Infinitely many sign-changing solutions for a semilinear elliptic equation with variable exponent

1.   Introduction and main result

Let $0\in\Omega\subset{\mathbb{R}}^N(N\geq 3)$ be a bounded domain with smooth boundary $\partial\Omega$. In this paper, we are interested in establishing the multiplicity of sign-changing solutions to the following semilinear elliptic equations with variable exponent

where $q(x)$ satisfies the following assumptions.

$(Q_1) \; q\in C(\overline{\Omega})$, $q(0) = 2$ and $2 < q(x)\leq \mathop {\max }\limits_{x\in \overline{\Omega}}\{q(x)\} = q^+ < 2^* = \frac{2N}{N-2}$ for $x\neq 0$;

$(Q_2)$ there exist $\alpha\in\left(0, \ \frac{N+2}{2}\right)$ and $B_{\delta_0} = \{x||x| < \delta_0\}\subset\Omega$ such that $q(x)\geq 2+|x|^{\alpha}$ for any $x\in B_{\delta_0}$.

In 1973, Ambrosetti and Rabinowitz in ^[2] obtained a positive and a negative solution to the following superlinear elliptic problem

The existence of the third solution to problem (1.2) was established by Wang in ^[17]. Castro, Cossio and Neuberger in ^[6] proved that the third solution to problem (1.2) obtained in ^[17] changes sign only once. Bartsch and Wang in ^[3] obtained the existence of sign-changing solution. In addition, Bartsch, Weth and Willem in ^[4] showed that problem (1.2) possesses a least energy sign-changing solution. In order to study the sign-changing critical points of even functionals, Li and Wang in ^[11] established a Ljusternik-Schnirelmann theory and showed that problem (1.2) possesses infinitely many sign-changing solutions. Subsequently, the existence of infinitely many sign-changing solutions to problem (1.2) was also obtained by some versions of the symmetric mountain pass lemma(see ^[15] and ^[19]).

In fact, these papers required $f(x, t)$ to satisfy the following condition ($(AR)$-condition, for short)

where $\theta > 2$ and $F(x, t) = \int_0^tf(x, s)\, ds$. It is well known that $(AR)$-condition is important to guarantee the boundedness of Palais-Smale sequence of the Euler-Lagrange functional associated to problem (1.2) which plays a crucial role in applying the critical point theory. For more than 40 years, several researchers studied problem (1.2) trying to drop the above $(AR)$-condition. For example, a weaker super-quadratic condition ($(SQ)$-condition, for short) is that

Under $(SQ)$-condition or some extra assumptions, the existence and multiplicity of nontrivial solution for problem (1.2) were obtained, see ^{[7,8,12,14,16]} and the references therein.

Recently, the special case of problem (1.1) as problem (1.2) is also concerned by some scholars(see ^{[1,5,9,10,13]}). They obtained the existence or multiplicity of the nontrivial solution of problem (1.1) from the discussion the compact embedding from $H^1_0(\Omega)$ to $L^{q(x)}(\Omega)$ with a variable critical or supercritical exponent. In particular, Cao, Li and Liu in ^[5] obtained that problem (1.1) has infinitely many nodal solutions when $q(x) = 2^{*}+|x|^{\alpha}-2$($0 < \alpha < \min\{\frac{N}{2}, N-2\}$) and $B_1$ is the unit ball in ${\mathbb{R}}^N$. In addition, Hashizume and Sano in ^[9] proposed that $ess \mathop {\inf }\limits_{x\in \Omega} \{q(x)\} = 2$ is another critical case. Indeed, if there exists $x_0\in \Omega$ such that $q(x_0) = \mathop {\inf }\limits_{x\in \Omega}\{q(x)\} = 2$, then the conditions $(AR)$ and $(SQ)$ do not hold. Therefore, the problem we intend to study is a new phenomenon. To the best of our knowledge, for either $p$-Laplacian equation(including semilinear elliptic equation) or $p(x)$-Laplacian equation, there are no results in this case. The main difficulty with problem (1.1) is that the corresponding functional may possess unbounded Palais-Smale sequences. To overcome this difficulty, we will use the perturbation technique and the Moser's iteration.

The main result of this paper reads as follows.

Theorem 1.1. Suppose that $(Q_1)$ and $(Q_2)$ hold. Then, for every integer $k\geq 1$, problem (1.1) has $k$ sign-changing solutions.

Remark 1.2. In ^[5] and ^[9], it is crucial to require the space is radially symmetric. However, we do not need the domain to be radial.

To end this section, we describe the basic ideas in the proof of Theorem 1.1. Noticing that $q(0) = 2$, inspired by ^[18], we first modify the nonlinear term to guarantee the boundedness of Palais-Smale sequence of the corresponding functional and obtain infinitely many sign-changing solutions of auxiliary problem by a version of the symmetric mountain pass lemma. Subsequently, we use the Moser iteration to obtain the existence of infinitely many sign-changing solutions for problem (1.1).

Throughout this paper, let $B_{\delta} = \{x||x| < \delta\}\subset \Omega$ and $\Omega_{\delta} = \Omega\backslash\ B_{\delta}$. We use $\|\cdot\|$ to denote the usual norms of $H^1_0(\Omega)$. The letter $C$ stands for positive constant which may take different values at different places.

2.   The modified problem

According to $q(0) = 2$, it seems to be difficult to confirm whether the energy function $I$ corresponding to (1.1) satisfies the Palais-Smale condition or not.To apply variational methods, the first step in proving Theorem 1.1 is modifying the nonlinear term to obtain the perturbation equation. Since $q(x)$ is a continuous function and $q^+ < 2^*$, we can choose $r > 0$ such that

Let $\psi(t)\in C^{\infty}_0({\mathbb{R}}, [0, 1])$ be a smooth even function with the following properties: $\psi(t) = 1$ for $|t|\leq 1$, $\psi(t) = 0$ for $|t|\geq 2$ and $\psi(t)$ is monotonically decreasing on the interval $(0, +\infty)$. Define

for $\mu\in (0, 1]$. We will deal with the modified problem

Theorem 2.1. Suppose that $(Q_1)$ and $(Q_2)$ hold. Then, for any $\mu\in(0, 1]$, problem (2.2) has infinitely many sign-changing solutions.

Let $E: = H^1_0(\Omega)$ be the usual Sobolev space endowed with the inner product $\langle u, v\rangle = { \int}_{\Omega}\nabla u\nabla v\, dx$ for $u, \ v\in E$ and the norm $\|u\|: = \langle u, u\rangle^{\frac{1}{2}}$. Let $\mathcal{P}$ be the positive cone of $E$, and $Y, M$ be two subspaces of $E$ with $\dim Y < \infty$, $\dim Y - \mbox{codim} M\geq 1$. For any $\delta > 0$, define $\pm \mathcal{D}(\delta): = \{u\in E:\mbox{dist}(u, \pm \mathcal{P}) < \delta\}$. Set $\mathcal{D}: = \mathcal{D}(\delta)\cup (-\mathcal{D}(\delta))$ and $\mathcal{S} = E\backslash \mathcal{D}$. Let $G\in C^1(E, {\mathbb{R}})$ and the gradient $G'$ be of the form $G'(u) = u-K_G(u)$, where $K_G: E\rightarrow E$ is a continuous operator. Let $\mathcal{K} = \{u\in E: G'(u) = 0\}$ and $\mathcal{K}[a, b] = \{u\in K: G(u)\in [a, b]\}$. We assume that there is another norm $\|\cdot\|_{*}$ of $E$ such that $\|u\|_{*}\leq C_{*}\|u\|$ for all $u\in E$, where $C_{*}$ is a positive constant. Moreover, we assume that $\|u_n-u^*\|_{*}\rightarrow 0$ whenever $u_n\rightharpoonup u^{*}$ weakly in $(E, \|\cdot\|)$. Write $E = M_1\oplus M$. Let

where $\rho > 0$, $D_{*} > 0$ and $p > 2$ are fixed constants. Let us assume that $Q^{**} = Q^{*}(\rho)\cap G^{\beta}\subset \mathcal{S}$ and $\gamma = \mathop {\inf }\limits_{Q^{**}}G$, where $\beta = \mathop {\sup }\limits_{Y}G$ and $G^{\beta} = \{u\in E: G(u)\leq \beta\}$. It is easy to see $\beta\geq \gamma$. In addition, we assume that

$(A) \; K_G(\pm \mathcal{D}(\delta))\subset \pm \mathcal{D}(\delta)$;

$(A_1^{*})$ Assume that for any $a, \ b > 0$, there is a $c_1 = c_1(a, b) > 0$ such that $G(u)\leq a$ and $\|u\|_{*}\leq b \Rightarrow \|u\|\leq c_1$;

$(A_2^{*}) \; \mathop {\lim }\limits_{u\in Y, \ \|u\|\rightarrow \infty}G(u) = -\infty$, $\mathop {\sup }\limits_{Y}G: = \beta$.

Now we recall the following Palais-Smale condition and abstract critical point theorem (see Definition 3.3 and Theorem 5.6 in ^[19]).

Definition 2.2. The functional $G$ is said to satisfy the $(w^*-PS)$ condition if for any sequence $\{u_n\}$ such that $\{G(u_n)\}$ is bounded and $G'(u_n)\rightarrow 0$, we have either $\{u_n\}$ is bounded and has a convergent subsequence or $\|G'(u_n)\|\|u_n\|\rightarrow\infty$. In particular, if $\{G(u_n)\}\rightarrow c$, we say that $(w^*-PS)_c$ is satisfied.

Theorem 2.3. Assume that $(A)$, $(A_1^{*})$ and $(A_2^{*})$ hold. If the even functional $G$ satisfies the $(w^*-PS)_c$ condition at level $c$ for each $c\in[\gamma, \beta]$, then $\mathcal{K}[\gamma-\varepsilon, \beta+\varepsilon]\cap (E\setminus (\mathcal{P}\cup(-\mathcal{P}))\neq \emptyset$ for all $\varepsilon > 0$ small.

Let $0 < \lambda_1 < \cdots < \lambda_k < \cdots$ denote the distinct Dirichlet eigenvalues of the eigenvalue problem

Then each $\lambda_k$ has finite multiplicity. In addition, the principal eigenvalue $\lambda_1$ is simple with a positive eigenfunction $\varphi_1$, and the eigenfunctions $\varphi_k$ corresponding to $\lambda_k(k\geq 2)$ are sign-changing. Let $N_k$ denote the eigenspace of $\lambda_k$. Then $\dim N_k < \infty$. We fix $k$ and let $E_{k}: = N_1\oplus N_2\oplus\cdots\oplus N_k$.

The formal energy functional $I_\mu: H^1_{0}(\Omega)\to{\mathbb{R}}$ associated with (2.2) is defined by

where $f_{\mu}(x, t) = \left(\frac{t}{m_{\mu}(t)}\right)^r|t|^{q(x)-2}t$, $F_{\mu}(x, t) = { \int}_0^tf_{\mu}(x, \tau)\, d\tau$. Then $I_{\mu}\in C^1(E, {\mathbb{R}})$ and $I'_{\mu} = id-(-\Delta)^{-1}f_{\mu} = id-K_{I_{\mu}}$. Obviously, the critical points of $I_{\mu}$ are just the weak solutions of problem (2.2).

Lemma 2.4. The function $F_{\mu}(x, t)$ defined above satisfies the following inequalities:

for $t > 0$, where $C_{\mu} > 0$ is a positive constant.

Proof. Since $b_{\mu}(t)$ to is monotonically decreasing on the interval $(0, +\infty)$, we have

for $t > 0$, where $\xi\in(0, t)$. Therefore, $\frac{t}{m_{\mu}(t)}$ is monotonically increasing on the interval $(0, +\infty)$. Hence, $\frac{f_{\mu}(x, t)}{t^{q(x)-1}} = \left(\frac{t}{m_{\mu}(t)}\right)^r$ is also monotonically increasing on the interval $(0, +\infty)$. It follows that

for $t > 0$.

By definition of the function $m_{\mu}$, we have $m_{\mu}(t) = \frac{A}{\mu}$ for $t\geq \frac{2}{\mu}$, where $A = 1+\int_1^2\psi(\tau)d\tau$. For $t > \frac{2}{\mu}$, one has

It implies from (2.3) and (2.4) that

for $t > 0$.

Lemma 2.5. Suppose that $(Q_1)$ and $(Q_2)$ hold. Then, for any $\mu\in (0, 1]$, $I_{\mu}$ satisfies the $(PS)$ condition.

Proof. Let $\{u_n\}$ be a $(PS)$ sequence of $I_{\mu}$ in $E$. This means that there exists $C > 0$ such that

From (2.1) and Lemma 2.4, we derive that

which implies that $\frac{r}{2(2+r)}\|u_{n}\|^{2}\leq C+C_{\mu}+o(\|u_{n}\|)$. We obtain $\{u_{n}\}$ is bounded in $E$. Up to a subsequence, we may assume that

For any integer pair $(i, j)$, one has

It follows from (2.5) that

It is easy to see that

Note that $2\leq q(x) < q(x)+r\leq q^++r < 2^*$. It implies that

as $i$ and $j$ tend to $+\infty$. From (2.6) and (2.7), we have $\|u_i-u_j\|\rightarrow 0$ as $i, \ j \rightarrow +\infty$, which implies that $\{u_n\}$ contains a strongly convergent subsequence in $E$. Hence $I_{\mu}$ satisfies the $(PS)$ condition.

$G$, $Y$ and $M$ are taken to be $I_{\mu}$, $E_k$ and $E_{k-1}^{\perp}$ in Theorem 2.3, respectively. Next we will complete the proof of Theorem 2.1 by verifying the conditions of Theorem 2.3 one by one.

Lemma 2.6. Suppose that $(Q_1)$ holds. If we replace $G$, $Y$ and $M$ with $I_{\mu}$, $E_k$ and $E_{k-1}^{\perp}$, respectively, then conditions $(A_1^{*})$ and $(A_2^{*})$ are satisfied.

Proof. Consider another norm $\|u\|_{*} = \|u\|_s$ of $E$, $s\in (2, 2^*)$. Then $\|u\|_s\leq C_{*}\|u\|$ for all $u\in E$, where $C_{*} > 0$ is a constant and $\|u_n-u^*\|_{*}\rightarrow 0$ whenever $u_n\rightharpoonup u^*$weakly in $(E, \|\cdot\|)$. Define $\beta_k = \mathop {\sup }\limits_{E_k}I_{\mu}$. Let

it is easy to obtain that there exists a constant $c_2 > 0$ such that $\|u\|_{s}\leq c_2$ for any $u\in Q_k^{*}(\rho)$. By assumption $(Q_1)$ and definition of the function $m_{\mu}$, we have

It implies that

By the Sobolev imbedding theorem, it implies from $2\leq q(x) < q(x)+r\leq q^++r < 2^*$ that

Set $\Omega_{\varepsilon} = \{x\in \Omega|2\leq q(x) < 2+\varepsilon\}$. By the Hölder inequality and the Sobolev imbedding theorem, we have

Since $\Omega_0 = \{0\}$, we obtain $|\Omega_{\varepsilon}|\rightarrow 0$ as $\varepsilon\rightarrow 0$. Therefore, there exists $\varepsilon_0 > 0$ such that

for any $\varepsilon\in (0, \varepsilon_0)$. From (2.8)-(2.11), for any $a, \ b > 0$, there is a $c_1 = c_1(a, b) > 0$ such that $I_{\mu}(u)\leq a$ and $\|u\|_{q^++r}\leq b \Rightarrow \|u\|\leq c_1$. That is, condition $(A_1^{*})$ is satisfied.

For $t > \max\left\{1, \frac{2}{\mu}\right\}$, one has

Set $Y = E_k$. Noticing that $\dim E_k < \infty$ and all norms of finite dimensional space are equivalent, it implies that

as $\|u\|\rightarrow\infty$, $u\in E_k$. Therefore, $\mathop {\lim }\limits_{u\in E_k, \ \|u\|\rightarrow \infty}I_{\mu}(u) = -\infty$. So condition $(A_2^{*})$ is satisfied.

Let $Q_k^{**} = Q_k^{*}(\rho)\cap I_{\mu}^{\beta_k}\subset \mathcal{S}$ and $\gamma_k = \mathop {\inf }\limits_{Q^{**}}I_{\mu}$. Set $\mathcal{P}: = \{u\in E: u(x)\geq 0 \ \mbox{for a.e} \ x\in \Omega$. Then, $\mathcal{P}(-\mathcal{P})$ is the positive(negative) cone of E and weakly closed. By Lemma 5.4 in ^[19], there is a $\eta = \eta(\beta_k) > 0$ such that $dist(Q^{**}, P) = \eta > 0$. We define $\pm \mathcal{D}_0(\delta_0): = \{u\in E:\mbox{dist}(u, \pm \mathcal{P}) < \delta_0\}$, where $\delta_0$ is determined by the following lemma.

Lemma 2.7. Under the assumption $(Q_1)$, there is a $\delta_0\in (0, \eta)$ such that $K_{I_{\mu}}(\pm \mathcal{D}_0(\delta_0))\subset \pm \mathcal{D}_0(\delta_0)$. Therefore, condition $(A)$ is satisfied.

Proof. Write $u^{\pm} = \max\{\pm u, 0\}$. For any $u\in E$ and each $s\in (2.2^{*}]$, there exists a $C_s > 0$ such that

Let $v = K_{I_{\mu}}(u)$. Similar to the derivation of (2.8), (2.9) and (2.10), we have

From (2.12) and (2.13), by the Hölder inequality and the Sobolev imbedding theorem, we obtain

That is,

It follows from (2.11) that there exists a $\delta_0\in (0, \eta)$ such that $\mbox{dist}(K_{I_{\mu}}(u), -\mathcal{P}) < \delta_0$ for every $u\in -\mathcal{D}_0(\delta_0)$. Similarly, $\mbox{dist}(K_{I_{\mu}}(u), \mathcal{P}) < \delta_0$ for every $u\in \mathcal{D}_0(\delta_0)$. The conclusion follows.

Now we are in a position to prove the main result of this section.

Proof of Theorem 2.1. By Theorem 2.3, Lemmas 2.5, 2.6 and 2.7, we obtain

for all $\varepsilon > 0$ small. That is, there exists a $u_{k, \mu}\in E\setminus (\mathcal{P}\cup(-\mathcal{P})$ (sign-changing critical point) such that

where $\gamma_k = \mathop {\inf }\limits_{Q_k^{**}}I_{\mu}$. Next we show the $\gamma_k\rightarrow \infty$ as $k\rightarrow \infty$. Recall the Gagliardo-Nirenberg inequality,

where $s\in (2, 2^{*})$ and $\alpha\in (0, 1)$ is defined by

In addition, for $u\in E_k^{\perp}$, we see that $\|u\|_2\leq \frac{1}{\lambda_k^{1/2}}\|u\|$. Combine (2.14) with (2.15), we have

For $u\in Q_k^{*}(\rho)$, by the Sobolev imbedding theorem, we deduce from (2.16) that

It implies that

where $\Lambda_s^{*} = \min\left\{C_{*}^{-1/2}, 2^{-1/(s-2)}C_{*}^{-2/(s-2)}c_s^{-1}\right\}$, $T_k = \min\left\{\lambda_k^{\beta_k/2}, \lambda_k^{(1-\alpha)/2}\right\}$ and $T = \min\left\{\rho, \rho^{1/(s-2)}\right\}$. From (2.8)-(2.11), we know that

We can choose that $\rho > 0$ such that $\rho < \frac{1}{8C}$. For any $u\in Q_k^{*}(\rho)$, we see that $\|u\|_s^s/\|u\|^2\leq \rho$. Therefore, for any $u\in Q_k^{*}(\rho)$, it implies from (2.18) that

Since $\lambda_k\rightarrow \infty$ as $k\rightarrow \infty$, we obtain $T_k = \min\left\{\lambda_k^{\beta_k/2}, \lambda_k^{(1-\alpha)/2}\right\}\rightarrow \infty$ as $k\rightarrow \infty$. Therefore, $\gamma_k\rightarrow \infty$ as $k\rightarrow \infty$. Hence, for any $\mu\in(0, 1]$, problem (2.2) has infinitely many sign-changing solutions. The proof is complete.

3.   A priori estimate and proof of Theorem 1.1

In this section, we will show that solutions of auxiliary problem (2.2) are indeed solutions of original problem (1.1). For this purpose, we need the following uniform $L^\infty$-estimate for critical points of the functional $I_\mu$.

Proposition 3.1. Suppose that $(Q_1)$ and $(Q_2)$ hold. If $v$ is a critical point of $I_\mu$ with $I_\mu(v)\leq L$, then there exists a positive constant $M = M(L)$ independent of $\mu$ such that $\|v\|_{L^{\infty}\left(\Omega\right)}\leq M$.

In order to prove Proposition 3.1, we need some preliminaries.

Lemma 3.2. Suppose that $(Q_1)$ and $(Q_2)$ hold. If $I_{\mu}(v)\leq L$ and $I_{\mu}'(v) = 0$, then, for any $\delta\in (0, \delta_0)$, there exists $C_{\delta} > 0$ independent of $\mu$ such that ${ \int}_{\Omega_{\delta}}|\nabla v|^{2}dx\leq C_{\delta}$.

Proof. By Lemma 2.4 and $(Q_1)$, we have

According to $(Q_1)$, for any $\delta\in (0, \delta_0)$, we know that there exists $m_{\delta} > 0$ such that $\frac{1}{2}-\frac{1}{q(x)}\geq m_{\delta}$ for any $x\in\Omega_{\delta}$. Therefore, we have

The proof is complete.

Lemma 3.3. Let $1 < p < \frac{N}{2}$ and $0 < r < R$. Suppose that the nonnegative functions $w(x)$ and $g(x)$ satisfy $g\in L^{p}(B_{R})$ and

Then, we have

where $C = C(N, p, R, r) > 0$.

Proof. Set $\xi = \frac{N(p-1)}{N-2p}$. Then, we have the following identity

Let $\varphi\in C^{\infty}_0 ({\mathbb{R}}^N, [0, 1])$ satisfies $\varphi(x) = 1$ for $|x|\leq r$ and $\varphi(x) = 0$ for $|x|\geq R$. For any $\theta > 0$, multiply inequality (3.2) by the test function $((w+\theta)^{\xi}-\theta^{\xi})\varphi^2$ and integrate to obtain

By the Young inequality, we hace

According to $1 < p < \frac{N}{2}$, we have $\frac{2p}{p-1} > \frac{2N}{N-2}$. It implies that

Letting $\theta\to 0$, we conclude from (3.5), (3.6) and (3.7) that

which implies that

or

From (3.4), (3.8) and (3.9), we have

or

Therefore, we obtain

The proof is complete.

Lemma 3.4. Suppose that $(Q_1)$ and $(Q_2)$ hold. If $I_{\mu}(v)\leq L$ and $I_{\mu}'(v) = 0$, then there exist $\delta_1\in(0, \ \delta_0)$ and $C > 0$ independent of $\mu$ such that ${ \int}_{B_{\delta}}|\nabla v|^{2}dx\leq C$ for any $\delta\in(0, \ \delta_1)$.

Proof. It follows from $I_{\mu}'(v) = 0$ that $v$ is a solution of problem (2.2). For any $\delta\in(0, \ \delta_0)$ and $B_{2\delta}\subset \Omega$, let $\phi\in C^{\infty}_0 (\Omega, [0, 1])$ satisfies $\phi(x) = 1$ for $|x|\leq \delta$, $\phi(x) = 0$ for $|x|\geq 2\delta$ and $|\nabla \phi|\leq C$ for $x\in \Omega$. Multiply equation (2.2) by $v\phi^2$ and integrate to obtain

By the Young inequality, we have

By definition of the function $m_{\mu}$, we know that $m_{\mu}(t) = t$ for $t\leq \frac{1}{\mu}$ and $m_{\mu}(t)\geq \frac{1}{\mu}$ for $t > \frac{1}{\mu}$. Therefore, we have

for any $\mu\in (0, \ 1]$ and $x\in \Omega$. From (3.10), (3.11) and (3.12), we obtain

In order to complete our proof, we just need to prove that there exists $\delta_2 > 0$ such that ${ \int}_{B_{\delta_2}}|v|^{q^++r}dx\leq C$.

By definition of the function $k_{\mu}$, we obtain $k_{\mu}(x, t)\geq t^{q(x)-1}$. According to Lemma 2.4, $(Q_1)$ and $(Q_2)$, for any $\delta\in(0, \ \delta_0)$, we have

Noticing that $N\geq 3$, from $(Q_2)$ and (2.1), we have

Therefore, we can choose $p\in\left(1, \frac{2N}{N+1}\right)$ satisfying

Let $q^+_{\delta} = \sup\{q(x)|x\in B_{\delta}\}$. It follows from $(Q_1)$ and (3.15) that there exists $\delta_3 < \min\{1, \ \delta_0\}$ such that

for any $\delta\in(0, \ \delta_3)$. Using the Young inequality, we deduce from (3.14) and (3.16) that

for any $\delta\in(0, \ \delta_3)$. According to (3.12) and (3.17), for any $\delta\in(0, \ \delta_3)$, we obtain

Since $-\triangle v = k_{\mu}(x, v)$ in $B_{\delta}$. By Lemma 3.2 and Lemma 3.3, for any $\delta'\in (0, \ \delta)$, it implies from (3.18) that

where $\delta\in(0, \ \delta_3)$ and $C = C(\delta', \delta, N, p) > 0$ is independent of $\mu$.

If $\zeta_1 = \frac{Np}{N-2p}\geq q^++r$, using the Hölder inequality, we are done. Otherwise, using the fact $r < \frac{1}{4N} < \frac{2}{N-2}$ provided by (2.1), we can choose $\sigma_1\in (0, \ \delta)\subset(0, \ \delta_3)$ such that $\tau_1 = q_{\sigma_1}^++r-1 < \frac{N}{N-2}$. It follows from (3.12) that

for any $\mu\in (0, \ 1]$ and $x\in B_{\sigma_1}$. Noticing that $\zeta_1 > \frac{N}{N-2}$, we have $p_1 = \frac{\zeta_1}{\tau_1} > 1$. According to (3.19), we obtain $k_{\mu}(x, v)\in L^{p_1}(B_{\sigma_1})$. Similar to (3.19), we can choose $\sigma_2\in (0, \sigma_1)$ to obtain

where $C = C(\sigma_1, \sigma_2, N, p_1) > 0$ is independent of $\mu$ and

here $d_1 = \frac{N}{(N-2)\tau_1} > 1$. If $\zeta_2\geq q^++r$, using the Hölder inequality, we are done. Otherwise, repeating the above process and using a finite number of iterations, we obtain that there exist $\zeta_k > 0$ and $\sigma_{k}\in (0, \sigma_{k-1})$ such that $\zeta_k\geq 2^* > q^++r$ and $\|v\|_{L^{\zeta_k}(B_{\sigma_k})}\leq C$, where $C > 0$ is independent of $\mu$. Using the Hölder inequality, we have

Let $\delta_1 = \frac{\zeta_k}{2}$. It implies from (3.13) and (3.20) that

for any $\delta\in(0, \delta_1)$.

Proof of Proposition 3.1. By Lemma 3.2 and Lemma 3.4, we obtain that there exists $C > 0$ independent of $\mu$ such that

Using the Sobolev embedding theorem, we have

Let $s > 0$ and $t = q^++r$. According to (3.12), multiply equation (2.2) by $v^{2s+1}$ and integrate to obtain

It implies that

On the one hand, by the Sobolev embedding theorem, we have

On the other hand, by the Hölder inequality and (3.22), we have

where $d = \frac{2^*-t+2}{2} > 1$. According to (3.23), (3.24) and (3.25), we obtain

which implies that

Now we carry out an iteration process. Set $s_k = d^{k}-1$ for $k = 1, 2, \cdots$. By (3.26), we have

Since $d > 1$, the series $\sum\limits_{j = 1}^\infty d^{-j}$ and $\sum\limits_{j = 1}^\infty jd^{-j}$ are convergent. Letting $k\to\infty$, we conclude from (3.22) and (3.27) that $\|v\|_{L^{\infty}(\Omega)}\leq M$. The proof is complete.

Proof of Theorem 1.1. By the proof of Theorem 2.1, for every integer $k\geq 1$, we know that problem (2.2) has $k$ sign-changing solutions $u_{k, \mu}$ satisfying $\gamma_k-1 < I_{\mu}(u_{k, \mu}) < \beta_k+1$. Consider the functional

By definition of the function $f_{\mu}$, we obtain $|f_{\mu}(x, t)|\geq |t|^{q(x)-1}$. It is easy to see that $I_{\mu}(u)\leq J(u)$. Therefore, there exists a sequence of positive numbers $\{\Upsilon_k\}$ independent of $\mu$ such that $\beta_k+1\leq \Upsilon_k$. Let $L_k = \max\{\Upsilon_1, \Upsilon_2, \cdots, \Upsilon_k\}$. By Proposition 3.1, there exists a positive constant $M_k = M_k(L_k)$ independent of $\mu$ such that $\|u_{k, \mu}\|_{L^{\infty}\left(\Omega\right)}\leq M_k$. By definition of the function $m_{\mu}$, we have $m_{\mu}(t) = t$ for $t\leq \frac{1}{\mu}$. Hence, problem (2.2) reduces to problem (1.1) for $|t|\leq \frac{1}{\mu}$. Let $\mu < \frac{1}{2M_k}$. It is easy to see that $u_{k, \mu}$ is indeed a sign-changing solution of problem (1.1).

Acknowledgments

Thanks to Professor Jiaquan Liu of Peking University for his great help and valuable advice in this paper. Supported by National Natural Science Foundation of China (No.11861021).

Conflict of interest

The authors declare no conflict of interest.