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A Kirchhoff-type problem involving concave-convex nonlinearities
Advances in Difference Equations volume 2021, Article number: 171 (2021)
Abstract
A Kirchhoff-type problem with concave-convex nonlinearities is studied. By constrained variational methods on a Nehari manifold, we prove that this problem has a sign-changing solution with least energy. Moreover, we show that the energy level of this sign-changing solution is strictly larger than the double energy level of the ground state solution.
1 Introduction
We study the following Kirchhoff-type equation with concave-convex nonlinearities:
where \(a>0\), \(b>0\), \(\lambda >0\), \(\kappa <0\), \(p\in (3,5)\), \(q\in (0,1)\), and \(Q,G\in C(\mathbb{R}^{3},\mathbb{R}^{+})\) satisfying the following conditions:
- \((Q_{1})\):
-
There exists \(\beta \in [0,p-2)\) such that \(\limsup_{x\rightarrow +\infty }\frac{Q(x)}{|x|^{\beta }}<+\infty \);
- \((G_{1})\):
-
\(G(x)\in L^{2}(\mathbb{R}^{3},\mathbb{R}^{+})\).
In recent years, the following elliptic problem has been investigated by many researchers [1, 3, 6, 9, 17, 20]:
where \(f\in C(\mathbb{R}^{3}\times \mathbb{R},\mathbb{R})\) and \(a>0\), \(b>0\). The term \(\int _{\mathbb{R}^{3}}|\nabla u|^{2}\) in (1.2) has an interesting physical application. Moreover, this problem is related to the stationary analogue of the following equation proposed by Kirchhoff [10]:
Inspired by the variational framework given by Lions [12], problem (1.3) has been investigated by many researchers, and the reader is referred to [5, 7, 11, 13, 19, 22] and the references therein for more details.
Shuai [16] studied the ground state sign-changing solution of problem (1.2) by using Brouwer degree theory, where \(f(x,u)\) is replaced with \(f(u)\) with the following hypotheses:
- \((f_{1}^{\prime })\)::
-
\(f(s)=o(|s|)\) as \(s\rightarrow 0\);
- \((f_{2}^{\prime })\)::
-
For some constant \(p\in (4,2^{*})\), \(\lim_{s\rightarrow \infty }\frac{f(s)}{s^{p-1}}=0\), where \(2^{*}=+\infty \) for \(N=1,2\) and \(2^{*}=6\) for \(N=3\);
- \((f_{3}^{\prime })\)::
-
\(\lim_{s\rightarrow \infty }\frac{F(s)}{s^{4}}=+\infty \), where \(F(s)=\int ^{s}_{0} f(t)\,dt\);
- \((f_{4}^{\prime })\)::
-
\(\frac{f(s)}{|s|^{3}}\) is an increasing function with respect to \(s\in \mathbb{R}\setminus \{0\}\).
Huang and Liu [8] obtained the ground state sign-changing solutions of problem (1.4) with accurately two nodal domains
where \(p\in (3,5)\), \(\lambda >0\) and \(V\in C(\mathbb{R}^{N},\mathbb{R})\) is to ensure the establishment of compactness.
Deng et al. [4] showed the existence of radial sign-changing solutions \(u_{k}^{b}\) of problem (1.5)
by constrained minimization on the Nehari manifold, where k is any positive integer. Ye [21] studied the existence of least energy sign-changing solutions for problem (1.5), where \(f(x,u)\) is replaced with \(f(u)\).
Shao and Mao [15] got at least one sign-changing solution of problem (1.6) with concave-convex nonlinearities
by using the method of invariant sets of descending flow.
Motivated by the aforementioned works, we prove the existence of sign-changing solutions with least energy for problem (1.1) with concave-convex nonlinearities and unbounded potential by constrained variational methods on a Nehari manifold.
Now we will give the main results by Theorems 1.1 and 1.2.
Theorem 1.1
Assume that \((Q_{1})\) and \((G_{1})\) hold, then, for \(a>0\), \(b>0\), \(\lambda >0\), and \(\kappa <0\), problem (1.1) has one least energy sign-changing solution with accurately two nodal domains.
Theorem 1.2
Assume that \((Q_{1})\) and \((G_{1})\) hold, then, for \(a>0\), \(b>0\), \(\lambda >0\), and \(\kappa <0\), problem (1.1) has one least energy solution. Moreover \(m_{\lambda }>2c_{\lambda }\), where \(m_{\lambda }\) and \(c_{\lambda }\) are defined by (2.3) and (2.5) respectively.
Remark 1.3
Comparing with Shuai [16], Huang and Liu [8], Deng et al. [4], and Ye [21], the difference is to consider Kirchhoff-type equation with concave and convex terms, where \(Q(x)\) is unbounded at infinity. Moreover, since \(H^{1}_{r}(\mathbb{R}^{3})\hookrightarrow L^{{q+1}}(\mathbb{R}^{3})\) is not compact for \({q}\in (0,1)\), this means that the appearance of concave and convex terms has greatly increased the difficulty of problem (1.1). Shao and Mao [15] got sign-changing solutions for Kirchhoff equation with concave and convex terms by using the method of invariant sets of descending flow. However, we want to obtain ground state sign-changing solutions of (1.1) by variational methods and constrained minimization on the sign-changing Nehari manifold. It should be addressed that our methods are different to those in [15].
The rest of the paper is organized as follows. In Sect. 2 we give some notations and the main lemmas related to the proof of our main results. Sections 3 and 4 give the proofs of Theorems 1.1 and 1.2, respectively.
2 Some notations and preliminary lemmas
Here are some notations to be used in this paper.
-
C denotes a positive constant;
-
\(H^{1}({\mathbb{R}}^{3})\) denotes the usual Sobolev space with the norm \(\|u\|^{2}=\int _{\mathbb{R}^{3}}(|\nabla u|^{2}+b |u|^{2})\);
-
\(|\cdot |\) denotes the usual norm \(L^{\bar{q}}(\mathbb{R}^{3})\) for \(\bar{q}\in [1,\infty )\);
-
\(H^{1}_{r}({\mathbb{R}}^{3}):=\{u:u\in H^{1}({\mathbb{R}}^{3}),u(x)=u(|x|) \}\);
-
\(u^{+}:=\max \{{u,0}\}\) and \(u^{-}:=\min \{{u,0}\}\).
Lemma 2.1
(see Berestycki and Lions [2])
Let \(N\geq 2\) and \(u\in H_{r}^{1}({\mathbb{R}}^{N})\), Then
where \(C_{0}>0\) is only related to N.
Remark 2.2
For any \(u\in H_{r}^{1}({\mathbb{R}}^{3})\), by \((Q_{1})\), \((G_{1})\), and Lemma 2.1, we have
and
The energy functional \(J_{\lambda }\in C^{1}(H^{1}_{r}({\mathbb{R}}^{3}),\mathbb{R})\) is well defined by
For each \(u,v\in H^{1}_{r}({\mathbb{R}}^{3})\),
In order to get a sign-changing solution \(u^{\pm }\neq 0\) of (1.1), the following functionals need to be established:
Let us define
and
In addition, we define
and
Lemma 2.3
Assume that \((Q_{1})\), \((G_{1})\), and \(u_{n}\rightharpoonup u\) in \(H^{1}_{r}(\mathbb{R}^{3})\) hold, then
In particular,
Proof
If \(u_{n}\rightharpoonup u\) in \(H^{1}_{r}(\mathbb{R}^{3})\), then \(u_{n}\rightarrow u\) in \(L^{\bar{q}}(\mathbb{R}^{3})\) for \(\bar{q}\in (2,6)\). According to [18, Theorem A.4, p. 134], we can obtain that \(|u_{n}|^{q+1}\rightarrow |u|^{q+1}\) in \(L^{2}(\mathbb{R}^{3})\). By the Hölder inequality, we have
Thus, \(\lim_{n\rightarrow \infty }\int _{\mathbb{R}^{3}}G(x)|u_{n}|^{q+1}= \int _{\mathbb{R}^{3}}G(x)|u|^{q+1}\). Similarly, \(\lim_{n\rightarrow \infty }\int _{\mathbb{R}^{3}}G(x)|u_{n}^{\pm }|^{q+1}= \int _{\mathbb{R}^{3}}G(x)|u^{\pm }|^{q+1}\). □
Lemma 2.4
Under the assumptions of Theorem 1.1. If \(u\in H^{1}_{r}({\mathbb{R}}^{3})\) with \(u^{\pm }\neq 0\), there exists a unique pair \((s_{u},t_{u})\in (0,+\infty )\times (0,+\infty )\) such that \(s_{u}u^{+}+t_{u}u^{-}\in \mathcal{M}_{\lambda }\). Moreover,
Proof
Let \(u\in H^{1}(\mathbb{R}^{3})\) with \(u^{\pm }\neq 0\). Define
According to Remark 2.2, for \(\kappa <0\), we have \(g_{i}(s,s)>0\) as \(s>0\) small and \(g_{i}(t,t)<0\) as \(t>0\) large, where \(i=1,2\). Then there exists \(0<\mu <\nu \) such that
By (2.6), (2.7), (2.8), we have that
From Miranda’s theorem [14], there exists a pair \((s_{u},t_{u})\) such that
Thus, \(s_{u}u^{+}+t_{u}u^{-}\in \mathcal{M}_{\lambda }\).
Secondly, we prove the uniqueness. Let both \((s_{1},t_{1})\) and \((s_{2},t_{2})\) satisfy \(u_{i}=s_{i}u^{+}+t_{i}u^{-}\in \mathcal{M}_{\lambda }\) (\(i=1,2\)) and \(u_{1}=s_{1}u^{+}+t_{1}u^{-}= m s_{2}u^{+}+n t_{2}u^{-}=mu_{2}^{+}+nu^{-}_{2}\), where \(m=\frac{s_{1}}{s_{2}}\), \(n=\frac{t_{1}}{t_{2}}\). By (2.6) and (2.7),
We only need to prove that \(m=n=1\). Now, assume that \(0< m\leq n\). By (2.9) and (2.10),
and
If \(m<1\), then
this is impossible for \(\kappa <0\). Then \(m\geq 1\). Similarly, if \(n>1\), (2.12) is impossible. Then \(n\leq 1\). Thus \(m=n=1\).
At last, let
Then, for \(\kappa <0\), we have \(H_{\lambda }(s,t)>0\) as \(|(s,t)|\rightarrow 0\), \(H_{\lambda }(s,t)<0\) as \(|(s,t)|\rightarrow \infty \), and \(H_{\lambda }\) cannot achieve the maximum point on \(\partial {\mathbb{R}^{+}}^{2}\). Without loss of generality, we only prove that \((0,t_{0})\) is not a maximum point of \(H_{\lambda }\). For \(s>0\) small enough,
this implies that \(H_{\lambda }(s,t_{0})\) is an increasing function with respect to s, where \(s>0\) is small enough, then \((0,t_{0})\) is not a maximum point of \(H_{\lambda }\). Thus, there exists \((s_{u},t_{u})\in {\mathbb{R}^{+}}^{2}\) such that
□
Lemma 2.5
Under the assumptions of Theorem 1.1. If \(\langle J^{\prime }_{\lambda }(u),u^{\pm }\rangle \leq 0\), there exists \((s_{u},t_{u})\in (0,1]\times (0,1]\) such that \(s_{u}u^{+}+t_{u}u^{-}\in \mathcal{M}_{\lambda }\) for \(u\in H^{1}_{r}({\mathbb{R}}^{3})\) with \(u^{\pm }\neq 0\).
Proof
Let \(u\in H^{1}_{r}({\mathbb{R}}^{3})\) with \(u^{\pm }\neq 0\), by Lemma 2.4, there exists a pair \((s_{u},t_{u})\) such that
Since \(\langle J^{\prime }_{\lambda }(u),u^{\pm }\rangle \leq 0\), we have that
Now, assume that \(0< t_{u}\leq s_{u}\). If \(s_{u}>1\), by (2.13) and (2.14),
which is contradictory for \(\kappa <0\). Then \(s_{u}\leq 1\). From \(0< t_{u}\leq s_{u}\), we obtain that \(0< t_{u}\leq s_{u}\leq 1\). □
Lemma 2.6
Under the assumptions of Theorem 1.1, \(m_{\lambda }>0\) can be achieved.
Proof
For all \(u\in \mathcal{M}_{\lambda }\), by the Sobolev embedding theorem, we have
Then there exists \(C\geq C_{1}\) such that \(\|u\|\geq (\frac{a}{C} )^{\frac{1}{p-1}}>0\). Since
for \(\kappa <0\). Then
Let \(\{u_{n}\}\subset \mathcal{M}_{\lambda }\) and \(J_{\lambda }(u_{n})\rightarrow m_{\lambda }\). By Remark 2.2, we have
This shows that \(\{u_{n}\}\) is bounded in \(H^{1}_{r}(\mathbb{R}^{3})\). Then there exists \(u_{\lambda }\in H^{1}_{r}(\mathbb{R}^{3})\) such that \(u_{n}^{\pm }\rightharpoonup u_{\lambda }^{\pm }\) in \(H^{1}_{r}(\mathbb{R}^{3})\), \(u_{n}^{\pm }\rightarrow u_{\lambda }^{\pm }\) in \(L^{q}(\mathbb{R}^{3})\) for \(q\in (2,6)\) and \(u_{n}^{\pm }(x) \rightarrow u_{\lambda }^{\pm }(x)\) a.e. on \(\mathbb{R}^{3}\). Since \(\{u_{n}\}\subset \mathcal{M}_{\lambda }\), we have
By Fatou’s lemma and Lemma 2.3,
this implies that
By Lemmas 2.4 and 2.5, there exists \((s_{{u}_{\lambda }},t_{{u}_{\lambda }})\in (0,1]\times (0,1]\) such that \(\widetilde{u}_{\lambda }=s_{{u}_{\lambda }}u^{+}_{\lambda }+t_{{u}_{ \lambda }}u^{-}_{\lambda }\in \mathcal{M}_{\lambda }\). Then
this implies that \(s_{u_{\lambda }}=t_{u_{\lambda }}=1\). Thus, \(\widetilde{u}_{\lambda }=u_{\lambda }\) and \(J_{\lambda }(u_{\lambda })=m_{\lambda }\). □
3 Sign-changing solutions
Lemma 3.1
Under the assumptions of Theorem 1.1. If \(u_{\lambda }\in \mathcal{M}_{\lambda }\) and \(J_{\lambda }(u_{\lambda })=m_{\lambda }\), then \(J^{\prime }_{\lambda }(u_{\lambda })=0\).
Proof
Suppose that \(J_{\lambda }^{\prime }(u_{\lambda })\neq 0\), then there are σ, \(\delta >0\) such that
Let \(D=(0.5,1.5)\times (0.5,1.5)\). By Lemma 2.4, we obtain that
For \(\varepsilon :=\min \{(m_{\lambda }-\iota )/2,\sigma \delta /8\}\) and \(S:=B(u_{\lambda },\delta )\), Willem [18, Lemma 2.3] produce a deformation η such that
-
(i)
\(\eta (1,u)=u\) if \(u\notin J_{\lambda }^{-1}([m_{\lambda }-2\varepsilon ,m_{\lambda }+2 \varepsilon ])\cap S_{2\delta }\);
-
(ii)
\(\eta (1,J_{\lambda }^{m_{\lambda }+\varepsilon }\cap S)\subset J_{ \lambda }^{m_{\lambda }-\varepsilon }\);
-
(iii)
\(J_{\lambda }(\eta (1,u))\leq J_{\lambda }(u)\) for all \(u\in H_{r}^{1}(\mathbb{R}^{3})\).
At first, we show that
For all \((s,t)\in \bar{D}\), by Lemma 2.4, we obtain \(J_{\lambda }(su_{\lambda }^{+}+tu_{\lambda }^{-})\leq m_{\lambda }< m_{ \lambda }+\varepsilon \), that is, \(su_{\lambda }^{+}+tu_{\lambda }^{-}\in J_{\lambda }^{m_{\lambda }+ \varepsilon }\). Therefore, \(J_{\lambda }(\eta (1,su_{\lambda }^{+}+tu_{\lambda }^{-}))\leq m_{ \lambda }-\varepsilon \).
Next, we prove that
Define \(h(s,t)=\eta (1,su_{\lambda }^{+}+tu_{\lambda }^{-})\) and \(\psi :[0,1]\times \bar{D}\rightarrow \mathbb{R}^{2}\), for any \(\vartheta \in [0,1]\), we have
Let
By a simple calculation, \(\operatorname{deg}(\psi _{0},D,0)=1\). According to (3.1), we obtain that \(u_{\lambda }=h\) on ∂D and from homotopy invariance that
Then there exists a pair \((s_{0},t_{0})\in D\) such that \(\psi _{1}(s_{0},t_{0})=0\) and \(\eta (1,s_{0}u_{\lambda }^{+}+t_{0}u^{-}_{\lambda })=h(s_{0},t_{0}) \in \mathcal{M}_{\lambda }\), which contradicts (3.1). Therefore, \(u_{\lambda }\) is a critical point of \(J_{\lambda }\), and so a sign-changing solution of (1.1). □
Proof of Theorem 1.1
Firstly, by the preceding lemmas, there exists \(u_{\lambda }\in \mathcal{M}_{\lambda }\) such that \(J_{\lambda }(u_{\lambda })=m_{\lambda }\) and \(J_{\lambda }^{\prime }(u_{\lambda })=0\). Thus, problem (1.1) has one least energy sign-changing solution \(u_{\lambda }\).
Secondly, we prove that \(u_{\lambda }\) has only two nodal domains. Assume that \(u_{\lambda }=u_{1}+u_{2}+u_{3}\) with
Setting \(w=u_{1}+u_{2}\) with \(w^{+}=u_{1}\) and \(w^{-}=u_{2}\), i.e., \(w^{\pm }\neq 0\). Since \(J_{\lambda }^{\prime }(u_{\lambda })=0\), we get
By Lemma 2.5, there exists \((s_{w},t_{w})\in (0,1]\times (0,1]\) such that
Note that \(\langle J_{\lambda }^{\prime }(u_{\lambda }),u_{\lambda }\rangle =0\) and \(\langle J_{\lambda }^{\prime }(s_{w}u_{1}+t_{w}u_{2}),s_{w}u_{1}+t_{w}u_{2} \rangle =0\), we have
which is a contradiction. □
4 Ground state solutions
Lemma 4.1
(Mountain pass theorem [18])
Let X be a Banach space, \(I \in C^{1}(X, \mathbb{R})\), \(e \in X\), and \(\rho > 0\) such that \(\|e\|> \rho \) and
If I satisfies the \((PS)_{c}\) condition with
then c is a critical value of I.
Lemma 4.2
Under the assumptions of Theorem 1.2, there exist \(e\in H^{1}_{r}(\mathbb{R}^{3})\) and \(\rho > 0\) such that \(\|e\|>\rho \) and \(\inf_{\|u\|=\rho }J_{\lambda }(u)>J_{\lambda }(0)> J_{\lambda }(e)\).
Proof
For all \(u\in H^{1}_{r}(\mathbb{R}^{3})\), by Remark 2.2,
then there exists \(\rho >0\) such that
Let \(t\geq 0\), we have
then there exists \(e:=tu\) such that \(\|e\|>\rho \) and \(J_{\lambda }(e)<0\). □
Lemma 4.3
Under the assumptions of Theorem 1.2. \(J_{\lambda }\) satisfies the \((P S)_{c}\) condition.
Proof
Let \(\{u_{n}\}\subset H_{r}^{1}(\mathbb{R}^{3})\) and \(J_{\lambda }(u_{n})\rightarrow c\), \(J_{\lambda }(u_{n})\rightarrow 0\) as \(n\rightarrow \infty \). By (2.15) in Lemma 2.6 above, it is easy to see that \(\{u_{n}\}\) is bounded in \(H_{r}^{1}(\mathbb{R}^{3})\). Going if necessary to a subsequence, \(u_{n}\rightharpoonup u\) in \(H^{1}_{r}(\mathbb{R}^{3})\), \(u_{n}\rightarrow u\) in \(L^{s}(\mathbb{R}^{3})\) for \(s\in (2,6)\), and \(u_{n}(x)\rightarrow u(x)\) a.e. on \(\mathbb{R}^{3}\), then by \((G_{1})\) we have
Since
and
Thus, \(u_{n}\rightarrow u\) in \(H^{1}_{r}(\mathbb{R}^{3})\). □
Set
Lemma 4.4
Under the assumptions of Theorem 1.2, we have \({c}=c_{\lambda }=c_{1}\).
Proof
Similar to the proof of Lemma 2.4, for all \(u\in H_{r}^{1}(\mathbb{R}^{3})\setminus \{0\}\), there exists unique \(t_{u}u\in \mathcal{N}\) such that \(J_{\lambda }(t_{u}u)=\max_{t\geq 0}J_{\lambda }(tu)\), this implies that \(c_{\lambda }\leq c_{1}\).
For each \(\gamma \in \Gamma \), it follows from the property of \(\mathcal{N}\) that \(\gamma (t)\) crosses \(\mathcal{N}\) as t varying over \([0,1]\). Since \(\gamma (0)=0\), \(J_{\lambda }(\gamma (1))<0\), then
Therefore \(c\geq c_{\lambda }\). On the other hand, for \(u\in H_{r}^{1}(\mathbb{R}^{3})\setminus \{0\}\), we have that \(J_{\lambda }(tu)<0\) for t large enough, and then
Therefore \(c_{1}\geq c\). □
Proof of Theorem 1.2
According to Lemmas 4.1, 4.2, 4.3, and 4.4, we obtain that problem (1.1) has one least energy solution.
Now we prove \(m_{\lambda }>2c_{\lambda }\). By the proof of Theorem 1.1, there exists \(u_{\lambda }\in \mathcal{M}_{\lambda }\) such that \(J_{\lambda }(u_{\lambda })=m_{\lambda }\). By Lemmas 2.4 and 4.4, we have
□
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References
Alves, C.O., Corrêa, F.J.S.A., Ma, T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)
Berestycki, H., Lions, P.L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82(4), 313–345 (1983)
Cheng, B.T., Wu, X.: Existence results of positive solutions of Kirchhoff problems. Nonlinear Anal. 71, 4883–4892 (2009)
Deng, Y.B., Peng, S.J., Shuai, W.: Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in \(\mathbb{R}^{3}\). J. Funct. Anal. 269, 3500–3527 (2015)
Duan, Y.L., Zhou, Y.G.: Existence of solutions for Kirchhoff-type equations with unbounded potential. Electron. J. Differ. Equ. 2017, 184 (2017)
He, H.M., Zou, W.M.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \(\mathbb{R}^{3}\). J. Differ. Equ. 2, 1813–1834 (2012)
He, Y., Li, G.B., Peng, S.J.: Concentrating bound states for Kirchhoff-type problems in \(\mathbb{R}^{3}\) involving critical Sobolev exponents. Adv. Nonlinear Stud. 14(2), 483–510 (2014)
Huang, Y.S., Liu, Z.: On a class of Kirchhoff type problems. Arch. Math. 102, 127–139 (2014)
Jeanjean, L.: On the existence of bounded Palais–Smale sequences and application to a Landsman–Lazer-type problem set on \(\mathbb{R}^{N}\). Proc. R. Soc. Edinb., Sect. A 129(2), 787–809 (1999)
Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)
Liang, S.H., Zhang, J.H.: Existence of solutions for Kirchhoff-type problems with critical nonlinearity in \(\mathbb{R}^{3}\). Nonlinear Anal., Real World Appl. 17, 126–136 (2014)
Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, 1977. North-Holland Math. Stud., vol. 30, pp. 284–346. North-Holland, Amsterdam (1978)
Mao, A.M., Luan, S.X.: Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems. J. Math. Anal. Appl. 383(1), 239–243 (2011)
Miranda, C.: Un’osservazione su un teorema di Brouwer. Boll. Unione Mat. Ital. 3(2), 5–7 (1940)
Shao, M.Q., Mao, A.M.: Signed and sign-changing solutions of Kirchhoff type problems. J. Fixed Point Theory Appl. 20(1), 2 (2018)
Shuai, W.: Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains. J. Differ. Equ. 259(4), 1256–1274 (2015)
Sun, J.T., Tang, C.L.: Existence and multiplicity of solutions for Kirchhoff type equations. Nonlinear Anal. 74, 1212–1222 (2011)
Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)
Wu, X.: Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in \(\mathbb{R}^{3}\). Nonlinear Anal., Real World Appl. 12, 1278–1287 (2011)
Xie, Q.L., Wu, X.P., Tang, C.L.: Existence and multiplicity of solutions for Kirchhoff type problems with critical exponent. Commun. Pure Appl. Anal. 12(6), 2773–2786 (2013)
Ye, H.Y.: The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations in \(\mathbb{R}^{N}\). J. Math. Anal. Appl. 431(2), 935–954 (2015)
Zhang, Z.T., Perera, K.: Sign-changing solutions of Kirchhoff-type problems via invariant sets of descent flow. J. Math. Anal. Appl. 317(2), 456–463 (2006)
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The authors would like to thank the referees for their useful suggestions which have significantly improved the paper.
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This work was supported financially by the National Natural Science Foundation of China (11871302).
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Gao, Y., Liu, L., Luan, S. et al. A Kirchhoff-type problem involving concave-convex nonlinearities. Adv Differ Equ 2021, 171 (2021). https://doi.org/10.1186/s13662-021-03331-x
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DOI: https://doi.org/10.1186/s13662-021-03331-x
MSC
- 35J20
- 35J65
- 35A15
- 35J60
Keywords
- Concave-convex nonlinearities
- Kirchhoff-type problem
- Nehari manifolds
- Ground state sign-changing solutions