Existence and multiplicity of sign-changing solutions for supercritical quasi-linear Schrödinger equations

1.   Introduction and main results

1.1.   Background

In the past two decades, many attentions have been devoted to the investigation of the quasi-linear Schrödinger equation:

where $z:\mathbb{R}^N\times \mathbb{R}\to\mathbb{C}$, $W:\mathbb{R}^N\to \mathbb{R}$ is a given potential, $\gamma$ is a real constant and $l$ is real functions.

Equation (1.1) appears in various fields of physics (see ^[1,2]), and is known to be more accurate in many physical phenomena compared with the semi-linear Schrödinger equation.

The additional term $\gamma z\Delta(|z|^{2})$ appears in various physical models and arises due to:

a) the non-locality of the nonlinear interaction for electron ^[3],

b) the weak nonlocal limit for nonlocal nonlinear Kerr media ^[4],

c) the surface term for superfluid film ^[5].

In particular, the standing wave solution of Eq (1.1) is also a solution of the form $z(t, x): = \exp(-iEt)u(x)$ with $E > 0$, we are led to study the following elliptic equation

where $V(x) = W(x)-E$ and $f(t): = l(|t|^2)t$. The parameter $\gamma$ represents the strength of each effect and can be assumed to be positive or negative in different situations.

Notice that, for $\tau = 0$, the Eq (1.2) is a Schrödinger-type equation, which is fundamental in modern physics and many other fields, see e.g., ^{[6,7,8,9,10,11]}.

1.2.   Motivation

In the last decades, scholars have obtained existence and multiplicity of solutions for Eq (1.2) with $\gamma < 0$, based on variational methods. To the best of our knowledge, Poppenberg, Schmitt and Wang proved the existence of positive solutions for the first time in ^[12] by means of a constrained minimization argument. By using a change of variable and converting the quasi-linear Eq (1.2) into a semi-linear one in an Orlicz space framework, Liu et al. in ^[13] obtained existence of solutions for a general case. Subsequently, Colin and Jeanjean chose the classical Sobolev space $H^{1}(\mathbb{R}^N)$ in ^[14], then they can use a simpler and shorter proof than ^[13] to get the same conclusion. We refer the readers to ^{[15,16,17,18,19,20,21,22]} for more results.

For $\gamma > 0$, in ^[23], Alves, Wang and Shen used the change of known variables $s = H^{-1}(t)$ for $t\in[0, M]$, where

and $H^{-1}(t) = -H^{-1}(-t)$ for $t\in[-M, 0)$. Since $\gamma > 0$ small enough, Eq (1.3) is well-defined and the inverse function $H^{-1}(t)$ exists. They established the existence of weak solutions for Eq (1.2) based on variational methods, for $\gamma > 0$ small enough and $f(u) = |u|^{p-2}u$ ($p\in(2, \frac{2N}{N-2})$).

For the case $\gamma = 1$, notice that $1-t^{2}$ may be negative, for this possibility, the change of variables Eq (1.3) is no longer suitable for dealing with such problems. Recently, in ^[24] we considered the existence of a positive solution for Eq (1.2) with $\gamma = 1$ and $\lambda$ large enough:

where $N\ge 3$, $f(t)\in C(\mathbb{R})$ and superlinear in a neighborhood of $t = 0$. There is a more in-depth study of this idea in ^[25], where treated the case that $\lambda = 1$. However in ^[24,25], we were mainly interested in obtaining the existence and multiplicity of solutions for Eq (1.4), leaving nodal properties of solutions unconsidered.

Motivated by ^[23,24,25] mentioned above, in this paper, we focus on the existence and multiplicity of sign-changing solutions for Eq (1.4).

Compared with ^[24], the aim of this paper is two-fold. The first purpose is to investigate the existence of a sign-changing solution for Eq (1.4). The second aim is to obtain infinite sign-changing solutions for Eq (1.4) with symmetric condition.

1.3.   Our problem and main results

In this paper, we try to consider sign-changing solutions for the following one-parameter supercritical quasi-linear Schrödinger equations:

where $N\geq3$, $\lambda > 0$ and $V\in C(\mathbb{R}^{N}, \mathbb{R})$ satisfying:

$(V_0)$: $V(x)\geq V_{0} > 0$ for all $x\in \mathbb{R}^{N}$;

$(V_1)$: $V(x)\leq V_{\infty}: = \lim\limits_{|x|\to\infty}V(x) = +\infty$.

We assume that the nonlinearity satisfies the following conditions: $f(t)\in C(\mathbb{R})$;

$(f_1)$: there exists $\alpha\in(2, 2^{\ast})$ such that

$(f_2)$: there exists $\beta\in(2, 2^{\ast})$ with $\beta > \alpha$ such that

where $2^{\ast} = \frac{2N}{N-2}$ is the critical Sobolev exponent and $F(t) = \int_{0}^{t}f(s)ds$;

$(f_3)$: there exists $\theta\in (2, 2^{\ast})$ such that

$(f_4)$: $f(-t) = -f(t)$, for $|t|$ small.

Remark 1.1. An example of the nonlinearity satisfying $(f_1)-(f_3)$ can be taken as

with $2 < \alpha < \beta < 2^{\ast} < q$ and $C_1$, $C_2$ are positive constants. Notice that $q > 2^{\ast}$, hence our method in this paper can be used to deal with the supercritical problems.

Inspired by Costa, Wang ^[26] and Huang, Jia ^[24], we establish a sign-changing solution for the following quasi-linear Schrödinger equation

where $h(t) = \sqrt{1-2t^2}$, for $|t|\leq\sqrt{1/6}$ and $\widetilde{f}(t)$ is a modified nonlinearity such that Eq (1.5) possess variational framework. Next, we show Eq (1.5) has a sign-changing solution by using the methods of invariant sets. Then, a regularity argument shows an $L^\infty$-estimate for this sign-changing solution which depends on parameter $\lambda$. Finally, take $\lambda$ large enough such that the solution of Eq $(1.5)$ is the solution of the original Eq (1.4).

Our main results are as follows.

Theorem 1.1. Assume that $(V_0)$, $(V_1)$, $(f_1)-(f_{3})$ hold. Then Eq (1.4) possesses at least one sign-changing solution $u\in E$ for all sufficiently large $\lambda$.

Theorem 1.2. Assume that $(V_0)$, $(V_1)$, $(f_1)-(f_{4})$ hold. For any given $n\geq1$, then Eq (1.4) possesses at least $n-1$ pairs sign-changing solutions $u\in E$ for all sufficiently large $\lambda$.

From our results, we obtain the existence and multiplicity of sign-changing solutions for supercritical problems.

1.4.   Outline of this paper

The outline of this paper is as follows. In Section 2, we describe the modified equation associated with the Eq (1.4). We are devoted to the proofs of Theorems 1.1 and 1.2 in Section 3.

2.   The modified problem

When viewed from the perspective of variational, one of the difficulties in treating Eq (1.4) lies in without the behavior of nonlinearity at infinity. Hence, we first give the precise definition of the modified problem.

The conditions $(f_{1})$ and $(f_{2})$ imply that there exist positive constants $\delta\in(0, \frac{1}{2})$, $A$ and $B$ such that for $-2\delta\leq t\leq2\delta$,

For fixed $\delta > 0$, let $d(t)\in C^{1}(\mathbb{R}, \mathbb{R})$ be a cut-off function satisfying:

$|td'(t)|\leq\frac{2}{\delta}$ and $0\leq d(t)\leq1$ for $t\in \mathbb{R}$. Using the truncation argument introduced by Costa and Wang ^[26], we define

where

And $\widetilde{f}(t) = \widetilde{F}'(t)$. In what follows, we recall the properties of $\widetilde{f}(t)$:

Lemma 2.1. ^[26] If $(f_1)$-$(f_3)$ are satisfied, then we get

(1) $\widetilde{f}\in C(\mathbb{R}, \mathbb{R})$ and $\widetilde{f}(t) = o(1)$ as $t\to 0$;

(2) $\lim\limits_{t\to+\infty}\frac{\widetilde{f}(t)}{t} = +\infty$;

(3) there exists $C > 0$ such that $|\widetilde{f}(t)|\leq C|t|^{\alpha-1}$, for all $t\in\mathbb{R}$;

(4) for all $\delta\in(0, 1)$, there exists a constant $C_\delta > 0$ such that $|\widetilde{f}(t)|\leq\delta|t|+C_\delta|t|^{2^{\ast}-1}$, where $C_\delta = C\delta^{\frac{\alpha-2^{\ast}}{\alpha-2}}$;

(5) for all $t\neq0$, it implies $0 < \kappa \widetilde{F}(t)\leq t\widetilde{f}(t)$, where $\kappa = \min\{\alpha, \theta\}$.

The technique to prove our main results deeply relies on the work of ^[23,24,26]. It should be pointed out that we need to modify the equation as follows in order to adapt to the variational method:

where $h(t):\mathbb{R}\to \mathbb{R}$ is given by

Next, we define

Then, we will state the properties of the variable $H^{-1}(t)$ after it changes, which plays an important role in proving our main conclusions.

Lemma 2.2. ^[23] (1) $\lim\limits_{t\to0}\frac{H^{-1}(t)}{t} = 1$;

(2) $\lim\limits_{t\to+\infty}\frac{H^{-1}(t)}{t} = \sqrt{6}$;

(3) $t\leq H^{-1}(t)\leq\sqrt{6}t$, for all $t\geq0$, $\sqrt{6}t\leq H^{-1}(t)\leq t$, for all $t\leq0$;

(4) $-\frac{1}{2}\leq\frac{t}{h(t)}h'(t)\leq0$, for all $t\in\mathbb{R}$.

Direct calculations show that if $|u|_{\infty} < \min\{\delta, \sqrt{1/6}\}$, then $h(u) = \sqrt{1-2u^{2}}$ and $\widetilde{f}(u) = f(u)$. Therefore, our mission is to prove the existence of sign-changing solution $u$ for Eq (2.2) satisfying $|u|_{\infty} < \min\{\delta, \sqrt{1/6}\}$.

Note that Eq (2.2) is the Euler-Lagrange equation associated to the natural energy functional

Taking the change variable

we observe that the functional $\widetilde{I}_{\lambda}(u)$ can be written by the following way

From Lemmas 2.1 and 2.2, we can get that $J_{\lambda}(v)$ is well-defined in $E$, $J_{\lambda}\in C^{1}(E, \mathbb{R})$ and

where

with the norm $\|u\|_{E} = \left(\int_{\mathbb{R}^N}(|\nabla u|^{2}+V(x)u^2) \mbox{d}x\right)^{\frac{1}{2}}$.

Remark 2.1. From condition $(V_1)$, it implies that embedding $E\hookrightarrow L^{q}(\mathbb{R}^N)(2\leq q < 2^{\ast})$ is compact. This compact result was firstly introduced by Bartsch, Pankov and Wang ^[27].

Lemma 2.3. If $v\in E$ is a critical point of $J_{\lambda}$, then $u = H^{-1}(v)\in E$ and this $u$ is a weak solution for Eq (2.2).

Proof. Using the fact that $H^{-1}(v)\in C^{2}$ and Lemma 2.2, we can show that $u = H^{-1}(v)\in E$ through a direct computation. If $v$ is a critical point for $J_{\lambda}$, we have that

Taking $\varphi = h(u)\psi$, where $\psi\in C^{\infty}_{0}(\mathbb{R}^N)$, in the above equation to get

or

This ends the proof.

To find the sign-changing solutions of Eq (2.2), it is sufficient to discuss the existence of the sign-changing solutions of the following equation

3.   Proofs of Theorem 1.1 and Theorem 1.2

In this section, we shall use two abstract critical point theorems based on classical Mountain Pass theorem and Symmetric Mountain Pass theorem to prove the existence and multiplicity of sign-changing solutions for Eq $(1.4)$. The two abstract critical point theorems are developed by Liu, Liu and Wang in ^[28]. In order to prove Theorem 1.1, we make use of the following notations. Let $E$ be a Banach space, $I\in C^{1}(E, \mathbb{R})$, $P, Q\subset E$ be open sets, $M = P\cap Q$, $\Sigma = \partial P\cap\partial Q$ and $W = P\cup Q$. For $c\in \mathbb{R}$, $K_{c} = \{u\in E: I(u) = c, I'(u) = 0\}$ and $I^{c} = \{u\in E: I(u)\leq c\}$.

Definition 3.1. ^[28] Suppose we have the following deformation properties: if $K_{c}\setminus W = \emptyset$, there exists $\varepsilon_{0} > 0$ such that for $\varepsilon\in(0, \varepsilon_{0})$, there exists $\sigma\in C(E, E)$ satisfying

(1) $\sigma(\overline{P})\subset \overline{P}$, $\sigma(\overline{Q})\subset \overline{Q}$;

(2) $\sigma|_{I^{c-2\varepsilon}} = id$;

(3) $\sigma(I^{c+\varepsilon}\setminus W)\subset I^{c-\varepsilon}$.

Then, $\{P, Q\}$ is called an admissible family of invariant sets with respect to $I$ at level $c$.

To obtain sign-changing solutions for Eq (2.4), the positive and negative cones as in many references such as ^[28,31] are defined:

For $\varepsilon > 0$, consider

Now, we are ready to prove that there exists a sign-changing solution for the modified Eq (2.4), and for this we take $P = P_{\varepsilon}^{+}$, $Q = P_{\varepsilon}^{-}$ and $I = J_{\lambda}$.

Lemma 3.1. Assume that $(f_1)-(f_3)$ and $(V_0)$ hold. Then the Palais-Smale sequence of $J_{\lambda}$ is bounded.

Proof. Since $\{v_n\}\subset E$ is a Palais-Smale sequence, then

and for any $\varphi\in E$, $\langle J_{\lambda}'(v_n), \varphi\rangle = o_{n}(1)\|\varphi\|$, that is

Fixing $\varphi = H^{-1}(v_n)h(H^{-1}(v_n))$, it follows from Lemma 2.2-(4) that

Notice that, Lemma 2.2-(3) implies that

Combining Eqs (3.3) and (3.4), we have

From $\langle J'_{\lambda}(v_n), H^{-1}(v_n)h(H^{-1}(v_n))\rangle = o_{n}(1)\|v_n\|$, we get

Therefore, by Eqs (3.1), (3.2) and (3.5), Lemma 2.1-(5) and Lemma 2.2-(3), we have

which implies $\|v_n\| < +\infty$.

Lemma 3.2. Up to subsequence, the Palais-Smale sequence $\{v_{n}\}$ converges to a critical point $v_{0}$ of $J_{\lambda}$ with $J_{\lambda}(v_{0}) = c_{0}$.

Proof. Since $\{v_{n}\}\subset E$ is bounded and the embedding $E\hookrightarrow L^{\alpha}(\mathbb{R}^N)$ is compact with $\alpha\in[2, 2^{\ast})$, up to a subsequence, we get

We rewrite

where

Using Lemmas 2.1 and 2.2, we have that for all $x\in\mathbb{R}^N$

where $\overline{f}(x, t) = \frac{d \overline{F}(x, t)}{dt}$. Thus, for all $\delta > 0$, there exists a constant $C_{\delta}$, such that

From Eq (3.6) and $v_{n}\to v_{0}$ strongly in $L^{\alpha}(\mathbb{R}^N)$, we have

Thus,

which implies $v_{n}\to v_{0}$ in $E$ and $v_{0}$ is critical point of $J_{\lambda}$.

3.1.   Properties of operator $\mathcal{A}$

We now define an auxiliary operator $\mathcal{A}$ as follows: for any $v\in E$, assuming $w = \mathcal{A}(v)\in E$ is the unique solution to the following equation

where $\overline{f}(x, v) = \lambda\frac{\widetilde{f}(H^{-1}(v))}{h(H^{-1}(v))}-V(x)\frac{H^{-1}(v)}{h(H^{-1}(v))}+V(x)v$.

We can use the auxiliary operator $\mathcal{A}$ to construct a descending flow for the functional $J_{\lambda}(v)$. Actually, the following three statements are equivalent:

● $v$ is a solution of Eq (2.4),

● $v$ is a critical point of $J_{\lambda}(v)$,

● $v$ is a fixed point of $\mathcal{A}$.

Lemma 3.3. The operator $\mathcal{A}$ is well defined as well as continuous and compact.

Proof. We firstly show that $\mathcal{A}$ is continuous. Assume that $v_n\to v$ in $E$. Up to a subsequence, suppose that $v_n\to v$ in $L^{s}(\mathbb{R}^N)$ with $s\in[2, 2^{\ast}]$. Set $\omega_n = \mathcal{A}(v_n)$ and $\omega = \mathcal{A}(v)$, we have

and

Testing with $\omega_n$ in Eq (3.8), by Eq (3.6) we have

Then $\{\omega_n\}$ is bounded in $E$. After passing to subsequence, suppose $\omega_n\rightharpoonup\omega^{\ast}$ weakly in $E$, $\omega_n\to\omega^{\ast}$ in strongly $L^{s}(\mathbb{R}^N)$ with $s\in[2, 2^{\ast})$. From $\omega_n\rightharpoonup\omega^{\ast}$ weakly in $E$, it is easy to see that $\omega^{\ast}$ is a solution of Eq (3.9) and then $\omega^{\ast} = \omega$ by the uniqueness. Moreover, testing with $\omega_n-\omega$ in Eqs (3.8) and (3.9), one has

Next, we are ready to estimate the right term of Eq (3.10). Let $\phi\in C^{\infty}_{0}(\mathbb{R})$ be a cut-off function such that $\phi(t)\in[0, 1]$ for $t\in\mathbb{R}$, $\phi(t) = 1$ for $|t|\leq1$ and $\phi(t) = 0$ for $|t|\geq2$. Setting

By Lemmas 2.1 and 2.2, there exists $C > 0$ such that $|h_{1}(t)|\leq C|t|$ and $|h_{2}(t)|\leq C|t|^{\alpha-1}$ for $t\in\mathbb{R}$. Then,

And it implies

Therefore, we can conclude that $\|\omega-\omega_n\|\to0$ as $n\to\infty$ by the dominated convergence theorem.

Finally, we give the proof of the compact of $\mathcal{A}$. Assume that $\{v_n\}$ is a bounded sequence, we can get the boundness of $\{\omega_n\}\subset E$ due to the continuous of $\mathcal{A}$. Passing to a subsequence, we may assume that $v_n\rightharpoonup v$ and $\omega_n\rightharpoonup\omega$ weakly in $E$ and strongly in $L^{s}(\mathbb{R}^N)$ with $s\in[2, 2^{\ast})$. From Eq (3.8), we have

Taking limit as $n\to\infty$ in Eq (3.11) yields

This means $\omega = \mathcal{A}(v)$ and thus

Using the similar method as before, we can get $\|\omega_n-\omega\|\to0$, i.e., $\mathcal{A}(v_n)\to \mathcal{A}(v)$ in $E$ as $n\to\infty$.

Lemma 3.4. There exists $\varepsilon_{0} > 0$ such that $\mathcal{A}(P^{\pm}_{\varepsilon})\subset P^{\pm}_{\varepsilon}$, for all $\varepsilon\in(0, \varepsilon_{0})$ and every nontrivial solution $v\in P^{-}_{\varepsilon}$ ($v\in P^{+}_{\varepsilon}$) is negative (positive).

Proof. Due to the similarity of the above two conclusions, we only prove $v\in P_{\varepsilon}^{-}$. Let $v\in E$ and $\omega = \mathcal{A}(v)$, for all $q\in[2, 2^{\ast}]$, there exists $S_{q} > 0$ such that

Since $\mbox{dist} (u, P^{-})\leq\|u^{+}\|$, we have

In consequence,

Therefore, if we choose $\delta$ small enough, there exists $\varepsilon_{0} > 0$ such that for $\varepsilon\in(0, \varepsilon_{0})$, it implies

for any $v\in P_{\varepsilon}^{-}.$ It implies that $\mathcal{A}(\partial P_{\varepsilon}^{-})\subset P_{\varepsilon}^{-}$. And if $v\in P_{\varepsilon}^{-}$ with $\mathcal{A}(v) = v$, then $v\in P^{-}$.

Lemma 3.5. (1) $\langle J'_{\lambda}(v), v-\mathcal{A}(v)\rangle\geq\|v-\mathcal{A}(v)\|^{2}$ for all $v\in E$;

(2) $\|J'_{\lambda}(v)\|\leq C\|v-\mathcal{A}(v)\|$ for some $C > 0$ and all $v\in E$.

Proof. Because $\mathcal{A}(v)$ is the solution of Eq (3.7), we have that

For any $\varphi\in E$, we get

Lemma 3.6. For $v\in E$, $a < b$ and $\alpha > 0$, if $J_{\lambda}(v)\in[a, b]$ and $\|J'_{\lambda}(v)\|\geq\alpha$, then there exists $\beta > 0$ such that $\|v-\mathcal{A}(v)\|\geq\beta$.

Proof. Otherwise, there exists a sequence $\{v_n\}\subset E$ such that

But, by Lemma 3.5-(2), we have a contradiction.

Following from ^[29] and ^[30], we can construct a locally Lipschitz continuous operator $\mathcal{B}$ on $E_{0}: = E\setminus K$ which inherits the main properties of $\mathcal{A}$.

Lemma 3.7. The locally Lipschitz continuous operator $\mathcal{B}:E_{0}\to E$ satisfying

(1) $\mathcal{B}(\partial P_{\varepsilon}^{+})\subset P_{\varepsilon}^{+}$ and $\mathcal{B}(\partial P_{\varepsilon}^{-})\subset P_{\varepsilon}^{-}$ for $\varepsilon\in(0, \varepsilon_0)$;

(2) $\frac{1}{2}\|v-\mathcal{B}(v)\|\leq\|v-\mathcal{A}(v)\|\leq2\|v-\mathcal{B}(v)\|$ for all $v\in E_{0}$;

(3) $\langle J'_{\lambda}(v), v-\mathcal{A}(v)\rangle\geq\frac{1}{2}\|v-\mathcal{A}(v)\|^{2}$ for all $v\in E_{0}$.

By the proof of Lemma 3.5 in ^[28] and Lemma 3.7, we have

Lemma 3.8. $\{P_{\varepsilon}^{+}, P_{\varepsilon}^{-}\}$ is an admissible family of invariant sets of the functional $J_{\lambda}$ at any level $c\in\mathbb{R}$.

3.2.   Existence of one sign-changing solution

Next, we are ready to construct $\varphi_0$ satisfying the hypotheses in the Theorem 2.4 in ^[28]. For $(t, s)\in\Delta$, $v_1, v_{2}\in C^{\infty}_{0}(\mathbb{R}^N)$ with supp$(v_1)\ \cap$ supp$(v_2) = \emptyset$ and $v_{1}\leq0, v_{2}\geq0$, define

here $R$ is a positive constant which be determined later. Actually, for $t, s\in[0, 1]$, $\varphi_{0}(0, s) = Rsv_{2}\in P_{\varepsilon}^{+}$ and $\varphi_{0}(t, 0) = Rsv_{1}\in P_{\varepsilon}^{-}$.

Lemma 3.9. Assume that $(V_0)$, $(V_1)$, $(f_1)$, $(f_{2})$ and $(f_{3})$ hold. Then, for $\lambda\geq1$, problem Eq (2.4) has a sign-changing solution.

Proof. We shall prove two claims as follows, which will be useful for us to prove Lemma 3.9.

Claim 1. For $q\in[2, 2^{\ast}]$, there exists $S_{q} > 0$ independence of $\varepsilon$ such that $\|v\|_{q}\leq 2S_{q}\varepsilon$ for $v\in M = P_{\varepsilon}^{+}\cap P_{\varepsilon}^{-}$.

In order to prove this claim, we consider

Claim 2. If $\varepsilon > 0$ is small enough then $J_{\lambda}(v)\geq\frac{\varepsilon^{2}}{2}$ for $v\in\Sigma = \partial P_{\varepsilon}^{+}\cap\partial P_{\varepsilon}^{-}$.

For $v\in\partial P_{\varepsilon}^{+}\cap\partial P_{\varepsilon}^{-}$, then

Since $\|v^{\pm}\|_{q}\leq S_{q}\varepsilon$ and $\|v\|^{2} = \|v^{+}\|^{2}+\|v^{-}\|^{2}$, for $\varepsilon > 0$ small enough, we have

Next, we are ready to verify the conditions (2) and (3) in Theorem 2.4 in ^[28]. Notice that $\rho = \min\{\|tv_{1}+(1-t)v_2\|_{2}:0\leq t\leq1\} > 0$. Then, from the above Claim 1, we have $\varphi_{0}(\partial_{0}\Delta)\cap M = \emptyset$. In fact, for $R$ large enough, if $v\in\varphi_{0}(\partial_{0}\Delta)$, we have $\|v\|_{2} > \rho R$.

By the definition of $\widetilde{F}$, for any $v\in\varphi_{0}(\partial_{0}\Delta)$, denote $A = \{x:\ |v|\geq2\delta\}$, $B = \{x:\ |v| < 2\delta\}$, and let $v_{A} = v|_{A}$, $v_{B} = v|_{B}$. Then, we have

And then

which togethers with the above Claim 2. One has for $R$ large enough and $\varepsilon$ small enough,

Finally, from the Theorem 2.4 in ^[28], there exists $v\in E\setminus(P^{+}_{\varepsilon}\cup P^{-}_{\varepsilon})$, which is a sign-changing solution of Eq (2.4).

We observe that the weak solutions of Eq (2.4) with $L^{\infty}$-norm less than $\min\{\sqrt{1/6}, \delta\}$ are equivalent to the weak solutions of Eq (1.4). Next, we turn to study the $L^{\infty}$ estimates of the critical points of $J_{\lambda}$.

Lemma 3.10. If $v\in E$ is a weak solution of problem Eq (2.4), then $v\in L^{\infty}(\mathbb{R}^N)$. Moreover,

where $C > 0$ only depends on $\alpha, N$.

Proof. Let $v\in E$ be a weak solution of $-\Delta v+V(x)\frac{H^{-1}(v)}{h(H^{-1}(v))} = \lambda \frac{\widetilde{f}(H^{-1}(v))}{h(H^{-1}(v))}$, i.e.,

Let $T > 0$, and define

Choosing $\varphi = v_{T}^{2(\eta-1)}v$ in Eq (3.15), where $\eta > 1$ will be determined later, we have

It follows from $\int_{\{x||v(x)| < T\}}v_{T}^{2(\eta-1)-1}v|\nabla v|^{2}\mbox{d}x\geq0$, $\int_{\mathbb{R}^N}V(x)\frac{H^{-1}(v)}{h(H^{-1}(v))}v_{T}^{2(\eta-1)}v\mbox{d}x\geq0$ and Lemma 2.1-(3), that

On the other hand, due to the Sobolev inequality, it implies

where we used that $(a+b)^{2}\leq 2(a^{2}+b^{2})$ and $\eta^{2}\geq(\eta-1)^{2}+1$.

From Eq (3.16), the Sobolev embedding theorem and the Hölder inequality, it implies

Next, taking $\zeta = \frac{ 22^{\ast}}{2^{\ast}-\alpha+2}$, we obtain

From the Fatou's lemma, it follows that

Let us define $\eta_{n+1}\zeta = 2^{\ast}\eta_{n}$ where $n = 0, 1, 2, ...$ and $\eta_{0} = \frac{2^{\ast}+2-\alpha}{2}$. By Eq (3.17) we have

It follows from Moser's iteration method that

Thus, we have

Lemma 3.11. Assume that $(f_1)-(f_3)$ and $(V_0)$ hold. Let $v_{\lambda}$ be a critical point of $J_{\lambda}$ with $J_{\lambda}(v_{\lambda}) = d_{\lambda}$. Then there exists $C > 0$ (independent of $\lambda$) such that

Proof. From Lemma 2.1-(5) and Eq (3.4), we obtain

It implies that $\|v_{\lambda}\|^{2}\leq Cd_{\lambda}$.

Proof of Theorem 1.1. Let $v_1, v_{2}\in C^{\infty}_{0}(\mathbb{R}^N)$, $v_{1}\leq0, v_{2}\geq0$ with supp$(v_1)\ \cap$ supp$(v_2) = \emptyset$ and $R > 0$ are large enough. Let $\varphi_{0}(t, s): = tRv_{1}+sRv_{2}$ for $(t, s)\in\Delta$. Define

where $\Gamma: = \{\varphi\in C(\Delta, E):\varphi(\partial_{1}\Delta)\subset P, \ \varphi(\partial_{2}\Delta)\subset Q, \ \varphi|_{\partial_{0}\Delta} = \varphi_{0}|_{\partial_{0}\Delta}\}$.

By Lemma 3.9, $J_{\lambda}$ has a sign-changing critical point $v_{\lambda}$ and $J_{\lambda}(v_{\lambda}) = d_{\lambda}$. Furthermore, from Eqs (3.12) and (3.13), we obtain

By Eqs (3.14), (3.18) and (3.19), we have

Hence, there exists $\lambda_{1} > 0$ such that for all $\lambda > \lambda_{1}$

where $\delta$ is fixed in Eq (2.1). Thus, for $\lambda > \lambda_1$, $u_\lambda = H^{-1}(v_\lambda)$ is a sign-changing solution of the original Eq (1.4).

3.3.   Multiplicity of sign-changing solutions

To prove Theorem 1.2, we make on further assumption, $G:E\to E$ is an isometric involution, i.e., $G^{2} = id$ and $d(Gu, Gv) = d(u, v)$ for $u, v\in E$. We assume $I$ is $G$-invariant on $E$ in the sense that $I(Gu) = I(u)$ for any $u\in E$. We also assume $Q = GP$. If for any $u\in F$, $Gu\in F$, then the subset $F\subset E$ is said to by symmetric. $\gamma(F)$ can be called the genus of a closed symmetric subset $F$ of $E\setminus\{0\}$.

Definition 3.2. If the following deformation property holds: there exist $\varepsilon_{0} > 0$ and a symmetric open neighborhood $N$ of $K_{c}\setminus W$ with $\gamma(\overline{N}) < +\infty$, such that for $\varepsilon\in(0, \varepsilon_{0})$, there exists $\sigma\in C(E, E)$ meet the following four conditions:

(1) $\sigma(\overline{P})\subset \overline{P}$, $\sigma(\overline{Q})\subset \overline{Q}$;

(2) $\sigma|_{I^{c-2\varepsilon}} = id$;

(3) $\sigma\circ G = G\circ\sigma$;

(4) $\sigma(I^{c+\varepsilon}\setminus (N\cup W))\subset I^{c-\varepsilon}$.

Then, we call $P$ is a $G$-admissible invariant set with respect to $I$ at level $c$.

We now assume that $f$ is odd and we turn to prove the existence of infinitely many sign-changing solutions to Eq (1.4). We plan to apply the Theorem 2.6 in ^[28], for this we take $G = -id$, $P = P_{\varepsilon}^{+}$, $Q = P_{\varepsilon}^{-}$ and $I = J_{\lambda}$. Next, lemma is used to prove $P$ is a $G$-admissible invariant set with respect to $J_{\lambda}\in C^1(E, \mathbb{R})$ at any level $c$.

Lemma 3.12. $P^+_{\varepsilon}$ is a $G$-admissible invariant set for the functional $J_{\lambda}$ at any level $c$.

Proof. The proof is similar to Lemma 3.8. Since $J_{\lambda}$ is even, thus $\sigma$ is odd in $u$. Here, we omit the details.

Proof of Theorem 1.2. Firstly, we shall use the Theorem 2.6 in ^[28] to get solutions for Eq (2.4) first. Making use of estimates on the critical values, for any fixed $n\in\mathbb{N}$ we shall show Eq (1.4) has $n-1$ pairs of sign-changing solutions for large $\lambda$.

For any $n\in\mathbb{N}$, let $\{v_{i}\}_{i = 1}^{n}\subset C^{\infty}_{0}(\mathbb{R}^N)\setminus\{0\}$ be such that supp$(v_i)\ \cap$ supp$(v_j) = \emptyset$ for $i\neq j$. Define $\varphi_n\in C(B_{n}, E)$ as

where $R_n > 0$ will be determined later. Actually, $\varphi_{n}(0) = 0\in P_{\varepsilon}^{+}\cap P_{\varepsilon}^{-}$ and $\varphi_{n}(-t) = -\varphi_{n}(t)$ for $t\in B_n$. Observe that

then $\|v\|_{2}^{2}\geq\rho_{n}^{2}R_{n}^{2}$ for $v\in\varphi_{n}(\partial B_n)$ and it follows from Claims 1 and 2 in Lemma 3.9 that $\varphi_n(\partial B_n)\cap(P_{\varepsilon}^{+}\cap P_{\varepsilon}^{-}) = \emptyset$. Similar to the proof of Theorem 1.1 (existence part), for large enough $R_n > 0$ independent on $\lambda$ we also have

For $j = 2, 3, \cdot\cdot\cdot, n$, let

where

and

Then, by the Theorem 2.6 in ^[28], we have that $0 < c_{2, \lambda}\leq c_{3, \lambda}\leq\cdot\cdot\cdot\leq c_{n, \lambda}$ are all critical values of $J_{\lambda}$ and there are at least $(n-1)$ pairs of sign-changing critical points at these critical values. Since $\varphi_{n}\in H_n$, we have

Due to supp$(v_i)\ \cap$ supp$(v_j) = \emptyset$ for $i\neq j$, similar with Eq (3.19), we have

Therefore, it follows from Lemmas 3.10 and 3.11, for $\lambda$ large that these $(n-1)$ pairs of sign-changing critical points of $J_{\lambda}$ are also solutions of the original Eq (1.4).

Acknowledgments

C. Huang is supported by Postdoctoral Science Foundation of China (2020M682065).

Conflict of interest

The authors declare there is no conflicts of interest.