In-silico based identification and functional analyses of miRNAs and their targets in Cowpea (Vigna unguiculata L.)

1.   Introduction and main results

In this paper, the following quasilinear Schrödinger-Poisson equations varying with time $t$ are considered

where $\varphi = \varphi(x, t)$ is the wave function, $k$ is a positive constant, $\rho$ is a real function. Here we focus on the case $k = 1$, $\rho(s) = s$. This class of Schrödinger type equations with a repulsive nonlocal Coulombic potential is obtained by approximation of the Hartree-Fock equation which has been used to describe a quantum mechanical system of many particles, more physical meanings can be found in references ^[1,2,3,4] and its references.

We are interested in whether the equations $E(i)$ have solutions in the following form

where $\lambda\in \mathbb{R}$ is the frequency of occurrence of $u(x)$. After standing wave transformation $\varphi = \varphi(x, t) = e^{-i\lambda t}u(x)$, the following steady-state equation is obtained

Therefore, if $e^{-i\lambda t}u(x)$ is the standing wave solution of equation $E(i)$, if and only if $u(x)$ is the solution of Eq (1.1). At this time, there are two research methods for finding the solution of Eq (1.1), one of them is to treat $\lambda$ as a fixed parameter and the another is to treat $\lambda$ as a Lagrange multiplier. When $\lambda$ is regarded as a fixed parameter, the solution of Eq (1.1) can be obtained by finding the critical point of the following functional

When $\lambda$ is a Lagrange multiplier, its value range is unknown, and the study of this situation will be more interesting. Therefore, in this paper, we regard $\lambda$ as a Lagrange multiplier. After giving the mass $\int_{\mathbb{R}^3} u^2dx = c$ in advance, we study the solution of Eq (1.1) satisfying $||u||^2_2 = c$. So, the solution of Eq (1.1) satisfying this condition can be obtained by the critical point of the following functional $K(u)$ under constraint $S_c$

where

In this case, the parameter $\lambda$ cannot be fixed but instead appears as a Lagrange multiplier, if $u\in S_c$ is a minimizer of problem

then there exists $\lambda\in \mathbb{R}$ such that $K^\prime(u) = \lambda u$, namely, $(u, \lambda)$ is the solution of Eq (1.1) and satisfies $||u||_2^2 = c$. Since in literature there are no results available for the quasilinear Schrödinger-Poisson equation studied in this paper, we can only provide references for similar problems.

In ^[5,6,7,8,9], many authors studied Schrödinger-Poisson equations similar to the following

where $f\in C(\mathbb{R}, \mathbb{R})$, $V(x, t)$ is the potential function, the unknown function $\psi = \psi(x, t):\mathbb{R}^3\times [0, T]\rightarrow \mathbb{C}$ is the wave function. The existence and nonexistence of normalized solutions of the general Schrödinger-Poisson equation were established, depending strongly on the value of $p\in (2, 6)$ and of the parameter $c > 0$. That is, it is precisely proved that when $p\in (2, 3)$ and $c > 0$ are sufficiently small, the energy functional corresponding to Schrödinger-Poisson equation has a global minimum solution on $S_c$. When $p\in (3, \frac{10}{3})$, there exists a $c_0 > 0$ such that a solution exists if and only if $c\geq c_0$. When $p\in (\frac{10}{3}, 6)$, since the energy functional corresponding to Schrödinger-Poisso equations have no lower bound on $S_c$, it is impossible to find the global minimum solution on $S_c$. But in ^[10], the authors proved that for any $C > 0$ is small enough, the following Schrödinger-Poisso equations

has an energy minimum solution on $S_c$. However, in ^[11], considering that $H_r^1(\mathbb{R}^3)$ is compact embedded in $L^q(\mathbb{R}^3)\; (q\in (2, 6))$, the author proved that the above equation has infinitely many normalized radial solutions when $p\in (\frac{10}{3}, 6)$ and $c > 0$ is sufficiently small.

In ^[12,13,14], different authors studied quasilinear Schrödinger-Poisson equations similar to the following

where $f\in C(\mathbb{R}, \mathbb{R})$, $V(x)$ is the potential function. Firstly, the quasilinear equation is transformed into a semilinear equation by using a change of variables, by conditionally limiting $f$, the existence results of positive solutions, ground state solutions and bound state solutions of the above equations were established by various analysis methods. For example, in ^[12], the authors studied the following quasilinear Schrödinger-Poisson equations

where $V\in C(\mathbb{R}^N, \mathbb{R})$, $h\in C(\mathbb{R}^+, \mathbb{R})$, is Hölder continuous and satisfy

$(V_0)$ there exists $V_0 > 0$ such that $V(x)\geq V_0 > 0$.

$(V_1)$ $\lim_{|x|\to \infty }V(x) = V(\infty)$ and $V(x)\leq V(\infty)$.

$(h_0)$ $\lim_{s\to 0 }\frac{h(s)}{s} = 0$.

$(h_1)$ for any $s\in\mathbb{R}$, $C > 0$, there exist $p < \frac{3N+2}{N-2}(when \; N = 1, 2, p < \infty)$ such that $|h(s)|\leq C(1+|s|^p).$ If one of the following conditions hold, then Eq (1.4) has a positive nontrivial solution:

$(h_2)$ There exists $\mu > 4$, such that, for any $s > 0$, $0 < \mu H(s)\leq h(s)s$ hold, where $H(s) = \int^s_0 h(t)dt.$

$(h_3)$ For any $s > 0$, $0 < 4 H(s)\leq h(s)s$ hold and when $N\geq4$, $p < \frac{3N+4}{N}(N = 3, p\leq 5)$, where $H(s) = \int^s_0 h(t)dt$.

Therefore, we are curious that does the quasilinear Schrödinger-Poisson equation like (1.1) has similar results as those in the above literature under some conditions. Compared with ^[5,6,7,8,9], we study the existence of infinitely many normalized radial solutions of Schrödinger-Poisson equation with quasilinear term. To our knowledge, there are very few results in this direction in the existing literature. Moreover, compared with references ^[12,13,14], in this paper, we establish the existence results of infinitely many normalized radial solutions for this kind of equation, this can be regarded as the supplement and generalization of quasilinear Schrödinger-Poisson equations in this research direction of gauge solution.

Our result is as follows.

Theorem 1.1. Assume that $p\in(\frac{10}{3}, 6)$. There exists $c_0 > 0$ sufficient small such that for any $c\in(0, c_0],$ (1.1) admits an unbounded sequence of distinct pairs of radial solutions $(\pm u_n, \lambda_n)\in S_c\times\mathbb{R}^-$ with $||u_n||_2^2 = c$ and $\lambda_n < 0$ for each $n\in\mathbb{N}$, and such that

Remark 1.1. Firstly, when $p\in (\frac{10}{3}, 6)$, the energy functionals $K(u)$ corresponding to Eq (1.1) have no lower bounds on $S_c,$ which will result in the absence of global minimum solution. Secondly, it can be easily checked that the functionasl $K(u),$ restricted to $S_c,$ do not satisfy the Palais-Smale condition.

Remark 1.2. Firstly, due to the existence of quasilinear term, the energy functionals corresponding to Eq (1.1) are not well defined in $H_r^1(\mathbb{R}^3)$, hence, the usual variational method can not be used directly. In order to overcome this difficulty, we have two methods: one is to re-establish an appropriate variational framework so that the energy functional corresponding to Eq (1.1) have a good definition, the another is to convert Eq (1.1) into semi-linear equations through a change of variables, and then we can use a general variational method to study it. In this paper, we select second method. Secondly, because of the existence of the nonlocal term $|x|^{-1}*|u|^2$, this will make the proof more complex.

Definition 1.1. For given $c > 0$, we say that $I(u)$ possesses a mountain pass geometry on $S_c$ if there exists $\rho_c > 0$ such that

where $I:H^1(\mathbb{R}^3)\to \mathbb{R}$, $\Gamma_c = \{g\in C([0, 1], S_c):||\nabla g(0)||_2^2\leq \rho_c, I(g(1)) < 0\}$, $S_c = \{u\in H^1(\mathbb{R}^3):||u||_2^2 = c\}.$

2.   Preliminaries and proof of Theorem 1.1

In the following, we will introduce some notations.

(1) $H^1(\mathbb{R}^3) = {\{u} \in L^2(\mathbb{R}^3):\nabla{u} \in L^2(\mathbb{R}^3)\}$.

(2) $H_r^1(\mathbb{R}^3) = {\{u} \in H^1(\mathbb{R}^3):u(x) = u(|x|)\}$.

(3) $||u|| = \Big(\int_{\mathbb{R}^3}|\nabla u|^2+u^2dx\Big)^\frac{1}{2}, \forall u \in H_r^1(\mathbb{R}^3).$

(4) $||u||_s = \Big({\int_{\mathbb{R}^3}|u|^sdx}\Big)^{\frac{1}{s}}, \forall s\in [1, + \infty).$

(5) $\langle u, v\rangle_{H_r^1} = \int_{\mathbb{R}^3}\nabla u\nabla v+uvdx.$

(6) $c, c_i, C, C_i$ denote various positive constants.

From the above description, we know that the energy functional $J_\lambda: H^1(\mathbb{R}^3)\rightarrow \mathbb{R}$ associated with problem (1.1) by $J_\lambda(u)$, where $J_\lambda(u)$ is given in (1.2).

Then such solutions of Eq (1.1) satisfying condition $||u||_2^2 = c$ and $u(x) = u(|x|)$ can be obtained by looking for critical points of the functionals $K$ limited to $S^\prime_c = \{u\in H_r^1(\mathbb{R}^3):||u||_2^2 = c\}$, where $K(u)$ is given in (1.3), $H_r^1(\mathbb{R}^3)$ is the space composed of the radial function of $H^1(\mathbb{R}^3)$, and the embedding $H_r^1(\mathbb{R}^3)\hookrightarrow L^q(\mathbb{R}^3)$ is compact for $q\in (2, 6)$.

In order to prove our results, we first do a appropriate change of variables, $\forall u, v\in H_r^1(\mathbb{R}^3)$, $v: = f^{-1}(u)$, $f$ satisfies the following conditions:

$(f_1)$ $f$ is uniquely defined, smooth and invertible;

$(f_2)$ $f^\prime(t) = \frac{1}{\sqrt{1+2f^2(t)}}$, $t\in[0, +\infty)$;

$(f_3)$ $f(0) = 0$ and $f(t) = -f(-t)$, $t\in(-\infty, 0]$.

In addition, we summarize some properties of $f$ which have been proved in ^[12,13,14].

$(f_4)$ $|f(t)|\leq |t|$, $\forall t\in \mathbb{R}$;

$(f_5)$ there exists a positive constant $C$ such that

By making a change of variables, we get the following functional

It can be seen that functional $I_\lambda$ is well defined in $H_r^1(\mathbb{R}^3)$ and the corresponding equation is

Then such solutions of Eq (1.1) satisfying condition $||u||_2^2 = c$ and $u(x) = u(|x|)$ can be obtained by looking for critical points of the following functional $F$ under the corresponding constraints

The corresponding constraint condition is

Then, if $u\in S^\prime_c$ is the solution of Eq (1.1), if and only if $v\in S_r(c)$ is the solution of Eq (2.1). Moreover we define, for short, the following quantities

We first establish some preliminary results. Let $\{V_n\}\subset H_r^1(\mathbb{R}^3)$ be a strictly increasing sequence of finite-dimensional linear subspaces in $H_r^1(\mathbb{R}^3)$, such that $\bigcup_nV_n$ is dense in $H_r^1(\mathbb{R}^3)$. We denote by $V_n^\bot$ the orthogonal space of $V_n$ in $H_r^1(\mathbb{R}^3)$. Then we have

Lemma 2.1. (See [5,Lemma 2.1]) Assume that $p\in (2, 6)$. Then there holds

Now for $c > 0$ fixed and for each $n\in \mathbb{N}$, we define

and

We also define

Lemma 2.2. For every $p \in (2, 6)$, we have $b_n \rightarrow \infty$ as $n\rightarrow \infty$. Particularly, there exists $n_0 \in \mathbb{N}$ such that $b_n \geq 1$ for all $n \geq n_0$, $n \in \mathbb{N}$.

Proof. For any $v\in B_n$, we have that

From this estimation and Lemma 2.1, it follows since $p > 2$, that $b_n\to +\infty$ as $n\to \infty$. Now, considering the sequence $V_n\in H_r^1(\mathbb{R}^3)$ only from an $n_0\in \mathbb{N}$ such that $b_n\geq 1$ for any $n\geq n_0$ it concludes the proof of the lemma. Next we start to set up our min-max scheme. First we introduce the map $\kappa:H: = H_r^1(\mathbb{R}^3)\times \mathbb{R}\to H_r^1(\mathbb{R}^3)$ by

where $H$ is a Banach space equipped with the product norm $||(v, \theta)||_H = \Big(||v||^2+|\theta|^2\Big)^{\frac{1}{2}}$. Observe that for any given $v\in S_r(c)$, we have $\kappa(v, \theta)\to H_r^1(\mathbb{R}^3)$ for all $\theta\in \mathbb{R}$. Also from [8,Lemma 2.1], we know that

Thus, using the fact that $V_n$ is finite dimensional, we deduce that for each $n\in \mathbb{N}$, there exists an $\theta_n > 0$, such that

satisfies

Now we define

Clearly $\overline{g}_n\in \Gamma_n$. Let

where $v^t(x) = t^\frac{3}{2}v(tx)$, then $||f(v^t)||_2^2 = ||f(v)||_2^2$, and so, for any $v\in S_r(c), \; t > 0$, we have $v^t\in S_r(c)$, and take into account

let

Now we give the key intersection result, due to ^[15].

Lemma 2.3. For each $n\in \mathbb{N}$,

Proof. The point is to show that for each $g\in \Gamma_n$ there exists a pair $(t, v)\in [0, 1]\times (S_r(c)\cap V_n)$, such that $g(t, v)\in B_n$ with $B_n$ defined in (2.2). but this proof is completely similar to the proof of Lemma 2.3 in ^[15], and this proof is omitted here.

According to Lemma 2.3 and (2.7), we derive that, $F$ satisfies mountain pass geometry, that is, for any $g \in \Gamma_n$, we have

Next, we shall prove that the sequence $\{\gamma_n(c)\}$ is indeed a sequence of critical values for $F$ restricted to $S_r(c)$. To this purpose, we first show that there exists a bounded Palais-Smale sequence at each level $\{\gamma_n(c)\}$. From now on, we fix an arbitrary $n\in \mathbb{N}$, then the following lemma can be obtained.

Lemma 2.4. For any fixed $c > 0$, there exists a sequence $\{v_k\}\subset S_r(c)$ satisfying

where

In particular $\{v_k\}\subset S_r(c)$ is bounded.

To find such a Palais-Smale sequence, we consider the following auxiliary functional

where $\kappa(v, \theta)$ is given in (2.4), it is checked easily that $Q(v) = \frac{\partial F(\kappa(v, \theta))}{\partial\theta}|_{\theta = 0}$, set

Clearly, for any $g \in \Gamma_n$, $\widetilde{g}: = (g, 0)\in \widetilde{\Gamma}_n$.

Define

According to ^[10,16,17], it is checked easily that $F$ and $\widetilde{F}$ have the same mountain pass geometry and $\widetilde{\gamma}_n(c) = \gamma_n(c)$.

Following ^[18], we recall that for any $c > 0$, $S_r(c)$ is a submanifold of $H_r^1(\mathbb{R}^3)$ with codimension 1 and the tangent space at $S_r(c)$ is defined as

The norm of the derivative of the $C^1$ restriction functional $F|_{S_r(c)}$ is defined by

Similarly, the tangent space at $(u_0, \theta_0)\in S_r(c)\times\mathbb{R}$ is given as

The norm of the derivative of the $C^1$ restriction functional $\tilde{F}|_{S_r(c)\times\mathbb{R}}$ is defined by

As in [19,Lemma 2.3], we have the following lemma, which was established by using Ekeland's variational principle.

Lemma 2.5. For any $\varepsilon > 0$, if $\widetilde{g}_0\in \widetilde{\Gamma}_n$ satisfies

Then there exists a pair of $(v_0, \theta_0)\in S_r(c)\times\mathbb{R}$ such that:

$(i)\; \; \tilde{F}(v_0, \theta_0)\in [\tilde{\gamma}_n(c)-\varepsilon, \tilde{\gamma}_n(c)+\varepsilon]$;

$(ii)\; \; \min_{t\in[0, 1], v\in S_r(c)\cap V_n}||(v_0, \theta_0)-\tilde{g}_k(t, v)||_H\leq\sqrt{\varepsilon}$;

$(iii)\; \; ||\tilde{F}^\prime|_{S_r(c)\times\mathbb{R}}(v_0, \theta_0)||\leq 2\sqrt{\varepsilon}$, i.e.,

Next, we will give the proof of Lemma 2.4.

Proof. From the definition of $\gamma_n(c)$, we know that for each $k\in \mathbb{N}$, there exists an $g_k\in \Gamma_n$ such that

Since $\widetilde{\gamma}_n(c) = \gamma_n(c)$, $\widetilde{g}_k = (g_k, 0)\in \widetilde{\Gamma}_n$ satisfy

Thus applying Lemma 2.5, we obtain a sequence $\{v_k, \theta_k\}\subset S_r(c)\times \mathbb{R}$ such that:

$(i)\; \; \tilde{F}(v_k, \theta_k)\in [\gamma_n(c)-\frac{1}{k}, \gamma_n(c)+\frac{1}{k}]$;

$(ii)\; \; \min_{t\in[0, 1], v\in S_r(c)\cap V_n}||(v_k, \theta_k)-(g_k(t, v), 0)||_H\leq\frac{1}{\sqrt{k}}$;

$(iii)\; \; ||\tilde{F}^\prime|_{S_r(c)\times\mathbb{R}}(v_k, \theta_k)||\leq 2\sqrt{k}$, that is

For each $k\in\mathbb{N}$, let $u_k = \kappa(v_k, \theta_k)$. We shall prove that $u_k\in S_r(c)$ satisfies (2.9). Indeed, firstly, from $(i)$ we have that $F(u_k)\to \gamma_n(c)(k\to \infty)$, since $F(u_k) = F(\kappa(v_k, \theta_k)) = \widetilde{F}(v_k, \theta_k)$. Secondly, note that

and $(0, 1)\in \widetilde{T}_{(v_k, \theta_k)}$. Thus $(iii)$ yields $Q(u_k)\to 0(k\to \infty)$. Finally, to verify that $F^\prime|_{S_r(c)(u_k)}\to 0(k\to \infty)$, it suffices to prove for $k\in \mathbb{N}$ sufficiently large, that

To this end, we note that, for $\omega\in T_{u_k}$, setting $\widetilde{\omega} = \kappa(\omega, -\theta_k)$, $v^\prime_k = f(v_k)$, one has

If $(\widetilde{\omega}, 0)\in \widetilde{T}_{(v_k, \theta_k)}$ and $||(\widetilde{\omega}, 0)||_H^2\leq 4||\omega||^2$ and $k\in \mathbb{N}$ is sufficiently large, then $(iii)$ implies (2.11). To verify these conditions, observe that $(\widetilde{\omega}, 0)\in \widetilde{T}_{(v_k, \theta_k)}\Leftrightarrow \omega \in T_{u_k}$. Also from $(ii)$ it follows that

by which we deduce that

holds for $k\in \mathbb{N}$ large enough. At this point, (2.11) has been verified. To end the proof of the lemma it remains to show that $\{u_k\}\in S_r(c)$ is bounded. But one notes that for any $v\in H^1(\mathbb{R}^3)$, there holds that

Thus we have

Since $p\in (\frac{10}{3}, 6)$ it follows immediately from (2.14) that $\{u_k\}\in S_r(c)$ is bounded in $H^1(\mathbb{R}^3)$.

Lemma 2.6. Assume that $(f_1)-(f_4)$ hold and $v$ is a weak solution of (2.1). Then $Q(v) = 0$. Furthermore, there exists a constant $c_0 > 0$ independing on $\lambda\in \mathbb{R}$ such that if $||f(v)||^2_2\leq c_0$ sufficiently small, then $\lambda < 0$.

Proof. Let $v$ be a weak solution of (2.1), the following Pohožaev-type identity holds

By multiplying (1.1) by $u$ and integrating. After a change of variables, we obtain the following identity

By multiplying (2.16) by $\frac{3}{2}$ and minus (2.15), we obtain

From the above equation $Q(v) = 0$. By multiplying (2.16) by $\frac{3}{p}$ and minus (2.15), we obtain

On the one hand, by $(f_4)$, Hardy-Littlewood-Sobolev inequality and Gagliardo-Nirenberg inequality, we have

On the other hand, by $(f_4)$, (2.17) and Gagliardo-Nirenberg inequality, we derive that, there exists a $C(p) > 0$ such that

implies

Due to $p\in (\frac{10}{3}, 6)$, then $\frac{10-3p}{2} < 0$. Note that (2.20) tells us that, for any solution $v$ of (2.1) with small $L^2$-norm, $||\nabla v||_2$ must be large. By the property of $f$, it is easy to see that $f$ is monotonically increasing on $(-\infty, +\infty)$. By this time, we just taking $||f(v)||_2^2 = c$ is small enough so as to $|v| < 1$, then by $(f_5)$, (2.18) and (2.19), we derive that, there exist $C > 0$ and $\widetilde{C}(p) > 0$ such that

It follws from (2.20) that when $||\nabla u||_2$ is sufficiently small, the left-hand side of (2.21) is negative, that is, $\lambda C^2||v||_2^2 < 0$($||\nabla u||_2\to 0$). Therefore, there exists a constant $c_0 > 0$ independing on $\lambda\in \mathbb{R}$, such that a solution $v$ of (2.1), if satisfies $||f(v)||^2_2\leq c_0$ sufficiently small, then $\lambda < 0$. Similar to ^[8,16,17], we have the following proposition.

Proposition 2.1. Let $\{v_k\}\subset S_r(c)$ be the Palais-Smale sequence obtained in Lemma 2.4. Then there exist $\lambda_n\in \mathbb{R}$ and $v_n\in H_r^1(\mathbb{R}^3)$, such that, up to a subsequence,

$(i)\; \; v_k\rightharpoonup v_n$, in $H_r^1(\mathbb{R}^3)$,

$(ii)\; \; -\Delta v_k-\lambda_n f(v_k)f^\prime(v_k)+(|x|^{-1}*|f(v_k)|^2)f(v_k)f^\prime(v_k)-$$|f(v_k)|^{p-2}f(v_k)f^\prime(v_k)\to 0$, in $H_r^{-1}(\mathbb{R}^3)$,

$(iii)\; \; -\Delta v_n-\lambda_n f(v_n)f^\prime(v_n)+(|x|^{-1}*|f(v_n)|^2)f(v_n)f^\prime(v_n)-$$|f(v_n)|^{p-2}f(v_n)f^\prime(v_n) = 0$, in $H_r^{-1}(\mathbb{R}^3)$. Moreover, if $\lambda_n < 0$, then we have

In particular, $||f(v_n)||_2^2 = c$, $F(v_n) = \gamma_n(c)$ and $F^\prime(v_n) = \lambda_nv_n$ in $H_r^{-1}(\mathbb{R}^3)$.

Proof. Since $\{v_k\}\subset S_r(c)$ is bounded, up to a subsequence, there exists a $v_n\in H_r^1(\mathbb{R}^3)$, such that

Now let's prove that $v_n\neq 0$. Suppose $v_n = 0$, then by the strong convergence of $v_k\to v_n$ in $L^p(\mathbb{R}^3)$, it follows that $C(v_k)\to 0$. Taking into account that $Q(v_k) \to 0$, it then implies that $A(v_k)\to 0$ and $B(v_k)\to 0$. Thus $F(v_k)\to 0$ and this contradicts the fact that $\gamma_n(c)\geq b_n\geq1$. Thus $(i)$ holds.

The proofs of $(ii)$ and $(iii)$ can be found in [10,Proposition 4.1]. Now using $(ii)$, $(iii)$ and the convergence $C(v_k)\to C(v_n)$, it follows that

If $\lambda_n < 0$, then we conclude from the weak convergence of $v_k\rightharpoonup v_n$ in $H_r^1(\mathbb{R}^3)$, that

Thus $v_k\to v_n(k\to \infty)$ in $H_r^1(\mathbb{R}^3)$, and in particular, $||f(v_n)||_2^2 = c$, $F(v_n) = \gamma_n(c)$ and $F^\prime(v_n) = \lambda_nv_n$ in $H_r^{-1}(\mathbb{R}^3)$.

Next, we will give the proof of Theorem 1.1.

Proof. We recall that in Lemma 2.6, it has been proved that if $(v, \lambda)\in S_r(c)\times \mathbb{R}$ solves (2.1), then it only need $\lambda < 0$ provided $c > 0$ is sufficiently small. Thus by Lemma 2.4 and Proposition 2.1, when $c > 0$ is small enough, for each $n\in \mathbb{N}$, we obtain a couple solution $(v_n, \lambda_n)\in H_r^1(\mathbb{R}^3)\times \mathbb{R}^-$ solving (2.1) with $||f(v_n)||_2^2 = c$ and $F(v_n) = \gamma_n(c)$. Note from Lemmas 2.2 and 2.3, we have $\gamma_n(c)\to \infty (n\to \infty)$ and then we deduce that the sequence of solutions $\{(v_n, \lambda_n)\}$ is unbounded. In conclusion, it is proved that there are infinitely many normalized radial solutions to Eq (2.1), that is, there are infinitely many normalized radial solutions for Eq (1.1) and such that

At this point, the proof of the theorem is completed. As the paper is about an elliptic PDE with a non-local term in the equation, for readers interested in non-local problems, there should also be a reference to recent articels about elliptic PDEs with non-local boundary conditions, e.g., ^[20].

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant Number 11661021 and 11861021); Science and Technology Foundation of Guizhou Province (Grant Number KY[2017]1084).

Conflict of interest

All authors declare no conflicts of interest in this paper.