Sign-changing solutions for Schrödinger system with critical growth

1.   Introduction

In this paper, we consider the Schrödinger system with critical growth

where $\Omega\subset\mathbb{R}^3$ is a bounded domain with smooth boundary. This type of coupled systems have applications in some physical problem. In physics literatures the signs of the coupling constants $\beta_{ij}, \, i\not = j$ being positive or negative determine the nature of the system being attractive or repulsive. We refer to ^{[1,2,3,4,5,6,7]} for further references on subcritical and critical problems therein. Our main result is the following.

Theorem 1.1. Assume $5 < q < 6$, $\lambda_j > 0, \, j = 1, \cdots, k$, $\beta_{jj} > 0, \, j = 1, \cdots, k, \, \, \beta_{ij} = \beta_{ji}, \, i, j = 1, \cdots, k$, and there exists $\beta_0 > 0$ such that

Then the problem (P) has infinitely many solutions with all components being sign-changing.

It is worth mentioning that our Theorem 1.1 allows some pairs of components to be attractive, and others repulsive.

Example 1.1. (attractive) $\beta_{ij}\geq0, \, i\not = j, \, i, j = 1, \cdots, k$.

Example 1.2. (repulsive) $\beta_{ij}\leq0, \, i\not = j, \, i, j = 1, \cdots, k$ and the matrix $B = (\beta_{ij})$ is positive definite.

The problem (P) has a variational structure given by the functional

for $U = (u_1, \cdots, u_k)\in X = H_{0}^1(\Omega)\times\cdots\times H_{0}^1(\Omega)$, the $k$-fold product of $H_{0}^1(\Omega)$. In order to prove Theorem 1.1, we shall use a subcritical approximation scheme together with the method of invariant sets of descending flow, in particular the abstract theorem from ^[8,9]. We briefly outline our approaches here.

When we try to apply the abstract theorem to the functional $I$, we are faced with some difficulties. Firstly being in dimension $3$ the problem (P) is of critical Sobolev growth, and the functional $I$ fails to satisfy the Palais-Smale condition. From the classical work of ^[10] (see also for p-Laplacian in ^[11] and for coupled systems in ^[9,12]), using a subcritical approximation is an effective method to overcome this difficulty. Since we also need to study the nodal property, we shall use an alternative scheme of subcritical approximation as done in ^[13]. Our approach avoids passing to the limit of the subcritical problems and is easier to deal with nodal property of the solutions. Secondly in order to deal with nodal property of the solutions we employ the method of invariant sets of descending flow which has become a well developed method for constructing multiple nodal solutions, we refer ^{[8,9,14,15,16,17,18,19,20]} for further references therein. In particular we rely on the recent developments in ^[8,9]. To use the method of invariant set of descending flow one needs to construct certain invariant sets. This usually requires some additional property of the gradient flow (or a pseudo gradient flow). Our approximation approach also accommodates this issue well.

More precisely, we consider the perturbed functionals

for $U = (u_1, \cdots, u_k)\in X,$ where $2 < p < 3, \, 0 < \varepsilon\leq1$ and $g_\varepsilon$ is a smooth function satisfying

here $c_\varepsilon$ is a constant depending on $\varepsilon$. The critical points of $I_{p}^{(\varepsilon)}$ will be used as approximate solutions of the problem $(P)$ and it turns out that the approximate solutions converge to the solutions of the problem $(P)$ as the parameters $\varepsilon \downarrow 0, \, p \uparrow 3$. Technically, we will first construct sign-changing critical points $U$ of $I_{p}^{(\varepsilon)}$, then send $\varepsilon$ to zero while holding $p$ fixed to get sign-changing critical points $U$ with $||U||_\infty < \frac{1}{\varepsilon}$ for the functionals $I_p$ defined by

Then using the profile decomposition of approximation solutions we obtain solutions of $(P)$ by passing limit of $p\to 3$.

The paper is organized as follows. In Section 2 we will work on the perturbed functionals and construct multiple nodal solutions as approximating solutions to the original problem. Section 3 is devoted to the convergence analysis of the approximating solutions, then the proof of our main result follows.

2.   Critical points of the perturbed functionals

First we give the exact definition of the function $g_\varepsilon$. Let $\varphi$ be a smooth function such that $\varphi(t) = 1$ for $0\leq t\leq 1$, $\varphi(t) = \frac{1}{2}$ for $t\geq 2$ and $\varphi(t)$ is decreasing in $t$. Define $g(t) = {\text exp}\Big\{\int_{1}^t\frac{\varphi(\tau)}{\tau}\, d\tau\Big\}$ and, for $\varepsilon > 0$, $g_\varepsilon(t) = \frac{1}{\varepsilon} g(\varepsilon t)$. Then $g_\varepsilon$ satisfies (1.1) and

To construct multiple nodal solutions for the subcritical approximating problems we will make use of the following abstract theorem from ^[8,9]. In the following, $\gamma(A)$ denotes the genus of a symmetric and closed subset $A$ (see ^[21] for properties of the genus theory).

Theorem 2.1. Let $X$ be a Banach space, $f$ be an even $C^1$-functional on $X$, $A$ be an odd mapping from $X$ to $X$, and $P_j, Q_j$, $j = 1, \cdots, k$ be open convex subsetsof $X$ with $Q_j = -P_j$. Denote $W = \bigcup\limits_{j = 1}^k(P_j\cup Q_j),$ $\Sigma = \bigcap\limits_{j = 1}^k(\partial P_j\cap \partial Q_j)$. Assume

($I_1$) $f$ is an even $C^1$-functional on $X$, and satisfies the Palais-Smale condition.

($I_2$) there exists $c^* > 0$ such that

($A_1$) given $b_0 > 0, \, c_0 > 0$, there exists $b = b(b_0, c_0)$ such that if $|f(x)|\leq c_0, \, \|Df(x)\|\geq b_0$, then

($A_2$) $A(\partial P_j)\subset P_j, \, A(\partial Q_j)\subset Q_j$, $j = 1, \cdots, k$.

Introduce a sequence of families of subsets of $X$.

Define

Assume

$(\Gamma)$ $\Gamma_l, \, l = 1, 2, \cdots$ are nonempty.

Then

$(1)$ $c_l\geq c^*, \, l = 1, 2, \cdots$ are critical values of $f$.

$(2)$ $c_1\leq c_2\leq\cdots$ and $c_l\rightarrow +\infty$ as $l\rightarrow \infty$.

$(3)$ The critical set $K_{c_l}^*$ is nonempty, where

$(4)$ If $c_l = c_{l+1} = \cdots = c_{l+k-1}$ for some integer $k$, then $\gamma(K_{c_l}^*)\geq k$.

$(5)$ If $X$ is a Hilbert space, $f$ is a $C^2$-functional and for every critical point $x$ of $f$, $D^2f(x)$ is a Fredholm operator, thenthere exists $x\in K_{c_l}^*$ with $m^*(x, f)\geq l$, where $m^*$ is the augmented Morse index of $x$.

Lemma 2.1. The functional $I_{p}^{(\varepsilon)}$ is of $C^2$-class, satisfies the Palais-Smale condition and $D^2I_{p}^{(\varepsilon)}(U)$ is a Fredholm operator for everycritical point $U$ of the functional $I_{p}^{(\varepsilon)}$.

Proof. Note that $2 < p < q < 6$. Thus it is easy to verify that $I_{p}^{(\varepsilon)}$ is a $C^2$-functional. Moreover

We have

In the above we have used the fact that $g_{\varepsilon}^\prime (t)t\geq \frac{1}{2}g_\varepsilon(t)$. By (2.3), any Palais-Smale sequence of $I_{p}^{(\varepsilon)}$ is bounded in $X$. Since the functional $I_{p}^{(\varepsilon)}$ is of subcritical growth, by a standard argument $I_{p}^{(\varepsilon)}$ satisfies the Palais-Smale condition.

Let $U$ be a critical point of $I_{p}^{(\varepsilon)}$. By the regularity theory, $U$ is continuous on $\bar \Omega$ and therefore is uniformly bounded. Then the operator $D^2I_{p}^{(\varepsilon)}(U)$ is a compact perturbation of the Laplacian operator, hence a Fredholm operator.

Lemma 2.2. Denote $\beta^{\pm}_{ij} = \max(\pm\beta_{ij}, \ 0)$. Define a compact and odd operator $A$: $U = (u_1, \cdots, u_k)\in X\mapsto V = (v_1, \cdots, v_k) = AU\in X$ by the equations

for $\phi = (\varphi_1, \cdots, \varphi_k)\in X$. Then the following property holds:if $\|I_{p}^{(\varepsilon)}(U)\|\leq c_0$ and $\|DI_{p}^{(\varepsilon)}(U)\|\geq b_0 > 0$, there exists $b = b(b_0, c_0)$ such that

Proof. By (2.2) and (2.4), we have

for $\phi = (\varphi_1, \cdots, \varphi_k)\in X$. Choose $\phi = U-V$, we obtain

Hence,

and

It follows from (2.1), (2.5) and (2.7) that

which implies that

Choose $s$ such that $2 < s < p < q$. Then we deduce from (2.1) and (2.5) that

Notice that the matrix $B = (\beta_{ij})$ is positively definite, we have

It implies from (2.1) that

From (2.7), (2.10) and (2.11), for sufficiently small $\sigma > 0$, we obtain

Therefore,

According to (1.1), there exists $C_{\varepsilon} > 0$ such that $t^{\frac{1}{2}} < C_{\varepsilon}(1+g_{\varepsilon}(t))$. Hence

Combining (2.9) with (2.13), we obtain

For $\sigma > 0$ small enough, we have

and

By the above inequalities, we have

If $\|I_{p}^{(\varepsilon)}(U)\|\leq c_0$ and $\|DI_{p}^{(\varepsilon)}(U)\|\geq b_0 > 0$, we deduce from (2.13) that there exists $b = b(b_0, c_0)$ such that $\|U-V\| > b$. It follows from (2.6) that

That $A$ is odd is obvious. The compactness of $A$ follows the regularity theory and the subcritical growth.

Lemma 2.3. Let $P_j, Q_j, \, j = 1, \cdots, k$ be open convex subsets of $X$, defined by

Then there exists a constant $\delta_1 > 0$ such that for $0 < \delta < \delta_1$ it holds that

Proof. Choose $\phi = V^+ = (v_{1}^+, \cdots, v_{k}^+)$ as test function in (2.4). Then we have

where $c_\varepsilon'$ is a constant depending on $\varepsilon$ but independent of $p\in [\frac{5}{2}, 3]$. By (2.15) we have

Choose $\delta_1$ such that $c'_\varepsilon(\delta_{1}^{p-2}+\delta_{1}^{q-2})\leq\frac{1}{2}$. Then for $0 < \delta < \delta_1$ and $U\in\partial Q_j$ we have $\|u_{j}^+\|_{L^6(\Omega)} = \delta$ and

That is for $U\in\partial Q_j$, we have $V = AU\in Q_j$ and $A(\partial Q_j)\subset Q_j$. Similarly $A(\partial P_j)\subset P_j, \, j = 1, \cdots, k$.

Lemma 2.4. There exists $\delta_2 > 0$ such that for $0 < \delta < \delta_2$ there exists $c^* > 0$ independent of $\varepsilon \in (0, 1], p\in [2,3]$ such that

Proof.

For $U\in\Sigma = \bigcap\limits_{j = 1}^k(\partial P_j\cap \partial Q_j),$

Hence

provided $c_2\big((\sqrt[6]{2k}\delta_2)^{2p-2}+(\sqrt[6]{2k}\delta_2)^{q-2}\big)\leq\frac{1}{2}c_1$ and $0 < \delta < \delta_2$.

Now we define a sequence of critical values of the perturbed functional $I_{p}^{(\varepsilon)}$

where $W = \bigcup\limits_{j = 1}^k(P_j\cup Q_j)$ and for $l = 1, 2, \cdots$

Proposition 2.1. $c_l(\varepsilon, p), \, l = 1, 2, \cdots$ are critical values of the functional $I_{p}^{(\varepsilon)}$. There exists $U_l(\varepsilon, p)\in X$ such that $I_{p}^{(\varepsilon)}(U_l(\varepsilon, p)) = c_l(\varepsilon, p),$ $DI_{p}^{(\varepsilon)}(U_l(\varepsilon, p)) = 0$, $U_l(\varepsilon, p)$ is sign-changing and the augmented Morse index$m^*(U_l(\varepsilon, p), I_{p}^{(\varepsilon)})\geq l$. Moreover there exists a constant $L_l$, independent of $\varepsilon, \, p$, such that

Proof. We apply Theorem 2.1 to our functional $I_{p}^{(\varepsilon)}$. We have verified the conditions $(I_1)$(Lemma 2.1), $(I_2)$(Lemma 2.4), $(A_1)$(Lemma 2.2) and $(A_2)$(Lemma 2.3). We need only to verify the condition $\Gamma$. Denote $n = l+k$. Choose $nk$ functions $v_i\in C_{0}^\infty(\Omega), \, i = 1, \cdots, nk$ with disjoint supports. Denote

By Lemma 4.2 in ^[9] for $R$ sufficiently large $F_l\in\Gamma_l$, $\Gamma_l$ is nonempty. Now we have

where

3.   Convergence of the approximate solutions

As we have mentioned that the critical points of the perturbed functional $I_{p}^{(\varepsilon)}$ will be used as approximate solutions of the original problem (P). Now we prove that these approximate solutions converge to solutions of the original problem. More precisely, we show for any given integer $k$, we can find $\epsilon > 0$ small so that the functional $I_{p}^{(\varepsilon)}$ has $k$ nodal critical points whose $L^\infty$ norm all less than $\frac{1}{\varepsilon}$ (therefore they are critical points of the functional $I_p$). Then we send $p$ to $3$ to get solutions of the original problem.

Lemma 3.1. Assume that $U\in X$ satisfies $I_{p}^{(\varepsilon)}(U)\leq L, \, DI_{p}^{(\varepsilon)}(U) = 0$, where $L$ is independent of $\varepsilon, p$. Then there exists a constant $\bar{\varepsilon} = \bar{\varepsilon}(L)$ such that if $0 < \varepsilon < \bar{\varepsilon}$, $I_p(U) = I_{p}^{(\varepsilon)}(U), \, DI_p(U) = DI_{p}^{(\varepsilon)}(U) = 0$, $U$ is a critical point of $I_p$.

Proof. By (2.3), we have

There exists a constant $M$, independent of $\varepsilon, \, p$, such that

Choose $\bar{\varepsilon} = \frac{1}{2M}$. Then for $0 < \varepsilon < \bar{\varepsilon}$,

Hence for $0 < \varepsilon < \bar{\varepsilon}$, $I_{p}(U) = I_{p}^{(\varepsilon)}(U),$ $DI_p(U) = DI_{p}^{(\varepsilon)}(U) = 0$.

Lemma 3.2. Assume $p_n\in(2, 3]$, $p_n\rightarrow 3$, $U_n\in X$, $n = 1, 2, \cdots$ such that $I_{p_n}(U_n)\leq L, DI_{p}(U_n) = 0$, where $L$ is independent of $p$, and $U_n$is sign-changing. Then $U_n$ is bounded in $X$. Assume $U_n\rightharpoonup U$ in $X$. Then $U_n\rightarrow U$ in $X$, $I(U)\leq L, \, DI(U) = 0$ and $U$ is sign-changing.

Proof. Again we have

So $U_n$ is bounded in $X$. Assume $U_n\rightharpoonup U$ in $X$. We have the following profile decomposition ^[22]

where $V_k\in\mathcal{D} = \mathcal{D}(\mathbb{R}^3), \, R_n\in\mathcal{D}, \, x_{n, k}\in\overline{\Omega}, \, \sigma_{n, k}\rightarrow +\infty$, $R_n\rightarrow0$ in $L^6(\Omega)$ as $n\rightarrow \infty$.

Assume $\sigma_n = \sigma_{n, 1} = \min\{\sigma_{n, k}|\, k\in\Lambda\}, \, x_n = x_{n, 1}$. The following claim can be proved as in ^[9,10,11].

Claim. There exist positive constant $c, \, \overline{c}$ such that

where $\mathcal{A}_n$ is called a safe region and defined by

Let $U_n = (u_1, \cdots, u_k)\in X$ be a critical point of $I_{p_n}$. The following local Pohožaev identity holds (e.g., ^[9]):

where $D_n = B_{(\bar{c}+3)\sigma_{n}^{-\frac{1}{2}}}(x_n)$, $\partial_{e}D_n = \partial D_n\cap \partial\Omega$, $n$ is the outward normal to $\partial\Omega$, $x^*\in\mathbb{R}^N, \, \eta\in C_{0}^\infty(\mathbb{R}^3)$ such that $\eta(x) = 1$ for $|x-x^*|\leq(\bar{c}+2)\sigma_{n}^{-\frac{1}{2}},$ $\eta(x) = 0$ for $|x-x^*|\geq(\bar{c}+3)\sigma_{n}^{-\frac{1}{2}}$ and $|\nabla \eta|\leq 2\sigma_{n}^{\frac{1}{2}}.$

Choose $x^*$ such that $|x^*-x|\leq (\bar{c}+8)\sigma_{n}^{-\frac{1}{2}}$ and $(x-x^*, n)\leq 0$ for all $x\in \partial_{e}D_n$. If $\partial_{e}D_n = \emptyset$ we simply choose $x^* = x_n$. With this choice of $x^*$ and the fact $2p_n\leq 6$ we have

The integrals of the right hand side of (3.4) are taken over the domain $\mathcal{A}_n$ due to the fact that $\nabla\eta = 0$ outside $\mathcal{A}_n$. Hence by the claim (3.2), we have

For the left hand side of (3.4), we have, keeping the profile decomposition (3.2) in mind

In the above we assume $\sigma_{n}^{-\frac{1}{2}}U_n(\sigma_{n}^{-1}\cdot+x_n)\rightharpoonup V$ in $\mathcal{D}$ and choose $L > 0$ such that $\int_{|y|\leq L}|V|^q\, dx > 0$. Because $q > 5$ we arrive at a contradiction for $n$ large in

Hence the index set $\Lambda$ in the profile decomposition (3.1) is empty and $U_n\rightarrow U$ in $L^6(\Omega)$, which implies that $U_n\rightarrow U$ in $X$ due to the fact $DI_{p_n}(U_n) = 0$. Therefore $I(U) = \lim\limits_{n\rightarrow \infty}I_{p_n}(U_n)\leq L$ and $DI(U) = \lim\limits_{n\rightarrow \infty}DI_{p_n}(U_n) = 0$.

Finally we prove that $U$ is sign-changing. Denote $U_n = (u_{1n}, \cdots, u_{kn})$, $U = (u_1, \cdots, u_k)$. We have

Choosing $\varphi_j = u_{jn}^+$, we have

in which we used that $\|U_n\|^2$ is bounded by $4L$. Hence there exists $\delta > 0$ such that $\|u_{jn}^+\|_{L^6(\Omega)}\geq\delta$ and $\|u_{j}\|_{L^6(\Omega)} = \lim\limits_{n\rightarrow \infty}\|u_{jn}^+\|_{L^6(\Omega)}\geq\delta > 0$. Similarly $\|u_{j}^{-}\|_{L^6(\Omega)}\geq\delta > 0, \, j = 1, \cdots, k$ and we have $U = (u_1, \cdots, u_k)$ is sign-changing.

Proof of Theorem 1.1. First, obviously functions in $X\backslash W$ are sign-changing. Given an integer $l$, by Proposition 2.1 the functional $I_{p}^{(\varepsilon)}, \, 0 < \varepsilon\leq 1, \, 2 < p < 3$ has a sign-changing critical point $U_l(\varepsilon, p)$ with the augmented Morse index $m^*(U_l(\varepsilon, l), \, I_{p}^{(\varepsilon)})\geq l$. Moreover, there exists a constant $L_l$, independent of $\varepsilon, \, p$, such that $I_{p}^{(\varepsilon)} (U_l(\varepsilon, p))\leq L_l$.

By Lemma 3.1 there exists $\bar{\varepsilon} = \bar{\varepsilon}(L_l)$, independent of $p$, such that $0 < \varepsilon < \bar{\varepsilon}, \, I_{p}(U_l(\varepsilon, p)) = I_{p}^{(\varepsilon)}(U_l(\varepsilon, p))\leq L_l$, $DI_{p}(U_l(\varepsilon, p)) = DI_{p}^{(\varepsilon)}(U_l(\varepsilon, p)) = 0$. Denote $U_l(p) = U_l(\varepsilon, p)$. Then $U_l(p)$ is a sign-changing critical point of the functional $I_p$ with the augmented Morse index $m^*(U_l(p), \, I_p)\geq l$. Moreover $I_p(U_l(p))\leq L_p$ for $2 < p < 3$.

Choose $p_n\in(2, 3), \, p_n\rightarrow 3$. By Lemma 3.2, $U_l(p_n)$ is bounded in $X$. Assume $U_l(p_n)\rightharpoonup U_l$ in $X$. Then $U_l(p_n)\rightarrow U_l$ in $X$, $I(U_l)\leq L_l, \, DI(U_l) = 0$, $U_l$ is sign-changing, and the augmented Morse index

$U_l$ is a sign-changing critical point of the functional $I$ with $m^*(U_l, \, I)\geq l$. Since the integer $l$ is arbitrary, $I$ has infinitely many sign-changing critical points, that is the problem (P) has infinitely many sign-changing solutions. Finally we prove $I(U_l)\rightarrow +\infty$ as $l\rightarrow \infty$. Otherwise $I(U_l)\leq L, \, DI(U_l) = 0$. By Lemma 3.2 $U_l$ is bounded in $X$. Assume $U_{l_n}\rightharpoonup U$ in $X$ as $l_n\rightarrow \infty$. Then by Lemma 3.2, $U_{l_n}\rightarrow U$ in $X$, $DI(U) = 0$. Therefore

we arrive at a contradiction.

Acknowledgments

The work is supported by NSFC (11771324, 11831009, 11861021, 12071438).

Conflict of interest

The authors declare there is no conflict of interest.