Crop yield prediction through machine learning: A path towards sustainable agriculture and climate resilience in Saudi Arabia

1.   Introduction

In the past decades, the following Schrödinger-Poisson system

has been studied extensively by many authors, where $V:\mathbb R^3\rightarrow \mathbb R$ and $f\in C(\mathbb R^3\times\mathbb R, \mathbb R)$. This system can be used to describe the interaction of a charged particle with the electrostatic field in quantum mechanics. In this context, the unknown $u$ and $\phi$ represent the wave functions related to the particle and electric potentials, respectively. Moreover, the local nonlinearity $f(x, u)$ models the interaction among particles. We refer the reader to ^[6,20] for more details on its physical background.

It is worth noting that system (1.1) is a nonlocal problem due to the appearance of the term $\phi u$, where $\phi = \phi_u$ is presented in (1.4) below. This fact states that problem (1.1) is no longer a pointwise identity and brings some essential difficulties. For example, the term $\int_{\mathbb{R}^{3}}\phi_u u^2\mathrm{d}x$ in the corresponding energy functional is homogeneous of degree four, then, compared with the local Schrödinger equation, it seems difficult to obtain the boundedness and compactness for any Palais-Smale sequence. In light of the previous observations, the existence of solutions for problem (1.1) have been widely studied and some open problems have been proposed ^{[3,11,15,16,19,25,28,30,35]}.

In what follows, we are particularly interested in the existence of sign-changing solutions (also known as nodal solutions) for problem (1.1). From this perspective, Wang and Zhou ^[29] were concerned with the existence and energy property of sign-changing solutions for problem (1.1) with $f(x, u) = |u|^{p-2}u$. By introducing appropriate compactness conditions on $V$, they used methods different from ^[5] to prove that the so-called sign-changing Nehari manifold is nonempty provided that $4 < p < 6$. Then, combining some analytical techniques and the Brouwer degree theory, the existence of least energy sign-changing solutions was established. After that, the authors in ^[21] investigated sign-changing solutions of problem (1.1) when $f\in C^1(\mathbb R, \mathbb R)$ satisfied super-cubic and subcritical growth at infinity, superlinear growth at origin, and a well-known Nehari-type monotonicity condition. In particular, they established the energy doubling ^[31]. Moreover, the authors in ^[10,38] obtained the similar existence results if the nonlinearity $f$ satisfied asymptotically cubic and three-linear growth, respectively. On the other hand, when $f$ satisfies three-sublinear growth, the existence and multiplicity of sign-changing solutions can be obtained by invariant sets of descending flow ^[13,18]. For more interesting results, such as the Sobolev critical exponent or bounded domains, we refer to ^{[1,24,27,34,36,37]} and the references therein.

According to the previous statements, we observe that the nonlinearities always satisfy superlinear growth or convexity (i.e. $f(x, u) = |u|^{p-2}u, \ 2 < p < 6$) provided that the sign-changing solution of Schrödinger-Poisson systems in the whole space $\mathbb R^3$ is considered. Once the nonlinearity is not constrained by the above forms, the methods mentioned previously cannot be directly used. Therefore, in present paper, we focus on a special type of nonlinearities; that is, the concave-convex type, such as $f(x, u) = |u|^{p-2}u+|u|^{q-2}u$ with $4 < p < 6$ and $1 < q < 2$. The concave-convex nonlinearities were introduced in ^[2], where the authors proved the existence of infinitely many solutions with negative energy for local elliptic problems in bounded domains. After this work, a great attention has been paid to the existence of solutions to elliptic problems with concave-convex nonlinearities. For example, see ^[7,8,17,32] for local Schrodinger equations, and ^{[9,14,22,23,26,33]} for Schrodinger-Poisson systems.

Note that only ^[7,8,17,33] involve the sign-changing solutions. More precisely, Bobkov ^[7] considered the following Schrödinger equation

where $\Omega\subset\mathbb R^N$ is a bounded connected domain with a smooth boundary, $N\geq1$, $1 < q < 2 < \gamma < 2^*$ and $2^*$ is the well-known Sobolev critical exponent. They proved the existence of a sign-changing solution on the nonlocal interval $\lambda\in(-\infty, \lambda_0^*)$, where $\lambda_0^*$ is determined by the variational principle of nonlinear spectral analysis through the fibering method. Moreover, the author in ^[8] obtained similar existence results and some interesting properties for the nodal solutions of the elliptic equation

where $1 < q < 2 < \gamma < 2^*$, $\lambda\in\mathbb{R}$ and the weight functions $k$, $h\in L^\infty(\Omega)$ satisfy the conditions $\rm ess\inf\limits_{\textit x\in\Omega}$$k(x) > 0$ and $\rm ess\inf\limits_{\textit x\in\Omega}$$h(x) > 0$. Note that the methods in ^[7,8] cannot be applied to the nonlocal elliptic problem (1.1). To this end, based on the setting of bounded domains, Yang and Ou ^[33] studied the following Schrödinger-Poisson system

where $\Omega$ is a bounded domain with smooth boundary $\partial \Omega$ in $\mathbb{R}^{3}$ and $1 < p < 2$, $4 < q < 6$, $\lambda$ is a constant. By constrained variational method and quantitative deformation lemma, they obtained that the problem (1.2) has a nodal solution $u_\lambda$ with positive energy when $\lambda < \lambda^*$, $\lambda^*$ is a constant. Here, we point out that if the bounded domain is involved, the embedding $H_0^1(\Omega)\hookrightarrow L^p(\Omega)$ is compact for $1\leq p < 2^*$, which not only avoids the verification of compactness but also ensures the boundedness of the concave term. However, once the whole space is considered, these points cannot be directly determined. Therefore, motivated by the works described above, in this paper we focus on the following Schrödinger-Poisson system in the whole space $\mathbb R^3$ with concave-convex nonlinearities

where $1 < q < 2$, $4 < p < 6$, $\lambda > 0$ and $V$, $K$ satisfy the assumptions:

$(V)$ $V\in C(\mathbb{R}^{3}, \mathbb{R})$ satisfies $\inf\limits_{x\in\mathbb{R}^{3}}V(x)\ge a > 0$ for each $A > 0$, $meas\{x\in\mathbb{R}^{3}:V(x)\le A\} < \infty$, where $a$ is a constant and $meas$ denotes the Lebesgue measure in $\mathbb{R}^{3}$;

$(K)$ $K$ is positive and $K\in L^{\frac{6}{6-q}}(\mathbb{R}^{3})$.

Here the condition $(V)$ is similar to ^[17]. This condition also ensures the compactness of embedding $H\hookrightarrow L^p(\mathbb R^3)$, $2\leq p < 2^*$, where $H$ is the Hilbert space

endowed with the norm

Meanwhile, we point out that the authors in ^[17] considered the local Schrödinger type equation in $\mathbb{R}^{N}$

where $1 < q < 2 < p < 2^*$, $N\geq 2$ and $\lambda, \mu, \nu$ are parameters. The above equation involves a combination of concave and convex terms. They obtained infinitely many nodal solutions by using the method of invariant sets. However, it seems that this method cannot be applied to problem (1.3). In order to overcome the previous difficulties, we introduce the condition $(K)$, which guarantees a weak continuity result (see Lemma 2.2 below). Moreover, conditions $(V)$ and $(K)$ allow us to construct a suitable nonempty closed subset of sign-changing Nehari manifold similar to ^[33], and then a least energy sign-changing solution can be obtained.

Before proceeding, we discuss the basic framework for dealing with our problem. The usual norm in the Lebesgue space $L^r({\mathbb{R}^{3}})$ is denoted by $|u|_r = \Big(\int_{\mathbb{R}^{3}}|u|^{r}\mathrm{d}x\Big)^{\frac{1}{r}}$, $r\in[1, +\infty)$. It is well known that, by the Lax-Milgram theorem, when $u\in H$, there exists a unique $\phi_u\in \mathcal{D}^{1, 2}(\mathbb{R}^{3})$ such that $-\Delta \phi_u = u^2$, where

Substituting (1.4) into (1.3), we can rewrite system (1.3) as the following equivalent form

Therefore, the energy functional associated with system (1.3) is defined by

The functional $I_\lambda(u)$ is well-defined for every $u\in H$ and belongs to $C^1(H, \mathbb{R})$. Furthermore, for any $v\in H$,

As is well known, the solution of problem (1.5) is the critical point of the functional $I_\lambda(u)$. Moreover, if $u\in H$ is a solution of problem (1.5) and $u^\pm\neq 0$, then $u$ is a sign-changing solution of system (1.3), where

Naturally, we introduce the Nehari manifold of $I_\lambda$ as

which is related to the behavior of the map $\varphi_u: r\rightarrow I_\lambda(ru)\ (r > 0)$ (see ^[12] for the introduction of this map). For $u\in H$, we have

It is well known that, for any $u\in H\backslash \{0\}$, $\varphi_u'(r) = 0$ if and only if $ru\in\mathcal N_\lambda$, which also implies that $\varphi_u'(1) = 0$ if and only if $u\in\mathcal N_\lambda$. This manifold is always used to find the positive ground state solution. In order to obtain sign-changing solutions of problem (1.3), it is necessary to consider the sign-changing Nehari manifold

Hoeever, this manifold cannot be directly applied due to appearance of concave term $\lambda k(x)|u|^{q-2}u$. As we will see, inspired by ^[33], we can construct the set $\mathcal{M}_\lambda^*\subset\mathcal{M}_\lambda$ and prove this set is a nonempty closed set, where $\mathcal{M}_\lambda^*$ is defined by (2.5) below. We then show that the minimization problem $m_\lambda: = \inf\limits_{u\in\mathcal{M}_\lambda^*}I_\lambda(u)$ is attained by some $u_\lambda\in\mathcal{M}_\lambda^*$ with positive energy. Finally, the classical deformation lemma [32, Lemma 2.3] states that the $u_\lambda$ is a weak solution of problem (1.3). Up to now, the main results can be stated as follows.

Theorem 1.1. Assume that $(V)$ and $(K)$ hold. Then there exists a constant $\lambda^* > 0$ (determined in (2.13)) such that for any $\lambda\in(-\infty, \lambda^*)$, problem (1.3) possesses a least energy sign-changing solution $u_\lambda$ with positive energy.

Remark 1.2. As mentioned previously, our Theorem 1.1 extends the result of ^[33] to the whole space $\mathbb R^3$. Moreover, for nonlinearities that do not involve concave terms, such as $f(x, u) = |u|^{p-2}u$ ($4 < p < 6$), with the aid of classic techniques in ^[29], one can obtain that the sign-changing Nehari manifold is nonempty. However, in our paper, $1 < q < 2$ and $4 < p < 6$ mean that $\lambda K(x)|u|^{q-2}u$ is concave and $|u|^{p-2}u$ is convex, which is different from ^[29]. At this point, it is difficult to directly prove that the set $\mathcal{M}_\lambda$ is nonempty. To this end, we carefully analyze the behavior of $f_u(s, t) = I_\lambda(su^++tu^-)$ and then introduce the set $\mathcal{M}_\lambda^*$. In particular, by determining some important lower bound estimates, we verify that $\mathcal{M}_\lambda^*\ne\emptyset$. On the other hand, we point out that the range of parameter $\lambda$ can be negative, which is similar to previous results.

The remainder of this paper is organized as follows. In section 2, we present some preliminary lemmas that are crucial for proving our main results. Section 3 is devoted to proving Theorem 1.1.

2.   Preliminears

In this section, we present some preliminary lemmas that are crucial for proving our main results. First, we recall some well-known properties of $\phi_u$ that are a collection of results in ^[11,19].

Lemma 2.1. For any $u\in H$, we get

($i$) there exists $C > 0$ such that $\int_{\mathbb{R}^{3}}\phi_u u^2\mathrm{d}x\le{C\|u\|^4}$;

($ii$) $\phi_u\ge0$, for any $u\in H$;

($iii$) $\phi_{tu} = t^2{\phi_u}$, for any $t > 0$ and $u\in H$;

($iv$) if $u_n\rightharpoonup u$ in $H$, then $\phi_{u_n}\rightharpoonup \phi_u$ in $\mathcal{D}^{1, 2}(\mathbb{R}^{3})$ and

Next, we verify a weak continuity of the concave term. The proof is similar to [32, Lemma 2.13], but we state the proof here for the readers convenience.

Lemma 2.2. Assume that $1 < q < 2$ and $(K)$ hold, then the functional

is weakly continuous.

Proof. Undoubtedly, it is sufficient to prove that if $u_n\rightharpoonup u$ in $H$, then $\int_{\mathbb{R}^{3}}K(x)|u_n|^q\mathrm{d}x\rightarrow \int_{\mathbb{R}^{3}}K(x)|u|^q\mathrm{d}x$ as $n\rightarrow \infty$. In fact, if $u_n\rightharpoonup u$ in $H$, going if necessary to a subsequence, we can assume that $u_n\rightarrow u$ a.e. on $\mathbb{R}^{3}$. Since $u_n\rightharpoonup u$ in $H$, we get that $\left\{u_n\right\}$ is bounded in $L^{6}(\mathbb{R}^{3})$ and $\left\{u_n^q\right\}$ is bounded in $L^{\frac{6}{q}}(\mathbb{R}^{3})$. Therefore, $u_n^q\rightharpoonup u^q$ in $L^{\frac{6}{q}}(\mathbb{R}^{3})$. Combining with $(K)$ and the definition of weak convergence, we obtain

Now, for any $u\!\in\! H$ with $u^\pm\neq 0$, we introduce the map $f_u:[0, +\infty)\!\times\![0, +\infty)\!\rightarrow \!\mathbb{R}$ defined by $f_u(s, t) = I_\lambda(su^++tu^-)$, i.e.,

Here we have used the fact that

Moreover, we have that

which implies that for any $u\in H$ with $u^\pm\neq 0$, $su^++tu^-\in\mathcal{M}_\lambda$ if and only if the pair $(s, t)$ is a critical point of $f_u$.

Lemma 2.3. Assume that $1 < q < 2$, $4 < p < 6$ and the assumption $(K)$ hold, then there exists a constant $\lambda_1 > 0$ such that for any $u\in H$ with $u^\pm\neq 0$, there holds that

(i) if $\lambda\in (0, \lambda_1)$, then for any fixed $t\ge 0$, $f_u(s, t)$ has exactly two critical points, $0 < s_1(t) < s_2(t)$; $s_1(t)$ is the minimum point and $s_2(t)$ is the maximum point; moreover, if $\lambda\le 0$, then for any fixed $t\ge 0$, $f_u(s, t)$ has exactly one critical point, $s_3(t) > 0$, and it is the maximum point;

(ii) if $\lambda\in (0, \lambda_1)$, then for any fixed $s\ge 0$, $f_u(s, t)$ has exactly two critical points, $0 < t_1(s) < t_2(s)$; $t_1(s)$ is the minimum point and $t_2(s)$ is the maximum point; moreover, if $\lambda\le 0$, then for any fixed $s\ge 0$, $f_u(s, t)$ has exactly one critical point, $t_3(s) > 0$, and it is the maximum point.

Proof. We define $f_\mu(s, t): [0, +\infty)\times[0, +\infty)\rightarrow \mathbb{R}$ by

where $\mu$ is a nonnegative parameter.

$(i)$ For any fixed $t\ge 0$, a direct calculation gives

Then, if $s > 0$, $\frac{\partial f_\mu}{\partial s}(s, t) = 0$ is equivalent to

For $\beta_\mu(s)$, we can obtain that

Clearly, for any fixed $t\ge 0$, $1 < q < 2$ and $4 < p < 6$, we can infer that $\beta_\mu$ has exactly one critical point $s_\mu > 0$, where $s_\mu$ is related to $t$ for any $\mu\ge 0$. Moreover, $\beta_\mu$ is strictly increasing in $(0, s_\mu)$ and strictly decreasing in $(s_\mu, +\infty)$.

Noting that if $\mu = 0$, we have

and

Hence,

and

Let

and

then it follows from Sobolev embedding that

Therefore, if $\lambda\in(0, \lambda_1^+)$, we can deduce $\beta_\mu(s_0) > \beta_0(s_0) > 0$ for any $u\in H$ with $u^\pm\neq 0$. For any fixed $t\ge 0$, if $\lambda\in(0, \lambda_1^+)$, there exist unique $s_1(t)$ and $s_2(t)$ with $0 < s_1(t) < s_2(t)$, such that $\beta_\mu(s) = 0$ and $\beta_\mu^\prime(s_1(t)) > 0$, $\beta_\mu^\prime(s_2(t)) < 0$. On the other hand, when $\lambda\le 0$, there is a unique $s_3(t) > 0$ such that $\beta_\mu(s) = 0$ and $\beta_\mu^\prime(s_3(t)) < 0$.

Finally, considering that

we can find that

Note that $\frac{\partial f_\mu}{\partial s}(s, t) = 0$ is equivalent to $\beta_\mu(s) = 0$, then we have

Thus, the facts that $\beta_\mu^\prime(s_1(t)) > 0, \; \beta_\mu^\prime(s_2(t)) < 0$ and $\beta_\mu^\prime(s_3(t)) < 0$ signify that

In particular, when $\mu = 1$, the results still hold.

$(ii)$ For any fixed $s\ge 0$, let

Analogously with the proof $(i)$, we can derive the conclusion.

At last, it is easy to see that $\lambda_1^+ = \lambda_1^-$. Indeed, $u^\pm\neq 0$ indicates that $(-u)^\pm\neq 0$, which yields that $\lambda_1^+ = \lambda_1^-$. Let $\lambda_1 = \lambda_1^+ = \lambda_1^-$, from $(i)$ and $(ii)$, then the proof is completed.

Let

Undoubtedly,

Here, the notation $S_p$ represents the embedding constant of $H\hookrightarrow L^{p}(\mathbb R^3)$, which has a value depending on $p\in [2, 6]$. According to Lemma 2.3, the following corollary is a direct result.

Corollary 2.4. Assume that $1 < q < 2$, $4 < p < 6$, the assumption $(K)$ and $0 < \lambda < \lambda_2$ hold, then for any $u\in H\backslash\{0\}$, $\varphi_u(r)$ has exactly two critical points, $0 < r_1(u) < r_2(u)$ and $\varphi_u^{\prime\prime}(r_1(u)) > 0$, $\varphi_u^{\prime\prime}(r_2(u)) < 0$. On the other hand, when $\lambda\le 0$, $\varphi_u(r)$ has exactly one critical point, $r_3(u) > 0$, and $\varphi_u^{\prime\prime}(r_3(u)) < 0$.

Lemma 2.5. Assume that $1 < q < 2$, $4 < p < 6$, the assumption $(K)$ and $\lambda < \lambda_1$ hold, then for any $u\in \mathcal{M}_\lambda$, $(\partial^2 f_u/\partial s^2)(1, 1)\neq 0$ and $(\partial^2 f_u/\partial t^2)(1, 1)\neq 0$. Moreover, $I_\lambda(u)\rightarrow +\infty$ as $||u||\rightarrow +\infty$, i.e., the functional $I_\lambda$ is coercive and bounded from below on $\mathcal{N}_\lambda$.

Proof. If $0 < \lambda < \lambda_1$, from Lemma 2.3, it follows that $f_u(s, 1)$ has exactly two critical points $s_1(1)$, $s_2(1)$ and $\frac{\partial ^2f_u}{\partial s^2}(s_1(1), 1) > 0$, $\frac{\partial ^2f_u}{\partial s^2}(s_2(1), 1) < 0$. Since $u\in \mathcal{M}_\lambda$, we have $\frac{\partial f_u}{\partial s}(1, 1) = 0$, which means that $s_1(1) = 1$ or $s_2(1) = 1$. Hence, $\frac{\partial ^2f_u}{\partial s^2}(1, 1)\neq 0$. Analogously, we can conclude that $\frac{\partial ^2f_u}{\partial t^2}(1, 1)\neq 0$.

If $\lambda\le0$, it follows from Lemma 2.3 that $f_u(s, 1)$ has exactly one critical point $s_3(1)$ and $\frac{\partial ^2f_u}{\partial s^2}(s_3(1), 1) < 0$. Combining with $u\in \mathcal{M}_\lambda$, we have $\frac{\partial f_u}{\partial s}(1, 1) = 0$, which shows that $s_3(1) = 1$. Therefore, we get $\frac{\partial ^2f_u}{\partial s^2}(1, 1)\neq 0$. Similarly, we can deduce that $t_3(1) = 1$, $\frac{\partial ^2f_u}{\partial t^2}(1, t_3(1)) < 0$ and the claim is clearly true.

Note that $u\in \mathcal{M}_\lambda\subset\mathcal{N}_\lambda$, then the Sobolev embedding indicates that

Hence, combining $1 < q < 2$ and $4 < p < 6$, we derive $I_\lambda(u)\rightarrow +\infty$ as $||u||\rightarrow +\infty$. That is, the functional $I_\lambda(u)$ is coercive and bounded from below on $\mathcal{N}_\lambda$. The proof is completed.

Similarly, we obtain the following result.

Corollary 2.6. Assume that $1 < q < 2$, $4 < p < 6$, the assumption $(K)$ and $\lambda < \lambda_2$ hold, then for any $u\in \mathcal{N}_\lambda$, $\varphi_u^{\prime\prime}(1)\neq 0$.

In what follows, we construct the following sets

and

Unquestionably, $\mathcal{M}_\lambda^*\subset\mathcal{M}_\lambda^-$. According to the properties of $f_u$ mentioned above, we can verify that the set $\mathcal{M}_\lambda^*$ is nonempty and $\mathcal{M}_{\lambda}^* = \mathcal{M}_{\lambda}^-$ (see Lemma 2.9). To this end, we first get the following fact.

Lemma 2.7. Assume that $1 < q < 2$, $4 < p < 6$ and the assumption $(K)$ hold, there exists $\sigma > 0$, which is independent of $u$ and $\lambda$, such that

for any $u\in \mathcal{M}_\lambda^-$.

Proof. For any $u\in \mathcal{M}_\lambda^-$, from $\frac{\partial ^2f_u}{\partial s^2}(1, 1) < 0$, $\frac{\partial ^2f_u}{\partial t^2}(1, 1) < 0$ and Sobolev embedding, it follows that

which implies

Hence, the proof is finished.

In order to prove that $\mathcal{M}_\lambda^*\ne\emptyset$, let us define

From Sobolev embedding and Lemma 2.7, it follows that

where $\sigma$ is given by (2.6). Therefore, we have

Furthermore, we compute that

and

For any $u\in\mathcal{M}_\lambda$, we obtain

and

Lemma 2.8. Assume that $1 < q < 2$, $4 < p < 6$, the assumption $(K)$ and $\lambda < \lambda_1$ hold, then for any $u\in H$ with $u^\pm\neq 0$, there exists a unique pair $(s_u, t_u)\in \mathbb{R}^+\times\mathbb{R}^+$ such that $s_uu^++t_uu^-\in \mathcal{M}_\lambda^-$. Moreover, if $\lambda < \lambda_3$, then $I_\lambda(s_uu^++t_uu^-) = \max\limits_{s, t > 0}I_\lambda(su^++tu^-)$.

Proof. First, we only prove the case of $0 < \lambda < \lambda_1$ since the proof of $\lambda\le 0$ is very similar. Let $u\in H$ with $u^\pm\neq 0$, from the proof of Lemma 2.3, then we have that $\frac{\partial f_u}{\partial s}(s, t)$ satisfies the conditions:

$(i)$ $\frac{\partial f_u}{\partial t}(s, t_2(s)) = 0$ for all $s\ge0$;

$(ii)$ $\frac{\partial f_u}{\partial t}(s, t)$ is continuous and has continuous partial derivatives in $[0, +\infty)\times[0, +\infty)$;

$(iii)$ $\frac{\partial ^2f_u}{\partial t^2}(s, t_2(s)) < 0$ for all $s\ge 0$.

Hence, we can obtain that if $0 < \lambda < \lambda_1$, $\frac{\partial f_u}{\partial t}(s, t) = 0$ determines an implicit function $t_2(s)$ with continuous derivative on $[0, +\infty)$ by using the implicit function theorem. Analogously, if $0 < \lambda < \lambda_1$, $\frac{\partial f_u}{\partial s}(s, t) = 0$ determines an implicit function $s_2(t)$ with continuous derivative on $[0, +\infty)$.

On the other hand, for every $s\ge 0$, from $\frac{\partial f_u}{\partial t}(s, t_2(s)) = 0$ and $\frac{\partial f_u}{\partial t}(s, t)\le 0$ for sufficiently large $t > 0$, we can show that

Otherwise, if $t_2(s)\ge s$, where $s$ is large enough, it follows from the definition of $\frac{\partial f_u}{\partial t}(s, t)$ that $\frac{\partial f_u}{\partial t}(s, t_2(s)) < 0$, which contradicts $\frac{\partial f_u}{\partial t}(s, t_2(s)) = 0$. Similarly, we get

Therefore, by (2.8), (2.9), $t_2(0) > 0, s_2(0) > 0$, the continuity of $t_2(s)$ and $s_2(t)$, we conclude that the curves of $t_2(s)$ and $s_2(t)$ must intersect at some point $(s_u, t_u)\in \mathbb{R}^+\times\mathbb{R}^+$. That is, $\frac{\partial f_u}{\partial t}(s_u, t_u) = \frac{\partial f_u}{\partial s}(s_u, t_u) = 0$. Additionally, noting that

for any $s > 0$, we obtain that the function $t_2(s)$ is strictly increasing in $(0, +\infty)$. Similarly, the function $s_2(t)$ is strictly increasing in $(0, +\infty)$. Consequently, there is a unique pair $(s_u, t_u)\in \mathbb{R}^+\times\mathbb{R}^+$ such that

and

that is, $s_uu^++t_uu^-\in \mathcal{M}_\lambda^-$.

Next, we prove that $(s_u, t_u)$ is the unique maximum point of $f_u(s, t)$ on $[0, +\infty)\times[0, +\infty)$. In fact, if $u\in\mathcal{M}_\lambda^-$, we only show that $(s_u, t_u) = (1, 1)$ is the pair of numbers such that $I_\lambda(s_uu^++t_uu^-) = \max\limits_{s, t > 0}I_\lambda(su^++tu^-)$. Define

If we verify that $H(u) > 0$, then $(1, 1)$ is a local maximum point of $f_u(s, t)$. Combining uniqueness of $(s_u, t_u)$, we have $(1, 1)$ as a global maximum point of $f_u(s, t)$. Let $u\in\mathcal{M}_\lambda^-$, then

From $\frac{\partial ^2f_u}{\partial s^2}(1, 1) < 0$, $\frac{\partial ^2f_u}{\partial t^2}(1, 1) < 0$, if $\lambda < \lambda_3$, we derive

and

which show that $H(u) > 0$.

If $u\not\in\mathcal{M}_\lambda^-$, then there exists a unique pair $(s_u^\prime, t_u^\prime)$ of positive numbers such that $s_u^\prime u^++t_u^\prime u^-\in\mathcal{M}_\lambda^-$. Let $v = s_u^\prime u^++t_u^\prime u^-$, i.e., $v\in\mathcal{M}_\lambda^-$. Repeat the above steps and we will get $H(v) > 0$. Hence, the proof is completed.

Lemma 2.9. If $1 < q < 2$, $4 < p < 6$, the assumption $(K)$ and $\lambda < \min{\{\lambda_1, \lambda_2, \lambda_3\}}$ hold, then $\mathcal{M}_\lambda^*\neq \emptyset$. Moreover, we get $\mathcal{M}_\lambda^* = \mathcal{M}_\lambda^-$.

Proof. By the definitions of $\mathcal{M}_\lambda^-$ and $\mathcal{M}_\lambda^*$, $\mathcal{M}_\lambda^*\subset\mathcal{M}_\lambda^-$ is obvious. Hence, we only need to prove that if $\lambda < \min{\{\lambda_1, \lambda_2, \lambda_3\}}$, then $\mathcal{M}_\lambda^-\subset\mathcal{M}_\lambda^*$. That is, for any $u\in \mathcal{M}_\lambda^-$, $\varphi_u$ reaches its maximum at point $r = 1$. It follows from $\lambda < \lambda_1$ and Lemma 2.8 that $\mathcal{M}_\lambda^-\neq \emptyset$, and from Lemma 2.8, for any $u\in\mathcal{M}_\lambda^-$, we obtain $H(u) > 0$ when $\lambda < \lambda_3$. Combining $f_u(r, r) = \varphi_u(r)$, it follows that $r = 1$ is a maximum of $\varphi_u$. Therefore, $\mathcal{M}_\lambda^-\subset\mathcal{M}_\lambda^*$. This completes the proof of Lemma 2.9.

Corollary 2.10. If $1 < q < 2$, $4 < p < 6$, the assumption $(K)$ and $\lambda < \min{\{\lambda_1, \lambda_2, \lambda_3\}}$ hold, for $u\in H$ and $u^\pm\neq 0$, then there exists a unique pair $(s_u, t_u)\in\mathbb{R}^+\times\mathbb{R}^+$ such that $s_uu^++t_uu^-\in\mathcal{M}_\lambda^*$ and $I_\lambda(s_uu^++t_uu^-) = \max\limits_{s, t > 0}I_\lambda(su^++tu^-).$

Lemma 2.11. If $1 < q < 2$, $4 < p < 6$, the assumption $(K)$ and $\lambda < \lambda_4$ hold, for all $u\in H\backslash\{0\}$, then there exists $r_u > 0$ such that $\varphi_u(r_u) > 0$, where $\lambda_4 > 0$.

Proof. Fixed $u\in H\backslash\{0\}$, let

for any $r\ge0$, then we have

Considering $E_u(r)$, we obtain that there is a unique $r_1(u) = \Big(\frac{||u||^2}{|u|_p^p}\Big)^{\frac{1}{p-2}} > 0$ such that $E_u(r)$ achieves its maximum at $r_1(u)$ and the maximum value is $E_u(r_1(u)) = \frac{p-2}{2p}\Big(\frac{||u||}{\; |u|_p}\Big)^{\frac{2p}{p-2}}$. Moreover, from Sobolev embedding and (2.10), it is clear to calculate that

Consequently, by taking

we conclude that if $\lambda < \lambda_4$, it holds

for any $u\in H\backslash\{0\}$. This together with (2.11) yields that $\varphi_u(r_1(u)) > 0$ for any $\lambda < \lambda_4$.

Let

then it follows from (2.2), (2.4), (2.7) and (2.12) that $\lambda^* > 0$. Now, we consider the properties of the set $\mathcal{M}_\lambda^*$.

Lemma 2.12. If $1 < q < 2$, $4 < p < 6$, $\lambda < \lambda^*$ and the assumptions $(V)$ and $(K)$ hold, then $\mathcal{M}_\lambda^*$ is a closed set.

Proof. Letting $\{u_n\}\subset\mathcal{M}_\lambda^*$ satisfy $u_n\rightarrow u_0$ as $n\rightarrow \infty$ in $H$, we now prove that $u_0\in\mathcal{M}_\lambda^*$. From $\{u_n\}\subset\mathcal{M}_\lambda^*$, we obtain

From Lemma 2.7, it follows that $||u_n^\pm|| > \sigma > 0$ for any $u_n\in\mathcal{M}_\lambda^-$ and hence $||u_0^\pm|| = \lim\limits_{n\rightarrow \infty}||u_n^\pm|| > \sigma > 0$, which indicates $u_0^\pm\neq 0$. Using this and (2.14), we obtain $u_0\in\mathcal{M}_\lambda$ and $r = 1$ is a critical point of $\varphi_{u_0}$. Consequently, by (2.15)-(2.17), Lemma 2.5 and Corollary 2.6, we derive that

Hence, $u_0\in\mathcal{M}_\lambda^*$ and $\mathcal{M}_\lambda^*$ is a closed set.

Lemma 2.13. If $1 < q < 2$, $4 < p < 6$, $\lambda < \lambda^*$ and assumptions $(V)$ and $(K)$ hold, then the infimum $m_\lambda: = \inf\limits_{u\in\mathcal{M}_\lambda^*}I_\lambda(u)$ can be achieved by some $u_\lambda\in\mathcal{M}_\lambda^*$ and $m_\lambda > 0$.

Proof. According to Lemma 2.5, $m_\lambda > -\infty$ when $\lambda < \lambda^*$. Let $\{u_n\}\subset\mathcal{M}_\lambda^*$ be a minimizing sequence for the functional $I_\lambda$, namely $I_\lambda(u_n)\rightarrow m_\lambda$ as $n\rightarrow \infty$. Since the functional $I_\lambda$ is coercive on $\mathcal{M}_\lambda^*$, then $\{u_n\}$ is bounded in $H$. Going if necessary to a subsequence, we may assume that

Now, we first claim that $u_\lambda^\pm\neq 0$. In fact, from Lemma 2.2, Lemma 2.7 and the convergence of $\{u_n\}$ in $L^p(\mathbb{R}^{3})$, for any $\lambda < \lambda^*$, we conclude that

This means that $u_\lambda^\pm\neq 0$ for any $\lambda < \lambda^*$.

Next, we proof that $u_n\rightarrow u_\lambda$ in $H$. Arguing by contradiction, suppose that

From Corollary 2.10, there exists a unique pair $(s_{u_\lambda}, t_{u_\lambda})$ such that $\tilde{u}_\lambda = s_{u_\lambda}u_\lambda^++t_{u_\lambda}u_\lambda^-\in\mathcal{M}_\lambda^*$ and $I_\lambda(u_n^++u_n^-) = \max\limits_{s, t > 0}I_\lambda(s_{u_\lambda}u_n^++t_{u_\lambda}u_n^-)$. Consequently,

That is, we get a contradiction. Therefore, $u_n\rightarrow u_\lambda$ in $H$ and $m_\lambda$ is achieved by $u_\lambda$. Combining the fact that $\mathcal{M}_\lambda^*$ is closed, so $u_\lambda\in\mathcal{M}_\lambda^*$.

Finally, it follows from $u_\lambda\in\mathcal{M}_\lambda^*$ that $\varphi_{u_\lambda}(r)$ reached its global maximum at $r = 1$. By this and Lemma 2.11, we can deduce that $\varphi_{u_\lambda}(1) > 0$, i.e., $m_\lambda > 0$. This finishes the proof of Lemma 2.13.

3.   Existence of sign-changing solutions

The main aim of this section is to prove our results. Thanks to Lemma 2.13, it suffices to check that the minimizer $u_\lambda$ for $m_\lambda$ is a sign-changing of problem (1.3).

Proof of Theorem 1.1. Since $u_\lambda\in\mathcal{M}_\lambda^*$, according to Corollary 2.10, we obtain that

Moreover, we get $I_\lambda(u_\lambda) > 0$, $\varphi_{u_\lambda}^{\prime\prime}(1, 1) < 0$, $(\partial ^2f_{u_\lambda}/\partial s^2)(1, 1) < 0$ and $(\partial ^2f_{u_\lambda}/\partial t^2)(1, 1) < 0$.

Let $D = (1-\delta, 1+\delta)\times(1-\delta, 1+\delta)$ and $h: D\rightarrow H$ by $h(s, t) = su_\lambda^++tu_\lambda^-$ for any $(s, t)\in D$. Then there is a constant $0 < \delta < 1$ such that

By the quantitative deformation lemma, we prove that $I_\lambda^\prime(u_\lambda) = 0$. Suppose by contradiction that $I_\lambda^\prime(u_\lambda)\neq 0$, then there exist $\lambda_1 > 0$ and $\xi > 0$ such that

Let $\varepsilon = \min\{\frac{m_\lambda-m}{3}, \frac{\lambda_1\xi}{8}\}$ and $s_\xi = \{u\in H:\; ||u-u_\lambda||\le \xi\}$, then the deformation lemma (see^[32], Lemma 2.3) shows that there is a deformation $\eta\in C([0, 1]\times H, H)$ such that

$(i)$ $\eta(d, u) = u$ if $u\not \in I_\lambda^-([m_\lambda-2\varepsilon, m_\lambda+2\varepsilon])\cap s_{2\xi}, \; d\in[0, 1]$;

$(ii)$ $I_\lambda(\eta(d, u))\le I_\lambda(u)$ for all $u\in H$, $d\in[0, 1]$;

$(iii)$ $I_\lambda(\eta(d, u)) < m_\lambda$, $\forall u\in I_\lambda^{m_\lambda}\cap s_\xi$, $\forall d\in(0, 1]$.

First, we need to prove that

In fact, for any $d\in(0, 1]$, it follows from Corollary 2.10 and $(ii)$ that

Moreover, Corollary 2.10 and $(iii)$ imply that

Hence, (3.3) holds. From the continuity of $\eta$ and (3.1)-(3.2), there exists a constant $d_0\in(0, 1]$ such that

In the following, we prove that $\eta(d_0, h(D))\cap\mathcal{M}_\lambda^*\neq\emptyset$, which contradicts the definition of $m_\lambda$. In fact, let $g(s, t) = \eta(d_0, h(s, t))$ and

Then Corollary 2.10 and the degree theory yield deg$(\psi_1, D, 0) = 1$. On the other hand, we know $\varepsilon < \frac{m_\lambda-m}{3}$, $m < m_\lambda-2\varepsilon$. Hence, from $(i)$ we have $\eta(d, h(s, t)) = h(s, t)$ for $d\in(0, 1]$, $(s, t)\in\partial D$, and it follows that

Combining the homotopy invariance property of the degree, we get deg$(\psi_2, D, 0)$ = deg$(\psi_1, D, 0) = 1$. That is, there exists $(s_0, t_0)\in D$ such that $\psi_2(s_0, t_0) = 0$. Therefore, using (3.4) and $\psi_2(s_0, t_0) = 0$, we have $\eta(d_0, h(s_0, t_0))\in\mathcal{M}_\lambda^*$, i.e., $\eta(d_0, h(D))\cap\mathcal{M}_\lambda^*\neq\emptyset$. From this, $u_\lambda$ is a critical point of $I_\lambda$, i.e., $I_\lambda^\prime(u_\lambda) = 0$.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research is supported by National Natural Science Foundation of China [No.11971393].

Conflict of interest

The authors declare there is no conflict of interest.