Multiple positive solutions for a class of Kirchhoff type equations in \(\mathbb{R}^{N}\)

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.

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.

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.

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

(see [13] or [14])

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. (1)

   \(\{u_{n}\}\) is bounded in X,

2. (2)

   \(\Phi_{\mu}(u_{n})\rightarrow c_{\mu}\),

3. (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

1. (i)

   \(\mu_{j}\rightarrow1^{-}\) as \(j\rightarrow\infty\), \(\{u_{n}^{j}\}\) is bounded in E;

2. (ii)

   \(I_{\mu_{j}}(u_{n}^{j})\rightarrow c_{\mu_{j}}\) as \(n\rightarrow \infty\);

3. (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 \(v_{0}>0\). □

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.

Acknowledgements

The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript.

Availability of data and materials

Not applicable.

Funding

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).

Affiliations

1. School of Mathematics and Information Engineering, Liupanshui Normal College, LiuPanshui, Guizhou, 553004, People’s Republic of China

   • Lian-bing She

2. Collect of Science, GuiZhou University of Engineering Science, BiJie, Guizhou, 551700, People’s Republic of China

   • Xin Sun
   •  & Yu Duan

Contributions

Each of the authors contributed to each part of this study equally. All authors read and approved the final vision of the manuscript.

Corresponding author

Correspondence to Yu Duan.

Competing interests

The authors declare that they have no competing interests.

Abbreviations

Not applicable.

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