Existence and stability of normalized solutions to the mixed dispersion nonlinear Schrödinger equations

1.   Introduction

We consider the biharmonic nonlinear Schrödinger equation (NLS) with the mixed dispersion and a general nonlinear term

where $N\geq1$, $\mathrm{i}$ denotes the imaginary unit, $\gamma > 0, \beta\in\mathbb{R}$ and the nonlinear term $f$ satisfies the following conditions throughout this paper:

(F1) $f \in C(\mathbb{C}, \mathbb{C}), f(0) = 0$.

(F2) $f(s) \in \mathbb{R}$ for $s \in \mathbb{R}, f\left(e^{i \theta} z\right) = e^{i \theta} f(z)$ for $\theta \in \mathbb{R}, z \in \mathbb{C}$.

(F3) $\lim_{z \rightarrow 0} f(z)/|z| = 0$.

(F4) $\lim_{|z| \rightarrow \infty} f(z)/|z|^{l-1} = 0$, where $l: = 2+8/N$.

(F5) There exists $s_{0} > 0$ such that $F\left(s_{0}\right) > 0,$ where $F(z) = \int_{0}^{|z|} f(\tau) d \tau$ for $z \in \mathbb{C}$.

In nonlinear optics, the NLS is usually derived from the nonlinear Helmholtz equation for the electric field by separating the fast oscillations from the slowly varying amplitude. In the so-called paraxial approximation, the NLS appears in the limit as the equation solved by the dimensionless electric-field amplitude, see [1,Section 2]. The fact that its solutions may blow up in finite time suggests that some small terms neglected in the paraxial approximation play an important role in preventing this phenomenon. Therefore, a small fourth-order dispersion term was proposed in ^[1] (see also ^[2,3,4]) as a nonparaxial correction, which eventually gives rise to (1.1). For more background, see ^[5,6,7] and references therein.

Under these conditions (F1)-(F5), for a solution $u$ of (1.1), it has been established that the following conservations laws:

where $L^q(\mathbb{R}^N)$ is the usual Lebesgue space with norm $|u|_q^q: = \int_{\mathbb{R}^N}|u|^q dx$, $1\leq q < \infty$, and $I$ is the energy functional associated with (1.1) defined by

for $\psi\in H^2(\mathbb{R}^N)$.

If $\psi$ is a standing wave, i.e., $\psi(t, x) = e^{\mathrm{i}\lambda t}u(x)$, then $u\in H^2(\mathbb{R}^N)$ and $\lambda\in\mathbb{R}$ satisfy the following equation:

In this paper, we look for solutions $(u, \lambda)$ with a priori prescribed $L^2$-norm. For a given $a > 0$, we put

In this way, the parameter $\lambda$ is unknown and appears as a Lagrange multiplier. We remark that this is natural, from a physical point view, to search for the normalized solutions which have prescribed mass.

When $\gamma = 0, \beta\neq0$, the problem (1.2)-(1.3) has attracted much attention in the last twenty years. The presence of the $L^2$-constraint makes several methods developed to deal with unconstrained variational problems unavailable, and new phenomena arise. If we set $f(u) = |u|^{p-2}u$, then a new critical exponent appears, i.e., the $L^2$-critical exponent

This is the threshold exponent for many dynamical properties such as global existence vs. blow-up, and the stability or instability of ground states. From the variational point of view, if the problem is purely $L^2$-subcritical, then $I$ is bounded from below on $S_a$. Thus, for every $a > 0$, a ground state can be found as a global minimizer of $I|_{S_a}$, and moreover, the minimizer would be orbitally stable, see ^[8,9] for homogeneous nonlinear term and ^[10,11] for general nonlinear term. And multiplicity results of normalized solutions in the $L^2$-subcritical case can be referred to ^[12,13] and the references therein. In the purely $L^2$-supercritical case, on the contrary, $I|_{S_a}$ is unbounded from below; however, exploiting the mountain pass lemma and a smart compactness argument, L. Jeanjean ^[14] could show that a normalized ground state does exist for every $a > 0$ also in this case. The associated standing wave is strongly unstable ^[15,16] for homogeneous nonlinear term, due to the supercritical character of the equation. We point out that, in ^{[14,17,18,19]}, more general nonlinearities are considered. With regard to the combined nonlinearities, we refer the reader to ^[20,21] for the existence and stability results.

For the case $\gamma\neq0, \beta\neq0$, there is only a few papers about the normalized solutions. As we know, this kind of problem would give rise to a new $L^2$-critical exponent, i.e.,

When $\gamma > 0, \beta > 0$, Bonheure et al. ^[5] have dealt with the $L^2$-subcritical case and obtained the existence of normalized solutions as energy minimizers, while for the $L^2$-critical and supercritical case, ^[6] is concerned with several questions including the existence of ground states and of positive solutions and the multiplicity of radial solutions, and the stability of the standing waves of the associated dispersive equation have also been discussed. Recently, in ^[22] the authors have improved some results to ^[5] and ^[6]. When $\gamma > 0, \beta < 0$, the problem is more involved, see ^[7,23] for the $L^2$-subcritical case and ^[24] for the $L^2$-supercritical case. We remark that all the aforementioned papers have only considered the homogeneous nonlinearity, i.e., $f(u) = |u|^{p-2}u$. For the general nonlinear term, as far as we know, the results are not there yet. With regard to the point, we attempt to study this kind of problem in this paper. We point out that, when dealing with general nonlinearity, we will face some extra difficulties. Such as the loss of homogeneity, which often plays an important role to use the scaling transformations. Besides, some inequalities about energy would be more difficult to obtain. But these inequalities are the key to obtain the compactness of the minimizing sequences. Finally, other types of normalized solution problems can be referred to ^{[25,26,27,28,29,30,31,32,33,34,35]} and the references therein.

In what follows, we give some notations. In $H^2(\mathbb{R}^N)$, when $\beta > 0$, we define its norm by

and for $\beta\leq0$, by

Recalling that the following interpolation inequality

we easily see these two norms above are equivalent in $H^2(\mathbb{R}^N)$. We define the energy functional associated with (1.2) by

for $u\in H^2(\mathbb{R}^N)$, and we consider a constrained variational problem as follows:

Denote the set of minimizers, called ground states for (1.1),

In this paper, we will study the orbital stability of standing waves for (1.1), in the following sense:

Definition 1.1. The set $\mathcal{M}_a$ is said to be orbitally stable if any given $\varepsilon > 0$, there exists $\delta > 0$ such that for any initial data $\psi_0$ satisfying

the corresponding solution $\psi(t, x)$ of the Cauchy problem (1.1) satisfies

According to the sign of $\beta$, we consider the following two cases respectively: (Ⅰ) $\beta\geq0$, (Ⅱ) $\beta < 0$.

(Ⅰ): $\beta\geq0$. In this case, we define

see Lemma 3.3 and (3.6) for more details about $a_0$. For the existence and stability of the minimizer of $m_a$, we have

Theorem 1.2. Under the case $\beta\geq0$, suppose (F1)–(F5) and that a constant $a_0\geq0$ which satisfies (1.6) is uniquely determined. If $a > a_0$,

(i) There exists a global minimizer with respect to $m_a$, i.e., $\mathcal{M}_a\neq\emptyset$.

(ii) Assume the local existence of the Cauchy problem (1.1), then $\mathcal{M}_a$ is orbitally stable, i.e., for any $\varepsilon > 0$, there exists $\delta > 0$ such that for any solution $u$ of (1.1) with $\mathit{\mbox{dist}}\left(u(0, \cdot), \mathcal{M}_a\right) < \delta$, it holds that

where $\mathit{\mbox{dist}}\left(\phi, \mathcal{M}_a\right) = \inf_{\psi\in\mathcal{M}_a}\left\|\phi-\psi \right\|_{H^2}$.

If $0 < a < a_0$, there is no global minimizer with respect to $m_a$.

Remark 1.3. Note that under assumption (F1)–(F5) it is not known if (1.1) is locally well posed. Thus, we need to assume the local existence of the Cauchy problem (1.1) in Theorem 1.2, a similar assumption also appears in Theorem 1.7. However, when $f(u) = |u|^{p-2}u, 2 < p < 2+8/N$, the local (even global) existence of the Cauchy problem (1.1) has been known, see ^[1] for $\beta\geq0$ and ^[40] for $\beta < 0$.

We briefly outline the proof of Theorem 1.2. As the celebrated paper ^[8] to prove the orbital stability of ground state, the main method we use is the Concentration Compactness Principle. However, our situation is far more complex because we deal with the operator $\gamma\Delta^2-\beta\Delta$ and the general nonlinearity. To rule out the vanishing case of the minimizing sequences, we need to know when the condition $m_a < 0$ holds for which mass $a$. This is the reason why we define the value of $a_0$ in (1.6). Besides, the second difficulty we face is to exclude the dichotomy. And we prove a strict subadditivity (conditional) inequality for $m_a$ to overcome this obstacle, see Lemma 3.3. In addition, since we often use the scaling transformations of functions, the general nonlinearity also causes some extra difficulties.

Next, we give the characterization of $m_a$. And it is a direct consequence of the definition of $a_0$ and Lemma 3.3.

Corollary 1.4. (i) If $a_0 = 0$, then $m_a < 0$ for any $a > 0$.

(ii) If $a_0 > 0$, then $m_a = 0$ for any $a\in(0, a_0]$, and $m_a < 0$ for any $a > a_0$.

It is a natural question that "When $a_0 > 0$ holds". To answer the question, the behavior of $f$ near 0 is important. We can show that the following results:

Theorem 1.5. If $\beta = 0$ and we assume $f$ satisfies (F1)-(F5).

(i) If $\liminf_{s\rightarrow0}F(s)/|s|^l = \infty$ holds, then $a_0 = 0$ holds.

(ii) If $\limsup_{s\rightarrow0}F(s)/|s|^l < \infty$ holds, then $a_0 > 0$ holds.

Theorem 1.6. If $\beta > 0$ and we assume $f$ satisfies (F1)-(F5).

(i) If $\liminf_{s\rightarrow0}F(s)/|s|^{2+4/N} = \infty$ holds, then $a_0 = 0$ holds.

(ii) If $\limsup_{s\rightarrow0}F(s)/|s|^{2+4/N} < \infty$ holds, then $a_0 > 0$ holds.

Let us explain why the conditions in Theorems 1.5 and 1.6 are different. For $\beta = 0$, the main effect for the integral $\int_{\mathbb{R}^N}F(u)dx$ is the semi-norm $|\Delta u|_2^2$. While for $\beta > 0$, both $|\Delta u|_2^2$ and $|\nabla u|_2^2$ affect the integral $\int_{\mathbb{R}^N}F(u)dx$. In particular, whether $m_a$ is negative or not greatly depends on the "small" $u$. For $u$ small, compared with $|\Delta u|_2^2$, the gradient norm $|\nabla u|_2^2$ (when it exists) dominates the effect.

(Ⅱ): $\beta < 0$. Under this case, the problem (1.2)-(1.3) is more involved since the term $\beta|\nabla u|_2^2$ in the energy functional $I$ can't be a part of $H^2$-norm but acts as an independent part which effects the behavior of the energy. At present, except for (F1)-(F5), we also assume $f$ satisfies:

(F6) Assume that $F(s)\geq0$ for every $s\geq0$ and there exists a constant $\eta > 2$ such that $F(\tau s)\geq \tau^\eta F(s)$ for every $\tau\geq1, s\geq0$.

We set

see Lemma 4.4 and (4.2) for more details about $a_1$. For the existence and stability of the minimizer of $m_a$, we have

Theorem 1.7. Under the case $\beta < 0$, suppose (F1)–(F6) and that a constant $a_1\geq0$ which satisfies (1.7) is uniquely determined. If $a > a_1$, we have

(i) there exists a global minimizer with respect to $m_a$, i.e., $\mathcal{M}_a\neq\emptyset$.

(ii) assume the local existence of the Cauchy problem (1.1), then $\mathcal{M}_a$ is orbitally stable, i.e., for any $\varepsilon > 0$, there exists $\delta > 0$ such that for any solution $u$ of (1.1) with $\mathit{\mbox{dist}}\left(u(0, \cdot), \mathcal{M}_a\right) < \delta$, it holds that

where $\mathit{\mbox{dist}}\left(\phi, \mathcal{M}_a\right) = \inf_{\psi\in\mathcal{M}_a}\left\|\phi-\psi \right\|_{H^2}$.

If $0 < a\leq a_1$, there is no global minimizer with respect to $m_a$.

Remark 1.8. A typical example of the nonlinear term satisfying (F1)–(F6) is $f(u) = |u|^{p-2}u+\mu|u|^{q-2}u, 2 < q < p < 2+8/N, \mu\geq0$.

In the proof of Theorem 1.7, the situation is more involved compared with the case $\beta\geq0$. To rule out the vanishing case of minimizing sequences, we need to analyse the spectrum of the operator $\gamma\Delta^2u-\beta\Delta u$. Thanks to a result in ^[23] (see Lemma 4.2), we can infer the behavior of $m_a$ and overcome the difficulty by defining the value of $a_1$ in (1.7). Besides, the dichotomy case is more hard to deal with. To this aim, we use suitable scaling transformations of functions to get the subadditivity condition for the minimizing energy and hence exclude this case. Once we get the precompactness of minimizing sequences, we can prove the existence and orbital stability of normalized solutions.

Next, we give the characterization of $m_a$. And it is a direct consequence of the definition of $a_1$ and Lemma 4.4.

Corollary 1.9. (i) If $a_1 = 0$, then $m_a < -\frac{\beta^2}{8\gamma}a$ for any $a > 0$.

(ii) If $a_1 > 0$, then $m_a = -\frac{\beta^2}{8\gamma}a$ for any $a\in(0, a_1]$, and $m_a < -\frac{\beta^2}{8\gamma}a$ for any $a > a_1$.

Finally, in the case $\beta < 0$, we consider the nonlinearity $f(u) = |u|^{p-2}u+\mu|u|^{q-2}u, 2 < q < p < 2+8/N, \mu < 0$. It is easy to see that $f$ satisfies the conditions (F1)–(F5), but does not satisfy (F6). However, with regard to the value of minimizing energy $m_a$, we can still give its partial characterization as follows.

Theorem 1.10. Let $\beta < 0$ and $f(u) = |u|^{p-2}u+\mu|u|^{q-2}u, 2 < q < p < 2+8/N, \mu < 0$. Then $m_a\leq-\frac{\beta^2}{8\gamma}a$ for any $a > 0$. Moreover, there exist two constants $a_*, a^*\in[0, \infty)$ with $a^*\geq a_*$ such that

(i) if $a^* = 0$, then $m_a < -\frac{\beta^2}{8\gamma}a$ for any $a > 0$.

(ii) if $a^* > 0$ and $a_* > 0$, then $m_a = -\frac{\beta^2}{8\gamma}a$ for any $a\in(0, a_*]$, and $m_a < -\frac{\beta^2}{8\gamma}a$ for any $a > a^*$.

(iii) if $a^* > 0$ and $a_* = 0$, then $m_a < -\frac{\beta^2}{8\gamma}a$ for any $a > a^*$.

Compared with the results of Corollary 1.9, the present situation is more involved and the behavior of $m_a$ is more difficult to figure out. Based on this point, we think the condition (F6) may be crucial to determine the behavior of $m_a$ and hence the existence of minimizers for $m_a$. Although we assume it holds that $a^* > a_* > 0$, we can't infer the behavior of $m_a$ in $(a_*, a^*]$ due to the combined effect of the nonlinear terms $|u|^{p-2}u$ and $\mu|u|^{q-2}u$. On the other hand, for $a > a^*$, we have $m_a < -\frac{\beta^2}{8\gamma}a$, but we still don't know whether $m_a$ can be achieved. At this case, ruling out the vanishing case of the minimizing sequence is easy, however, the dichotomy case is difficult to deal with because the strict subadditivity condition is unclear. Thus, we can't deduce the precompactness of the minimizing sequence. All the facts above show that there is a sharp contrast between the conditions containing (F6) and these conditions lacking (F6).

This paper is organized as follows. In Section 2, we give some preliminaries which will be used later. Section 3 is devoted to studying the existence and orbital stability of normalized solutions which belong to the ground state set $\mathcal{M}_a$ under the case $\beta\geq0$. The main method we use is the concentration compactness principle. And we rule out the vanishing case according to the negative of energy and exclude the dichotomy by proving a strict additivity inequality for $m_a$, see Lemma 3.4. Also in this section, we give the proofs of Theorems 1.2, 1.5 and 1.6. In Section 4, we consider the case $\beta < 0$. This situation is more involved. To obtain the existence and orbital stability of normalized solutions, we propose an extra condition on $f$, i.e., (F6). But it is more hard to rule out the vanishing case. We make use of the spectral analysis for the operator $\gamma\Delta^2u-\beta\Delta u$ which was given in ^[23] to rule out the vanishing case. The proof of Theorem 1.7 is finished in this section. Finally, to reveal the effect of the condition (F6), we consider a special nonlinearity which does not satisfy (F6) and investigate the behavior of the energy $m_a$, i.e., Theorem 1.10.

2.   Preliminaries

We begin by recall two well-known Gagliardo-Nirenberg interpolation inequalities for functions $u\in H^2(\mathbb{R}^N)$, namely,

where

and

where

See, for instance, ^[36,37].

In what follows, we give a concentration-compactness lemma for the sequence in $H^2(\mathbb{R}^N)$.

Lemma 2.1. Let $\{u_n\}_{n\in\mathbb{N}}$ be a bounded sequence in $H^2(\mathbb{R}^N)$ which satisfies

Then, for $p\in(2, 4^*)$,

holds, where $4^* = 2N/(N-2)_+$ is the critical Sobolev exponent.

Proof. For the proof, we can take a similar argument as that of the classical concentration-compactness lemma by Lions and omit the details. See, for instance, [38,Lemma 1.21].

3.   Existence and stability under the case $\beta\geq0$

Throughout this section, unless otherwise stated, we always assume $f$ satisfies (F1)–(F5). First, we show that $m_a$ is bounded from blow for any $a > 0$.

Lemma 3.1. (i) Let $\{u_n\}_{n\in\mathbb{N}}$ be a bounded sequence in $H^2(\mathbb{R}^N)$. If either $\lim_{n\rightarrow \infty}|u_n|_2 = 0$ or $\lim_{n\rightarrow \infty}|u_n|_l = 0$ holds, then it is true that $\lim_{n\rightarrow \infty}\int_{\mathbb{R}^N}F(u_n)dx = 0$.

(ii) There exists a positive constant $C = C(f, N, a, \gamma)$ depending $f, N$ and $a$ such that

holds for any $u\in S_a$. Specifically, $m_a\geq -C > -\infty$.

Proof. (ⅰ): By the assumptions $(F1)$–$(F4)$, for any $\varepsilon > 0$, there exists a positive constant $C(f, \varepsilon)$ which depends on $\varepsilon$ and $f$ such that

where $l = 2+8/N$. For $u\in H^2(\mathbb{R}^N)$, we have

The Gagliardo-Nirenberg inequality implies that

where $B_N$ is a positive constant which depends on $N$. Thus, we obtain

We take the case where $\{u_n\}_{n\in\mathbb{N}}$ is a bounded sequence in $H^2(\mathbb{R}^N)$ satisfying $\lim_{n\rightarrow \infty}|u_n|_2 = 0$. By (3.4), we have $\lim_{n\rightarrow \infty}\int_{\mathbb{R}^N}F(u_n)dx = 0$. Alternatively, we can take the case where $\{u_n\}_{n\in\mathbb{N}}$ is a bounded sequence in $H^2(\mathbb{R}^N)$ satisfying $\lim_{n\rightarrow \infty}|u_n|_l = 0$. By (3.3), we have

Since we can choose $\varepsilon > 0$ arbitrary, we obtain $\lim_{n\rightarrow \infty}\int_{\mathbb{R}^N}F(u_n)dx = 0$.

(ⅱ): In (3.4), we choose $\varepsilon > 0$ satisfying $B_Na^{\frac{4}{N}}\varepsilon = \frac{\gamma}{4}$. Then, for $u\in S_a$, we have

where $C = C(f, N, a, \gamma)$ is a positive constant which depends on $f, N, \gamma$ and $a$. This implies (3.1).

Lemma 3.2. Let $\{u_n\}_{n\in\mathbb{N}}$ be a bounded sequence in $H^2(\mathbb{R}^N)$ satisfying $\lim_{n\rightarrow \infty}|u_n|_2^2 = a > 0$. Let $\alpha_n = \sqrt{a}/|u_n|_2$ and $\tilde{u}_n = \alpha_n u_n$. Then the following holds:

Proof. Clearly, $\tilde{u}_n\in S_a$ and $\lim_{n\rightarrow \infty} \alpha_n = 1$ hold. We can compute

We have $0\leq |u_n|+(|\alpha_n-1|)\theta|u_n|\leq\left(|\alpha_n+2|\right)|u_n|$. Under the assumptions (F1)–(F4), we have $|f(s)|\leq|s|+C|s|^{l-1}$. Hence, we obtain

Since $\{u_n\}_{n\in\mathbb{N}}$ is bounded in $H^2(\mathbb{R}^N)$, we achieve our conclusion.

In what follows, we give some properties on $m_a$.

Lemma 3.3. (i) $m_a\leq0$ for any $a > 0$.

(ii) $m_{a+b}\leq m_a+m_b$ for any $a, b > 0$.

(iii) $a\mapsto m_a$ is nonincreasing.

(iv) For sufficiently large $a$, $m_a < 0$ holds.

(v) $a\mapsto m_a$ is continuous.

Proof. (ⅰ): Let $u\in S_a$. For $\tau > 0$, we set $u_\tau(x) = \tau^{N/2}u(\tau x)$, giving $u_\tau\in S_a$. Moreover, $|u_\tau|_l^l = \tau^4|u|_l^l\rightarrow0$ as $\tau\rightarrow0$. By Lemma 3.1(ⅰ), we have

As

we see that $\lim_{\tau\rightarrow0}I(u_\tau) = 0$ holds. By the definition of $m_a$, we have $m_a\leq I(u_\tau)$. Thus, we obtain $m_a\leq0$.

(ⅱ): We fix $\varepsilon > 0$. By the definition of $m_a$ and $m_b$, there exist $u\in S_a\cap C_0^\infty(\mathbb{R}^N)$ and $v\in S_b\cap C_0^\infty(\mathbb{R}^N)$ such that

Since $u$ and $v$ have compact support, by using parallel translation, we can assume $\mbox{supp}\; u \cap\mbox{supp}\; v = \emptyset$. Therefore, we have $u+v\in S_{a+b}$. Thus, we find

Since $\varepsilon$ is arbitrary, we have $m_{a+b}\leq m_a+m_b$.

(ⅲ): By (ⅰ) and (ⅱ), we have

for any $a, b > 0$. This gives (ⅲ).

(ⅳ): We set

where $s_0$ is a constant determined in $(F5)$. By $(F5)$, we know $M_0 > 0$. And then we choose a constant $\alpha > 1$ such that

We take a cut-off function $\varphi\in C_0^\infty(\mathbb{R}^N)$ such that

For $R > 0$, we set $\varphi_R(x) = \varphi(x/R)$, then there exist two constants $C_1, C_2 > 0$ such that

We write $|S^{N-1}|$ for the surface area of the unit sphere. If $N = 1$, set $|S^0| = 2$. Now we estimate $I(\varphi_R)$ as follows:

Since

for a sufficiently large $R$, we have $I(\varphi_R) < 0$. By choosing such a $R$ and setting $a_R = |\varphi_R|_2^2$, we obtain $m_{a_R}\leq I(\varphi_R) < 0$. By (ⅲ), we have $m_b\leq m_{a_R} < 0$ if $b\geq a_R$.

(ⅴ): We fix $a > 0$. By (ⅲ), $m_{a-h}$ and $m_{a+h}$ are monotonic and bounded as $h\rightarrow 0^+$, so therefore they has limits. Moreover, $m_{a-h}\geq m_a\geq m_{a+h}$ holds due to (ⅲ). Thus, we obtain

Claim: $\lim_{h\rightarrow 0^+}m_{a-h}\leq m_a$.

By (ⅰ), this is clear if $m_a = 0$. So we consider the case $m_a < 0$. Take $u\in S_a$ and let $u_h(x) = \sqrt{1-h/a}\; u(x)$ for $0 < h < < 1$. Since $|u_h|_2^2 = (1-h/a)a = a-h$, we have $u_h\in S_a$. On the other hand, we have

Thus, we obtain $\lim_{h\rightarrow 0^+}I(u_h) = I(u)$. By $m_{a-h}\leq I(u_h)$, we have

As we choose $u\in S_a$ arbitrarily, we obtain $\lim_{h\rightarrow 0^+}m_{a-h}\leq m_a$.

Claim: $\lim_{h\rightarrow 0^+}m_{a+h}\geq m_a$.

Since the left hand side converges, it is sufficient to consider the case $h = 1/n$, where $n\in\mathbb{N}$. Choose a $u_n\in S_{a+1/n}$ which satisfies $I(u_n)\leq m_{a+1/n}+1/n$ for each $n\in\mathbb{N}$. By (ⅰ), $I(u_n)\leq1/n$. Lemma 3.1(ⅱ) asserts that $\{u_n\}_{n\in\mathbb{N}}$ is a bounded sequence in $H^2(\mathbb{R}^N)$. By the definition of $u_n$, we have

which implies

Let $v_n = u_n/\sqrt{1+1/(an)}$ for $n\in\mathbb{N}$. Then, $\{v_n\}_{n\in\mathbb{N}}$ is also a bounded sequence in $H^2(\mathbb{R}^N)$. Moreover, we have

Hence, $v_n\in S_a$ holds. By Lemma 3.2, we obtain

By (3.5), the claim holds.

We define

By Lemma 3.3, $a_0$ is well-defined. Moreover, if $a_0 > 0$, by Lemma 3.3 (ⅴ), we know

Under certain conditions, we can further prove the strict subadditivity for $m_a$.

Lemma 3.4. (i) Assume that there exists a global minimizer $u\in S_a$ with respect to $m_a$ for some $a > 0$. Then $m_b < m_a$ for any $b > a$. In particular, we have $m_b < 0$ for any $b > a$.

(ii) Assume that there exist global minimizers $u\in S_a$ and $v\in S_b$ with respect to $m_a$ and $m_b$ respectively for some $a, b > 0$. Then $m_{a+b} < m_a+m_b$.

Proof. (ⅰ): By Lemma 3.3(ⅰ), we have $I(u)\leq0$. Now setting $\tau = b/a > 1$ and $\tilde{u}(x) = u(\tau^{-1/N}x)$, by the assumption, we have $|\tilde{u}|_2^2 = b$ and

Noticing that $I(u) = m_a$ and the definition of $m_b$, we obtain $m_b\leq I(\tilde{u}) < \tau m_a\leq m_a$.

(ⅱ): By the assumption and the argument as above, we have

Noting that we can assume $0 < b\leq a$ without loss of generality, taking $\eta = (a+b)/a$ and $\tau = a/b$, we obtain

It completes the lemma.

With regard to the minimizing sequence for $m_a$, we have

Theorem 3.5. Suppose (F1)–(F5) and that $a > 0$. If $\{u_n\}_{n\in\mathbb{N}}\subset S_a$ is a minimizing sequence with respect to $m_a$, then one of the following holds:

(i)

(ii) Taking a subsequence if necessary, there exist $u\in S_a$ and a family $\{y_n\}_{n\in\mathbb{N}}\subset \mathbb{R}^N$ such that $u_n(\cdot-y_n)\rightarrow u$ in $H^2(\mathbb{R}^N)$ as $n\rightarrow \infty$. Specifically, $u$ is a global minimizer.

Proof. Suppose that $\{u_n\}_{n\in\mathbb{N}}\subset S_a$ is a minimizing sequence which does not satisfy (3.8). It is sufficient to show that (ⅱ) holds. Since (3.8) does not hold and $\{u_n\}_{n\in\mathbb{N}}\subset S_a$, we have

Taking a subsequence if necessary, there exists a family $\{y_n\}_{n\in\mathbb{N}}\subset \mathbb{R}^N$, such that

Since $\{u_n\}_{n\in\mathbb{N}}\subset S_a$ is a minimizing sequence, Lemma 3.1(ⅱ) asserts that $\{u_n\}_{n\in\mathbb{N}}$ is a bounded sequence in $H^2(\mathbb{R}^N)$. Hence $\{u_n(\cdot-y_n)\}_{n\in\mathbb{N}}$ is a bounded sequence in $H^2(\mathbb{R}^N)$. Using the weak compactness of a Hilbert space and the Rellich compactness, for some subsequence, there exists $u\in H^2(\mathbb{R}^N)$ such that

Equations (3.9) and (3.11) assert that $|u|_2 > 0$. We put $v_n = u_n(\cdot-y_n)-u$. By (3.10), $v_n\rightharpoonup 0$ weakly in $H^2(\mathbb{R}^N)$. Thus, we have

Using (3.12), the Brezis-Lieb theorem(see ^[39] or [13,Lemma 3.2]) implies that

Since $I(u_n) = I(u_n(\cdot-y_n)) = I(u+v_n)$, we can obtain

We will show the following claim.

Claim.

Suppose that (3.17) does not hold. Since $\{v_n\}_{n\in\mathbb{N}}$ is bounded in $H^2(\mathbb{R}^N)$, similarly as above, for some subsequence, there exist a family $\{z_n\}_{n\in\mathbb{N}}\subset \mathbb{R}^N$ and $v\in H^2(\mathbb{R}^N)$ satisfying $|v|_2 > 0$ such that

We put $w_n = v_n(\cdot-z_n)-v$. Then, similarly as above, we can obtain

Consequently, we have

Here, we set $\eta = |u|_2^2, \zeta = |v|_2^2$ and $\delta = a-\eta-\zeta$. Then, we have $\lim_{n\rightarrow \infty}|w_n|_2^2 = \delta\geq0$. We will consider cases $\delta > 0$ and $\delta = 0$.

In the case $\delta > 0$, we set $\tilde{w}_n = \alpha_n w_n$ and $\alpha_n = \sqrt{\delta}/|w_n|_2$. By Lemma 3.2, we have $\tilde{w}_n\in S_\delta$ and $I(w_n) = I(\tilde{w}_n)+o(1)$. Thus, by (3.18) and the definition of $m_\delta$, we have

As $n\rightarrow \infty$, Lemma 3.3 implies that

Hence $u$ and $v$ are global minimizers with respect to $m_\eta$ and $m_\zeta$ respectively. Here, we can apply Lemma 3.4 (ⅱ) to obtain

It contradicts to (3.20).

In the case $\delta = 0$, the equations $a = \eta+\zeta$ and $\lim_{n\rightarrow \infty}|w_n|_2 = 0$ hold. By Lemma 3.1(ⅰ), we have

Thus, we obtain

As $n\rightarrow$ in (3.18), we have

Hence $u$ and $v$ are global minimizers with respect to $m_\eta$ and $m_\zeta$ respectively. Here, we can apply Lemma 3.4 (ⅱ) to obtain

which is a contradiction. It completes the proof of the claim.

By (3.17) and Lemma 2.1, we have $\lim_{n\rightarrow \infty}|v_n|_l = 0$. Lemma 3.1(ⅰ) asserts that

Next, we estimate the $L^2$ norm of $v_n$.

Claim. $\lim_{n\rightarrow \infty}|v_n|_2 = 0$. In particular, $|u|_2^2 = a$.

By (3.16) and $\eta = |u|_2^2$, it is sufficient to show that $\eta = a$. Otherwise, $\eta < a$ holds because $\eta\leq a$. By (3.21), we have

Taking the limit in (3.16), we obtain $m_a\geq I(u)$. Using Lemma 3.3(ⅲ) along with $u\in S_\eta$, we have

This requires $m_\eta = m_a$. Moreover, $u$ is a global minimizer with respect to $m_\eta$. By Lemma 3.4(ⅰ), we obtain $m_\eta > m_a$ because $\eta < a$. It contradicts to (3.22).

Finally, we estimate the $H^2$-norm of $v_n$. Using the above claim, $u\in S_a$. This gives $I(u)\geq m_a$. Therefore, we have

As $n\rightarrow \infty$, we obtain

while (3.21) asserts that

Since $\lim_{n\rightarrow \infty}|v_n|_2 = 0$, we have $\lim_{n\rightarrow \infty}\left\|v_n\right\|_{H^2} = 0$. Hence $\lim_{n\rightarrow \infty} u_n(\cdot-y_n) = u$ in $H^2(\mathbb{R}^N)$.

For any minimizing sequence of $m_a$, Theorem 3.5 shows that the dichotomy case can't occur. To rule out the vanishing case, we will use the condition $m_a < 0$. Thus, for $a > a_0$ ($a_0$ is given by (3.6)), we can obtain the compactness of the minimizing sequence for $m_a$.

Proposition 3.6. Suppose that $a > a_0$. If $\{u_n\}_{n\in\mathbb{N}}\subset S_a$ is a minimizing sequence with respect to $m_a$, i.e., $\lim_{n\rightarrow \infty}I(u_n) = m_a$. Then, taking a subsequence if necessary, there exist a family $\{y_n\}_{n\in\mathbb{N}}\subset \mathbb{R}^N$ and $u\in S_a$ such that $\lim_{n\rightarrow \infty} u_n(\cdot-y_n) = u$ in $H^2(\mathbb{R}^N)$. In particular, $u$ is a global minimizer, i.e., $u\in \mathcal{M}_a$.

Proof. By the assumption of the proposition and (3.6), we have $m_a < 0$. Let $\{u_n\}_{n\in\mathbb{N}}\subset S_a$ be a minimizing sequence with respect to $m_a$. It is sufficient to show that $\{u_n\}_{n\in\mathbb{N}}$ satisfies (ⅱ) in Theorem 3.5. Otherwise, by Theorem 3.5, $\{u_n\}_{n\in\mathbb{N}}$ satisfies (3.8). By Lemma 3.1(ⅱ), $\{u_n\}_{n\in\mathbb{N}}$ is bounded in $H^2(\mathbb{R}^N)$, so (3.8) and Lemma 2.1 imply that $u_n\rightarrow0$ in $L^l(\mathbb{R}^N)$. By Lemma 3.1(ⅰ), we have

Since $I(u_n)\geq-\int_{\mathbb{R}^N} F(u_n)dx$, we can obtain

contradicting to $m_a < 0$.

After the above preparations have been done, we are now in position to prove our main results.

Proof of Theorem 1.2. First, we consider the case $0 < a < a_0$ and suppose by contradiction that there exists a global minimizer with respect to $m_a$. By the assumption, we have $m_a = 0$. Here, Lemma 3.4 (ⅰ) asserts that

It contradicts to (3.7).

Next, we consider the case $a > a_0$. Proposition 3.6 asserts Theorem 1.2 (ⅰ). For (ⅱ), we assume it does not hold by contradiction. Then there exists $\varepsilon_0 > 0$ such that for a sequence of solutions $u_n$ of (1.1) with $\mbox{dist}\left(u_n(0, \cdot), \mathcal{M}_a\right) < 1/n$, it holds that

which implies that

Let $\alpha_n = \sqrt{a}/|u_n(t_n, \cdot)|_2$ and $\tilde{u}_n(x) = \alpha_n u_n(t_n, x)$. Then by Lemma 3.2 the following holds:

By Proposition 3.6, there exist a family $\{y_n\}_{n\in\mathbb{N}}\subset \mathbb{R}^N$ and $u\in \mathcal{M}_a$ such that $\lim_{n\rightarrow \infty} \tilde{u}_n(\cdot-y_n) = u$ in $H^2(\mathbb{R}^N)$. Thus, we also get $\lim_{n\rightarrow \infty} \left\|u_n(t_n, \cdot-y_n)-u\right\|_{H^2} = 0$. We can deduce a contradiction from the following inequalities:

In what follows, we prove Theorems 1.5 and 1.6, which answer the question that "When $a_0 > 0$ holds".

Proof of Theorem 1.5. (ⅰ): We fix $a > 0$ and take some function $u\in S_a\cap C_0^\infty(\mathbb{R}^N)\setminus\{0\}$. For $\tau > 0$, let $u_\tau(x) = \tau^{N/2}u(\tau x)$. Then, we see that $u_\tau\in S_a$. By the assumption of (ⅰ), there exists a positive constant $\delta$ such that

where $C$ is a constant determined by

Hence $F(u_\tau)\geq C|u_\tau|^l$ holds for a sufficiently small $\tau$. Thus we have

It concludes that $m_a\leq I(u_\tau) < 0$ for any $a > 0$.

(ⅱ): By the assumption of (ⅱ), there exists a positive constant $C = C(f)$ such that $F(s)\leq C|s|^l$ holds for any $s\geq0$. For $u\in S_a$, using the Gagliardo-Nirenberg inequality, we have

For a sufficiently small $a > 0$, it can be shown that $C B_Na^{4/N}\leq\gamma/2$ holds. After choosing an appropriately small $a$, we have

This together with Lemma 3.3 (ⅰ) implies $m_a = 0$ for a small $a > 0$. Hence, we obtain $a_0 > 0$.

Proof of Theorem 1.6. (ⅰ): We fix $a > 0$ and take some function $u\in S_a\cap C_0^\infty(\mathbb{R}^N)\setminus\{0\}$. For $\tau > 0$, let $u_\tau(x) = \tau^{N/2}u(\tau x)$. Then, we see that $u_\tau\in S_a$. By the assumption of (ⅰ), there exists a positive constant $\delta$ such that

where $C$ is a constant determined by

Hence $F(u_\tau)\geq C|u_\tau|^{2+4/N}$ holds for a sufficiently small $\tau$. Thus we have

If necessary, we take a smaller $\tau$, then we conclude that $m_a\leq I(u_\tau) < 0$ for any $a > 0$.

(ⅱ): By (F3)-(F4), there exist two positive constants $C_1 = C_1(f)$ and $C_2 = C_2(f)$ such that $F(s)\leq C_1|s|^{2+4/N}+C_2|s|^l$ holds for any $s\geq0$. For $u\in S_a$, using the Gagliardo-Nirenberg inequality, we have

For a sufficiently small $a > 0$, it can be shown that $C_1 C_Na^{2/N}\leq\beta/2$ and $C_2 B_Na^{4/N}\leq\gamma/2$ hold. After choosing an appropriately small $a$, we have

This together with Lemma 3.3 (ⅰ) implies $m_a = 0$ for a small $a > 0$. Hence, we obtain $a_0 > 0$.

4.   Existence and stability under the case $\beta < 0$

Throughout this section, unless otherwise stated, we always assume $f$ satisfies (F1)-(F6).

Lemma 4.1. (i) Let $\{u_n\}_{n\in\mathbb{N}}$ be a bounded sequence in $H^2(\mathbb{R}^N)$. If either $\lim_{n\rightarrow \infty}|u_n|_2 = 0$ or $\lim_{n\rightarrow \infty}|u_n|_l = 0$ holds, then it is true that $\lim_{n\rightarrow \infty}\int_{\mathbb{R}^N}F(u_n)dx = 0$.

(ii) There exist two positive constants $C_1 = C_1(a, \beta)$ and $C_2 = C_2(f, N, a, \gamma)$ such that

holds for any $u\in S_a$. Specifically, $m_a > -\infty$.

Proof. (ⅰ): the proof can be proceeded as that of Lemma 3.1 (ⅰ) and is omitted.

(ⅱ): First, notice that by (1.4), we have

and we set $C_1 = -\beta\sqrt{a}/2$. In addition, recalling that (3.4) in the proof of Lemma 3.1, we choose $\varepsilon > 0$ satisfying $B_Na^{\frac{4}{N}}\varepsilon = \frac{\gamma}{4}$. Then, for $u\in S_a$, we have

where $C_2 = C_2(f, N, a, \gamma)$ is a positive constant which depends on $f, N, \gamma$ and $a$. Together with the two inequalities above, we get (4.1).

To character the properties of $m_a$, we will use some results from ^[23].

Lemma 4.2. (see ^[23])

(i) For any $\gamma > 0, \beta\in\mathbb{R}$ and $u\in H^2(\mathbb{R}^N)$, it follows that

Thus

(ii) When $\beta < 0$, we introduce

and consider the constrained minimization problem

Then for all $a > 0$, it follows that:

(J1) $m_a^J = -\frac{\beta^2}{8\gamma}a$.

(J2) $m_a^J$ is never achieved.

(J3) All minimizing sequences present vanishing, i.e., if $\{u_n\}_{n\in\mathbb{N}}\subset S_a$ is a minimizing sequence with respect to $m_a^J$, then $\{u_n\}_{n\in\mathbb{N}}$ satisfies (3.8).

Lemma 4.3. Let $\{u_n\}_{n\in\mathbb{N}}$ be a bounded sequence in $H^2(\mathbb{R}^N)$ satisfying $\lim_{n\rightarrow \infty}|u_n|_2^2 = a > 0$. Let $\alpha_n = \sqrt{a}/|u_n|_2$ and $\tilde{u}_n = \alpha_n u_n$. Then the following holds:

Proof. Since the proof is similar as that of Lemma 3.2, we omit it.

In what follows, we give some properties about $m_a$.

Lemma 4.4. (i) $m_a\leq-\frac{\beta^2}{8\gamma}a$ for any $a > 0$.

(ii) $m_{a+b}\leq m_a+m_b$ for any $a, b > 0$.

(iii) $a\mapsto m_a$ is decreasing.

(iv) $m_{a\tau}\leq\tau m_a$ for any $a > 0$ and $\tau\geq1$.

(v) For sufficiently large $a$, $m_a < -\frac{\beta^2}{8\gamma}a$ holds.

(vi) $a\mapsto m_a$ is continuous.

Proof. (ⅰ): By (F5) and (F6), we know $F(z) = F(|z|)\geq0$ for any $z\in \mathbb{C}$. Thus, we get $I(u)\leq J(u)$ for any $u\in S_a$. By Lemma 4.2 (ⅱ), it holds that $m_a\leq m_a^J = -\frac{\beta^2}{8\gamma}a$.

(ⅱ): The proof is similar as that of Lemma 3.3 (ⅱ) and omitted.

(ⅲ): For any $0 < a < b$, we get from (ⅱ) that $m_b\leq m_a+m_{b-a}$. By (ⅰ), we have $m_{b-a}\leq-\frac{\beta^2}{8\gamma}(b-a) < 0$, which implies $m_b < m_a$.

(ⅳ): For any $u\in S_a$ and $\tau\geq1$, we set $u_\tau(x) = \tau^{1/2}u(x)$, then $u_\tau\in S_{a\tau}$. Moreover, by (F5)-(F6), we obtain

Since $u$ is arbitrary, we get (ⅳ).

(ⅴ): By (F5), we can choose a function $u\in H^2(\mathbb{R}^N)$ with $|u|_2^2 = 1$ such that $\int_{\mathbb{R}^N}F(u)dx > 0$. In fact, we take a cut-off function $\varphi\in C_0^\infty(\mathbb{R}^N)$ such that

where $s_0 > 0$ is a constant given by (F5). For $R > 0$, we set $\varphi_R(x) = \varphi(x/R)$, then $|\varphi_R|_2^2 = R^N|\varphi|_2^2$. Thus, we can choose a $R_0 > 0$ such that $|\varphi_{R_0}|_2 = 1$. Now we take $u(x): = \varphi_{R_0}(x)$, then, by (F5)-(F6), we see

For the $u$ above, we set $u_a(x) = a^{1/2}u(x), a\geq1$, then $u_a\in S_{a}$. Moreover, by (F5)-(F6), we obtain

Since $\eta > 2$, we have

For $a > 0$ large enough, we deduce that

(ⅵ): The proof is similar as that of Lemma 3.3 (ⅴ) and omitted.

Next we define

By Lemma 4.4 (ⅴ), we see that $a_1 < \infty$. And again by Lemma 4.4 (ⅳ) and (ⅵ), we know $m_a < -\frac{\beta^2}{8\gamma}a$ for $a > a_1$. Moreover, if $a_1 > 0$, then it concludes from Lemma 4.4 (ⅰ) and (ⅵ) that

Under certain conditions, we can further prove the strict subadditivity for $m_a$.

Lemma 4.5. (i) Assume $\mathcal{M}_a\neq \emptyset$ for some $a > 0$. Then $m_{\tau a} < \tau m_a$ for any $\tau > 1$.

(ii) Assume that there exists a global minimizer $u\in S_a$ with respect to $m_a$ for some $a > 0$ and let $b > 0$. Then $m_{a+b} < m_a+m_b$.

Proof. (ⅰ): First, if $u\in\mathcal{M}_a$, then we claim that $\int_{\mathbb{R}^N}F(u)dx > 0$. Otherwise, by Lemma 4.2(ⅱ), $m_a = I(u) = \int_{\mathbb{R}^N}\left|\Delta u\right|^2dx+\frac{\beta}{2}\int_{\mathbb{R}^N}\left|\nabla u\right|^2dx\geq-\frac{\beta^2}{8\gamma}a$. Thus, we conclude together with Lemma 4.4 (ⅰ) that $m_a = -\frac{\beta^2}{8\gamma}a$, which implies $u$ is also a minimizer for $m_a^J$. This contradicts to (J2) of Lemma 4.2(ⅱ).

Next, for $u\in\mathcal{M}_a$, we set $u_\tau(x) = \tau^{1/2}u(x), \tau > 1$, then $u_\tau\in S_{a\tau}$. Moreover, by (F5)-(F6), we obtain

Therefore, we get $m_{\tau a}\leq I(u_\tau) < \tau I(u) = \tau m_a$.

(ⅱ): Assume first that $0 < b\leq a$. Then, by Lemma 4.4(ⅳ) and Lemma 4.5 (ⅰ), we have

If $0 < a < b$, by again Lemma 4.4(ⅳ) and Lemma 4.5 (ⅰ), we obtain

With regard to the minimizing sequence for $m_a$, we have

Theorem 4.6. Suppose (F1)–(F6) and that $a > 0$. If $\{u_n\}_{n\in\mathbb{N}}\subset S_a$ is a minimizing sequence with respect to $m_a$, then one of the following holds:

(i)

(ii) Taking a subsequence if necessary, there exist $u\in S_a$ and a family $\{y_n\}_{n\in\mathbb{N}}\subset \mathbb{R}^N$ such that $u_n(\cdot-y_n)\rightarrow u$ in $H^2(\mathbb{R}^N)$ as $n\rightarrow \infty$. Specifically, $u$ is a global minimizer.

Proof. Suppose that $\{u_n\}_{n\in\mathbb{N}}\subset S_a$ is a minimizing sequence which does not satisfy (4.4). It is sufficient to show that (ⅱ) holds. Since (4.4) does not hold and $\{u_n\}_{n\in\mathbb{N}}\subset S_a$, we have

Taking a subsequence if necessary, there exists a family $\{y_n\}_{n\in\mathbb{N}}\subset \mathbb{R}^N$, such that

Since $\{u_n\}_{n\in\mathbb{N}}\subset S_a$ is a minimizing sequence, Lemma 4.1(ⅱ) asserts that $\{u_n\}_{n\in\mathbb{N}}$ is a bounded sequence in $H^2(\mathbb{R}^N)$. Hence $\{u_n(\cdot-y_n)\}_{n\in\mathbb{N}}$ is a bounded sequence in $H^2(\mathbb{R}^N)$. Using the weak compactness of a Hilbert space and the Rellich compactness, for some subsequence, there exists $u\in H^2(\mathbb{R}^N)$ such that

Equations (4.5) and (4.7) assert that $|u|_2 > 0$. We put $v_n = u_n(\cdot-y_n)-u$. By (4.6), $v_n\rightharpoonup 0$ weakly in $H^2(\mathbb{R}^N)$. Thus, we have

Using (4.8), the Brezis-Lieb theorem(see ^[39] or [13,Lemma 3.2]) implies that

Since $I(u_n) = I(u_n(\cdot-y_n)) = I(u+v_n)$, we can obtain

Claim: $|v_n|_2^2\rightarrow0$ as $n\rightarrow \infty$.

In order to prove this, we set $\zeta = |u|_2^2 > 0$. By (4.12), if we show that $\zeta = a$, the claim follows. We assume by contradiction that $\zeta < a$ and we define

By Lemma 4.3 and (4.12), it follows that

Let $n\rightarrow \infty$, and by Lemma 4.4 (ⅱ), we have

and so, $I(u) = m_\zeta$, i.e., $u\in S_\zeta$ is a global minimizer with respect to $m_\zeta$. Thus, by Lemma 4.5 (ⅱ), we get

which contradicts (4.13). Hence, the claim follows and $|u|_2^2 = a$.

At this point, since $\left\{v_n\right\}$ is a bounded sequence in $H^2(\mathbb{R}^N)$, it follows from (1.4) that $|\nabla v_n|_2\rightarrow0$ as $n\rightarrow \infty$. By Lemma 4.1 (ⅰ), we obtain that

On the other hand, since $|u|_2^2 = a$, we deduce from (4.12) that

and so, that

From (4.14) and (4.15) we deduce that $|\Delta v_n|_2^2\rightarrow0$ as $n\rightarrow \infty$ and so, that $u_n(\cdot-y_n)\rightarrow u$ in $H^2(\mathbb{R}^N)$.

For any minimizing sequence of $m_a$, Theorem 4.6 shows that the dichotomy case can't occur. To rule out the vanishing case, we will use the condition $m_a < -\frac{\beta^2}{8\gamma}a$. Thus, for $a > a_1$ ($a_1$ is given by (4.2)), we can obtain the compactness of the minimizing sequence for $m_a$.

Proposition 4.7. Suppose that $a > a_1$. If $\{u_n\}_{n\in\mathbb{N}}\subset S_a$ is a minimizing sequence with respect to $m_a$, i.e., $\lim_{n\rightarrow \infty}I(u_n) = m_a$. Then, taking a subsequence if necessary, there exist a family $\{y_n\}_{n\in\mathbb{N}}\subset \mathbb{R}^N$ and $u\in S_a$ such that $\lim_{n\rightarrow \infty} u_n(\cdot-y_n) = u$ in $H^2(\mathbb{R}^N)$. In particular, $u$ is a global minimizer, i.e., $u\in \mathcal{M}_a$.

Proof. By the assumption of the proposition and (4.2), we have $m_a < -\frac{\beta^2}{8\gamma}a$. Let $\{u_n\}_{n\in\mathbb{N}}\subset S_a$ be a minimizing sequence with respect to $m_a$. It is sufficient to show that $\{u_n\}_{n\in\mathbb{N}}$ satisfies (ⅱ) in Theorem 4.6. Otherwise, by Theorem 4.6, $\{u_n\}_{n\in\mathbb{N}}$ satisfies (4.4). By Lemma 4.1(ⅱ), $\{u_n\}_{n\in\mathbb{N}}$ is bounded in $H^2(\mathbb{R}^N)$, so (4.4) and Lemma 2.1 imply that $u_n\rightarrow0$ in $L^l(\mathbb{R}^N)$. By Lemma 4.1(ⅰ), we have

Since $u_n\in S_a$, by Lemma 4.2(ⅱ), we can obtain

contradicting to $m_a < -\frac{\beta^2}{8\gamma}a$ for $a > a_1$.

With the above preparations at hand, we prove our Theorem 1.7.

Proof of Theorem 1.7. First, we consider the case $0 < a\leq a_1$ and suppose by contradiction that there exists a global minimizer $u$ with respect to $m_a$. By (4.3), we have

From (4.16), we deduce that $u\neq0$. We choose a sequence $\left\{u_n\right\}$ in $H^2(\mathbb{R}^N)$ such that $u_n\rightarrow u$ in $H^2(\mathbb{R}^N)$. Then $|u_n|_2\rightarrow |u|_2$ as $n\rightarrow \infty$. We define

then we easily see that

as $n\rightarrow \infty$. Thus, $\tilde{u}_n$ is a minimizing sequence of $m_a$. By (J1) of Lemma 4.2 (ⅱ), we know $m_a = m_a^J$ for $0 < a\leq a_1$. So $\tilde{u}_n$ is also a minimizing sequence of $m_a^J$. By (J3) of Lemma 4.2 (ⅱ), it must be vanishing, i.e., it satisfies (4.4). Combining with Lemma 2.1, we infer that $u = 0$ a.e. in $\mathbb{R}^N$. This contradicts to (4.16).

Next, we consider the case $a > a_1$. Proposition 4.7 asserts Theorem 1.7 (ⅰ).

(ⅱ): The proof is similar as that of Theorem 1.2 (ⅱ) and omitted.

Finally, in the case $\beta < 0$, we consider the nonlinearity $f(u) = |u|^{p-2}u+\mu|u|^{q-2}u, 2 < q < p < 2+8/N, \mu < 0$. We give the partial characterization of the value of minimizing energy $m_a$.

Proof of Theorem 1.10. Let $\left\{u_n\right\}\in S_a$ be a minimizing sequence for $m_a^J$, then by (J3) of Lemma 4.2 (ⅱ), it must be vanishing. Thus, we have

which implies $m_a\leq -\frac{\beta^2}{8\gamma}a$ for any $a > 0$. On the other hand, let $u\in S_1$ and $u_a(x) = \sqrt{a}u(x)$, then we see

Since $p > q > 2$, we have

as $a\rightarrow \infty$, and we conclude that

for $a$ large enough.

In what follows, we set

if the $a$ above does not exist, we set $a_* = 0$. Besides, we define

then we know $a^* < \infty$ from (4.17). Noticing that $m_a$ is continuous (the continuity of $m_a$ can be proved as that of Lemma 3.3 (ⅴ)), together with the definitions of $a_*$ and $a^*$, we get the conclusion.

Acknowledgments

The authors would like to thank the reviewers for the valuable suggestions and comments to improve the manuscript.

H. Luo is supported by National Natural Science Foundation of China, No. 11901182, by Natural Science Foundation of Hunan Province, No. 2021JJ40033, and by the Fundamental Research Funds for the Central Universities, No. 531118010205. Z. Zhang is supported by National Natural Science Foundation of China, No. 12031015, 11771428, 12026217.

Conflict of interest

The authors declare there is no conflicts of interest.