Multiple solutions for the fourth-order Kirchhoff type problems in $\mathbb{R}^N$ involving concave-convex nonlinearities

1.   Introduction and main results

In this paper we study the following fourth-order Kirchhoff type problem:

where $u\in H^2(\mathbb{R}^N) (4 < N < 8)$, $1 < q < 2, 4 < p < 2_\ast(2_\ast = 2N/(N-4))$. The parameter $\lambda > 0$, $a$ and $b$ are positive constants, the potential function $V$ and the weight function $f$ satisfy the following conditions:

$(V)\quad V(x)\in C(\mathbb{R}^N, \mathbb{R}), V_0: = \inf_{ \mathbb{R}^N}V(x) > 0$ and there exists a constant $l_0 > 0$ such that

where $meas(\cdot)$ denotes the Lebesgue measure in $\mathbb{R}^N$.

$(F)\quad f\in C(\mathbb{R}^N)\cap L^{r_q}(\mathbb{R}^N)$, where $r_q = r/(r-q)$ for some $r\in(2, 2_\ast)$.

The general form of (1.1) can be written as

where $u\in H^2(\mathbb{R}^N)$, $a$ and $b$ are positive constants, $V(x): \mathbb{R}^N\rightarrow \mathbb{R}$ is a continuous potential. Different forms of the nonlinearity $g(x, u)$ will lead to different difficulties, such as the existence of critical sequence in sublinear case, the boundedness of critical sequence in superlinear case or the compactness in critical case. The nonlinearity $g(x, u)$ is superlinear, sublinear and critical growth, which has been widely studied by many scholars, see ^[1,2,3] and their references therein.

Let $V(x) = 0$, replace $\mathbb{R}^N$ by a smooth bounded domain $\Omega\subset \mathbb{R}^N$ and set $u = \nabla u = 0$ on $\partial\Omega$, the problem (1.2) is reduced to the following fourth-order Kirchhoff type problem:

which is related to the following stationary analogue of the Kirchhoff type problem:

where $\Delta^2$ is the biharmonic operator. In low dimensions, (1.4) is often used to describe the phenomenon of nonlinear vibration of beam or plate in physics and engineering (see ^[4,5]). Because of the existence of integral term $\int_{\Omega}|\nabla u|^2dx$, this kind of problem is nonlocal, which indicates that Eq (1.4) is no longer pointwise identity. This phenomenon has caused some difficulties for mathematical research, so it has attracted the attention of a large number of scholars.

Here we focus on $g(x, u)$ with concave-convex nonlinearities. Semilinear elliptic equations with concave-convex nonlinearities in bounded domains are extensively researched. Ambrosetti et al. ^[6], for example, considered the following equation:

where $\lambda > 0$, $1 < q < 2 < p < 2^\ast = 2N/(N-2)$. They proved that there exists $\lambda_0 > 0$ such that (1.5) admits at least two positive solutions for all $\lambda\in(0, \lambda_0)$ and has one positive solution for $\lambda = \lambda_0$ and no positive solution for $\lambda > \lambda_0$. Actually, many scholars have also obtained this result in the unit ball $B^N(0;1)$, see ^[7,8,9]. In addition, Chen, Kuo and Wu ^[10] investigated the following Kirchhoff type problem:

where $a, \ b > 0$, $1 < q < 2 < p < 2^\ast$ and $f, \ g\in C(\bar\Omega)$ are sign-changing weight functions. Using the Nehari manifold and fibering maps, the authors examined the existence of multiple positive solutions for three cases: $p > 4$, $p = 4$ and $p < 4$ when $b$ and $\lambda$ belong to specific intervals. For more results of problems involving concave-convex nonlinearities and sign-changing weights in bounded domain, the reader may see ^[11,12] and the references therein. Furthermore, this kind of question in $\mathbb{R}^N$ also arouses the scholar's interest. Wu ^[13] has researched the following equation involving sign-changing weight functions:

where $u\in H^1(\mathbb{R}^N)$, $1 < q < 2 < p < 2^\ast$, the parameters $\lambda, \mu > 0$. He assumed that $a_\lambda(x) = \lambda a_+(x)+a_-(x)$ is sign-changing and $b_\mu(x) = c(x)+\mu d(x)$, where $c(x)$ and $d(x)$ satisfy appropriate hypotheses, and obtained the multiplicity of positive solutions for the problem (1.7).

Inspired by the above work, the main aim of this paper is to study the Kirchhoff problem (1.1) in $\mathbb{R}^N$ involving conave-convex nonlinearities and sign-changing weight function. In addition, from the condition $(F)$, we can see that $f$ is allowed to be sign-changing. As far as we know, there are few articles to deal with the fourth-order Kirchhoff type problem (1.1). We are going to discuss the Nehari manifold and thoroughly check the relation between the Nehari manifold and the fibering maps; then using methods similar to those used in ^[14], we will prove the existence of two solutions by using Ekeland variational principle ^[15].

Set

and $0 < \lambda_2 = \frac{q}{p-2}\lambda_1 < \lambda_1$, where $|f|_{r_q} = \left(\int_{ \mathbb{R}^N}|f|^{r_q}dx\right)^{1/{r_q}}$ and $S_p$ is described below. Now, we state the main result about the multiciplity of solution of (1.1) in $\mathbb{R}^N$.

Theorem 1.1. Assume that $(V)$ and $(F)$ hold. If $\lambda\in(0, \lambda_1)$, then $(1.1)$ admits at least two nontrivial solutions, one of which has negative energy. Furthermore, if $\lambda\in(0, \lambda_2)$, then $(1.1)$ has at least one negative energy ground state solution and one positive energy solution.

Theorem 1.2. Assume $(V)$ holds. Then $(1.1)$ has a sign-changing solution for $f(x)\equiv0$.

In Eq (1.1), the unboundedness of the whole space $\mathbb{R}^N$ leads to no compactness, therefore we consider condition $(V)$ to recover the compactness. The condition $(V)$ was first mentioned by Bartsch and Wang in ^[11]. At the same time, we also have $(V_1)$ and $(V_2)$ conditions in the following remark to repair compactness. Therefore, using $(V_1)$ or $(V_2)$ instead of $(V)$ can also get the same result. But the following two conditions are stronger than $(V)$.

Remark 1.2. These conditions are usually used to restore compactness.

$(V_1)\quad V(x)\in C(\mathbb{R}^N, \mathbb{R}), V_0: = \inf_{ \mathbb{R}^N}V(x) > 0, V(x)\rightarrow +\infty$ as $|x|\rightarrow +\infty$ (see ^[16]).

$(V_2)\quad V(x)\in C(\mathbb{R}^N, \mathbb{R}), V_0: = \inf_{ \mathbb{R}^N}V(x) > 0$, for each $M > 0$, $meas(\{x\in \mathbb{R}^N|V(x)\leq M\}) < \infty$ (see ^[17]).

This paper is organized as follows. In Section 2, some notations and preliminaries are given, including lemmas that are required in proving the main theorem. In Section 3, we are concerned with the proof of Theorem 1.1. In Section 4, we are concerned with the proof of Theorem 1.2.

2.   Preliminaries

Define our working space

with the inner product and norm

where $H^2(\mathbb{R}^N)$ is the well known Sololev space.

Throughout this paper, under assumption $(V)$, the embedding $E\hookrightarrow L^r(\mathbb{R}^N)$ is continuous for $r\in[2, 2_\ast]$ and compact for $r\in[2, 2_\ast)$ ^[18]. We denote by $S_r$ the best Sobolev constant for the embedding of $E$ in $L^r(\mathbb{R}^N)$ with $r\in[2, 2_\ast)$. In particular,

where $L^r(\mathbb{R}^N)$ is the usual Lebesgue space endowed with the standard norm $|u|_r = \left(\int_{ \mathbb{R}^N}|u|^rdx\right)^{1/r}$ for $1\leq r < \infty$.

The energy functional $I_\lambda$ we consider that corresponds to (1.1) is given by, for each $u\in E$,

It is well known that the functional $I_\lambda$ is of class $C^1$ in $E$ and the solutions of (1.1) are the critical points of energy functional $I_\lambda$ ^[19] and thus, by taking $(V)(F)$ and using a direct computation, we have

for any $u, v\in E$, and where $\langle\cdot, \cdot\rangle$ denotes the usual scalar product in $H^2(\mathbb{R}^N)$. Moreover, it is clear that $lim_{t\rightarrow \infty}I_\lambda(tu) = -\infty$ and so $I_\lambda$ is not bounded below on $E$. In order to obtain critical points, we consider the $I_\lambda$ on the Nehari manifold

Thus, $u\in \mathcal{N}_\lambda$ if and only if

Moreover, $\mathcal{N}_\lambda$ comprises all nontrivial solutions of problem (1.1). And we have the following lemma.

Lemma 2.1. The energy functional $I_\lambda$ is coercive and bounded below on $\mathcal{N}_\lambda$.

Proof. For $u\in \mathcal{N}_\lambda$, by the Hölder and Sobolev inequalities,

This ends the proof due to $1 < q < 2$.

Distinctly, $\mathcal{N}_\lambda$ is a much smaller set than $E$ and so it is simpler to discuss $I_\lambda$ on $\mathcal{N}_\lambda$. The Nehari manifold $\mathcal{N}_\lambda$ is closely related to the character of functions of the form $\varphi_{\lambda, u}:t\rightarrow I_\lambda(tu)$ for $t > 0$. Such functions are known as fibering maps, which were studied by Brown and Wu in ^[20]. If $u\in E$, we have

Evidently,

and so, for $u\in E\backslash\{0\}$ and $t > 0$, $\varphi'_{\lambda, u}(t) = 0$ if and only if $tu\in \mathcal{N}_\lambda$, that is, the critical points of $\varphi_{\lambda, u}$ correspond to points on the Nehari manifold, In particular, $u\in \mathcal{N}_\lambda$ if and only if $\varphi'_{\lambda, u} = 0$. Thus, it is inartificial to split $\mathcal{N}_\lambda$ into three parts ^[14]:

It is easy to see that

Thus, for each $u\in \mathcal{N}_\lambda$, we have

and

We now derive some basic properties of $\mathcal{N}_\lambda^+$, $\mathcal{N}_\lambda^0$ and $\mathcal{N}_\lambda^-$.

Lemma 2.2. Assume that $u$ is a local minimizer for $I_\lambda$ on $\mathcal{N}_\lambda$ and $u\notin \mathcal{N}_\lambda^0$. Then $I'_\lambda(u) = 0$.

Proof. The details of the proof can be referred to Brown and Zhang ^[12].

Lemma 2.3. If $\lambda\in(0, \lambda_1)$, then $\mathcal{N}_\lambda^0 = \emptyset$.

Proof. Suppose the contrary. There exist $u\in \mathcal{N}_\lambda$ such that $\varphi''_{\lambda, u}(1) = 0$. From (2.5) and the Sobolev inequality, we have

and so

Similarly, using (2.4) and Hölder and Sobolev inequalities, we have

which implies that

Combining (2.6) and (2.7) we deduce that

which is a contradiction.This completes the proof.

Lemma 2.4. If $\lambda\in(0, \lambda_1)$, then the set $\mathcal{N}_\lambda^-$ is closed in $E$.

Proof. Let $\{u_n\}\subset \mathcal{N}_\lambda^-$ such that $u_n\rightarrow u$ in $E$. In the following we show $u\in \mathcal{N}_\lambda^-$. In fact, by $\langle I'_\lambda(u_n), u_n\rangle = 0$ and

we have $\langle I'_\lambda(u), u\rangle = 0$. So $u\in \mathcal{N}_\lambda$. For any $u\in \mathcal{N}_\lambda^-$, from (2.5) we have

Then by Sobolev inequality, we have

that is,

Hence $\mathcal{N}_\lambda^-$ is bounded away from $0$. Obviously, by (2.4), it follows that $\varphi''_{\lambda, u_n}(1)\rightarrow \varphi''_{\lambda, u}(1)$ as $n\rightarrow +\infty$. From $\varphi''_{\lambda, u_n}(1) < 0$, we have $\varphi''_{\lambda, u}(1)\leq0$. By Lemma 2.3, for $\lambda\in(0, \lambda_1)$, $\mathcal{N}_\lambda^0 = \emptyset$, then $\varphi''_{\lambda, u}(1) < 0$. Thus we deduce $u\in \mathcal{N}_\lambda^-$. This completes the proof.

In order to obtain a better comprehension of the Nehari manifold and fibering maps, we consider the function $\psi_b: \mathbb{R}^+\rightarrow \mathbb{R}$ defined by

Clearly $tu\in \mathcal{N}_\lambda$ if and only if $\psi_b(t) = \lambda\int_{ \mathbb{R}^N}f(x)|u|^qdx$. Moreover,

and so it is easy to see that, if $tu\in \mathcal{N}_\lambda$, then $t^{q-1}\psi'_b(t) = \varphi''_{\lambda, u}(t)$. Hence, $tu\in \mathcal{N}_\lambda^+$ (or $tu\in \mathcal{N}_\lambda^-$) if and only if $\psi'_b(t) > 0$ (or $\psi'_b(t) < 0$). Furthermore, from $1 < q < 2$, $4 < p < 2_\ast$, $\psi'_b(t) = 0$ and $\psi_b(0) = 0$, we can deduce that there is a unique $t_{b, max} > 0$ such that $\psi_b(t)$ achieves its maximum at $t_{b, max}$, increasing for $t\in[0, t_{b, max})$ and decreasing for $t\in(t_{b, max}, +\infty)$ with $\lim\limits_{t\rightarrow +\infty}\psi_b(t) = -\infty$.

The next lemma allows us to assume that $\mathcal{N}_\lambda^+$ and $\mathcal{N}_\lambda^-$ are nonempty under the hypothesis.

Lemma 2.5. Suppose that $\lambda\in(0, \lambda_1)$, $u\in E\backslash\{0\}$. Then

${\rm(i)}$ if $\lambda\int_{ \mathbb{R}^N}f(x)|u|^qdx\leq0$, then there is a unique $t^- > t_{b, max}$ such that $t^-u\in \mathcal{N}_\lambda^-$ and

${\rm(ii)}$ if $\lambda\int_{ \mathbb{R}^N}f(x)|u|^qdx > 0$, then there are unique $t^+$ and $t^-$ with $0 < t^+ < t_{b, max} < t^-$ such that $t^+u\in \mathcal{N}_\lambda^+$, $t^-u\in \mathcal{N}_\lambda^-$ and

Proof. (i) if $\lambda\int_{ \mathbb{R}^N}f(x)|u|^qdx\leq0$, noting that $\psi_b(t)$ achieves its maximum at $t_{b, max}$, increasing for $t\in[0, t_{b, max})$ and decreasing for $t\in(t_{b, max}, +\infty)$ with $\lim\limits_{t\rightarrow +\infty}\psi_b(t) = -\infty$, then there is a unique $t^- > t_{b, max}$ such that $\psi_b(t^-) = \lambda\int_{ \mathbb{R}^N}f(x)|u|^qdx$, that is $t^-u\in \mathcal{N}_\lambda$. Moreover by $\psi'_b(t) < 0$, we obtain that $t^-u\in \mathcal{N}_\lambda^-$. And by

we have $I_\lambda(t^-u) = \sup_{t\geq0}I_\lambda(tu)$.

(ii) Since $b > 0$, $t > 0$, we have

where $\psi_0(t) = \psi_b(t)|_{b = 0}$. Clearly, $\psi_0(t)$ has a unique critical point at $t_{0, max} = t_{0, max}(u)$, where

Moreover, by Sobolev inequality, we obtain

Thus, $\psi_b(t_{b, max}) > \psi_0(t_{0, max}) > 0$.

From $\lambda\in(0, \lambda_1)$, (2.8), Hölder and Sobolev inequalities we also have

If $\lambda\int_{ \mathbb{R}^N}f(x)|u|^qdx > 0$. Since (2.9), the equation $\psi_b(t) = \lambda\int_{R^N}f(x)|u|^qdx$ has exactly two solutions $0 < t^+ < t_{b, max} < t^-$ such that

and

Thus, there exist exactly two multiples of $u$ lying in $\mathcal{N}_\lambda$, that is, $t^+u\in \mathcal{N}_\lambda^+$ and $t^-u\in \mathcal{N}_\lambda^-$. Finally, by analyzing $\frac{dI_\lambda(tu)}{dt} = t^{q-1}\left(\psi_b(t)-\lambda\int_{ \mathbb{R}^N}f(x)|u|^qdx\right)$, $I_\lambda(tu)$ is decreasing for $t\in(0, t^+)$ and increasing for $t\in(t^+, t_{b, max})$. Moreover, $I_\lambda(tu)$ is increasing for $t\in(t_{b, max}, t^-)$ and decreasing for $t\in(t_{b, max}, +\infty)$. therefore,

3.   Proof of Theorem 1.1

First, we remark that it follows from Lemma 2.3 that

for all $\lambda\in(0, \lambda_1)$. Furthermore, by Lemma 2.5 it follows that $\mathcal{N}_\lambda^+$ and $\mathcal{N}_\lambda^-$ are nonempty, and by Lemma 2.1 we may define

Then we get the following result.

Lemma 3.1. One has the following.

${\rm(i)}$ If $\lambda\in(0, \lambda_1)$, then one has $\alpha_\lambda^+ < 0$.

${\rm(ii)}$ If $\lambda\in(0, \lambda_2)$, then one has $\alpha_\lambda^- > d_0$ for some $d_0 > 0$.

In particular, for each $\lambda\in(0, \lambda_2)$, one has $\alpha_\lambda^+ = \alpha_\lambda$.

Proof. (i) Let $u\in \mathcal{N}_\lambda^+$. By (2.4)

and so

Therefore, $\alpha_\lambda^+ < 0$.

(ii) Let $u\in \mathcal{N}_\lambda^-$. By Lemma 2.4, we have

Furthermore, by Hölder and Sobolev inequalities, we have

Thus, if $\lambda\in(0, \lambda_2)$, then

for some positive constant $d_0$. This completes the proof.

From Lemma 2.1 we can obtain the minimizing sequence of the $I_\lambda(u)$ on the Nehari manifold $\mathcal{N}_\lambda$. To gain a $(PS)_c$ sequence from the minimizing sequence of the $I_\lambda(u)$ on Nehari manifold $\mathcal{N}_\lambda$, we require the following three lemmas:

Lemma 3.2. If $\lambda\in(0, \lambda_1)$, then for every $u\in \mathcal{N}_\lambda^+$, there exist $\epsilon > 0$ and a differentiable function $g^+:B_{\epsilon}(0)\subset E\rightarrow \mathbb{R}^+: = (0, +\infty)$ such that

and

for all $v\in E$. Moreover, if $0 < C_1\leq||u||\leq C_2$, then there exists $C > 0$ such that

Proof. We define $F: \mathbb{R}\times E\rightarrow \mathbb{R}$ by

it is easy to see $F$ is differentiable. Since $F(1, 0) = \langle I'_\lambda(u), u\rangle = 0$ and $F_t(1, 0) = \varphi''_{\lambda, u}(1) > 0$, we apply the implicit function theorem at point $(1, 0)$ to get the existence of $\epsilon > 0$ and differentiable function $g^+:B_{\epsilon}(0)\rightarrow \mathbb{R}^+$ such that $g^+(0) = 1$ and $F(g^+(\omega), \omega) = 0$ for $\forall\omega\in B_\epsilon(0)$. Thus,

Next, we show $g^+(\omega)(u-\omega)\in \mathcal{N}_\lambda^+$, $\forall\omega\in B_\epsilon(0)$. By $u\in \mathcal{N}_\lambda^+$ and (2.3), we have

Since $g^+(\omega)(u-\omega)$ is continuous with respect to $\omega$, when $\epsilon$ is small enough, we know for $\omega\in B_\epsilon(0)$

Thus, $g^+(\omega)(u-\omega)\in \mathcal{N}_\lambda^+$, $\forall\omega\in B_\epsilon(0)$.

Also by the differentiability of the implicit function theorem, we know that

Note that

and $F_t(1, 0) = \varphi''_{\lambda, u}(1)$. So we prove (3.1).

Moreover, by (3.1), $0 < C_1\leq||u||\leq C_2$ and Hölder inequality, we have

for some $\widetilde C > 0$. Therefore, in order to prove (3.2), we only need to show that $|\varphi''_{\lambda, u}(1)| > d$ for some $d > 0$. We argue by contradiction. Assume that there exists a sequence $\{u_n\}\in \mathcal{N}_\lambda^+$, $C_1\leq||u_n||\leq C_2$, we have $\varphi''_{\lambda, u_n}(1) = o_n(1)$, where $o_n(1)\rightarrow0$ as $n\rightarrow +\infty$. Then for $C_1\leq||u_n||\leq C_2$ by (2.5) and Sobolev inequality, we have

and so

Similarly, using (2.4), Hölder and Sobolev inequalities, we have

which implies

Combining (3.3) and (3.4) as $n\rightarrow +\infty$, we deduce

which is a contradiction. Thus if $0 < C_1\leq||u||\leq C_2$, there exists $C > 0$ such that

This completes the proof.

Analogously, we establish the following lemma.

Lemma 3.3. If $\lambda\in(0, \lambda_1)$, then for every $u\in \mathcal{N}_\lambda^-$, there exist $\epsilon > 0$ and a differentiable function $g^-:B_{\epsilon}(0)\subset E\rightarrow \mathbb{R}^+$ such that

and

for all $v\in E$. Moreover, if $0 < C_1\leq||u||\leq C_2$, then there exists $C > 0$ such that

Lemma 3.4. If $\lambda\in(0, \lambda_1)$, one has the following:

${\rm(i)}$ there exists a minimizing sequence $\{u_n\}\subset \mathcal{N}_\lambda^+$ such that

${\rm(ii)}$ there exists a minimizing sequence $\{u_n\}\subset \mathcal{N}_\lambda^-$ such that

Proof. (i) By Lemma 2.1 and the Ekeland variational principle on $\mathcal{N}_\lambda^+$, there exists a minimizing sequence $\{u_n\}\subset \mathcal{N}_\lambda^+$ such that

and

And we can show that there exists $C_1, C_2 > 0$ such that $0 < C_1\leq||u_n||\leq C_2$. Indeed, if not, that is, $u_n\rightarrow0$ in $E$, then $I_\lambda(u_n)$ would converge to zero, which contradict with $I_\lambda(u_n)\rightarrow \alpha_\lambda^+ < 0$. Moreover, by Lemma 2.1 we know that $I_\lambda$ is coercive on $\mathcal{N}_\lambda^+$, $\{u_n\}$ is bounded in $\mathcal{N}_\lambda^+$.

Now, we show that

Applying Lemma 3.2 with $u_n$ to obtain the functions $g_n^+(\omega):B_{\epsilon_n}(0)\rightarrow \mathbb{R}^+$ for some $\epsilon_n > 0$, such that

We choose $0 < \rho < \epsilon_n$. Let $u\in E\backslash\{0\}$ and $\omega_\rho = \rho u/||u||$. Since $g_n^+(\omega_\rho)(u_n-\omega_\rho)\in \mathcal{N}_\lambda^+$, we deduce from (3.8) that

Note that

If we divide the ends of (3.9) by $\rho$ and let $\rho\rightarrow0^+$, we have

that is,

By the boundedness of $||u_n||$ and Lemma 3.2, there exists $\hat{C} > 0$ such that

Hence we have

that is, $I'_\lambda(u_n) = o(1)$ as $n\rightarrow +\infty$. This completes the proof of (i).

(ii) Similarly, by using Lemma 3.3, we can prove (ii). We will omit detailed proof here.

Now, we establish the existence of a minimum for $I_\lambda$ on $\mathcal{N}_\lambda^+$.

Theorem 3.5. If $\lambda\in(0, \lambda_1)$, the functional $I_\lambda$ has a minimizer $u_0^+$ in $\mathcal{N}_\lambda^+$ and it satisfies $I_\lambda(u_0^+) = \alpha_\lambda^+$.

Proof. By Lemma 3.4, there exist a minimizing sequence $\{u_n\}\subset \mathcal{N}_\lambda^+$ such that

Then by Lemma 2.1 and the compact embedding theorem, there exist a subsequence $\{u_n\}$ and $u_0^+\in E$ such that

Next we prove $u_n\rightarrow u_0^+$ in $E$. Note that

then we can deduce that $||u_n-u_0^+||\rightarrow0$ as $n\rightarrow \infty$. Indeed, from the boundedness of $\{u_n\}$ in $E$ and the continuous embedding, $\{u_n\}$ is bounded in $L^r(\mathbb{R}^N)$, $r\in[2, 2_\ast]$. Using Hölder inequality we see that

where $C$ is a positive constant. Similarly, we obtain

From

and

we have $||u_n-u_0^+||\rightarrow0$ as $n\rightarrow \infty$.

In addition, from the proof of Lemma 3.4 we know that there exists $C_1, C_2 > 0$ such that $0 < C_1\leq||u_n||\leq C_2$, then $0 < C_1\leq||u_0^+||\leq C_2$. Thus $u_0^+\neq0$.

Next we prove $u_0^+\in \mathcal{N}_\lambda^+$. In fact, it follows from (2.4) that

From $\varphi''_{\lambda, u_n}(1) > 0$, we have $\varphi''_{\lambda, u_0^+}(1)\geq0$. By Lemma 2.3, we know $\varphi''_{\lambda, u_0^+}(1) > 0$. Thus we deduce

This completes the proof. Next, we establish the existence of a minimum for $I_\lambda$ on $\mathcal{N}_\lambda^-$.

Theorem 3.6. If $\lambda\in(0, \lambda_1)$, the functional $I_\lambda$ has a minimizer $u_0^-$ in $\mathcal{N}_\lambda^-$ and it satisfies $I_\lambda(u_0^-) = \alpha_\lambda^-$.

Proof. By Lemma 3.4, there exist a minimizing sequence $\{u_n\}\subset \mathcal{N}_\lambda^-$ such that

Then by Lemma 2.1 and the conpact embedding theorem, there exist a subsequence $\{u_n\}$ and $u_0^-\in E$ such that

In view of the proof of Lemma 3.4 we know that there exists $C_1, C_2 > 0$ such that $0 < C_1\leq||u_n||\leq C_2$, then $0 < C_1\leq||u_0^-||\leq C_2$. Thus $u_0^-\neq0$. Moreover, in the same way as Theorem 3.5, we still have $u_n\rightarrow u_0^-$ in $E$. By Lemma 2.4 the set $\mathcal{N}_\lambda^-$ is closed in $E$, we know $u_0^-\in \mathcal{N}_\lambda^-$. Thus,

This completes the proof. Now we can give the proof of the main result.

Proof of Theorem 1.1. From Theorems 3.5, 3.6 and Lemma 2.2, we know if $\lambda\in(0, \lambda_1)$, then Eq (1.1) has at least two solutions $u_0^-$, $u_0^+$ and $I_\lambda(u_0^+) < 0$. Since $u_0^+\in \mathcal{N}_\lambda^+$, $u_0^-\in \mathcal{N}_\lambda^-$ and $\mathcal{N}_\lambda^+\cap \mathcal{N}_\lambda^- = \emptyset$, this implies that $u_0^+$ and $u_0^-$ are different. In addition, if $\lambda\in(0, \lambda_2)$, by Lemma 3.1 we have $I_\lambda(u_0^+) < 0$ and $I_\lambda(u_0^-) > 0$, which implies $\alpha_\lambda = \alpha_\lambda^+ = I_\lambda(u_0^+)$. So $u_0^+$ is a ground state solution of Eq (1.1). It completes the proof of Theorem 1.1.

4.   Proof of Theorem 1.2

In this section, we denote $^+u = \max\{u(x), 0\}$ and $^-u = \min\{u(x), 0\}$, then $u = ^+u+^-u$. Define working space

where $H = \{u\in H^1(\mathbb{R}^N)|\int_{ \mathbb{R}^N}V(x)u^2dx < +\infty\}$. Moreover, the functional $I:\bar{E}\to \mathbb{R}$ by

In order to obtain a sign-changing solution of $(1.1)$, we consider the minimization of the following manifold

Define $\alpha = \inf_{u\in^\pm \mathcal{N}}I(u)$. Similar to Lemma $2.2$, if there exists $u\in^\pm \mathcal{N}$ such that $I(u) = \alpha$, then $u$ is a solution of $(1.1)$.

Proof of Theorem 1.2. Without loss of generality, we can assume $b = 1$. Let $\{u_n\}\subset^\pm \mathcal{N}$ be a minimizing sequence of $\alpha$. Going if necessary to a subsequence, one has

that is, $\{u_n\}$ is a bounded sequence of $\bar{E}$. Then by the compact embedding theorem, there exist a subsequence $\{u_n\}$ and $u\in \bar{E}$ such that

as $n\to\infty$. We assert that there exists $C > 0$ such that $|^\pm u_n|_p\geq C$, which implies that $^\pm u\neq0$. In fact, for any $u\in^\pm \mathcal{N}$, there exists $C > 0$ such that $|u|_p\geq C$. Suppose to the contrary that there exists a sequence $\{u_n\}\subset^\pm \mathcal{N}$ such that $|u_n|_p\to 0$ as $n\to\infty$. From $\langle I'(u_n), ^+u_n\rangle = 0$, there holds

Therefore, there exists $C > 0$ such that $||^+u_n||\geq C$. Moreover, it follows from $|^+u_n|_p\leq|u_n|_p\to 0$ and $\langle I'(u_n), ^+u_n\rangle = 0$ that $||^+u_n||\to 0$ as $n\to\infty$, which contradicts $||^+u_n||\geq C > 0$. Therefore, there exists $C > 0$ such that $|u|_p\geq C$ for any $u\in^\pm \mathcal{N}$. Similar to the discussion of Lemma $2.5$, there exists $0 < ^-t\leq ^+t$ such that $^-t^-u+^+t^+u\in^\pm \mathcal{N}$, which implies that

Since $^-t\leq ^+t$, there holds

Moreover, it follows from $\{u_n\}\subset^\pm \mathcal{N}$ that

and by the weakly lower semicontinuity of norm, one has

It follows from $(4.1)$ and $(4.2)$ that

which implies $^+t\leq1$. Therefore, $0 < ^-t\leq ^+t\leq1$. By $^-t^-u+^+t^+u\in^\pm \mathcal{N}$, one has

which implies that $^+t = ^-t = 1$, $u = ^+u+^-u\in^\pm \mathcal{N}$ and $I(u) = I(^+u+^-u) = \alpha$. Then, we conclude that $u = ^+u+^-u$ is a sign-changing solution of $(1.1)$. It completes the proof of Theorem 1.2.

Acknowledgments

The authors thank the anonymous referees for their valuable suggestions and comments.

Conflict of interest

The authors declare there is no conflicts of interest.