1. Introduction

In the last decades, the stable and finite Morse index sign-changing solutions of weighted p-Laplace equations on unbounded domains of $\mathbb{R}^N$ have received a lot of attention (see e.g., [Reference Cowan and Fazly1–Reference Dancer, Du and Guo3, Reference Farina6, Reference Farina, Mercuri and Willem7, Reference Le9, Reference Le11, Reference Le12, Reference Rahal and Harrabi15, Reference Rahal16] and the references therein). The definition of stability is motivated by a phenomenon in physical sciences, which states that a system is in a stable state if it can recover from small perturbations. We refer to the monograph [Reference Dupaigne5] for more discussions on the physical motivation and mathematical background of stable solutions.

Liouville theorems for stable solutions, which concern about nonexistence of this particular type of solutions, have drawn much attention in the last decade. In his celebrated article [Reference Farina6], Farina established a sharp Liouville theorem for stable classical solutions to the problem

(1)\begin{align} -\text{div}\left(\omega_1(x)\nabla u\right)=\omega_2(x)|u|^{q-1}u \quad \text{in}\ \mathbb{R}^N \end{align}

with $\omega_1(z)=\omega_2(z)\equiv1$ and q > 1. He showed that the problem does not admit any non-trivial stable C ² solution if and only if $1 \lt q \lt q_c(N), $ where

\begin{equation*} {q_c}(N): = \left\{ {\matrix{{ + \infty \;\;} \hfill & {{\rm{if N}} \le {\rm{10}},} \hfill \cr {{{{{(N - 2)}^2} - 4N + 8\sqrt {N - 1} } \over {(N - 2)(N - 10)}}\;\;} \hfill & {{\rm{if N}} \ge {\rm{11}}.} \hfill \cr } } \right.\end{equation*}

After that, the above results have been extended to the weighted case $\omega_1(x)\not\equiv1$ or $\omega_2(x)\not\equiv1$ in [Reference Cowan and Fazly1, Reference Dancer, Du and Guo3, Reference Rahal16, Reference Wang and Ye19]. In [Reference Dancer, Du and Guo3], under the restriction that the solutions are locally bounded, the authors presented the non-existence of non-trivial stable weak solutions of problem (1). Later, this restriction was removed in [Reference Wang and Ye19]. In [Reference Cowan and Fazly1], under various assumptions on $\omega_1(x)$ and $\omega_2(x)$, Cowan and Fazly established several Liouville-type theorems for stable positive classical solutions of problem (1). In particular, they examined a specific class of weights $\omega_1(x)= (|x|^2+1)^{\frac{\gamma_1}{2}}g(x)$ and $\omega_2(x)=(|x|^2+1)^{\frac{\gamma_2}{2}}g(z)$, where g is a positive function with a finite limit at $\infty$. For this class of weights, non-existence results are optimal.

Recently, Zhao [Reference Zhao20] studied the non-existence of finite Morse index solution for the equation

(2)\begin{equation} -\Delta u= \left(|z|^{-\gamma} \ast |u|^{q+1}\right) \left|u\right|^{q-1}u \quad \mbox{in}\; \mathbb{R}^N, \; N \gt 2. \end{equation}

In [Reference Zhao20], the author showed that problem (2) has no non-trivial solution with finite Morse index if $0 \lt \gamma \lt \min\{4, N\}$ and $1 \lt q \lt \frac{N+2-\gamma}{N-2} $. Notice that the right-hand side of Equation (2) is a non-local term which is usually referred to as the Hartree-type non-linearity in the literature. This kind of equation is usually called the Choquard-type equation since, in 1976, a similar equation as Equation (2) was used by P. Choquard to describe an electron trapped in its hole, in a certain approximation to Hartree–Fock theory of one component plasma [Reference Lieb13]. In some contexts, equation of type (2) is also called the non-linear Schrödinger–Newton equation or the stationary Hartree equation. The second author [Reference Le10] proved that this equation does not possess a positive solution for $1 \lt q \lt \frac{N+2-\gamma}{N-2}$ by using the moving plane method. In [Reference Le9], with the help of Farina’s approach, the second author showed that Equation (2) has no non-trivial stable weak solution if $0 \lt \gamma \lt \min\{4, N\}$, N > 2 and q > 1. This phenomenon is quite different from that of the Lane–Emden equation studied by Farina [Reference Farina6], where such a result only holds for low exponents in high dimensions.

Liouville theorems for $C^{1,\alpha}$ solutions of the p-Laplace Hartree equation

\begin{equation*} -\Delta_p u = \left(\frac{1}{|x|^{N-\alpha}}*|u|^q\right) |u|^{q-2}u \quad\text{in}\ \mathbb{R}^N, \end{equation*}

was also examined by the second author [Reference Le12]. He proved that if $2\le p \lt N$, $\max\{0,N-2p\} \lt \alpha \lt N$, $p \lt q \lt q_c$ and u is stable, then $u\equiv0$. Here q_c is a new critical exponent, which equals infinity when $\frac{N+\alpha}{N-p} \ge \frac{p+1}{2}$. He also showed that if $p \lt q \lt \frac{p(N+\alpha)}{2(N-p)}$ and u is stable outside a compact set or has a finite Morse index, then $u\equiv0$. The results in [Reference Le12] cover the ones in [Reference Le9, Reference Zhao20] when p = 2.

Besides the standard quasilinear operators, the so-called double phase problem

(3)\begin{equation} -\text{div} (|\nabla u|^{p-2} \nabla u + \gamma_1(x)|\nabla u|^{q-2} \nabla u) = \gamma_2(x)|u|^{r-1}u \quad\text{in}\ \mathbb{R}^N \end{equation}

gets a lot of attention in recent years. The operator on the left-hand side of Equation (3) is called the double phase operator since its behaviour switches between two different elliptic situations depending on the values of the weight function w. This kind of problem and the associated energy functionals arise in many applications. In the non-linear elasticity theory, the modulating coefficient w deforms the geometry of composites made of two different materials with distinct power hardening exponents q and p. Zhikov et al. [Reference Jikov, Kozlov and Oleinik8, Reference Zhikov21] used double phase functionals to describe models of strongly anisotropic materials in the context of homogenization. Double phase functionals also play an important role in the study of duality theory and the context of the Lavrentiev phenomenon [Reference Zhikov22].

Recently, the second author [Reference Le11] obtained classification for stable sign-changing solutions to problem (3) as follows.

Theorem A. (see [Reference Le11, Theorem 1]). Let u be a stable solution of Equation (3), where $q \ge p\ge2$, $r \gt q-1$ and $\gamma_1,\gamma_2 \in L^1_{\rm loc}(\mathbb{R}^N)$ are two non-negative functions such that $\gamma_1(x) \le C_1|x|^a$ and $\gamma_2(x) \ge C_2|x|^b$ for all $|x| \gt R_0$, with $R_0,C_1,C_2 \gt 0$ and $a,b\in\mathbb{R}$. Assume that

\begin{equation*} N \lt N^\sharp := \min\left\{\frac{p(\beta_0+r) + b(\beta_0+p-1)}{r-p+1}, \frac{(q-a)(\beta_0+r) + b(\beta_0+q-1)}{r-q+1}\right\}, \end{equation*}

where

\begin{equation*} \beta_0 := \frac{2r -q + 1 + 2\sqrt{r(r-q+1)}}{q-1}. \end{equation*}

Then $u \equiv 0$.

Theorem B. (see [Reference Le11, Theorem 2]). Let u be a solution of Equation (3) with $\gamma_1(x) =|x|^a$ and $\gamma_2(x) =|x|^b$ such that u is stable outside a compact set, where $q \gt p\ge2$ and $r \gt q-1$. Assume furthermore that $|\nabla u|^{p-2} \nabla u + |x|^a|\nabla u|^{q-2} \nabla u \in W^{1,2}_{\rm loc}(\mathbb{R}^N,\mathbb{R}^N)$.

1. (i) If $\frac{N+b}{r+1} \gt \max\left\{\frac{N-p}{p}, \frac{N-q+a}{q}\right\}$, then $u \equiv 0$.

2. (ii) If $\frac{N+b}{r+1} = \max\left\{\frac{N-p}{p}, \frac{N-q+a}{q}\right\}$, then we have the identity

   \begin{equation*} \frac{N-p}{p} \int_{\mathbb{R}^N} |\nabla u|^p\, {\rm d}x + \frac{N-q+a}{q} \int_{\mathbb{R}^N} |x|^a |\nabla u|^q\, {\rm d}x = \frac{N+b}{r+1} \int_{\mathbb{R}^N} |x|^b |u|^{r+1} \,{\rm d}x \lt \infty. \end{equation*}

In this paper, we prove analogous results for the double phase problem involving a non-local term

(4)\begin{equation} -\text{div} (|\nabla u|^{p-2} \nabla u + w(x)|\nabla u|^{q-2} \nabla u) = \left(\frac{1}{|x|^{N-\mu}}*f|u|^r\right) f(x)|u|^{r-2}u \quad\text{in}\ \mathbb{R}^N, \end{equation}

where $q\ge p\ge2$, r > q, $0 \lt \mu \lt N$ and $w,f \in L^1_{\rm loc}(\mathbb{R}^N)$ are two non-negative functions such that $w(x) \le C_1|x|^a$ and $f(x) \ge C_2|x|^b$ for all $|x| \gt R_0$, where $R_0,C_1,C_2 \gt 0$, $a,b\in\mathbb{R}$ and $\max\{0,N+2(a-q-b)\} \lt \mu \lt N$. The main feature of problem (4) is that it combines the double phase phenomenon on the left-hand side and the non-local phenomenon on the right-hand side. This causes some difficulty, which makes the study of such a problem interesting. The existence of solutions to problem (4) in bounded domains was obtained recently in [Reference Sun and Chang17]. More precisely, Sun and Chang [Reference Sun and Chang17] studied the problem

(5)\begin{equation} \begin{cases} -\text{div} (|\nabla u|^{p-2} \nabla u + w(x)|\nabla u|^{q-2} \nabla u) = \left(\frac{1}{|x|^{N-\mu}}*|u|^r\right) |u|^{r-2}u &\text{in}\ \Omega,\\ u=0 & \text{on}\ \partial\Omega, \end{cases} \end{equation}

where Ω is a bounded domain of $\mathbb{R}^N$ and $1 \lt p \lt q \lt N$. Using the constrained variational method and Brouwer degree theory, they proved the existence of least energy nodal solutions to Equation (5) under a subcritical assumption on r. Unlike problem (5), it is usually unfavourable for elliptic problems in the whole space such as Equation (4) to have positive solutions under some subcritical assumption. In fact, our main Liouville-type theorem holds for problem (4), with r being less than some critical exponent r_c (see Theorem 1 below).

We recall functional settings for double phase problems. Let $H: \Omega \times [0, \infty) \to [0, \infty)$ be the function $H(x,t) = t^p + w(x) t^q$, where $\Omega\subset\mathbb{R}^N$ is a domain of $\mathbb{R}^N$. We define

\begin{equation*} \rho_H(u) = \int_{\Omega} H(x, |u|) {\rm d}x = \int_{\Omega} (|u|^p + w(x) |u|^q) {\rm d}x \end{equation*}

and

\begin{equation*} L^H(\Omega) = \{u:\Omega\to\mathbb{R} \mid u\ \text{is measurable and}\ \rho_H(u) \lt \infty\}, \end{equation*}

which is called the Musielak–Orlicz space. This space is equipped with the Luxemburg norm

\begin{equation*} \|u\|_H = \inf\left\{\tau \gt 0 \mid \rho_H\left(\frac{u}{\tau}\right) \le 1 \right\}. \end{equation*}

Then we define the Musielak–Orlicz Sobolev space

\begin{equation*} W^{1,H}(\Omega) = \left\{u \in L^H(\Omega) \mid |\nabla u| \in L^H(\Omega)\right\}, \end{equation*}

which admits the norm

\begin{equation*} \|u\|_{1,H} = \|\nabla u\|_H + \|u\|_H. \end{equation*}

As usual, we define $W^{1,H}_0(\Omega)$ as the closure of $C_c^1(\Omega)$ with respect to the norm in $W^{1,H}(\Omega)$. Moreover, we set

\begin{equation*} W^{1,H}_{\rm loc}(\Omega) = \left\{u:\Omega\to\mathbb{R} \mid u\varphi \in W^{1,H}_0(\Omega)\ \text{for all}\ \varphi \in C_c^1(\Omega)\right\}. \end{equation*}

Although the $C^{1,\alpha}_{\rm loc}$ regularity of the solutions for p-Laplace equations is well known (see, for instance, [Reference DiBenedetto4, Reference Tolksdorf18]), the $C^{1,\alpha}_{\rm loc}$ regularity cannot be guaranteed for double phase problems due to the behaviour of the weight w. Therefore, it is more natural to work with the notion of weak local solutions as follows.

Definition 1. A function $u\in W^{1,H}_{\rm loc}(\mathbb{R}^N)$ is said to be a weak solution of Equation (4) if

\begin{equation*} \left(\frac{1}{|x|^{N-\mu}}*f|u|^r\right) f(x)|u|^{r-1} \in L^1_{\rm loc}(\mathbb{R}^N) \end{equation*}

and

(6)\begin{align} \int_{\mathbb{R}^N}& \left(|\nabla u|^{p-2} \nabla u + w(x)|\nabla u|^{q-2} \nabla u\right) \cdot \nabla\varphi \,{\rm d}x\\&\qquad = \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{f(x)|u(x)|^{r-2} u(x) \varphi(x) f(y)|u(y)|^r}{|x-y|^{N-\mu}} {\rm d}x\,{\rm d}y \end{align}

for all $\varphi \in C_c^1(\mathbb{R}^N)$. Furthermore, u is called a finite energy solution if

(7)\begin{equation} \int_{\mathbb{R}^N} \left(|\nabla u|^p + w(x)|\nabla u|^q\right) {\rm d}x + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{f(x)|u(x)|^{r} f(y)|u(y)|^r}{|x-y|^{N-\mu}} {\rm d}x\,{\rm d}y \lt \infty. \end{equation}

Notice that condition (7) is also used in the literature to characterize Coulomb–Sobolev spaces, see [Reference Mercuri, Moroz and Van Schaftingen14].

Motivated by [Reference Damascelli, Farina, Sciunzi and Valdinoci2, Reference Dancer, Du and Guo3, Reference Le11, Reference Le12, Reference Rahal and Harrabi15], in this paper, we are interested in Liouville theorems for stable and finite Morse index solutions of Equation (4), which are defined as follows.

Remark 1. By Schwartz’s inequality, if u is a stable solution to Equation (4), then

(8)\begin{equation} \begin{split} &(p-1) \int_{\mathbb{R}^N} |\nabla u|^{p-2} |\nabla \varphi|^2\, {\rm d}x +(q-1) \int_{\mathbb{R}^N} w(x) |\nabla u|^{q-2} |\nabla \varphi|^2 \,{\rm d}x \\ &- (r-1) \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{f(x)|u(x)|^{r-2} \varphi(x)^2 f(y)|u(y)|^r}{|x-y|^{N-\mu}} {\rm d}x\,{\rm d}y\\ &- r \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{f(x)|u(x)|^{r-2} u(x) \varphi(x) f(y)|u(y)|^{r-2} u(y) \varphi(y)}{|x-y|^{N-\mu}} {\rm d}x\,{\rm d}y \ge 0 \end{split} \end{equation}

for all $\varphi \in C^1_c(\mathbb{R}^N)$. Moreover, Equations (6) and (8) hold for all $\varphi \in W^{1,H}_0(\Omega)$ by density arguments.

As far as we know, there are no non-existence results on stable and finite Morse index solutions for Equation (4) with $q\geq p\geq2$ and $w\not\equiv0$. In this paper, we will establish some Liouville-type theorems for such solutions. Our first result reads as follows.

Remark 2. Clearly, assumptions (i) and (ii) imply $a\ge0$ and $q\le p+a \lt N+a$.

When $\frac{N+\mu +2b}{N+a-q} \lt \frac{q+1}{2}$, one can verify that $\alpha_0 \gt 1$ and α ₀ is the largest solution of the equation

\begin{equation*} \frac{1}{2} \left[\frac{(q+\alpha-1)(N+\mu+2b)}{N +a- q} - (\alpha-1)\right]= \frac{(q-1)(\alpha+1)^2}{4\alpha} + 1, \end{equation*}

see the proof of Lemma 3 in the next section for more detail. We also remark that when $w=f\equiv1$ and q = p, Theorem 1 basically reduces to Theorem 1 in [Reference Le12]. We stress, however, that only C ¹ solutions are considered in [Reference Le12]. Hence, the statement of Liouville theorems in our paper is more general even in this specific case.

To prove Theorem 1, we follow the approach in [Reference Damascelli, Farina, Sciunzi and Valdinoci2, Reference Le11, Reference Le12]. As in these references, we test Equations (6) and (8) with suitable truncated functions of powers of u and exploit the Young inequality several times to obtain a Caccioppoli-type estimate (see Equation (16) below). Then we can control a term $L_R(u)$, which depends on the radius R > 0 and contains double integrals on $|\nabla u|$ and $|u|$, by its powers (see Equation (25)). Here a new idea is presented to show that this term goes to 0 as $R\to+\infty$, which implies $u\equiv0$. In order for the last step to work, we need that θ in the inequality (27) is negative. A sufficient condition to ensure that is that (ii), (iii) and (iv) hold. More precisely, assumptions (ii) and (iii) are rather technical, so that Equation (29) and Lemma 3 can be proved and used in the last step. However, we expect that r_c in (iv) is sharp, which is the case when $q\le3$ as we mentioned in Remark 3.

In this paper, we also study solutions stable outside a compact set of the problem

(9)\begin{equation} -\text{div} (|\nabla u|^{p-2} \nabla u + |x|^a|\nabla u|^{q-2} \nabla u) = \left(\frac{1}{|x|^{N-\mu}}*|\cdot|^b |u|^r\right) |x|^b |u|^{r-2}u \quad\text{in}\ \mathbb{R}^N, \end{equation}

which is a special case of problem (4) with $w(x)=|x|^a$ and $f(x)=|x|^b$. We will prove the following result by exploiting a Pohožaev-type identity.

The rest of this paper is devoted to the proof of our main results. In § 2, we prove Theorem 1 by exploiting the method of integral estimates with some ideas from the works [Reference Le11, Reference Le12] of the second author. In § 3, we prove a Pohožaev-type identity and obtain some energy estimates. Then we use them to prove Theorem 2.

As usual, we use C to denote different positive constants whose values may change from line to line or even in the same line. Furthermore, we may append subscripts to C to specify its dependence on the subscript parameters. We also denote by B_R the ball centred at the origin with radius R > 0. We will drop notion ${\rm d}x$ in the integrals in $\mathbb{R}^N$ for brevity.