Multiple solutions of Kirchhoff type equations involving Neumann conditions and critical growth

1.   Introduction and main results

We study the following Neumann problem of Kirchhoff type equation with critical growth

where $\Omega$ $\subset$ $\mathbb{R}^3$ is a bounded domain with a smooth boundary, $a, b > 0$, $1 < q < 2$, $\lambda > 0$ is a real parameter. We assume that $Q(x)$ and $P(x)$ satisfy the following conditions:

$(Q_1)$ $Q(x)\in C(\bar{\Omega})$ is a sign-changing;

$(Q_2)$ there exists $x_M\in \Omega$ such that $Q_M = Q(x_M) > 0$ and

$(Q_3)$ there exists $0\in \partial\Omega$ such that $Q_m = Q(0) > 0$ and

$(P_1)$ $P(x)$ is positive continuous on $\bar{\Omega}$ and $P(x_0) = \max_{x\in\bar{\Omega}}P(x)$;

$(P_2)$ there exist $\sigma > 0$, $R > 0$ and $3-q < \beta < \frac{6-q}{2}$ such that $P(x)\geq \sigma |x-y|^{-\beta}$ for $|x-y|\leq R$, where $y$ is $x_M\in\Omega$ or $0\in\partial\Omega$.

In recent years, the following Dirichlet problem of Kirchhoff type equation has been studied extensively by many researchers

which is related to the stationary analogue of the equation

proposed by Kirchhoff in ^[13] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations. In (1.2) and (1.3), $u$ denotes the displacement, $b$ is the initial tension and $f(x, u)$ stands for the external force, while $a$ is related to the intrinsic properties of the string (such as Young's modulus). We have to point out that such nonlocal problems appear in other fields like biological systems, such as population density, where $u$ describes a process which depends on the average of itself (see Alves et al. ^[2]). After the pioneer work of Lions ^[18], where a functional analysis approach was proposed. The Kirchhoff type Eq (1.2) with critical growth began to call attention of researchers, we can see ^{[1,9,14,17,23,24,28,30]} and so on.

Recently, the following Kirchhoff type equation has been well studied by various authors

There has been much research regarding the concentration behavior of the positive solutions of (1.4), we can see ^{[10,11,12,25,33]}. Many papers studied the existence of ground state solutions of (1.4), for example ^{[5,8,15,16,21,22,24]}. In addition, the authors established the existence of sign-changing solutions of (1.4) in ^[20,31]. In papers ^[27,32] proved the existence and multiplicity of nontrivial solutions of (1.4) by using mountain pass theorem.

In particular, Chabrowski in ^[6] studied the solvability of the Neumann problem

where $\Omega$ $\subset$ $\mathbb{R}^N$ is a smooth bounded domain, $2^* = \frac{2N}{N-2}(N\geq3)$ is the critical Sobolev exponent, $\lambda > 0$ is a parameter. Assume that $Q(x)\in C(\overline{\Omega})$ is a sign-changing function and $\int_{\Omega}Q(x)dx < 0$, under the condition of $f(x, u)$. Using the space decomposition $H^1(\Omega) = span 1\oplus V$, where $V = \{v\in H^1(\Omega): \int_{\Omega}vdx = 0\}$, the author obtained the existence of two distinct solutions by the variational method.

In ^[14], Lei et al. considered the following Kirchhoff type equation with critical exponent

where $\Omega$ $\subset$ $\mathbb{R}^3$ is a smooth bounded domain, $a, b > 0$, $1 < q < 2$, $\lambda > 0$ is a parameter. They obtained the existence of a positive ground state solution for $0\leq\beta < 2$ and two positive solutions for $3-q\leq\beta < 2$ by the Nehari manifold method.

In ^[34], Zhang obtained the existence and multiplicity of nontrivial solutions of the following equation

where $\Omega$ is an open bounded domain in $\mathbb{R}^3$, $a, b > 0$, $1 < q < 2$, $\lambda\geq0$ is a parameter, $f(x, u)$ and $Q(x)$ are positive continuous functions satisfying some additional assumptions. Moreover, $f(x, u)\thicksim|u|^{p-2}u$ with $4 < p < 6$.

Comparing with the above mentioned papers, our results are different and extend the above results to some extent. Specially, motivated by ^[34], we suppose $Q(x)$ changes sign on $\Omega$ and $f(x, u)\equiv0$ for (1.5). Since (1.1) is critical growth, which leads to the cause of the lack of compactness of the embedding $H^1(\Omega)\hookrightarrow L^{6}(\Omega)$, we overcome this difficulty by using P.Lions concentration compactness principle ^[19]. Moreover, note that $Q(x)$ changes sign on $\Omega$, how to estimate the level of the mountain pass is another difficulty.

We define the energy functional corresponding to problem (1.1) by

A weak solution of problem (1.1) is a function $u\in H^1(\Omega)$ and for all $\varphi\in H^1(\Omega)$ such that

Our main results are the following:

Theorem 1.1. Assume that $1 < q < 2$ and $Q(x)$ changes sign on $\Omega$. Then there exists $\Lambda_{0} > 0$ such that for every $\lambda\in(0, \Lambda_{0})$, problem $(1.1)$ has at least one nontrivial solution.

Theorem 1.2. Assume that $1 < q < 2$, $3-q < \beta < \frac{6-q}{2}$ and $Q(x)$ changes sign on $\Omega$, there exists $\Lambda_{*} > 0$ such that for all $\lambda\in(0, \Lambda_{*})$. Then problem $(1.1)$ has at least two nontrivial solutions.

Throughout this paper, we make use of the following notations:

● The space $H^1(\Omega)$ is equipped with the norm $\|u\|_{H^1(\Omega)}^2 = \int_{\Omega}(|\nabla u|^2+u^2)dx$, the norm in $L^p(\Omega)$ is denoted by $\|\cdot\|_p$.

● Define $\|u\|^2 = \int_{\Omega}(a|\nabla u|^2+u^2)dx$ for $u\in H^1(\Omega)$. Note that $\|\cdot\|$ is an equivalent norm on $H^1(\Omega)$ with the standard norm.

● Let $D^{1, 2}(\mathbb{R}^3)$ is the completion of $C_0^{\infty}(\mathbb{R}^3)$ with respect to the norm $\|u\|_{D^{1, 2}(\mathbb{R}^3)}^2 = \int_{\mathbb{R}^3}|\nabla u|^2dx$.

● $0 < Q_M = \max_{x\in \bar{\Omega}}Q(x),$ $0 < Q_m = \max_{x\in \partial\Omega} Q(x)$.

● $\Omega^+ = \{x\in \Omega:Q(x) > 0\}$ and $\Omega^- = \{x\in \Omega:Q(x) < 0\}$.

● $C, C_1, C_2, \dots$ denote various positive constants, which may vary from line to line.

● We denote by $S_\rho$ (respectively, $B_\rho$) the sphere (respectively, the closed ball) of center zero and radius $\rho$, i.e. $S_\rho = \{u\in H^{1}(\Omega): \|u\| = \rho\},$ $B_\rho = \{u\in H^{1}(\Omega): \|u\|\leq \rho\}$.

● Let $S$ be the best constant for Sobolev embedding $H^1(\Omega)\hookrightarrow L^{6}(\Omega)$, namely

● Let $S_0$ be the best constant for Sobolev embedding $D^{1, 2}(\mathbb{R}^3)\hookrightarrow L^{6}(\mathbb{R}^3)$, namely

2.   Proofs of theorems

In this section, we firstly show that the functional $I_\lambda(u)$ has a mountain pass geometry.

Lemma 2.1. There exist constants $r, \rho, \Lambda_0 > 0$ such that the functional $I_{\lambda}$ satisfies the following conditions for each $\lambda\in (0, \Lambda_0)$:

$(\mathrm{i})$ $I_{\lambda}|_{u\in S_\rho}\geq r > 0$; $\inf_{_{u\in B_\rho}}I_{\lambda}(u) < 0$.

$(\mathrm{ii})$ There exists $e\in H^1(\Omega)$ with $\|e\| > \rho$ such that $I_{\lambda}(e) < 0$.

Proof. $(\mathrm{i})$ From $(P_1)$, by the H$\ddot{\mathrm{o}}$lder inequality and the Sobolev inequality, for all $u\in H^1(\Omega)$ one has

and there exists a constant $C > 0$, we get

Hence, combining (2.1) and (2.2), we have the following estimate

Set $h(t) = \frac{1}{2}{t}^{2-q}-\frac{C}{6}{S^{-3}t}^{6-q}$ for $t > 0$, then there exists a constant $\rho = \left(\frac{3(2-q)S^3}{C(6-q)}\right)^{\frac{1}{4}} > 0$ such that $\max_{t > 0}h(t) = h(\rho) > 0$. Letting $\Lambda_0 = \frac{qS^{\frac{q}{2}}}{P(x_0)|\Omega|^{\frac{6-q}{6}}}h(\rho)$, there exists a constant $r > 0$ such that $I_{\lambda}|_{u\in S_{\rho}}\geq r$ for every $\lambda\in (0, \Lambda_0)$. Moreover, for all $u\in H^1(\Omega)\backslash\{0\}$, we have

So we obtain $I_{\lambda}(tu) < 0$ for every $u\neq 0$ and $t$ small enough. Therefore, for $\|u\|$ small enough, one has

$(\mathrm{ii})$ Let $v\in H^1(\Omega)$ be such that supp $v\subset\Omega^+$, $v\not\equiv 0$ and $t > 0$, we have

as $t\rightarrow\infty$, which implies that $I_{\lambda}(tv) < 0$ for $t > 0$ large enough. Therefore, we can find $e\in H^1(\Omega)$ with $\|e\| > \rho$ such that $I_{\lambda}(e) < 0$. The proof is complete.

Denote

Then we have the following compactness result.

Lemma 2.2. Suppose that $1 < q < 2$. Then the functional $I_{\lambda}$ satisfies the $(PS)_{c_{\lambda}}$ condition for every $c_{\lambda} < c_* =$ min $\{\Theta_1-D\lambda^{\frac{2}{2-q}}, \Theta_2-D\lambda^{\frac{2}{2-q}}\}$, where $D = \frac{2-q}{3q}(\frac{6-q}{4}P(x_0)S^{-\frac{q}{2}}|\Omega|^{\frac{6-q}{6}})^{\frac{2}{2-q}}$.

Proof. Let $\{u_n\}\subset H^1(\Omega)$ be a $(PS)_{c_\lambda}$ sequence for

It follows from (2.1), (2.3) and the H$\ddot{\mathrm{o}}$lder inequality that

Therefore $\{u_n\}$ is bounded in $H^1(\Omega)$ for all $1 < q < 2$. Thus, we may assume up to a subsequence, still denoted by $\{u_n\}$, there exists $u\in H^{1}(\Omega)$ such that

as $n\rightarrow\infty$. Next, we prove that $u_n\rightarrow u$ strongly in $H^1(\Omega)$. By using the concentration compactness principle (see ^[19]), there exist some at most countable index set $J$, $\delta_{x_j}$ is the Dirac mass at $x_j\subset \bar{\Omega}$ and positive numbers $\{\nu_j\}$, $\{\mu_j\}$, $j\in J$, such that

Moreover, numbers $\nu_{j}$ and $\mu_{j}$ satisfy the following inequalities

For $\varepsilon > 0$, let $\phi_{\varepsilon, j}(x)$ be a smooth cut-off function centered at $x_j$ such that $0\leq\phi_{\varepsilon, j}\leq 1,$ $|\nabla \phi_{\varepsilon, j}|\leq\frac{2}{\varepsilon}$, and

There exists a constant $C > 0$ such that

Since $|\nabla \phi_{\varepsilon, j}|\leq\frac{2}{\varepsilon}$, by using the H$\ddot{\mathrm{o}}$lder inequality and $L^2(\Omega)$-convergence of $\{u_n\}$, we have

where $C_1 > 0$, and we also derive that

Noting that $u_n\phi_{\varepsilon, j}$ is bounded in $H^1(\Omega)$ uniformly for $n$, taking the test function $\varphi = u_n\phi_{\varepsilon, j}$ in (2.3), from the above information, one has

so that

which shows that $\{u_n\}$ can only concentrate at points $x_{j}$ where $Q(x_{j}) > 0$. If $\nu_{j} > 0$, by (2.5) we get

From (2.5) and (2.6), we have

To proceed further we show that (2.7) is impossible. To obtain a contradiction assume that there exists $j_0\in J$ such that $\mu_{j_0}\geq\frac{bS_0^3+\sqrt{b^2S_0^6+4aS_0^3Q_M}}{2Q_M}$ and $x_{j_0}\in \Omega$. By (2.1), (2.3) and (2.4), one has

Set

then

we can deduce that $\min_{t\geq0}g(t)$ attains at $t_0 > 0$ and

Consequently, we obtain

where $D = \frac{2-q}{3q}\left(\frac{6-q}{4}P(x_0)S^{-\frac{q}{2}}|\Omega|^{\frac{6-q}{6}}\right)^{\frac{2}{2-q}}$. If $\mu_{j_0}\geq\frac{bS_0^3+\sqrt{b^2S_0^6+16aS_0^3Q_m}}{8Q_m}$ and $x_{j_0}\in \partial\Omega$, then, by the similar calculation, we also get

Let $c_* = \min\{\Theta_1-D\lambda^{\frac{2}{2-q}}, \Theta_2-D\lambda^{\frac{2}{2-q}}\}$, from the above information, we deduce that $c_\lambda\geq c_*$. It contradicts our assumption, so it indicates that $\nu_j = \mu_j = 0$ for every $j\in J$, which implies that

as $n\rightarrow\infty$. Now, we may assume that $\int_{\Omega}|\nabla u_n|^2dx\rightarrow A^2$ and $\int_{\Omega}|\nabla u|^2dx\leq A^2$, by (2.3), (2.4) and (2.8), one has

Then, we obtain that $u_n\rightarrow u$ in $H^1(\Omega)$. The proof is complete.

As well known, the function

satisfies

and

Let $\phi\in C^1(\mathbb{R}^3)$ such that $\phi(x) = 1$ on $B(x_M, \frac{R}{2})$, $\phi(x) = 0$ on $\mathbb{R}^3-B(x_M, R)$ and $0\leq\phi(x)\leq 1$ on $\mathbb{R}^3$, we set $v_\varepsilon(x) = \phi(x)U_{\varepsilon, x_M}(x)$. We may assume that $Q(x) > 0$ on $B(x_M, R)$ for some $R > 0$ such that $B(x_M, R)\subset \Omega$. From ^[4], we have

Moreover, by ^[28], we get

Then we have the following Lemma.

Lemma 2.3. Suppose that $1 < q < 2$, $3-q < \beta < \frac{6-q}{2}$, $Q_M > 4Q_m$, $(Q_1)$ and $(Q_2)$, then $\sup_{t\geq0}I_{\lambda}(tv_\varepsilon) < \Theta_1-D\lambda^{\frac{2}{2-q}}.$

Proof. By Lemma 2.1, one has $I_{\lambda}(tv_{\varepsilon})\rightarrow-\infty$ as $t\rightarrow\infty$ and $I_{\lambda}(tv_{\varepsilon}) < 0$ as $t\rightarrow 0$, then there exists $t_{\varepsilon} > 0$ such that $I_{\lambda}(t_{\varepsilon} v_{\varepsilon}) = \sup_{t > 0}I_{\lambda}(tv_{\varepsilon})\geq r > 0$. We can assume that there exist positive constants $t_1, t_2 > 0$ and $0 < t_1 < t_{\varepsilon} < t_2 < +\infty$. Let $I_{\lambda}(t_{\varepsilon}v_{\varepsilon}) = \beta(t_{\varepsilon}v_{\varepsilon})-\lambda\psi(t_{\varepsilon}v_{\varepsilon})$, where

and

Now, we set

It is clear that $\lim_{t\rightarrow0}h(t) = 0$ and $\lim_{t\rightarrow\infty}h(t) = -\infty$. Therefore there exists $T_{1} > 0$ such that $h(T_{1}) = \max_{t\geq0}h(t)$, that is

from which we have

By (2.11) we obtain

In addition, by $(Q_2)$, for all $\eta > 0$, there exists $\rho > 0$ such that $|Q(x)-Q_M| < \eta|x-x_M|$ for $0 < |x-x_M| < \rho$, for $\varepsilon > 0$ small enough, we have

where $C_1, C_2 > 0$ (independent of $\eta$, $\varepsilon$). From this we derive that

Then from the arbitrariness of $\eta > 0$, by (2.9) and (2.12), one has

Hence, it follows from (2.9), (2.10) and (2.13) that

where the constant $C_3 > 0$. According to the definition of $v_\varepsilon$, from ^[29], for $\frac{R}{2} > \varepsilon > 0$, there holds

where $C_4 > 0$ (independent of $\varepsilon, \lambda$). Consequently, from the above information, we obtain

Here we have used the fact that $\beta > 3-q$ and let $\varepsilon = \lambda^{\frac{2}{2-q}}$, $0 < \lambda < \Lambda_1 = \min\{1, (\frac{C_3+D}{C_4})^{\frac{2-q}{6-2q-2\beta}}\}$, then

The proof is complete.

We assume that $0\in\partial\Omega$ and $Q_m = Q(0)$. Let $\varphi\in C^1(\mathbb{R}^3)$ such that $\varphi(x) = 1$ on $B(0, \frac{R}{2})$, $\varphi(x) = 0$ on $\mathbb{R}^3-B(0, R)$ and $0\leq\varphi(x)\leq 1$ on $\mathbb{R}^3$, we set $u_\varepsilon(x) = \varphi(x)U_\varepsilon(x)$, the radius $R$ is chosen so that $Q(x) > 0$ on $B(0, R)\cap \Omega$. If $H(0)$ denotes the mean curvature of the boundary at $0$, then the following estimates hold (see ^[6] or ^[26])

where $A_3 > 0$ is a constant. Then we have the following lemma.

Lemma 2.4. Suppose that $1 < q < 2$, $3-q < \beta < \frac{6-q}{2}$, $Q_M\leq4Q_m$, $H(0) > 0$, Q is positive somewhere on $\partial\Omega$, $(Q_1)$ and $(Q_3)$, then $\sup_{t\geq0}I_{\lambda}(tu_\varepsilon) < \Theta_2-D\lambda^{\frac{2}{2-q}}.$

Proof. Similar to the proof of Lemma 2.3, we also have by Lemma 2.1, there exists $t_{\varepsilon} > 0$ such that $I_{\lambda}(t_{\varepsilon} u_{\varepsilon}) = \sup_{t > 0}I_{\lambda}(tu_{\varepsilon})\geq r > 0$. We can assume that there exist positive constants $t_1, t_2 > 0$ such that $0 < t_1 < t_{\varepsilon} < t_2 < +\infty$. Let $I_{\lambda}(t_{\varepsilon}u_{\varepsilon}) = A(t_{\varepsilon}u_{\varepsilon})-\lambda B(t_{\varepsilon}u_{\varepsilon})$, where

and

Now, we set

Therefore, it is easy to see that there exists $T_{2} > 0$ such that $f(T_{2}) = \max_{f\geq0}f(t)$, that is

From (2.17) we obtain

By the assumption $(Q_3)$, we have the expansion formula

Hence, combining (2.16) and (2.18), there exists $C_5 > 0$, such that

Consequently, by (2.14) and (2.15), similarly, there exists $\Lambda_2 > 0$ such that $0 < \lambda < \Lambda_2$, we get

where $C_6 > 0$ (independent of $\varepsilon, \lambda$). The proof is complete.

Theorem 2.5. Assume that $0 < \lambda < \Lambda_0$ ($\Lambda_0$ is as in Lemma 2.1) and $1 < q < 2$. Then problem $(1.1)$ has a nontrivial solution $u_\lambda$ with $I_\lambda(u_\lambda) < 0$.

Proof. It follows from Lemma 2.1 that

By the Ekeland variational principle ^[7], there exists a minimizing sequence $\{u_n\}\subset {\overline{B_\rho(0)}}$ such that

Therefore, there holds $I_\lambda(u_n)\rightarrow m$ and $I_\lambda'(u_n)\rightarrow 0$. Since $\{u_n\}$ is a bounded sequence and ${\overline{B_\rho(0)}}$ is a closed convex set, we may assume up to a subsequence, still denoted by $\{u_n\}$, there exists $u_{\lambda}\in{\overline{B_\rho(0)}}\subset H^{1}(\Omega)$ such that

By the lower semi-continuity of the norm with respect to weak convergence, one has

Thus $I_{\lambda}(u_\lambda) = m < 0$, by $m < 0 < c_\lambda$ and Lemma 2.2, we can see that $\nabla u_n\rightarrow \nabla u_\lambda$ in $L^2(\Omega)$ and $u_\lambda\not\equiv0$. Therefore, we obtain that $u_\lambda$ is a weak solution of problem (1.1). Since $I_\lambda(|u_\lambda|) = I_\lambda(u_\lambda)$, which suggests that $u_\lambda\geq0$, then $u_\lambda$ is a nontrivial solution to problem (1.1). That is, the proof of Theorem 1.1 is complete.

Theorem 2.6. Assume that $0 < \lambda < \Lambda_{*}$$(\Lambda_{*} = \min\{\Lambda_0, \Lambda_1, \Lambda_2\})$, $1 < q < 2$ and $3-q < \beta < \frac{6-q}{2}$. Then the problem (1.1) has a nontrivial solution $u_{1}\in H^1(\Omega)$ such that $I_{\lambda}(u_{1}) > 0$.

Proof. Applying the mountain pass lemma ^[3] and Lemma 2.2, there exists a sequence $\{u_n\}\subset H^1(\Omega)$ such that

where

and

According to Lemma 2.2, we know that $\{u_n\}\subset H^1(\Omega)$ has a convergent subsequence, still denoted by $\{u_n\}$, such that $u_n\rightarrow u_{1}$ in $H^1(\Omega)$ as $n\rightarrow\infty$,

which implies that $u_{1}\not\equiv0$. Therefore, from the continuity of $I'_\lambda$, we obtain that $u_{1}$ is a nontrivial solution of problem (1.1) with $I_{\lambda}(u_{1}) > 0$. Combining the above facts with Theorem 2.5 the proof of Theorem 1.2 is complete.

3.   Conclusions

In this paper, we consider a class of Kirchhoff type equations with Neumann conditions and critical growth. Under suitable assumptions on $Q(x)$ and $P(x)$, using the variational method and the concentration compactness principle, we proved the existence and multiplicity of nontrivial solutions.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant Nos. 11661021 and 11861021). Authors are grateful to the referees for their very constructive comments and valuable suggestions.

Conflict of interest

The authors declare no conflict of interest in this paper.