Abstract
In this paper, we study the following nonlinear Kirchhoff type equation:
where a, b, V are positive constants, \(N=2\) or 3. Under appropriate assumptions on f and h, we get that the equation has two positive solutions by using variational methods.
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1 Introduction and main results
We consider the following nonlinear Kirchhoff type equation:
where a, b, V are positive constants, \(N=2\) or 3.
In recent years, the existence or multiplicity of solutions for the following Kirchhoff type equation
where a, b are positive constants, \(N=1,2,3\), has been widely investigated by many authors, for example [1–6], etc. But in those papers, the nonlinearity f satisfies 3-superlinear growth at infinity, which assures the boundedness of any Palais-Smale sequence or Cerami sequence.
Very recently, Guo [7], Li and Ye [8], Liu and Guo [9], Tang and Chen [10] studied respectively the following equation:
where \(a,b\) are positive constants, f only needs to satisfy superlinear growth at infinity. By using the Pohozaev equality, it is easy to obtain a bounded Palais-Smale sequence. Thus they obtained the existence of positive solution.
Inspired by [7–10], we study equation (1.1); in here, very weak conditions are assumed on f. Exactly, \(f\in C(\mathbb {R}^{+},\mathbb{R})\) satisfies
- \((f_{1})\) :
-
when \(N=2\), there exists \(p\in(2,+\infty)\) such that \(\lim_{t\rightarrow+\infty}\frac{f(t)}{t^{p-1}}=0\); when \(N=3\), \(\lim_{t\rightarrow+\infty}\frac{f(t)}{t^{5}}=0\);
- \((f_{2})\) :
-
\(\lim_{t\rightarrow0^{+}}\frac{f(t)}{t}=m\in(-\infty,V)\);
- \((f_{3})\) :
-
\(\lim_{t\rightarrow+\infty}\frac{f(t)}{t}=+\infty\).
On h, we make the following hypotheses:
- \((h_{1})\) :
-
\(h\in L^{2}(\mathbb{R}^{N})\cap C^{1}(\mathbb{R}^{N})\) is nonnegative and \(h\not\equiv0\);
- \((h_{2})\) :
-
when \(N=2\), \(0\leq(\nabla h(x),x)\in L^{2}(\mathbb{R}^{2})\); when \(N=3\), \((\nabla h(x),x)\in L^{2}(\mathbb{R}^{3})\);
- \((h_{3})\) :
-
h is radially symmetric.
By using Ekeland’s variational principle [11] and Struwe’s monotonicity trick [12], we get the following.
Theorem 1.1
Suppose that \((f_{1})\)-\((f_{3})\) and \((h_{1})\)-\((h_{3})\) hold. Then there exists \(m_{0}>0\) such that, when \((\int_{\mathbb{R}^{N}}h^{2}\,dx )^{\frac{1}{2}}< m_{0}\), equation (1.1) has two positive solutions.
When \(f(t)<0\), by \((f_{2})\) and \((f_{3})\), there exists \(l>0\) such that \(f(t)+lt\geq0\) for all \(t\geq0\). Thus equation (1.1) is equivalent to the following equation:
where \(W=V+l>0\) and \(k(t)=f(t)+lt\in C(\mathbb{R}^{+},\mathbb{R}^{+})\) satisfies
- \((k_{1})\) :
-
when \(N=2\), there exists \(p\in(2,+\infty)\) such that \(\lim_{t\rightarrow+\infty}\frac{k(t)}{t^{p-1}}=0\); when \(N=3\), \(\lim_{t\rightarrow+\infty}\frac{k(t)}{t^{5}}=0\);
- \((k_{2})\) :
-
\(\lim_{t\rightarrow0^{+}}\frac{k(t)}{t}=m+l:=d\in[0,W)\);
- \((k_{3})\) :
-
\(\lim_{t\rightarrow+\infty}\frac{k(t)}{t}=+\infty\).
Hence in order to prove Theorem 1.1, we only need to prove the following.
Theorem 1.2
Suppose that \((k_{1})\)-\((k_{3})\) and \((h_{1})\)-\((h_{3})\) hold. Then there exists \(m_{0}>0\) such that when \((\int_{\mathbb{R}^{N}}h^{2}\,dx )^{\frac {1}{2}}< m_{0}\), equation (1.2) has two positive solutions.
Remark 1.3
Under hypotheses on k, we are not able to obtain directly the boundedness of the Palais-Smale sequences. Inspired by Jeanjean’s idea in [13] and [14], we will use an indirect approach, i.e., Struwe’s monotonicity trick developed by Jeanjean. It is worth pointing out that comparing with \(N=3\), when \(N=2\), it is more complex to prove the boundedness of the Palais-Smale sequences, which will be seen in Lemma 3.8.
2 Preliminaries
From now on, we will use the following notations.
-
\(E:=\{u\in H^{1}(\mathbb{R}^{N}):u(x)=u( \vert x \vert )\}\) is the usual Sobolev space endowed with the norm
$$\begin{aligned} \Vert u \Vert = \biggl( \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}+u^{2}\,dx \biggr)^{\frac{1}{2}}. \end{aligned}$$ -
\(D^{1,2}(\mathbb{R}^{N})\) is completion of \(C_{0}^{\infty}(\mathbb {R}^{N})\) with respect to the norm
$$\begin{aligned} \Vert u \Vert _{D^{1,2}(\mathbb{R}^{N})}= \biggl( \int_{\mathbb {R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr)^{\frac{1}{2}}. \end{aligned}$$ -
For any \(1\leq p<\infty\), \(L^{p}(\mathbb{R}^{N})\) denotes the Lebesgue space and its norm is denoted by
$$\begin{aligned} \vert u \vert _{p}= \biggl( \int_{\mathbb{R}^{N}} \vert u \vert ^{p}\,dx \biggr)^{\frac{1}{p}}. \end{aligned}$$ -
\(\langle\cdot,\cdot\rangle\) denotes the action of dual, \((\cdot,\cdot)\) denotes the inner product in \(\mathbb{R}^{N}\).
-
C, \(C_{i}\) denote various positive constants.
Since we are looking for positive solution, we may assume that \(k(t)=0\) for all \(t<0\). Under the assumptions on k and h, it is obvious that the functional \(I:E\rightarrow\mathbb{R}\) defined by
is of class \(C^{1}\), where \(K(t)=\int_{0}^{t}k(s)\,ds\) and
for all \(u, v\in E\). As is well known, the weak solution of equation (1.2) is the critical point of I in E.
3 Proof of the main results
Next lemma can be viewed as a generalization of Struwe’s monotonicity trick [12] and is the main tool for obtaining a bounded Palais-Smale sequence.
Lemma 3.1
Let X be a Banach space equipped with a norm \(\Vert \cdot \Vert _{X}\), and let \(J\subset\mathbb{R}^{+}\) be an interval. We consider a family \(\{\Phi_{\mu}\}_{\mu\in J}\) of \(C^{1}\)-functionals on X of the form
where \(B(u)\geq0\) for all \(u\in X\) and such that either \(A(u)\rightarrow +\infty\) or \(B(u)\rightarrow+\infty\) as \(\Vert u \Vert _{X}\rightarrow+\infty\). We assume that there are two points \(v_{1}\), \(v_{2}\) in X such that
where
Then, for almost every \(\mu\in J\), there is a bounded \((PS)_{c_{\mu}}\) sequence for \(\Phi_{\mu}\), that is, there exists a sequence \(\{u_{n}\}\subset X\) such that
-
(1)
\(\{u_{n}\}\) is bounded in X,
-
(2)
\(\Phi_{\mu}(u_{n})\rightarrow c_{\mu}\),
-
(3)
\(\Phi'_{\mu}(u_{n})\rightarrow0\) in \(X^{*}\), where \(X^{*}\) is the dual of X.
Remark 3.2
In [13], it is also proved that, under the assumptions of Lemma 3.1, the map \(\mu\mapsto c_{\mu}\) is left-continuous.
In the paper, we set \(X:=E\), \(\Vert \cdot \Vert _{X}:= \Vert \cdot \Vert \) and \(J:=[\frac{1}{2},1]\). Let us define \(I_{\mu}:E\rightarrow\mathbb{R}\) by \(I_{\mu}(u)=A(u)-\mu B(u)\), where
Then \(I_{1}(u)=I(u)\). By \((k_{1})\)-\((k_{3})\) and \((h_{1})\), it is obvious that \(I_{\mu}\in C^{1}(E,\mathbb{R})\), \(B(u)\geq0\) for all \(u\in E\) and \(A(u)\geq\frac{\min\{a,W\}}{2} \Vert u \Vert ^{2}-C \vert h \vert _{2} \Vert u \Vert \rightarrow+\infty\) as \(\Vert u \Vert \rightarrow+\infty\).
Lemma 3.3
Assume that \((k_{1})\)-\((k_{3})\) and \((h_{1})\) hold. Then there exist \(\rho >0\), \(\alpha>0\) and \(m_{0}>0\) such that \(I_{\mu}(u)\vert_{\Vert u\Vert=\rho}\geq\alpha\) for all h satisfying \(\vert h\vert_{2}< m_{0}\) and for all \(\mu\in J\).
Proof
First, we consider \(N=2\). It follows from \((k_{1})\) and \((k_{2})\) that, for all \(t\in\mathbb{R}\), we have
By (3.1), the Hölder inequality and the Sobolev inequality, for all \(\mu\in J\) and \(u\in E\), one has
Let \(g_{1}(t)=\frac{\min\{2a,W-d\}}{4}t-C_{1}t^{p-1}\) for \(t\geq0\). Since \(p>2\), we know that there exists a constant \(\rho>0\) such that \(\max_{t\geq 0}g_{1}(t)=g_{1}(\rho)>0\). Choose \(m_{0}=\frac{1}{2C_{2}}g_{1}(\rho)\), then there exists \(\alpha>0\) such that \(I_{\mu}(u)\vert_{\Vert u\Vert=\rho}\geq\alpha\) for all h satisfying \(\vert h\vert_{2}< m_{0}\).
Next when \(N=3\), it follows from \((k_{1})\) and \((k_{2})\) that, for all \(t\in\mathbb{R}\), we have
By (3.2), the Hölder inequality and the Sobolev inequality, for all \(\mu\in J\) and \(u\in E\), one has
Let \(g_{2}(t)=\frac{\min\{2a,W-d\}}{4}t-C_{3}t^{5}\) for \(t\geq0\), we know that there exists a constant \(\rho>0\) such that \(\max_{t\geq0}g_{2}(t)=g_{2}(\rho)>0\). Choose \(m_{0}=\frac {1}{2C_{4}}g_{2}(\rho)\), then there exists \(\alpha>0\) such that \(I_{\mu}(u)\vert_{\Vert u\Vert =\rho}\geq\alpha\) for all h satisfying \(\vert h\vert_{2}< m_{0}\). □
Lemma 3.4
Assume that \((k_{1})\)-\((k_{3})\) and \((h_{1})\) hold. Then \(-\infty< c:=\inf\{ I(u): \Vert u \Vert \leq\rho\}<0\), where ρ is given by Lemma 3.3.
Proof
Since \(h\in L^{2}(\mathbb{R}^{N})\) and \(h\not\equiv0\), then for \(\varepsilon=\frac{ \vert h \vert _{2}}{2}\), there exists \(\phi \in C^{\infty}_{0}(\mathbb{R}^{N})\) such that \(\vert h-\phi \vert _{2}<\varepsilon\). Thus
and then
Hence
for \(t>0\) small enough. Then we get \(c=\inf\{I(u): \Vert u \Vert \leq\rho\}<0\). \(c>-\infty\) is obvious. □
In order to prove the compactness, we define \(g(t)=k(t)-dt\), \(\forall t\in\mathbb{R}\). Then, by \((k_{1})\) and \((k_{2})\), we get that
and when \(N=2\),
when \(N=3\),
Lemma 3.5
Suppose that \((k_{1})\)-\((k_{3})\), \((h_{1})\) and \((h_{3})\) hold. Assume that \(\{ u_{n}\}\subset E\) is a bounded Palais-Smale sequence of \(I_{\mu}\) for each \(\mu\in J\). Then \(\{u_{n}\}\) has a convergent subsequence in E.
Proof
Since \(\{u_{n}\}\) is bounded in E and \(E\hookrightarrow L^{s}(\mathbb{R}^{3})\), \(\forall s\in(2,6)\), \(E\hookrightarrow L^{s}(\mathbb {R}^{2})\), \(\forall s\in(2,+\infty)\) are compact (see [15]), up to a subsequence, we can assume that there exists \(u\in E\) such that \(u_{n}\rightharpoonup u\) in E, \(u_{n}\rightarrow u\) in \(L^{s}(\mathbb{R}^{3})\), \(\forall s\in(2,6)\), \(u_{n}\rightarrow u\) in \(L^{s}(\mathbb{R}^{2})\), \(\forall s\in(2,+\infty)\), \(u_{n}(x)\rightarrow u(x)\) a.e. in \(\mathbb{R}^{N}\).
By (3.3) and (3.4), for any \(\varepsilon>0\), we have
Then, by (3.6) and the Hölder inequality, one has
Similarly, we can obtain that
By (3.3) and (3.5), for any \(\varepsilon>0\), we have
Hence, by (3.7) and the Hölder inequality, one has
Similarly, we can obtain that
Hence when \(N=2\) or 3, one has
It is clear that
and
Note that
Therefore we get that \(\Vert u_{n}-u \Vert \rightarrow0\) as \(n\rightarrow\infty\). □
Proof of the first solution of Theorem 1.2
By Lemma 3.4 and Ekeland’s variational principle [11], there exists a sequence \(\{u_{n}\}\subset E\) such that \(\Vert u_{n} \Vert \leq\rho\), \(I(u_{n})\rightarrow c\) and \(I'(u_{n})\rightarrow 0\) as \(n\rightarrow\infty\). From Lemma 3.5 with \(\mu=1\), there exists \(u_{0}\in E\) such that \(u_{n}\rightarrow u_{0}\) in E and then \(I'(u_{0})=0\) and \(I(u_{0})=c<0\). Put \(u_{0}^{-}:=\max\{-u_{0},0\}\), one has
which implies that \(u_{0}^{-}=0\) and then \(u_{0}\geq0\). By the strong maximum principle, we get \(u_{0}>0\). □
For ρ and α in Lemma 3.3, we have following result.
Lemma 3.6
Assume that \((k_{1})\)-\((k_{3})\) and \((h_{1})\) hold. Then
- \((*)\) :
-
\(\exists v_{2}\in E\) with \(\Vert v_{2} \Vert >\rho\) such that \(I_{\mu}(v_{2})<0\), \(\forall\mu\in J\).
- \((**)\) :
-
\(c_{\mu}=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}I_{\mu}(\gamma (t))>\max\{I_{\mu}(0),I_{\mu}(v_{2})\}\), \(\forall\mu\in J\), where
$$\begin{aligned} \Gamma=\bigl\{ \gamma\in C\bigl([0,1],E\bigr):\gamma(0)=0,\gamma(1)=v_{2} \bigr\} . \end{aligned}$$
Proof
It follows from \((k_{3})\) that, for any \(L>0\), there exists \(C_{L}>0\) such that, for all \(t\geq0\), one has
Fix \(0\leq w\in C_{0}^{\infty}(\mathbb{R}^{N})\) with \(\operatorname {supp}w\subset B_{1}:=\{x\in\mathbb{R}^{N}: \vert x \vert <1\}\) and \(w\not\equiv0\). Define \(w_{t}(x)=tw(\frac{x}{t^{2}})\) for \(t>0\), then
By direct computation, we have
and, by (3.9),
Therefore
for all \(\mu\in J\). When \(N=2\), we choose \(L=2W\). When \(N=3\), we choose \(L=2W+b\frac{ (\int_{\mathbb{R}^{N}} \vert \nabla w \vert ^{2}\,dx )^{2}}{\int_{\mathbb{R}^{N}}w^{2}\,dx}\). Then \(I_{\mu}(w_{t})\rightarrow -\infty\) as \(t\rightarrow+\infty\). Hence there exists \(t'>0\) such that \(v_{2}:=w_{t'}\) with \(\Vert v_{2} \Vert >\rho\) and \(I_{\mu}(v_{2})<0\), \(\forall\mu\in J\). This completes the proof of \((*)\).
By Lemma 3.3 and the definition of \(c_{\mu}\), for all \(\mu\in J\), we have
Therefore, by \(I_{\mu}(0)=0\) and \(I_{\mu}(v_{2})<0\), we obtain the proof of \((**)\). □
So far we have verified all the conditions of Lemma 3.1. Then there exists \(\{\mu_{j}\}\subset J\) such that
-
(i)
\(\mu_{j}\rightarrow1^{-}\) as \(j\rightarrow\infty\), \(\{u_{n}^{j}\}\) is bounded in E;
-
(ii)
\(I_{\mu_{j}}(u_{n}^{j})\rightarrow c_{\mu_{j}}\) as \(n\rightarrow \infty\);
-
(iii)
\(I'_{\mu_{j}}(u_{n}^{j})\rightarrow0\) as \(n\rightarrow\infty\).
Using (i)-(iii) and Lemma 3.5, there exists \(u_{j}\in E\) such that \(u_{n}^{j}\rightarrow u_{j}\) in E as \(n\rightarrow\infty\) and then \(I_{\mu_{j}}(u_{j})=c_{\mu_{j}}\) and \(I'_{\mu_{j}}(u_{j})=0\). Hence, from \(I_{\mu_{j}}(u_{j})=c_{\mu_{j}}\) and \(\langle I'_{\mu _{j}}(u_{j}),u_{j}\rangle=0\), we get respectively
Next, for obtaining \(\{u_{j}\}\) is bounded in E, we need the following lemma (Pohozaev type identity). The proof is similar to Lemma 2.6 in [16], and we omit its proof in here.
Lemma 3.7
Suppose that \((h_{1})\) and \((h_{2})\) hold. If \(I'_{\mu}(u)=0\), we have
Since \(I'_{\mu_{j}}(u_{j})=0\), by Lemma 3.7, we get that
Lemma 3.8
Assume that \((k_{1})\)-\((k_{3})\) and \((h_{1})\)-\((h_{3})\) hold. Then \(\{u_{j}\}\) is bounded in E.
Proof
It follows from (3.10) and (3.12) that
Be similar to (3.8), by \(I'_{\mu_{j}}(u_{j})=0\), we obtain \(u_{j}\geq0\).
Firstly, we consider \(N=2\). From (3.13) and \(c_{\mu_{j}}\leq c_{\frac{1}{2}}\), we get
Since \((\nabla h(x),x)\geq0\), by (3.14) and \(u_{j}\geq0\), one has \(\{\int_{\mathbb{R}^{2}} \vert \nabla u_{j} \vert ^{2}\,dx\}\) is bounded. Next we prove \(\{\int_{\mathbb{R}^{2}}u_{j}^{2}\,dx\}\) is bounded. Inspired by [14], we suppose by contradiction that \(\lambda _{j}:= \vert u_{j} \vert _{2}\rightarrow+\infty\). Define \(w_{j}:=u_{j}(\lambda_{j}x)\), then
and
Hence \(\{w_{j}\}\) is bounded in E. Up to a subsequence, we may assume that \(w_{j}\rightharpoonup w\) in E, \(w_{j}\rightarrow w\) in \(L^{s}(\mathbb {R}^{2})\), \(\forall s\in(2,+\infty)\), \(w_{j}\rightarrow w\) in \(L_{\mathrm{loc}}^{s}(\mathbb{R}^{2})\), \(\forall s\in[1,+\infty)\), \(w_{j}(x)\rightarrow w(x)\) a.e. in \(\mathbb{R}^{2}\). By \(I'_{\mu_{j}}(u_{j})=0\), one has
For any \(v\in C_{0}^{\infty}(\mathbb{R}^{2})\), one has
and by the Lebesgue dominated convergence theorem, we have
Hence by (3.16)-(3.18), we have \((W-d)w=g(w)\) in \(\mathbb{R}^{2}\), from which we get that \(w=0\). Indeed, since 0 is an isolated solution of \((W-d)z=g(z)\), \(w=0\). Therefore by (3.6), (3.15) and (3.16), one has
which implies a contradiction. Hence \(\{\int_{\mathbb{R}^{2}} \vert u_{j} \vert ^{2}\,dx\}\) is bounded and then \(\{u_{j}\}\) is bounded in E.
Secondly, for \(N=3\), we have a simple proof. From (3.13), \((h_{2})\) and \(c_{\mu_{j}}\leq c_{\frac{1}{2}}\), we get
We prove directly \(\{\int_{\mathbb{R}^{3}}u_{j}^{2}\,dx\}\) is bounded. Similar to (3.19), we obtain
By the Hölder inequality, we have
By (3.3) and (3.5), for all \(t\in\mathbb{R}\), one has
From (3.11), (3.21), (3.22), \(\mu_{j}\leq 1\) and \(D^{1,2}(\mathbb{R}^{3})\hookrightarrow L^{6}(\mathbb{R}^{3})\), it follows that
which implies that \(\{\int_{\mathbb{R}^{3}}u_{j}^{2}\,dx\}\) is bounded. Combining with (3.19), we get that \(\{u_{j}\}\) is bounded in E. □
Proof of the second solution of Theorem 1.2
By \(I_{\mu_{j}}(u_{j})=c_{\mu_{j}}\), \(I'_{\mu_{j}}(u_{j})=0\), \(\mu_{j}\rightarrow 1^{-}\) and Remark 3.2, we get \(I(u_{j})\rightarrow c_{1}\) and \(I'(u_{j})\rightarrow0\) as \(n\rightarrow +\infty\). By Lemmas 3.5 and 3.8, there exists \(v_{0}\in E\) such that \(u_{j}\rightarrow v_{0}\) in E as \(n\rightarrow+\infty\) and then \(I(v_{0})=c_{1}>0\), \(I'(v_{0})=0\). Be similar to (3.8), we get \(v_{0}\geq0\). By the strong maximum principle, one has \(v_{0}>0\). □
4 Conclusions
The goal of this paper is to study the multiplicity of positive solutions for the following nonlinear Kirchhoff type equation:
where a, b, V are positive constants, \(N=2\) or 3. Under very weak conditions on f, we get that the equation has two positive solutions by using variational methods.
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Acknowledgements
The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript.
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This work was supported by the Natural Science Foundation of Education of Guizhou Province (No. KY[2016]103, KY[2016]281, KY[2017]297); the Science and Technology Foundation of Guizhou Province (No. LH[2015]7595, LH[2016]7054).
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She, Lb., Sun, X. & Duan, Y. Multiple positive solutions for a class of Kirchhoff type equations in \(\mathbb{R}^{N}\). Bound Value Probl 2018, 10 (2018). https://doi.org/10.1186/s13661-018-0928-8
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DOI: https://doi.org/10.1186/s13661-018-0928-8