such that
since \(p\in (4,6)\). Then, by using Miranda lemma [23], we conclude that there exists \((s_u,t_u)\in {\mathbb {R}}_+\times {\mathbb {R}}_+\) such that
which implies that \(s_uu^+ +t_uu^-\in \mathcal {M}_{b,\lambda }\).
Step 2: We prove \((s_u,t_u)\) is unique.
Case 1: \(u\in \mathcal {M}_{b,\lambda }\). Suppose \((\bar{s},\bar{t})\ne (1,1)\) be another pair of positive numbers such that \(\bar{s}u^+ +\bar{t}u^-\in \mathcal {M}_{b,\lambda }\), then
Without loss of generality, we assume \(\bar{s}\ge \bar{t}>0\), then
Since \(u\in \mathcal {M}_{b,\lambda }\), we have
Thus, we conclude that
which implies \(1\ge \bar{s}\ge \bar{t}\ge 1\). Thus, \((\bar{s},\bar{t})=(1,1)\).
Case 2: \(u\not \in \mathcal {M}_{b,\lambda }\) but \(u^{\pm }\in \mathcal {A}\), then by Step 1, we know that there exists \((s_u, t_u)\in {\mathbb {R}}_+\times {\mathbb {R}}_+\) such that \(s_u u^+ +t_u u^-\in \mathcal {M}_{b,\lambda }\). Assume that \((s_u', t_u')\in {\mathbb {R}}_+\times {\mathbb {R}}_+\) also satisfying \(s_u' u^+ +t_u' u^-\in \mathcal {M}_{b,\lambda }\). Hence we have
Since \(s_u u^+ +t_u u^-\in \mathcal {M}_{b,\lambda }\), by the arguments of case 1, we deduce that
Thus, \(s'_u=s_u\) and \(t'_u=t_u\). \(\square \)
Lemma 2.3
Assume \(h:\overline{\Omega }\rightarrow {\mathbb {R}}\) is a sign-changing continuous function and \(p\in (4,6)\), suppose that \(u^{\pm }\in \mathcal {A}\) such that
Then the unique pair \((s_u,t_u)\) of positive numbers obtained in Lemma 2.2 satisfies \(0<s_u,t_u\le 1\).
Proof
Suppose that \(s_u\ge t_u>0\), since \(s_u u^+ +t_u u^-\in \mathcal {M}_b\), then we have
On the other hand,
Combine (2.6) and (2.7), we then get
Therefore, we must have \(s_u\le 1\). Then the proof is completed. \(\square \)
Lemma 2.4
Assume \(h:\overline{\Omega }\rightarrow {\mathbb {R}}\) is a sign-changing continuous function and \(p\in (4,6)\). If \(u^{\pm }\in \mathcal {A}\), then the vector \((s_u,t_u)\) which obtained in Lemma 2.2 is the unique maximum point of the function \(\phi :({\mathbb {R}}_+\times {\mathbb {R}}_+)\rightarrow {\mathbb {R}}\) defined by \(\phi (s,t):=I_{b,\lambda }(su^+ + tu^-)\).
Proof
From the proof of Lemma 2.2, \((s_u,t_u)\) is the unique critical point of \(\phi \) in \({\mathbb {R}}_+\times {\mathbb {R}}_+\). Since \(p\in (4,6)\), we deduce that \(\phi (s,t)\rightarrow -\infty \) uniformly as \(|(s,t)|\rightarrow +\infty \), so it is sufficient to check that the maximum point is not achieved on the boundary of \({\mathbb {R}}_+\times {\mathbb {R}}_+\).
Fix \(\bar{t}>0\), since
is an increasing function with respect to s if \(s>0\) small enough, therefore the pair \((0,\bar{t})\) is not a maximum point of \(\phi \) in \({\mathbb {R}}_+\times {\mathbb {R}}_+\). \(\square \)
By Lemma 2.2, we now define
Lemma 2.5
Assume \(h:\overline{\Omega }\rightarrow {\mathbb {R}}\) is a sign-changing continuous function and \(p\in (4,6)\), then \(m_{b,\lambda }>0\) is achieved.
Proof
For every \(u\in \mathcal {M}_{b,\lambda }\), we have \(\langle I'_{b,\lambda }(u),u\rangle =0\). Then, by using Sobolev embedding theorem, one gets
Thus, there exists a constant \(\alpha >0\) such that \(\Vert u\Vert ^2\ge \alpha \). Therefore
which implies \(m_{b,\lambda }\ge \left( \frac{1}{2}-\frac{1}{p}\right) a\alpha >0\).
Let \(\{u_n\}\subset \mathcal {M}_{b,\lambda }\) be a sequence such that \(I_{b,\lambda }(u_n)\rightarrow m_{b,\lambda }\). Then \(\{u_n\}\) is bounded in \(H^1_0(\Omega )\), up to a subsequence, still denote by \(\{u_n\}\), such that \(u_n^{\pm }\rightharpoonup u_{b,\lambda }^{\pm }\) weakly in \(H^1_0(\Omega )\). Since \(u_n\in \mathcal {M}_{b,\lambda }\), we have \(\langle I'_{b,\lambda }(u_n),u_n^\pm \rangle =0\), that is
Similar as (2.9) there exist a constant \(\mu >0\) such that \(\Vert u_n^\pm \Vert ^2\ge \mu \) for all \(n\in {\mathbb N}\). Since \(u_n\in \mathcal {M}_{b,\lambda }\), thus
By the compactness of the embedding \(H^1_0(\Omega )\hookrightarrow L^{q}(\Omega )\) for \(2\le q <6\), we get
Hence, \(u^\pm _{b,\lambda }\in \mathcal {A}\). By the weak semicontinuity of norm, we have
It follows from (2.10) that
From (2.13) and Lemma 2.3, there exists \((\bar{s},\bar{t})\in (0,1]\times (0,1]\) such that
Hence
which implies that \(\bar{s}=\bar{t}=1\). Thus, \(\overline{u}_{b,\lambda }=u_{b,\lambda }\) and \(I_{b,\lambda }(u_{b,\lambda })=m_{b,\lambda }\). \(\square \)
3 Proof of Theorems 1.1–1.3.
The main aim of this section is to prove Theorems 1.1–1.3. We first prove that the minimizer \(u_{b,\lambda }\) to the minimization problem (2.8) is indeed a sign-changing solution of Eq. (1.6), that is, \(I'_{b,\lambda }(u_{b,\lambda })=0\).
Proof of Theorem 1.1
Using the quantitative deformation lemma, we prove that \(I'_{b,\lambda }(u_{b,\lambda })=0\).
It is clear that \(\langle I'_{b,\lambda }(u_{b,\lambda }),u^+_{b,\lambda }\rangle =0=\langle I'_{b,\lambda }(u_{b,\lambda }),u^-_{b,\lambda }\rangle \). If \((s,t)\in {\mathbb {R}}_+\times {\mathbb {R}}_+\) and \((s,t)\ne (1,1)\), it follows from Lemma 2.4 that
If \(I'_{b,\lambda }(u_{b,\lambda })\ne 0\), then there exist \(\delta >0\) and \(\rho >0\) such that
Let \(D:=(\frac{1}{2},\frac{3}{2})\times (\frac{1}{2},\frac{3}{2})\) and \(g(s,t):=su_{b,\lambda }^+ +tu_{b,\lambda }^-\). It follows from Lemma 2.4 again that
For \(\varepsilon :=\min \{(m_{b,\lambda }-\bar{m}_{b,\lambda })/2, \rho \delta /8\}\) and \(S:=B(u_{b,\lambda },\delta )\), [see [31], Lemma 2.3] yields a deformation \(\eta \) such that
(a) \(\eta (1,u)=u\) if \(u\not \in I_{b,\lambda }^{-1}([m_{b,\lambda }-2\varepsilon ,m_{b,\lambda }+2\varepsilon ])\cap S_{2\delta }\);
(b) \(\eta (1,I_{b,\lambda }^{m_{b,\lambda }+\varepsilon }\cap S)\subset I_{b,\lambda }^{m_{b,\lambda }-\varepsilon }\);
(c) \(I_{b,\lambda }\left( \eta (1,u)\right) \le I_{b,\lambda }(u)\) for all \(u\in H_0^1(\Omega )\).
It is clear that
We now prove that \(\eta (1,g(D))\cap \mathcal {M}_{b,\lambda }\ne \varnothing \), contradicting to the definition of \(m_{b,\lambda }\). Let us define \(h(s,t):=\eta (1,g(s,t))\) and
Lemma 2.2 and the the degree theory now yields deg\((\Psi _0,D,0)=1\). It follows from (3.2) that \(g=h\) on \(\partial D\). Consequently, we obtain deg\((\Psi _1,D,0)=\)deg\((\Psi _0,D,0)=1\). Therefore, \(\Psi _1(s_0,t_0)=0\) for some \((s_0,t_0)\in D\), so that \(\eta (1,g(s_0,t_0))=h(s_0,t_0)\in \mathcal {M}_{b,\lambda }\), which is a contradiction. From this, \(u_{b,\lambda }\) is a critical point of \(I_{b,\lambda }\), and so, a sign-changing solution for equation (1.6).
Now, we show that \(u_{b,\lambda }\) has exactly two nodal domains. The proof on the number of nodal domains follows the arguments in Bartsch [3] and Castro et al. [7]. To this end, we assume by contradiction that
with
and
Setting \(v:=u_1+u_2\), we see that \(v^+ =u_1\) and \(v^- =u_2\), i.e. \(v^\pm \ne 0\). Then, we can conclude \(v^\pm \in \mathcal {A}\). By Lemma 2.2, there exists a unique pair \((s_v,t_v)\) of positive numbers such that
or equivalently,
And so,
Moreover, using the fact that \(\langle I_{b,\lambda }'(u_{b,\lambda }),u_i\rangle =0\) for \(i=1,2,3\), it follows that
From Lemma 2.3, we have that
On the other hand,
Then, by using (3.4), we can calculate that
Then, from (3.5), (3.6) and (3.7), we have
which is a contradiction. This way, \(u_3=0\), and \(u_{b,\lambda }\) has exactly two nodal domains.
Recall that \(c_{b,\lambda }\) and \(\mathcal {N}_{b,\lambda }\) are defined by (1.11) and (1.12), respectively. Then, similar as the proof of Lemma 2.5, for each \(b>0\), we can deduce that there exists \(v_{b,\lambda }\in \mathcal {N}_{b,\lambda }\) such that \(I_{b,\lambda }(v_{b,\lambda })=c_{b,\lambda }>0\). By Corollary 2.9 in [15], the critical points of the functional \(I_{b,\lambda }\) on \(\mathcal {N}_{b,\lambda }\) are critical points of \(I_{b,\lambda }\) in \(H_0^1(\Omega )\), we conclude that \(I'_{b,\lambda }(v_{b,\lambda })=0\). Thus, \(v_{b,\lambda }\) is a ground state solution of (1.6).
On the other hand, suppose that \(u_{b,\lambda }=u_{b,\lambda }^+ +u^-_{b,\lambda }\) is a least energy sign-changing solution for Eq. (1.6). By Lemma 2.1, there is unique \(\bar{s}>0\), \(\bar{t}>0\) such that
Then, by Lemma 2.4, we get
that is \(m_{b,\lambda }>2c_{b,\lambda }\). This completes the proof. \(\square \)
Now, we are in a situation to prove Theorem 1.2. In the following, we regard \(b>0\) as a parameter in equation (1.6). We shall analyze the convergence property of \(u_{b,\lambda }\) as \(b\rightarrow 0^+\).
Proof of Theorem 1.2
For any \(b>0\) and \(\lambda >0\), denote \(u_{b,\lambda }\in H^1_0(\Omega )\) the least energy sign-changing solution of (1.6) obtained in Theorem 1.1, which changes sign only once.
Step 1. We claim that, for any sequence \(\{b_n\}\) with \(b_n \rightarrow 0^+\) as \(n\rightarrow \infty \), \(\{u_{b_n,\lambda }\}\) is bounded in \(H^1_0(\Omega )\).
Choose a nonzero function \(\varphi \in \mathcal {C}_0^\infty (\Omega )\) with \(\varphi ^\pm \in \mathcal {A}\). Since \(p\in (4,6)\), then, for any \(b\in [0,1]\), there exists a pair \((\tau _1,\tau _2)\) of positive numbers, which does not depend on b, such that
where \(B_{\varphi }=\int \limits _{\Omega }|\nabla \varphi ^+|^2\mathrm{{d}}x\int \limits _{\Omega }|\nabla \varphi ^-|^2\mathrm{{d}}x\). In view of Lemma 2.2 and Lemma 2.3, for any \(b\in [0,1]\), there exists a unique pair \(\left( s_\varphi (b),t_\varphi (b)\right) \in (0,1]\times (0,1]\) such that
Thus, for any \(b\in [0,1]\), we have
where \(C_0\) does not depend on b. For n large enough, it follows that
which implies \(\{u_{b_n,\lambda }\}\) is bounded in \(H_0^1(\Omega )\).
Step 2. There exists a subsequence of \(\{b_n\}\), still denoted by \(\{b_n\}\), such that
Then, \(u_{0,\lambda }\) is a weak solution of (1.13). Since \(u_{b_n,\lambda }\) is the least energy sign-changing solution of (1.6) with \(b=b_n\), then by the compactness of the embedding \(H_0^1(\Omega )\hookrightarrow L^q(\Omega )\) for \(2\le q<6\), we deduce that \(u_{b_n,\lambda }\rightarrow u_{0,\lambda }\) strongly in \(H_0^1(\Omega )\) as \(n\rightarrow \infty \). In fact,
and the right hand of last equality tend to zero as \(n\rightarrow \infty \). Then, by the same arguments as (2.11), we conclude \(u_{0,\lambda }^\pm \ne 0\), hence \(u_{0,\lambda }\) is sign-changing solution of equation (1.13).
Step 3. Suppose that \(v_0\) is a least energy sign-changing solution of (1.13), the existence of \(v_0\) was proved by Vladimir in [32]. By Lemma 2.2, for each \(b_n>0\), there is a unique pair \(\left( s_{b_n},t_{b_n}\right) \) of positive numbers such that
Then, we have
and
Recall that \(v_0^\pm \) satisfies
and
Up to a subsequence, one can easily deduce that
It follows from (3.13) and Lemma 2.4 that
which implies \(u_{0,\lambda }\) is a least energy sign-changing solution of Eq. (1.13). This completes the proof of Theorem 1.2. \(\square \)
Proof of Theorem 1.3
For arbitrary \(b>0\), let \(u_{b, \lambda _n}\in H^1_0(\Omega )\) is a least energy sign-changing solution for Eq. (1.6) with \(\lambda =\lambda _n\), which is obtained by Theorem 1.1. Obviously,
Therefore
which implies that \(\{u_{b,\lambda _n}\}\) is bounded in \(H^1_0(\Omega )\). Up to a subsequence, we may suppose there exists \(u_{b,0}\in H^1_0(\Omega )\) such that \(u_{b,\lambda _n}\rightharpoonup u_{b,0}\) weakly in \(H^1_0(\Omega )\).
Since \(\{u_{b,\lambda _n}\}\) is bounded in \(H^1_0(\Omega )\), it follows from (3.15) that
Therefore
which implies \(u_{b,0}=0\) on \(\Omega ^-\).
On the other hand, since \(\langle {I}'_{b,\lambda _{n}}(u_{b,\lambda n})-I'_{b,0}(u_{b,0}), u_{b,\lambda _n}-u_{b,0}\rangle =0\), then
the right hand of (3.16) tend to zero as \(n\rightarrow \infty \) since \(u_{b,\lambda _{n}}\rightharpoonup u_{b,0}\) weakly in \(H_{0}^{1}(\Omega )\), which implies \(u_{b,n} \rightarrow u_{b,0}\) strongly in \(H_{0}^{1}\left( \Omega \right) \). Therefore
which implies \(u_{b,0}\) is a solution of Eq. (1.14). By a similar method that used in [25], one can prove the existence of least energy sign-changing solution for equation (1.14). Suppose \(v_{b,0}\) is a least energy sign-changing solution for Eq. (1.14), by Lemma 2.2, for each \(\lambda _{n}>0\), there exist a unique pair of positive numbers \(\left( s_{\lambda _{n}}, t_{\lambda _{n}}\right) \) such that
That is
and
Recall that \(v_{b,0}^{\pm }\) satisfying
It follows from (3.17)–(3.19) that
Therefore, by (3.20) and Lemma 2.4, we can deduce that
Therefore, we conclude that \(u_{b,0}\) is a least energy sign-changing solution for Eq. (1.14), which changes sign once. The proof is completed. \(\square \)
4 A special minimax value for the energy functional
In this section, we assume \(h:\overline{\Omega }\rightarrow {\mathbb {R}}\) is a sign-changing continuous function and \((h_1)\)–\((h_2)\) hold.
We first state a result on the existence of solutions for Eq. (1.17).
Theorem 4.1
(Theorem1.2, [10]) Suppose that \(4<p<6\) and \((h_1)\)–\((h_2)\) hold. Then, for any non-empty subset \(\Gamma \subset \{1,2,\ldots ,k\}\) satisfies (1.16), Eq. (1.17) has a nontrivial solution \(u\in H^1_0(\Omega )\) with \(u|_{\Omega _i}\) is positive for \(i\in \Gamma _1\), \(u|_{\Omega _i}\) is negative for \(i\in \Gamma _2\), \(u|_{\Omega _i}\) changes sign exactly once for \(i\in \Gamma _3\), and \(u\equiv 0\) on \(\Omega \setminus \Omega _\Gamma \). Furthermore, u is the least energy solution among all solutions with these sign properties, that is, u achieves the following extremum
The functional \(I_{\Gamma }:H^1_0(\Omega _\Gamma )\rightarrow {\mathbb {R}}\) is defined by
Without loss of generality, we next only consider the case \(\Gamma _1=\{1\}\), \(\Gamma _2=\{2\}\), \(\Gamma _3=\{3\}\) for simplicity. In this case, \(\Gamma =\cup _{i=1}^3\Gamma _i=\{1,2,3\}\) and
We can choose open sets \(\Omega ^\rho _{i}:=\big \{x\in \Omega \,\ dist(x,\Omega _{i})<\rho \big \}\) for \(i=1,2,3\) with smooth boundary such that
We denote \(\Omega ^\rho :=\cup ^3_{i=1}\Omega ^\rho _{i}\) and define
Now, we consider the following constraint minimization problem
where
Combining the approach applied in Sect. 2 in [10] and that used in the proof of Theorem 1.1, we deduce that there exists \(v_{\lambda }\in H_0^1(\Omega ^\rho )\) such that
Proposition 4.2
Suppose \(\lambda _n\rightarrow +\infty \) as \(n\rightarrow \infty \) and \(\{v_{\lambda _n}\}\subset H_0^1(\Omega ^\rho )\) satisfying
then, up to a subsequence, there exists \(v\in H_0^1(\Omega ^\rho )\) such that
(i) \(v_{n} \rightarrow v\) strongly in \(H_0^1(\Omega ^\rho )\), where we write \(v_{\lambda _n}\) as \(v_n\) for simplicity;
(ii) \(v=0\) in \(\Omega ^\rho {\setminus } \Omega _\Gamma \) and v is a solution to Eq. (1.17);
(iii) \(\displaystyle \widehat{I}_{b,\lambda _n}(v_n)\rightarrow \widehat{I}_{b,0}(v)=\frac{a}{2}\int \limits _{\Omega _\Gamma }|\nabla v|^{2}\mathrm{{d}}x +\frac{b}{4}\left( \,\,\int \limits _{\Omega _\Gamma }|\nabla v|^2\mathrm{{d}}x\right) ^2 -\int \limits _{\Omega _\Gamma } h^+(x)|v|^p \mathrm{{d}}x\).
Proof
It is easy to prove that \(\{v_n\}\) is bounded in \(H_0^1(\Omega ^\rho )\), since \(\widehat{m}_{\lambda _n}\le m_\Gamma \). Then, up to a subsequence, there exists \(v \in H_0^1(\Omega ^\rho )\) such that
We first prove \(v=0\) in \(\Omega ^\rho \setminus \Omega _\Gamma \). Set \(\Omega ^\rho _-=\big \{x\in \Omega ^\rho : h(x)< 0 \big \}\), since \(\{v_{\lambda _n}\}\) is bounded in \(H_0^1(\Omega ^\rho )\), then
Therefore
which indicates that \(v=0\) on \(\Omega ^\rho _-\). Thus, we conclude \(v=0\) in \(\Omega ^\rho \setminus \Omega _\Gamma \).
By using the fact \(\langle \widehat{I}_{b,\lambda _n}'(v_n) -\widehat{I}_{b,0}'(v),v_n-v\rangle =0\) that
Obviously, the right hand of the last equality tend to zero as \(n\rightarrow \infty \), since \(\{v_n\}\) is bounded in \(H_0^1(\Omega ^\rho )\) and \(v=0\) in \(\Omega ^\rho {\setminus } \Omega _\Gamma \). Thus, \(v_n \rightarrow v\) strongly in \(H^{1}_0(\Omega ^\rho )\), and hence v is a solution of (1.17).
Finally, it is easy to conclude that (iii) from (i)–(ii). \(\square \)
Moreover, we have the following asymptotic behavior for \(\widehat{m}_{\lambda }\) as \(\lambda \rightarrow +\infty \).
Lemma 4.3
There holds that
(i) \(0< \widehat{m}_{\lambda } \le m_{\Gamma }\), for all \(\lambda \ge 0\);
(ii) \(\widehat{m}_{\lambda } \rightarrow m_{\Gamma }\), as \(\lambda \rightarrow +\infty \).
Proof
The proof of point (i) is trivial, so we omit the detail.
Now, we are going to prove point (ii). Let \(\{\lambda _n\}\) be a sequence with \(\lambda _n \rightarrow +\infty \) as \(n\rightarrow +\infty \). For each \(\lambda _n\), there exists \(v_{\lambda _n} \in H^1_0(\Omega ^\rho )\) with
We suppose, up to a subsequence, \(\{\widehat{I}_{b,\lambda _n}(v_{\lambda _n})\}\) converges, since \(\widehat{m}_{b,\lambda _n} \le m_{\Gamma }\). By using similar arguments as in Proposition 4.2, we know that there exists \(v\in H^1_0(\Omega ^\rho )\) such that
and \((v|_{\Omega _1})^+\), \((v|_{\Omega _2})^-\), \((v|_{\Omega _3})^\pm \ne 0\). Moreover,
and
By the definition of \(m_{\Gamma }\), we have that
By conclusion (i) of this Lemma, we know that \(\widehat{m}_{b,\lambda _n}\rightarrow m_{\Gamma }\) as \(n\rightarrow \infty \). \(\square \)
Next, we denote the solution of (1.17) given in Theorem 4.1 by \(v\in H_0^1(\Omega )\), that is
and \(v_1=v|_{\Omega _1}\) is positive, \(v_2=v|_{\Omega _2}\) is negative, \(v_3=v|_{\Omega _3}\) changes sign exactly once. Obviously, there exist constants \(\tau _2>\tau _1>0\) such that
We now define \(\gamma _0:[\frac{1}{2},\frac{3}{2}]^4\rightarrow H_0^1(\Omega )\) by
and
where
Obviously, \(\gamma _0\in \Sigma _\lambda \), so \(\Sigma _\lambda \ne \varnothing \). Thus \(m_\lambda \) is well-defined.
Lemma 4.4
For any \(\gamma \in \Sigma _\lambda \), there exists an 4-tuple \(\textbf{t}^*=(t_1^*,t_2^*,t_3^*,t_4^*)\in D=(\frac{1}{2},\frac{3}{2})^4\) such that
where \(\gamma _{i}(\textbf{t})=\gamma (\textbf{t})|_{\Omega ^\rho _{i}}\) for \(i=1,2,3\).
Proof
For each \(\gamma \in \Sigma _\lambda \), let us define \(\Psi :[\frac{1}{2},\frac{3}{2}]^4\rightarrow {\mathbb {R}}^4\) given by
Denote
Obviously,
Therefore, we can verify that
This implies that there exists \(\textbf{t}^*\in (\frac{1}{2},\frac{3}{2})^4\) such that \(\Psi (\textbf{t}^*)=0\). \(\square \)
Lemma 4.5
There holds that
(i) \(\widehat{m}_\lambda \le m_\lambda \le m_{\Gamma }\) for all \(\lambda \ge 1\);
(ii) \(m_\lambda \rightarrow m_{\Gamma }\) as \(\lambda \rightarrow +\infty \);
(iii) There exists \(\varepsilon _0>0\) such that \(I_{b,\lambda }(\gamma (\textbf{t}))<m_{\Gamma }-\varepsilon _0\) for all \(\lambda > 0\), \(\gamma \in \Sigma _\lambda \) and \(\textbf{t}=(t_1,t_2,t_3,t_4)\in \partial [\frac{1}{2},\frac{3}{2}]^4\).
Proof
(i) Since \(\gamma _0\in \Sigma _\lambda \), we have
where we have used Lemma 2.2 in [10]. Recall that
For each \(\gamma \in \Sigma _\lambda \), fix \(\textbf{t}^*\in (\frac{1}{2},\frac{3}{2})^4\) given by Lemma 4.4, then
Therefore,
Thus,
(ii) Since \(\widehat{m}_\lambda \rightarrow m_{\Gamma }\) by Lemma 4.3 (ii), we have
(iii) For \(\textbf{t}=(t_1,t_2,t_3,t_4)\in \partial [\frac{1}{2},\frac{3}{2}]^4\), it holds \(\gamma (\textbf{t})=\gamma _0(\textbf{t})\) and hence
By Lemma 2.2 in [10], we know that (1, 1, 1, 1) is the unique maximum point of \(\varphi (\textbf{t})=I_{b,0}(\gamma _0(\textbf{t}))\), which gives that
where \(\varepsilon _0>0\) is a small constant. \(\square \)
5 Proof of Theorem 1.4.
In this section, we prove Theorem 1.4. More precisely, we show that the existence of sign-changing multi-bump solutions to Eq. (1.6) for large \(\lambda \), which converges to solutions of (1.17) with prescribed sign properties as \(\lambda \rightarrow +\infty \).
Define
where
Obviously, \(\mathcal {S}\) contains all least energy solutions of (1.17) with \(u|_{\Omega _1}\) is positive, \(u|_{\Omega _2}\) is negative, \(u|_{\Omega _3}\) changes sign exactly once. Moreover, we have the following Lemma.
Lemma 5.1
\(\mathcal {S}\) is compact in \(H_0^1(\Omega _\Gamma )\).
Proof
Let \(\{u_n\}\subset \mathcal {S}\), then \(\{u_n\}\) is a bounded \((PS)_{m_{\Gamma }}\) sequence of \(I_{\Gamma }\). Since \(I_{\Gamma }\) satisfies (PS)-condition, up to a subsequence, we may suppose \(u_n\rightarrow u_\infty \) strongly in \(H_0^1(\Omega _\Gamma )\). It follows that \(u_\infty \in \mathcal {M}_{\Gamma }\) and \(I_{\Gamma }(u_\infty )=\lim \limits _{n\rightarrow \infty } I_{\Gamma }(u_n)=m_{\Gamma }\). Therefore, \(u_\infty \in \mathcal {S}\). \(\square \)
Lemma 5.2
Let \(d>0\) be a fixed number and let \(\{u_n\}\subset \mathcal {S}^d\) be a sequence. Then, up to a subsequence, \(u_n\rightharpoonup u_0\in \mathcal {S}^{2d}\) weakly in \(H^1_0(\Omega )\) as \(n\rightarrow \infty \), where
and dist denotes the distance in \(H^1_0(\Omega )\).
Proof
Since \(\mathcal {S}\) is compact in \(H_0^1(\Omega )\), then there exists a sequence \(\{\bar{u}_n\}\subset \mathcal {S}\) such that
By Lemma 5.1, there exists \(\bar{u}\in \mathcal {S}\) such that, up to a subsequence, \(\bar{u}_n\rightarrow \bar{u}\) strongly in \(H_0^1(\Omega )\). Hence, \(dist\left( \bar{u}_n,\bar{u}\right) \le d\) for n large enough. Thus, \(\{u_n\}\) is bounded and, up to a subsequence, \(u_n\rightharpoonup u_0\) weakly in \(H^1_0(\Omega )\). Since \(B_{2d}(\bar{u})\) is weakly closed in \(H^1_0(\Omega )\), therefore, \(u_0\in B_{2d}(\bar{u})\subset \mathcal {S}^{2d}\). \(\square \)
Lemma 5.3
Let \(d\in (0,\tau _1)\), where \(\tau _1\) is given by (4.11). Suppose that there exist a sequence \(\lambda _n>0\) with \(\lambda _n\rightarrow +\infty \), and \(\{u_n\}\subset \mathcal {S}^d\) satisfying
Then, up to a subsequence, \(\{u_n\}\) converges strongly in \(H^1_0(\Omega )\) to an element \(u\in \mathcal {S}\).
Proof
Since \(\lim \limits _{n\rightarrow \infty } I_{b,\lambda _n}(u_n)\le m_\Gamma \) and \(\lim \limits _{n\rightarrow \infty }I'_{b,\lambda _n}(u_n)=0\), we deduce that \(\{\Vert u_n\Vert \}\) and \(\{I_{\lambda _n}(u_n)\}\) are bounded. Up to a subsequence, we may assume that
By using Proposition 4.2, there exists \(u \in H^1_0(\Omega )\) such that
Moreover, u is a solution to the following equation
Since \(\{u_n\}\subset \mathcal {S}^d\) and \(d\in (0,\tau _1)\), we deduce that \((u|_{\Omega _1})^+\ne 0\), \((u|_{\Omega _2})^-\ne 0\) and \((u|_{\Omega _3})^\pm \ne 0\). Consequently, \(I_\Gamma (u)\ge m\). The conclusion \(I_{\Gamma }(u)=m\) follows from the fact that \(I_{b,\lambda _n}(u_n)\rightarrow I_{\Gamma }(u)\le m_{\Gamma }\), Thus, \(u \in \mathcal {S}\) is proved. \(\square \)
Lemma 5.4
Let \(\tau _1>0\) be as in Lemma 5.3. Then, for \(\delta \in (0,d)\), there exist constants \(0<\sigma <1\) and \(\Lambda _1>0\) such that \(\Vert I_{b,\lambda }'(u)\Vert _{H^{-1}}\ge \sigma \) for any \(u\in I_{b,\lambda }^{m_\lambda }\cap (\mathcal {S}^\delta {\setminus } \mathcal {S}^{\frac{\delta }{2}})\) and \(\lambda \ge \Lambda _1\).
Proof
We argue by contradiction. Suppose that there exist a number \(\delta _0\in (0,d)\), a positive sequence \(\{\lambda _j\}\) with \(\lambda _j\rightarrow 0\), and a sequence of function \(\{u_j\}\subset I_{b,\lambda _j}^{m_{\lambda _j}}\cap (\mathcal {S}^{\delta _0}{\setminus } \mathcal {S}^{\frac{\delta _0}{2}})\) such that
Up to a subsequence, we obtain
Hence, we can apply Lemma 5.3 and assert that there exists \(u\in \mathcal {S}\) such that \(u_j\rightarrow u\) strongly in \(H^1_0(\Omega )\). As a consequence, \(dist\left( u_j,\mathcal {S}\right) \rightarrow 0\) as \(j\rightarrow +\infty \). This contradict the fact that \(u_j\not \in \mathcal {S}^{\frac{\delta _0}{2}}\). \(\square \)
From now on, we fix a small constant \(\delta \in (0,d)\) and corresponding constants \(0<\sigma <1\) and \(\Lambda _1>0\) such that our Lemma 5.4 hold. For convenient, we next denote \(Q:=[\frac{1}{2},\frac{3}{2}]^4\).
Lemma 5.5
There exist \(\Lambda _2\ge \Lambda _1\) and \(\alpha >0\) such that for any \(\lambda \ge \Lambda _2\),
Proof
Assume by contradiction that there exist \(\lambda _n\rightarrow \infty \), \(\alpha _n\rightarrow 0\) and \((t^{(n)}_1,t^{(n)}_2,t^{(n)}_3,t^{(n)}_4)\in Q\) such that
Passing to a subsequence, we may assume that \((t^{(n)}_1,t^{(n)}_2,t^{(n)}_3,t^{(n)}_4)\rightarrow (\bar{t}_1,\bar{t}_2,\bar{t}_3,\bar{t}_4)\in Q\). Then, Lemma 4.5 implies that
From Lemma 2.2 in [10], we can deduce that \((\bar{t}_1,\bar{t}_2,\bar{t}_3,\bar{t}_4)=(1,1,1,1)\) and hence
However, \(\gamma _0(1,1,1,1)=v\in \mathcal {S}\), which contradicts to (5.4). \(\square \)
Next, we set
where \(\delta \), \(\sigma \) are given in Lemma 5.4, \(\alpha \) is from Lemma 5.5, \(\varepsilon _0\) is from Lemma 4.5 (iii). By Lemma 4.4, there exists \(\Lambda _3\ge \Lambda _2\) such that
Proposition 5.6
For each \(\lambda \ge \Lambda _3\), there exists a critical point \(u_\lambda \) of \(I_{b,\lambda }\) with \(u_\lambda \in \mathcal {S}^\delta \cap I_{b,\lambda }^{m_{\Gamma }}\).
Proof
Fix \(\lambda \ge \Lambda _3\). Assume by contradiction that there exists \(0<\rho _\lambda <1\) such that \(\Vert I_{b,\lambda }'(u)\Vert \ge \rho _\lambda \) on \(\mathcal {S}^\delta \cap I_\lambda ^{m_{\Gamma }}\). Then there exists a pseudo-gradient vector field \(T_\lambda \) in \(H^1_0(\Omega )\) which is defined on a neighborhood \(Z_\lambda \) of \(\mathcal {S}^\delta \cap I_{b,\lambda }^{m_{\Gamma }}\) such that for any \(u\in Z_\lambda \) there holds
Let \(\psi _\lambda \) be a Lipschitz continuous function on \(H^1_0(\Omega )\) such that \(0\le \psi _\lambda \le 1\), \(\psi _\lambda \equiv 1\) on \(S^\delta \cap I_{b,\lambda }^{m_{\Gamma }}\) and \(\psi _\lambda \equiv 0\) on \(H^1_0(\Omega ){\setminus } Z_\lambda \). Let \(\xi _\lambda \) be a Lipschitz continuous function on \({\mathbb {R}}\) such that \(0\le \xi _\lambda \le 1\), \(\xi _\lambda (t)\equiv 1\) if \(|t-m_{\Gamma }|\le \frac{\alpha }{2}\) and \(\xi _\lambda (t)\equiv 0\) if \(|t-m_{\Gamma }|\ge \alpha \). Define
Then there exists a global solution \(\eta _\lambda : H^1_0(\Omega ) \times [0,+\infty )\rightarrow H^1_0(\Omega )\) for the initial value problem
It is easy to see that \(\eta _\lambda \) has the following properties:
(1) \(\eta _\lambda (u,\theta )=u\) if \(\theta =0\) or \(u\in H^1_0(\Omega ){\setminus } Z_\lambda \) or \(|I_{b,\lambda }(u)-m_\Gamma |\ge \alpha \).
(2) \(\Vert \frac{\textrm{d}}{\textrm{d}\theta }\eta _\lambda (u,\theta )\Vert \le 2\).
(3) \(\frac{\textrm{d}}{\textrm{d}\theta }I_{b,\lambda }(\eta _\lambda (u,\theta ))=\langle I_{b,\lambda }'(\eta _\lambda (u,\theta )), e_\lambda (\eta _\lambda (u,\theta ))\rangle \le 0\). \(\square \)
Claim 1
For any \((t_1,t_2,t_3,t_4)\in Q\), there exists \(\overline{\theta }=\theta (t_1,t_2,t_3,t_4)\in [0,+\infty )\) such that \(\eta _\lambda (\gamma _0(t_1,t_2,t_3,t_4),\overline{\theta }) \in I_{b,\lambda }^{m_{\Gamma }-\alpha _0}\), where \(\alpha _0\) is given by (5.5).
Assume by contradiction that there exists \((t_1,t_2,t_3,t_4)\in Q\) such that
for any \(\theta \ge 0\). Note that \(\alpha _0<\alpha \), we see, from Lemma 5.5, that \(\gamma _0(t_1,t_2,t_3,t_4)\in \mathcal {S}^{\frac{\delta }{2}}\). Moreover, since \(I_{b,\lambda }(\gamma _0(t_1,t_2,t_3,t_4))\le m_{\Gamma }\), we have, from the property (3) of \(\eta _\lambda \), that
for \(\theta \ge 0\). This implies that \(\xi _\lambda (I_{b,\lambda } (\eta _\lambda (\gamma _0(t_1,t_2,t_3,t_4),\theta )))\equiv 1\). If \(\eta _\lambda (\gamma _0(t_1,t_2,t_3,t_4),\theta )\in \mathcal {S}^\delta \) for all \(\theta \ge 0\), we can deduce that
for all \(\theta >0\). It follows that
which is a contradiction. Thus, there exists \(\theta _3>0\) such that \(\eta _\lambda (\gamma _0(t_1,t_2,t_3,t_4),\theta _3)\not \in \mathcal {S}^\delta \). Note that \(\gamma _0(t_1,t_2,t_3,t_4)\in \mathcal {S}^{\frac{\delta }{2}}\), there exist \(0<\theta _1<\theta _2\le \theta _3\) such that
and
By Lemma 5.4, we have that
By using property (2) of \(\eta _\lambda \) we have
This implies that
which is a contradiction. Thus, we finish the proof of Claim 1.
Now, we can define
and
Then \(\Phi _\lambda (\widetilde{\gamma }(t_1,t_2,t_3,t_4))\le m_{\Gamma }-\alpha _0\) for all \((t_1,t_2,t_3,t_4)\in Q\).
Claim 2
\(\widetilde{\gamma }(t_1,t_2,t_3,t_4)=\eta _\lambda (\gamma _0(t_1,t_2,t_3,t_4),T(t_1,t_2,t_3,t_4)) \in \Sigma _\lambda \).
For any \((t_1,t_2,t_3,t_4)\in \partial Q\), by (5.5)–(5.6), we have
which implies that \(T(t_1,t_2,t_3,t_4)=0\) and thus \(\widetilde{\gamma }(t_1,t_2,t_3,t_4)=\gamma _0(t_1,t_2,t_3,t_4)\) for \((t_1,t_2,t_3,t_4)\in \partial Q\).
By the definition of \(\Sigma _\lambda \) in (4.14), it suffices to prove that \(\Vert \widetilde{\gamma }(t_1,t_2,t_3,t_4)\Vert \le 6\tau _2+\tau _1\) for all \((t_1,t_2,t_3,t_4)\in Q\) and \(T(t_1,t_2,t_3,t_4)\) is continuous with respect to \((t_1,t_2,t_3,t_4)\).
For any \((t_1,t_2,t_3,t_4)\in Q\), we have \(T(t_1,t_2,t_3,t_4)=0\) if \(I_{b,\lambda } (\gamma _0(t_1,t_2,t_3,t_4))\le m_{\Gamma }-\alpha _0\), and hence \(\widetilde{\gamma }(t_1,t_2,t_3,t_4)=\gamma _0(t_1,t_2,t_3,t_4)\). By (4.11), we deduce that \(\Vert \widetilde{\gamma }(t_1,t_2,t_3,t_4)\Vert \le 6\tau _2< 6\tau _2+\tau _1\).
On the other hand, if \(I_{b,\lambda } (\gamma _0(t_1,t_2,t_3,t_4))>m_{\Gamma }-\alpha _0\), we can deduce that
thus \(\gamma _0(t_1,t_2,t_3,t_4)\in \mathcal {S}^{\frac{\delta }{2}}\) and
This implies that
Now, we are going to prove that \(\widetilde{\gamma }(t_1,t_2,t_3,t_4)\in \mathcal {S}^\delta \). Otherwise, if \(\widetilde{\gamma }(t_1,t_2,t_3,t_4)\not \in \mathcal {S}^\delta \), similar to the proof of Claim 1, we can find two constants \(0<\theta _1<\theta _2<T(t_1,t_2,t_3,t_4)\) such that (5.9) hold. This implies that \(I_{b,\lambda } (\eta _\lambda (\gamma _0(t_1,t_2,t_3,t_4),\theta _2))<m_{\Gamma }-\alpha _0\) which contradicts to the definition of \(T(t_1,t_2,t_3,t_4)\). Therefore,
Thus there exists \(u\in \mathcal {S}\) such that \(\Vert \widetilde{\gamma }(t_1,t_2,t_3,t_4)-u\Vert \le \delta \le \tau _1\). It follows from (4.11) that
To prove the continuity of \(T(t_1,t_2,t_3,t_4)\), we fix arbitrarily \((t_1,t_2,t_3,t_4)\in Q\). First, we assume that \(I_{b,\lambda }(\widetilde{\gamma }(t_1,t_2,t_3,t_4))<m_{\Gamma }-\alpha _0\). In this case, we deduce directly that \(T(t_1,t_2,t_3,t_4)=0\) by the definition of \(T(t_1,t_2,t_3,t_4)\), which gives that
By the continuity of \(\gamma _0\), there exists \(r>0\) such that for any \((s_1,s_2,s_3,s_4)\in B_r(t_1,t_2,t_3,t_4)\cap Q\), we have \(I_{b,\lambda }(\gamma _0(s_1,s_2,s_3,s_4))<m_{\Gamma }-\alpha _0\). Thus, \(T(s_1,s_2,s_3,s_4)=0\), and hence T is continuous at \((t_1,t_2,t_3,t_4)\).
Now, we assume that \(I_{b,\lambda } (\widetilde{\gamma }(t_1,t_2,t_3,t_4))=m_{\Gamma }-\alpha _0\). From the previous proof we see that \(\widetilde{\gamma }(t_1,t_2,t_3,t_4)=\eta _\lambda (\gamma _0(t_1,t_2,t_3,t_4),T(t_1,t_2,t_3,t_4))\in \mathcal {S}^\delta \), and so
Thus for any \(\omega >0\), we have
By the continuity of \(\eta _\lambda \), there exists \(r>0\) such that
for any \((s_1,s_2,s_3,s_4)\in B_r(t_1,t_2,t_3,t_4)\cap Q\). Thus, \(T(s_1,s_2,s_3,s_4)\le T(t_1,t_2,t_3,t_4)+\omega \). It follows that
If \(T(t_1,t_2,t_3,t_4)=0\), we immediately implies that
If \(T(t_1,t_2,t_3,t_4)>0\), we can similarly deduce that
for any \(0<\omega <T(t_1,t_2,t_3,t_4)\).
By the continuity of \(\eta _\lambda \) again, we see that
It follows from (5.10)–(5.11) that T is continuous at \((t_1,t_2,t_3,t_4)\). This completes the proof of Claim 2.
Thus, we have proved that \(\widetilde{\gamma }(t_1,t_2,t_3,t_4)\in \Sigma _\lambda \) and
which contradicts the definition of \(m_{\Gamma }\). This completes the proof. \(\square \)
Proof of Theorem 1.4
We still prove Theorem 1.4 with \(\Gamma _1=\{1\}\), \(\Gamma _2=\{2\}\) and \(\Gamma _3=\{3\}\). For the general \(\Gamma \) verifying (1.16), the proof is very similar and just needs a slight modification.
By Proposition 5.6, there exists a solution \(u_\lambda \) for Eq. (1.6) with \(u_\lambda \in \mathcal {S}^{\delta }\cap I_{b,\lambda }^{m_{\Gamma }}\) for all \(\lambda \ge \Lambda _3\). Therefore, for any sequence \(\{\lambda _n\}\) with \(\lambda _n \rightarrow +\infty \) as \(n\rightarrow \infty \), there exists a sequence \(\{u_n\}\subset H^1_0(\Omega )\) such that
By using Lemma 5.3, we can deduce that \(u_{\lambda _n}\rightarrow u\in \mathcal {S}\) strongly in \(H^1_0(\Omega )\). Thus, we complete the proof of Theorem 1.4. \(\square \)
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Cui, Z., Shuai, W. Sign-changing solutions for Kirchhoff-type equations with indefinite nonlinearities. Z. Angew. Math. Phys. 74, 150 (2023). https://doi.org/10.1007/s00033-023-02031-8
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DOI: https://doi.org/10.1007/s00033-023-02031-8