Existence and asymptotic behavior of normalized solutions for the mass supercritical fractional Kirchhoff equations with general nonlinearities

1.   Introduction and main results

The purpose of this paper is to study the existence and blow-up behavior of positive solutions for the following fractional Kirchhoff equation:

having prescribed mass

where $a, b, c$ are positive constants, $1\leqslant N < 4s$, $s\in (0, 1)$, $V\in C(\mathbb{R}^{N}, \mathbb{R})$, $f\in C(\mathbb{R}, \mathbb{R})$, $\omega\in \mathbb{R}$ is a Lagrange multiplier, and $(-\Delta)^{s}$ denotes the fractional Laplacian operator defined as

where $\mathcal{F}$ denotes the Fourier transform on $\mathbb{R}^{N}$. It is well-known that it can also be computed by

if $v$ is smooth enough, where $C_{N, s}$ is the normalization constant, and $P.V.$ denotes a Cauchy principle value, see ^[13,15,18].

When $s = 1, f(u) = |u|^{p-2}u$, $V(x) = 0$, ($\mathcal{A}$) reduces to the following case, i.e.,

The sharp existence and the concentration behavior of the normalized solution of (1.2) in the mass subcritical, supercritical and critical cases were established in ^[7,14,22]. In fact, the author obtained the solutions by looking for critical points of the following functional:

constrained on the $L^{1}$ sphere in $H^{1}(\mathbb{R}^{N})$,

In ^[22], Ye showed that the constrained minimization problem

admits a minimizer if, and only if, $c > c_{p}^{*}$ with $p\in (0, 2+\frac{4}{N}]$ or $c\geqslant c_{p}^{*}$ with $p\in (2+\frac{4}{N}, 2+\frac{8}{N})$, and there is no minimizer of (1.3) if $p\in [2+\frac{8}{N}, +\infty)$. In ^[23], Ye studied (1.3) with $p = 2+\frac{8}{N}$ and obtained that there exists a mountain pass critical point of $I(u)$ on $S_{c}$ if $c > c^{*}$. In the mass subcritical case, a complete classification with respect to the exponent $p$ for its $L^{2}$ normalized critical points can be deduced from some simple energy estimates in ^[24]. To be precise, they gained existence and the uniqueness of the mountain pass type minimum for (1.3) with $p\in (2+\frac{8}{N}, 2^{*})$ or $p = 2+\frac{8}{N}$ and $c > c^{*}$. Moreover, if $f(u)$ satisfies some suitable conditions, the authors of ^[8] studied the blow-up behavior of minimizers (1.3).

To the best of our knowledge, if $b\equiv 0$, then Eq ($\mathcal{A}$ ) with a prescribed mass has been studied in ^[16,17,21]. Frank-Lenzmann-Silvestre ^[4] established the uniqueness result of the positive ground state solution of the equation

which is an important foundation for blow-up analysis. In ^[2], Du et al. explored the existence, nonexistence and mass concentration of $L^{2}$-normalized solutions for nonlinear fractional Schrödinger equations with nonnegative potentials

under the following assumptions:

$(f_{1})$: $f\in C(\mathbb{R}, \mathbb{R}), |f(t)|\leqslant c_{1}(|t|+|t|^{p-1})$ for some $c_{1} > 0$ and $2 < p < 2+\frac{4s}{N}$.

$(f_{2})$: $f\in C(\mathbb{R}, \mathbb{R}), |f(t)|\leqslant c_{1}(|t|+|t|^{p-1})$ for some $c_{1} > 0$ and $2+\frac{4s}{N} < p < 2_{s}^{*} = \frac{2N}{N-2s}$.

$(f_{3})$: there exist $\nu > 2+\frac{4s}{N}$ and $r_{0} > 0$ such that

For the case of power function $f(t) = t^{p-2}t$ with $2 < p < 2_{s}^{*}$, Du et al. conducted a complete classification of the existence and nonexistence of minimizers, except that $p = 2+4s/N$. Very recently, Bao-Lv-Ou ^[1] investigated the following fractional Schrödinger equation with prescribed mass:

where $s\in (0, 1), 2+\frac{4s}{N} < p < 2_{s}^{*}$. The existence of the bounded state normalized solution under various conditions on $a(x)$ was demonstrated in ^[1]. For more recent works about the fractional Schrödinger or Kirchhoff equation, see ^[10,12,20] and the references therein.

It is worth pointing out that $p = 2+4s/N$ is the mass critical exponent related to ($\mathcal{A}$). However, Eq ($\mathcal{A}$) involving the general potential and nonlinearities has not yet been resolved. An interesting question now is whether the same existence or nonexistence results occurs for the nonhomogeneous nonlinearities and mass supercritical case of ($\mathcal{A}$). On the other hand, there have been no previous articles studying the asymptotic behavior of solutions to ($\mathcal{A}$). Therefore, our goal is to fill the gaps in these areas. More precisely, in the first part of the paper, we prove the existence of solutions for ($\mathcal{A}$) with $V(x)\neq 0$.

Before describing more details, let's introduce the following fractional Gagliardo-Nirenberg-Sobolev inequality in ^[2] and Hardy inequality in ^[5].

Lemma 1.1. ^[2] For $u\in H^s( \mathbb{R}^{N})$ and $q\in (0, 2_s^*-2)$, the fractional Gagliardo-Nirenberg-Sobolev inequality

is attained at a function $\phi_q(x)$ with the following properties:

(i) $\phi_q(x)$ is radial, positive, and strictly decreasing in $|x|$.

(ii) $\phi_q(x)$ belongs to $H^{2s+1} (\mathbb{R}^{N})\cap C^\infty (\mathbb{R}^{N})$ and satisfies

where $C_i (i = 1, 2)$ are positive constants.

(iii) $\phi_q(x)$ is the unique solution of the fractional Schrödinger equation

(iv) $C_{opt} = \frac{q+2}{2\|\phi_q\|_{2}^q}$.

Lemma 1.2. ^[5] Let $s\in (0, 1)$ and $N > 2s$. Then, for all $u\in D^{s, 2}(\mathbb{R}^{N})$,

where

Lemma 1.3. ^[13] Let $s\in(0, 1)$ and $N > 2s$. Then, there exists a constant $S > 0$ such that for any $u\in D^{s, 2} (\mathbb{R}^{N})$,

In this paper, we will require $f(x)$ to satisfy the following conditions:

$(H_{1})$ $f\in C(\mathbb{R}, \mathbb{R})$ and odd.

$(H_{2})$ There exist some $\lambda, \gamma\in \mathbb{R}^{+}\times \mathbb{R}^{+}$ with

such that

$(H_3)$ The function $\widetilde{F}(t) : = \frac{1}{2}f(t)t-F(t)$ is of class $C^{1}$ and

where $\lambda$ is given in $(H_{2})$.

We assume that $V(x)$ is a radial function and satisfies the following assumptions:

$(V_{1})$ $V(x)\in C^{1}(\mathbb{R}^{N})\cap L^{p}(\mathbb{R}^{N}), p\in (\frac{N}{2}, \infty), \; \lim_{|x|\to \infty}V(x) = 0, \; \inf_{x\in\mathbb{R}^{N}}V(x) = 0$.

$(V_{2})$ There exists $\kappa_{1}\in [0, s)$ such that one of the following two conditions holds

(ⅰ)$\nabla V(x)\cdot x \leqslant \frac{2a\kappa_{1}}{H_{N, s}|x|^{2s}}$, for any $x\in \mathbb{R}^{N}\setminus\{0\}$, where $H_{N, s}$ is given in Lemma 1.2;

(ⅱ)$\|\max\{\nabla V(x)\cdot x, 0\}\|_{L^{\frac{N}{2s}}(\mathbb{R}^{N})} \leqslant 2a\kappa_{1}S$, where $S$ is given in Lemma 1.3.

$(V_{3})$ There exists $\kappa_{2}\in \left(0, \frac{N(\lambda-2)-4s}{4}\right)$ such that

for any $x\in \mathbb{R}^{N}\setminus\{0\}$.

Let us introduce the space of radial functions in $H^{s}(\mathbb{R}^{N})$ defined by

It is standard to see that critical points of the energy functional

restricted to the (mass) constraint

are normalized solutions of ($\mathcal{A}$).

From $(H_{1})$ and $(H_{2})$, there exist $C_{1}, C_{2}>0$ such that for each $t\in\mathbb{R}$,

where $C_{2} = F(1)$. It follows from (1.4) that there exists $C_{3} > 0$, for any $u\in H^{s}(\mathbb{R}^{N})$,

where $\lambda' = \frac{(\lambda-2)N}{2s} \in (2, \lambda), \gamma' = \frac{(\gamma-2)N}{2s} \in (2, \gamma)$. Let

$(V_{4})$ $\sup_{x\in \mathbb{R}^{N}} \left(V(x)+\frac{1}{2s}\nabla V(x)\cdot x \right) < K$.

Our first result is as follows.

Theorem 1. Let $s\in(0, 1)$ and $N > 2s$. Assume that $(V_{1})$–$(V_{4})$ and $(H_{1})$–$(H_{3})$ hold. Then, Eq ($\mathcal{A}$) admits at least a radial solution.

The second purpose of this article is to establish the existence results of ground state solutions for ($\mathcal{A}$) wih $V(x) = 0$.

Theorem 2. Let $1\leqslant N<4s$, $s\in (0, 1)$, $V(x) = 0$, and suppose that $f$ satisfies $(H_{1})$–$(H_{3})$. Then, for all $c>0$ fixed, Eq ($\mathcal{A}$) admits a ground state normalized solution $(\omega_{c}, u_{c})$ with $\omega_{c}>0$ and $u_{c}\in H^{s}_{rad}(\mathbb{R}^{N})$.

Remark 1. Theorems 1 and 2 extend and complement the previous results on the fractional Kirchhoff equation with a mass super critical general nonlinearities. Particularly, unlike ^[8], in which $V(x) = 0$, we apply a new deformation argument for the constrained functional on $\mathcal{S}_{c}$.

Remark 2. We now point out some difficulties faced in Theorems 1 and 2.

(i) When $V(x)>0$ and the nonlinearity $f$ is general mass supercritical, it prevents us from obtaining the compactness. We will apply a new deformation argument for the constraint functional with a new type of Palais-Smale condition denoted by $(PSP)_{m}$.

(ii) The simplest case of the function $f$ satisfying the assumptions $(H_{1})$–$(H_{3})$ is $f(t) = |t|^{p-2}t$ with $2$$+8s/N<p<2_{s}^{*}$. Naturally, the class of general nonlinearities satisfying these assumptions is much more difficult than this homogeneous case.

(iii) Due to the appearance of the Kirchhoff nonlocal term

Eq ($\mathcal{A}$) is no longer point by point identity. Compared with ^[8], it is worth noting that ($\mathcal{A}$) is a double nonlocal equation, and the decay estimates of test function near infinity are different from those in the case of the classical local problem; we thus borrow ideas of ^[3] for nonlocal operators to establish the decay estimates. This phenomenon has caused some mathematical difficulties, making research on such problems particularly interesting.

Our other aim is to study the behavior of the normalized solution $u_{c}$ given in Theorem 2 as $c\to 0$ and $c\to +\infty$. In this direction, we need to assume that

$(H_{4})$ $\lim_{t\to 0^{+}}\frac{f(t)}{t^{\lambda-1}} = \mu_{1}>0$.

$(H_{5})$ $\lim_{t\to +\infty}\frac{f(t)}{t^{\gamma-1}} = \mu_{2}>0$.

The following results demonstrate the asymptotic behavior of $u_{c}$ in the sense of $C_{loc}^{2, \alpha}(\mathbb{R}^{N})$ as well as $H^{s}(\mathbb{R}^{N})$.

Theorem 3. Let $1\leqslant N<4s$, $s\in (\frac{1}{2}, 1)$, $V(x) = 0$, and suppose that $f$ satisfies $(H_{1})$–$(H_{5})$. For $c>0$, let $(\omega_{c}, u_{c})$ be given by Theorem 2, then

where $Q$ is the unique radial positive solution of

Theorem 4. Let $1\leqslant N < 4s$, $s\in (\frac{1}{2}, 1)$, $V(x) = 0$, and suppose that $f$ satisfies $(H_{1})$–$(H_{5})$. For $c > 0$, let $v_{c}$ and $Q$ be given by Theorem 3, then

Theorem 5. Let $1\leqslant N < 4s$, $s\in (\frac{1}{2}, 1)$, $V(x) = 0$, and suppose that $f$ satisfies $(H_{1})$–$(H_{5})$. For $c > 0$, let $(\omega_{c}, u_{c})$ be given by Theorem 2, then

where $U$ is the unique radial positive solution of

Theorem 6. Let $1\leqslant N < 4s$, $s\in (\frac{1}{2}, 1)$, $V(x) = 0$, and suppose that $f$ satisfies $(H_{1})$–$(H_{5})$. For $c > 0$, let $\tilde{v}_{c}$ and $U$ be given by Theorem 5, then

Throughout the paper, we use the following notations:

● $L^{q}(\mathbb{R}^{N})$ denotes the Lebesgue space with the norm

● For any $x\in \mathbb{R}^{N}$ and $R > 0$, $B_{R}(x): = \{y\in \mathbb{R}^{N}:|y-x| < R\}$.

● $C$ indicates positive numbers that may be different in different lines.

The rest of this paper is organized as follows. Section 2 is dedicated to some preliminary notations and lemmas. In Section 3, we obtain the radial solutions for Eq ($\mathcal{A}$) with $V(x)\neq 0$ and Theorem 1 will be proved there. In Section 4, we derive the existence of ground state normalized solution for problem ($\mathcal{A}$) and give the proof of Theorems 2. In Section 5, we deal with asymptotic property of minimizers to problem ($\mathcal{A}$) by proving Theorems 3–6.

2.   Preliminaries

In this section, we provide some lemmas that will be frequently used in the rest of this article.

We claim that the condition $(V_{2})$ yields that for any $u\in H^{s}(\mathbb{R}^{N})$,

Indeed, if ${\rm(i)}$ of $(V_{2})$ holds, from Lemma 1.2, we deduce that

If ${\rm(ii)}$ of $(V_{2})$ holds, by the Sobolev embedding inequality, we observe that

Define

and

Then

Set

Lemma 2.1. Let $s\in(0, 1)$ and $N > 2s$. Assume that $(H_{1})$–$(H_{3})$ and $(V_{1})$–$(V_{4})$ hold, then there exists $\bar{m} > 0$ such that $m_{c}\geqslant \bar{m} > 0$. Moreover,

where $\varrho$ is given in (1.10).

Proof. For any $u\in \partial \mathcal{M}_{c}$, applying (1.7), (1.8), (2.1), and (2.2), we obtain that

Set

Since $g(t)$ is decreasing on $[0, +\infty)$, there exists a unique $t_{0} > 0$ such that

and $t_{0}$ is dependent on $\kappa_{1}, \lambda, \gamma, C_{2}, C_{3}, c$. It follows from (2.4) that $g(\|u\|_{D^{s, 2}(\mathbb{R}^{N})})\leqslant 0$. Therefore, for any $u\in \partial \mathcal{M}_{c}$, we observe that $\|u\|_{D^{s, 2}(\mathbb{R}^{N})}\geqslant t_{0}$ and

due to (1.7), (2.2), $(V_{3})$, and Lemma 1.2. Let

then, $m_{c}\geqslant \bar{m} > 0$. From (1.9), (2.5), and (2.6), we conclude that

which yields that

If $L < c^{\frac{\gamma-\gamma'}{2}}+c^{\frac{\lambda-\lambda'}{2}}$, thus $g(1) < 0$, and we conclude that $0 < t_{0} < 1$ and

that is, $t_{0} > \left(\frac{L^{2}/2}{c^{\lambda-\lambda'} +c^{\gamma-\gamma'}} \right)^{\frac{1}{2(\lambda'-2)}}$. If $L\geqslant c^{\frac{\gamma-\gamma'}{2}}+c^{\frac{\lambda-\lambda'}{2}}$, thus $g(1)\geqslant 0$, and $t_{0}\geqslant 1$. Therefore, $t_{0}^{2}\geqslant \varrho$ and (2.3) holds. □

3.   Nontrivial solutions for ($\mathcal{A}$ ) with $V(x)\neq 0$

As in ^[9], we define

where

We will show that $m_{\Gamma}$ is well-defined.

Lemma 3.1. Let $s\in(0, 1)$ and $N > 2s$. Assume that $(H_{1})$–$(H_{3})$ and $(V_{1})$–$(V_{4})$ hold, then $\Gamma\neq \emptyset$.

Proof. For any $u\in \mathcal{S}_{c}$, define

It's not difficult to see $u_{\tau}(x)\in \mathcal{S}_{c}$. From (1.6), for any $\tau\geqslant1$,

and for any $0 < \tau < 1$,

According to (3.1) and

we observe that

which yields that $J(u_{\tau})\to-\infty$, as $\tau\to+\infty$. On the other side, from (3.2), we conclude that

and

which leads to $J(u_{\tau})\to 0^{+}$, as $\tau\to 0^{+}$, recalling $\lim_{|x|\to\infty}V(x) = 0$. From $(V_{2})$, $(V_{3})$, and (1.5), we see that

Combining with (1.7), (2.2), and (3.1), we deduce that

and

which yields that $\mathcal{P}(u_{\tau})\to -\infty$ as $\tau\to+\infty$, and $\mathcal{P}(u_{\tau})\to 0^{+}$ as $\tau\to 0^{+}$. Therefore, for any given $u\in \mathcal{S}_{c}$, we can take $\tau_{0} > 0$ large enough, and $\tau_{1}\in (0, 1)$ small enough such that

Thus, taking $\gamma_{0}(t): = u_{\tau_{0}(1-t)+\tau_{1}t}$, we see that $\gamma_{0}(t)\in \Gamma$. □

Lemma 3.2. Let $s\in(0, 1)$ and $N > 2s$. Assume that $(H_{1})$–$(H_{3})$ and $(V_{1})$–$(V_{4})$ hold, then $m_{\Gamma} \geqslant m_{c} \geqslant \bar{m}.$

Proof. For any $\gamma\in \Gamma$, there exists $t_{\gamma}\in [0, 1]$ such that $\gamma(t_{\gamma})\in \partial \mathcal{M}_{c}$. Therefore,

Together with Lemma 2.1, the conclusion holds. □

Now in view of Lemma 3.1, we can apply a new deformation argument for the constraint functional on $\mathcal{S}_{c}$ with a new type of Palais-Smale condition denoted by $(PSP)_{m}$. The functional $J$ satisfies the $(PSP)_{m}$ condition on $\mathcal{S}_{c}$, if, and only if, any $(PSP)_{m}$ sequence $\{u_{n}\}\subset \mathcal{S}_{c}$ satisfying

has a strongly convergent subsequence.

Lemma 3.3. Let $s\in(0, 1)$ and $N > 2s$. Assume that $(H_{1})$–$(H_{3})$ and $(V_{1})$–$(V_{4})$ hold. If $\{u_{n}\}\subset \mathcal{S}_{c}$ is a $(PSP)_{m}$ sequence satisfying (3.3) with $m\geqslant \bar{m}$, then

(i) $\{u_{n}\}$ is bounded in $H^{s}(\mathbb{R}^{N})$.

(ii) there exists $\omega > 0$ such that the sequence of Lagrange multipliers $\omega_{n}$ satisfying

converges to $\omega$ in the sense of subsequence.

Proof. (ⅰ) For every $m\in \mathbb{R}$, let $\{u_{n}\}\subset \mathcal{S}_{c}$ be a $(PSP)_{m}$ sequence satisfying (3.3). From (3.3), $(V_{1})$, $(V_{3})$ and (1.7), we conclude that

which implies that $\{u_{n}\}$ is bounded in $H^{s}(\mathbb{R}^{N})$, recalling that $\kappa_{2}\in \left(0, \frac{N(\lambda-2)-4s}{4}\right)$ and $\lambda > 2+\frac{8s}{N}$. Then, up to a subsequence, $u_{n}\rightharpoonup u$ weakly in $H^{s}(\mathbb{R}^{N})$. From (3.3), there exists $\{\omega_{n}\}\subset \mathbb{R}$ such that

(ⅱ) By (3.3), we see that

Combining with (1.7), we observe that

Set

Obviously, $y_{n}\geqslant 0$, $\bar{\gamma}$, $\bar{\alpha} > 0$, and

Applying $(V_{4})$, (1.11), and Lemma 2.1, there exists $\delta > 0$ such that

Noting that $m\geqslant \bar{m}$, we obtain

Let

and from (3.5), we deduce that

Using (3.4)–(3.6), we deduce that

where $\hat{m} = \bar{\alpha}\bar{y}-\bar{z}$, and $(\bar{y}, \bar{z})$ satisfies

Therefore,

Gathering (1.7), (2.2), (3.7), and (3.8), we conclude that

Meanwhile, from (3.9), $(H_{2})$, $(V_{1}), (V_{3})$, and Lemmas 1.1 and 1.2, we infer that

(3.9) and (3.10) imply that $\omega_{n}\to \omega > 0$. □

In what follows, we consider that $V(x)$ is radial, and then we can choose $H^{s}_{rad}(\mathbb{R}^{N})$ as the workspace. We take $\mathcal{S}_{c}^{rad} : = \mathcal{S}_{c}\cap H^{s}_{rad}(\mathbb{R}^{N}).$

Lemma 3.4. Assume that $(V_{1})$–$(V_{4})$ and $(H_{1})$–$(H_{3})$ hold. Then, $J(u)$ satisfies the $(PSP)_{m}$ condition on $\mathcal{S}_{c}^{rad}$ for $m\geqslant \bar{m}$.

Proof. Let $\{u_{n}\}\subset \mathcal{S}^{rad}_{c}$ be a $(PSP)_{m}$ sequence satisfying (3.3) with $m\geqslant \bar{m}$. Due to Lemma 3.3, there exist $u\in H^{s}_{rad}(\mathbb{R}^{N})$ and $\omega > 0$ such that, up to a subsequence, $u_{n}\rightharpoonup u$ weakly in $H^{s}_{rad}(\mathbb{R}^{N})$ and

Then, using the compact embedding $H_{rad}^{s}(\mathbb{R}^{N}) \hookrightarrow\hookrightarrow L^{q}(\mathbb{R}^{N})$ for all $q\in (2, 2_{s}^{*})$, we deduce that $u_{n}\to u$ strongly in $L^{q}(\mathbb{R}^{N})$ and

Due to $(V_{1})$, for $p\in (\frac{N}{2s}, +\infty)$, $\|u_{n}-u\|_{L^{2p{'}}(\mathbb{R}^{N})} \to 0$ as $n\to\infty$, where $p^{'} = \frac{p}{p-1}$. Hence, by $u_{n}\rightharpoonup u$ weakly in $H^{s}_{rad}(\mathbb{R}^{N})$, we observe that

Therefroe, by (3.11)–(3.13), we see that

which implies that $u_{n}\to u$ strongly in $H^{s}_{rad}(\mathbb{R}^{N})$. □

To prove the existence of nontrivial solutions, we use the following deformation result given by ^[9].

Define

Lemma 3.5. ([9, Proposition 4.5]) Assume that $J$ satisfies the $(PSP)_{m}$ condition on $\mathcal{S}_{c}$. For any neighborhood $D$ of $\mathcal{K}_{m}$ (if $\mathcal{K}_{m} = \emptyset$, $D = \emptyset$) and any $\tilde{\varepsilon} > 0$, there exists $\varepsilon\in (0, \tilde{\varepsilon})$ and $\eta\in C([0, 1]\times \mathcal{S}_{c}, \mathcal{S}_{c})$ such that

(i) $\eta(0, u) = u$, for $u\in \mathcal{S}_{c}$;

(ii) $\eta(t, u) = u$, for $t\in [0, 1]$ if $u\in [J\leqslant m-\tilde{\varepsilon}]_{\mathcal{S}_{c}}$;

(iii) $t\longmapsto J(\eta(t, u))$ is nonincreasing for $u\in \mathcal{S}_{c}$;

$(iv)$ $\eta(1, [J\leqslant m-\varepsilon]_{\mathcal{S}_{c}}\setminus D) \subset [J\leqslant m-\varepsilon]_{\mathcal{S}_{c}}, \eta(1, [J\leqslant m+\varepsilon]_{\mathcal{S}_{c}}) \subset [J\leqslant m-\varepsilon]_{\mathcal{S}_{c}} \cup D$.

Proof of Theorem 1. Let

then $U = \emptyset$. It follows from Lemmas 3.2 and 3.4 that $J$ satisfies the $(PSP)_{m_{\Gamma}}$ condition. Due to $m_{\Gamma}\geqslant m_{c} > 0$, taking $\tilde{\varepsilon} = m_{\Gamma}-\frac{m_{c}}{2}$, and using Lemma 3.5, we obtain that there exist $\varepsilon\in (0, \tilde{\varepsilon}), \eta\in C([0, 1]\times\mathcal{S}_{c}^{rad}, \mathcal{S}_{c}^{rad})$ satisfying

According to the definition of $m_{\Gamma}$ and Lemma 3.1, there exists $\gamma\in \Gamma$ such that

Let us define $\tilde{\gamma}: = \eta(1, \gamma(t))$, and claim that $\tilde{\gamma}\in \Gamma$. Indeed, $J(\gamma(0)) < \frac{1}{2}m_{c} = m_{\Gamma}-\tilde{\varepsilon}$, which implies that $\gamma(0)\in [J\leqslant m_{\Gamma}-\tilde{\varepsilon}]_{\mathcal{S}^{rad}_{c}}$. By (3.14), we observe that $\tilde{\gamma}(0) = \eta(1, \gamma(0)) = \gamma(0)$ and $\tilde{\gamma}(1) = \gamma(1)$ analogously. By (3.15), we conclude that

Therefore, $J(\tilde{\gamma}(t))\leqslant m_{\Gamma}-\varepsilon$ for any $t\in [0, 1]$. We have

which yields a contradiction. Hence, $\mathcal{K}_{m_{\Gamma}}\neq \emptyset$. Thie implies that ($\mathcal{A}$) admits a radial solution. □

4.   Ground satate solution for ($\mathcal{A}$) with $V(x) = 0$

In this section, we consider the existence of ground state solutions for ($\mathcal{A}$) with $V(x) = 0$. From now on, in this article, we always assume that $(H_ {1})$–$(H_ {3})$ hold and will not further mention it.

Lemma 4.1. Let $1\leqslant N<4s$, $s\in (0, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{3})$. If $u\in H^{s}(\mathbb{R}^{N})$ is a nontrivial solution of Eq ($\mathcal{A}$), then $u\in\mathcal{M}$, where

and

Proof. Let $u$ be a solution to Eq ($\mathcal{A}$), and we derive that

Meanwhile, $u$ satisfies the following Pohožaev identity:

Combining (4.1) and (4.2), we conclude that

□

Define

where $u_{\tau}(x): = \tau^{\frac{N}{2}}u(\tau x), \tau > 0$. We observe that

Particularly, it holds that $\mathcal{P}(u) = (J_{u})^{'}(1)$. By exploiting it, we can deduce the following result.

Lemma 4.2. Let $u\in \mathcal{S}_{c}$. Then, $\tau > 0$ is a critical point of $J_{u}(\tau)$ if, and only if, $u_{\tau}\in \mathcal{M}_{c}$, where

Lemma 4.3. For each critical point of $J|_{\mathcal{M}_{c}}$, if $(J_{u})^{''}(1)\neq 0$, then there exists a $\omega\in \mathbb{R}$ such that

Proof. Let $u$ be a critical point of $J(u)$ constrained on $\mathcal{M}_{c}$, then there exist $\omega, \nu\in \mathbb{R}$ such that

Therefore, we need to show that $\nu = 0$. Let

and it is the corresponding energy functional of (4.5). Thanks to (4.5), one sees that $u$ satisfies the corresponding Pohozaev identity

By (4.4), we observe that

which yields

Together with (4.6), one gets

According to $(J_{u})^{''}(1)\neq 0$, we derive that $\nu = 0$. □

Lemma 4.4. Let $1\leqslant N < 4s$, $s\in (0, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{3})$. Then if $\omega\leqslant 0$, Eq ($\mathcal{A}$) has no nontrivial solution.

Proof. Assume by contradiction that $u\in H^{s}(\mathbb{R}^{N})$ is a nontrivial solution of Eq ($\mathcal{A}$) with $\omega\leqslant0$. It follows from (4.1) and (4.2), $\omega\leqslant0$, and $(H_{2})$ that

Therefore, we derive that $\omega = 0$ and

For $\omega = 0$, by Lemma 4.1, we deduce that $u\in \mathcal{M}$, that is, (4.3) holds. This yields

which contradicts $u\not\equiv 0$ in $H^{s}(\mathbb{R}^{N})$. □

Lemma 4.5. Let $1\leqslant N < 4s$, $s\in (0, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{3})$. Then, for any $c > 0$ fixed, there exists a $\sigma_{c} > 0$ such that

i.e.,

Proof. Since $u_{\tau}\in \mathcal{M}_{c}$, we obtain that $\mathcal{P}(u_{\tau}) = 0$. According to (4.3), we derive that

Together with $(H_{2})$, for $\|u\|_{D^{s, 2}(\mathbb{R}^{N})} = 1$, we get that

From $(H_{2})$, there exists some $C > 0$ such that

Note that for $u\in \mathcal{M}_{c}$ with $\|u\|_{D^{s, 2}(\mathbb{R}^{N})} = 1$, by Lemma 1.1, there exists $C > 0$ such that

Combining (4.7) and (4.8), we deduce that

which implies that $\sigma_{c} > 0$. □

Lemma 4.6. Let $1\leqslant N < 4s$, $s\in (0, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{3})$. Then, $(J_{u})^{''}(1) < 0$, for each $u\in\mathcal{M}_{c}$, and $\mathcal{M}_{c}$ is a natural constraint of $J|_{\mathcal{S}_{c}}$.

Proof. From (4.4), by a direct caculation, we derive that

Then,

Together with $(H_{3})$ and $\mathcal{P}(u) = 0$, we conclude that

According to $\lambda > 2+\frac{8s}{N}$ and Lemma 4.5, one can see that

Recalling Lemma 4.3, we conclude that $\mathcal{M}_{c}$ is a natrual contraint of $J|_{\mathcal{S}_{c}}$. □

Corollary 4.1. Let $1\leqslant N < 4s$, $s\in (0, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{3})$. Then, for each $u\in H^{s}(\mathbb{R}^{N}) \setminus \{0\}$, there exists a unique $\tau_{u} > 0$ such that $u_{\tau_{u}}\in \mathcal{M}$. Moreover,

Proof. Set $c: = \|u\|^{2}_{L^{2}(\mathbb{R}^{N})}$. From $(H_{1})$ and $(H_{2})$, we observe that for each $\tau\geqslant 0$ and $t\in \mathbb{R}$,

Hence, we obtain

which implies that

noting that $\lambda > 2+\frac{8s}{N}$. Recalling that

we can deduce that $\mathcal{P}(u_{\tau}) > 0$ for $\tau > 0$ small enough. Meanwhile, according to (4.4), one can see that $(J_{u})^{'}(\tau) > 0$ for $\tau > 0$ small enough. Then, there exists a $\tau_{0} > 0$ such that $J_{u}(\tau)$ is increasing in $\tau\in (0, \tau_{0})$.

On the other hand, from $\int_{\mathbb{R}^{N}} F(u) \mathrm{d}x > 0$, (4.9), and $\lambda > 2+\frac{8s}{N}$, we derive that

which yields that

Hence, there exist some $\tau_{1} > \tau_{0}$ such that

and $(J_{u})^{'}(\tau_{1}) = 0$. Then, by Lemma 4.2, we conclude that $u_{\tau_{1}}\in \mathcal{M}$. Next, we will prove that $\tau_{1}$ is unique. Assume by contradiction that there exists $\tau_{2} > 0$ such that $u_{\tau_{2}}\in \mathcal{M}$. From Lemma 4.6, we know that $\tau_{1}$ and $\tau_{2}$ are strict local maximum points of $J_{u}(\tau)$. Without loss of generality, we suppose that $\tau_{1} < \tau_{2}$. Thus, there exist some $\tau_{3}\in (\tau_{1}, \tau_{2})$ such that

which indicates that $\tau_{3}$ is a local minimum point of $J_{u}(\tau)$. Then, $(J_{u})^{'}(\tau_{1}) = 0$ and $u_{\tau_{3}}\in \mathcal{M}$ with $(J_{u_{\tau_{3}}})^{''}(1) = (J_{u})^{''}(\tau_{3}) \geqslant 0$, which contradicts Lemma 4.6. □

Corollary 4.2. Let $1\leqslant N < 4s$, $s\in (0, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{3})$. For each $u\in H^{s}(\mathbb{R}^{N})\setminus \{0\}$, let $\tau_{u}$ be given in Corollary 4.1. Then,

Proof. By Corollary 4.1, we obtain that

Moreover,

The conclusion follows from (4.4). □

Lemma 4.7. Let $1\leqslant N < 4s$, $s\in (0, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{3})$. Then $J|_{\mathcal{M}_{c}}$ is coercive.

Proof. For $u\in \mathcal{M}_{c}$, from $(H_{2})$, we observe that

Therefore,

Thanks to $\lambda > 2+\frac{8s}{N}$, we complete the proof. □

For given $c > 0$, let us define

Since $u$ is a solution to ($\mathcal{A}$) satisfying (1.1), $u$ must belong to $\mathcal{M}_{c}$. If $u$ attains $m_{c}$, we can assert that $u$ is the least energy solution, i.e., ground state solution. It follows from (4.12) and Lemma 4.5 that $m_{c}>0$ for each $c>0$.

For each $u\in H^{s}(\mathbb{R}^{N})$, let $u^{*}$ be the symmetric radial decreasing rearrangement of $u$. By $(H_{1})$, without loss of generality, we assume that $u$ is nonnegative. Then, we can obtain that

From [11, Lemma 2.3], one can see that

Thus, we derive that

Set

Define

and we can obtain that

Moreover, we find the following.

Lemma 4.8. Let $1\leqslant N < 4s$, $s\in (0, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{3})$. Then,

Proof. Since $\mathcal{S}_{c}^{rad} \subset \mathcal{S}_{c},$ it is clear that $m_{c}^{rad} \geqslant m_{c}.$ On the other side, for each $t > 0$,

Hence, it holds true that

As a consequence of (4.13), for each $u\in\mathcal{M}_{c}$, we obtain that

which yields that $m_{c}^{rad} \leqslant m_{c}$, by the arbitrary of $u\in \mathcal{M}_{c}$, and this ends the proof. □

Proof of Theorem 2. Thanks to Lemma 4.8, let $\{u_{n}\}\subset \mathcal{M}_{c}^{rad}$ be such that $J(u_{n})\to m_{c} > 0$. From Lemma 4.7, we obtain that $\{u_{n}\}$ is bounded in $H^{s}(\mathbb{R}^{N})$. We may suppose that up to a subsequence, $u_{n}\rightharpoonup u$ in $H^{s}(\mathbb{R}^{N})$. Then, for $N = 2, 3$, using the compact embedding $H_{rad}^{s}(\mathbb{R}^{N}) \hookrightarrow\hookrightarrow L^{q}(\mathbb{R}^{N})$ for all $q\in (2, 2_{s}^{*})$, we deduce that

For $N = 1$, we may suppose that $u_{n} = u_{n}^{*}, \forall n\in \mathbb{N}$. Then, (4.14) also holds true.

Now, we claim that $u\neq 0$. Suppose by contradiction that $u = 0$. Then, $\int_{\mathbb{R}^{N}} \widetilde{F}(u_{n}) \mathrm{d}x = o_{n}(1)$, and taking into account of $\{u_{n}\}\subset \mathcal{M}_{c}^{rad}$, we obtain that

which contradicts Lemma 4.5.

Since $\{u_{n}\}$ is bounded in $H^{s}(\mathbb{R}^{N})$, it is obvious that

is a bounded sequence. Particularly, applying (4.3), the definition of $\widetilde{F}$, and $(H_{2})$, we derive that

On the other hand, from $(H_{2})$, it follows that

Due to $\gamma > 2+\frac{8s}{N}$, by Lemma 4.5, there exist some $\delta_{c} > 0$ such that for any $n\in \mathbb{N}$,

Therefore, together with (4.15), there exist some $\eta_{c} > 0$ such that for any $n\in\mathbb{N}$,

Hence, we can assume that $\omega_{n}\to \omega_{c} > 0$. Up to a subsequence, if necessary, we can assume that $\|u_{n}\|_{D^{s, 2}(\mathbb{R}^{N})}^{2} \to A\geqslant 0$. It is easily seen that $u_{c}\in H^{s}_{rad}(\mathbb{R}^{N})$ is a solution of

Then, $u_{c}\in \mathcal{M}^{rad}$ and

Hence, $s(a+bA)(A-\|u_{c}\|_{D^{s, 2}(\mathbb{R}^{N})}^{2}) = 0$. By $s > 0, a, b > 0$, we get $\|u_{c}\|_{D^{s, 2}(\mathbb{R}^{N})}^{2} = A$, which yields that $u_{n}\to u_{c}$ in $D^{s, 2}_{0}(\mathbb{R}^{N})$. So, (4.17) gives that $u_{c}$ is a solution of

As a consequence, we get that

which yields that

So, $u_{c}\in \mathcal{S}_{c}$, and then $u_{c}\in \mathcal{M}_{c}$. Therefore,

that is, $(\omega_{c}, u_{c})$ is a ground state normalized solution to Eq ($\mathcal{A}$). Recalling Lemma 4.4, we complete the proof. □

5.   Proofs of Theorems 3–6

The aim of this section is to consider the continuity and the limit behavior of $m_{c}$ and $\omega_{c}$ as $c\to 0^{+}$ as well as $c\to+\infty$.

For $c>0$, let $(\omega_{c}, u_{c})$ be the solution to ($\mathcal{A}$), which is given by Theorem 2. We remark that $\omega_{c}>0$ and $u_{c}\in \mathcal{S}_{c}^{rad}$ satisfy

and

5.1.   Preliminary results on $m_{c}$ and $\omega_{c}$

Lemma 5.1. Let $1\leqslant N < 4s$, $s\in (0, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{3})$. Then, $m_{c}$ is continuous with respect to $c\in (0, +\infty)$.

Proof. Thanks to Theorem 2, we note that $m_{c}$ is attained by a symmetric decreasing function $u_{c}\in H^{s}_{rad}(\mathbb{R}^{N})$. Let $c > 0$ be fixed, and for each $\{c_{n}\}\subset \mathbb{R}^{+}$ with $c_{n}\to c$ as $n\to+\infty$, for the sake of simplicity, we denote $(\omega_{c_{n}}, u_{c_{n}})$ by $(\omega_{n}, u_{n})$. Without loss of generality, we may suppose that for any $n\in \mathbb{N}$, $\frac{c}{2} < c_{n} < 2c$. Taking into account of (4.11), (4.16), and (5.1), we deduce that

Let $u_{c/2}$ attain $m_{c/2}$. For $\theta\in (1, 4), \theta u_{c/2}\in \mathcal{S}_{\theta^{2}c/2}$. From Corollary 4.1, there exists a unique $\tau_{\theta} > 0$ such that $(\theta u_{c/2})_{\tau_{\theta}}\in \mathcal{M}_{\theta^{2}c/2}$ and

From (4.10), we conclude that

which leads to that $\tau_{\theta}$ is bounded due to $\frac{N\lambda}{2}-4s-N \geqslant \frac{N\gamma}{2}-4s-N > 0$. It easily follows from $m_{\theta^{2}c/2} \leqslant J((\theta u_{c/2})_{\tau_{\theta}})$ that $\{m_{\theta^{2}c/2}: \theta\in (1, 4)\}$ is bounded. Together with (5.2), we conclude that $\{u_{n}\}$ is bounded in $H^{s}(\mathbb{R}^{N})$.

Similar as the proof of Theorem 2, we can suppose that $\omega_{n}\to \omega_{c} > 0$ and $u_{n}\rightharpoonup u_{c}$ in $H^{s}(\mathbb{R}^{N})$, where $u_{c}\in \mathcal{S}_{c}$ is a solution of

Therefore,

If $J(u_{c})\neq m_{c}$, there exist some $\delta > 0$ such that for $n$ large enough,

Recall that $\sqrt{\rho}u_{c}\in\mathcal{S}_{\rho c}$, and let $\tau_{\rho} > 0$ be the unique number such that $(\sqrt{\rho}u_{c})_{\tau_{\rho}} = \sqrt{\rho}(u_{c})_{\tau_{\rho}} \in \mathcal{M}_{\rho c}$, that is,

By Lemma 4.5, we can derive that $\tau_{\rho}$ is bounded away from $0$ as $\rho$ approaches to 1. Thus, the uniqueness indicates that $\tau_{\rho}\to 1$ as $\rho\to 1$. Therefore, for $\rho$ close to 1 enough, we conclude that

Hence, there exists $N_{0}\in \mathbb{N}$ such that for $n\geqslant N_{0}$,

which contradicts (5.3). Consequently, we obtain that $J(u_{c}) = m_{c}$ and

□

Lemma 5.2. Let $1\leqslant N < 4s$, $s\in (0, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{3})$. Then,

Proof. Using $(H_{2})$ and Lemma 1.1, there exists $C > 0$ such that

Therefore,

which yields that

as $c\to 0^{+}$, since $\frac{N(\lambda-2)-4s}{2s}, \frac{\lambda}{2}-\frac{N(\lambda-2)}{4s}, \frac{N(\gamma-2)-4s}{2s}, \frac{\gamma}{2}-\frac{N(\gamma-2)}{4s} > 0.$ Together with (4.12), we observe that

For the case $c\to +\infty$, we choose a positive $v\in H^{s}(\mathbb{R}^{N})$ with $\|v\|_{L^{2}(\mathbb{R}^{N})} = 1$. For $c > 0$, it is easy to check that $u = \sqrt{c}v\in \mathcal{S}_{c}$. From Corollary 4.1, there exists a unique $\tau_{c} > 0$ such that $u_{\tau_{c}}\in \mathcal{M}_{c}$. We observe that $c\tau_{c}^{2s}\to 0$ as $c\to+\infty$. If not, there exists a sequence $\{c_{n}\}$ with $c_{n}\to+\infty$ as $n\to+\infty$ and $c_{n}\tau_{c_{n}}^{2s}\geqslant \delta > 0$, for all $n\in \mathbb{N}$. Applying (4.9), we obtain that

According to

and

we conclude that

Therefore,

for $n$ large enough, which is a contradiction. Hence, our claim $c\tau_{c}^{2s}\to 0$ as $c\to+\infty$ holds true. Consequently, combining with $w: = u_{\tau_{c}} = \sqrt{c}v_{\tau_{c}} \in \mathcal{M}_{c}$, we deduce that

On the other hand, it follows from (4.12) and Lemma 4.5 that $m_{c} > 0$ for each $c > 0$. This completes the proof. □

Lemma 5.3. Let $1\leqslant N < 4s$, $s\in (0, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{3})$. Then,

Proof. Combining (5.2) and Lemma 5.2, we derive that

Moreover, from $\mathcal{P}(u_{c}) = 0$, we conclude that

Together with (5.4), we see that

Similar as (4.15), we derive that

By $(H_{2})$, there exist $C_{1}, C_{2} > 0$ such that

which clearly means

recalling (5.5). □

Remark 3. For the two quantities $\zeta(c), \varphi(c)$, if there exist $C_{1}, C_{2} > 0$ independent of $c$ such that

we say $\zeta(c)$ and $\varphi(c)$ are comparable. Thus, by the proofs above in Section 5, it is easy to see that any two elements in the set

are comparable.

5.2.   The case of $c\to +\infty$

Let $c_{n}\to+\infty$. By Lemma 5.3, we know that $\omega_{c_{n}}\to 0$. For convenience, we denote $(\omega_{c_{n}}, u_{c_{n}})$ as $(\omega_{n}, c_{n})$. Set

From (5.4), one sees that $e_{n}\to 0$ as $n\to +\infty$.

Lemma 5.4. Let $1\leqslant N < 4s$, $s\in (\frac{1}{2}, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{5})$. Let $(\omega_{c}, u_{c})$ be the solution given by Theorem 2. Then,

Proof. Let us argue by contradiction. Assume that there exists a sequence $\{c_{n}\}\to +\infty$ such that

Take $x = y/m_{n}^{\frac{\gamma-2}{2s}}$ and define

Thus, $\max_{y\in \mathbb{R}^{N}} Q_{n}(y) = 1$ and

It follows from $(H_{1})$ and $(H_{2})$ that there exists a $C > 0$ such that

Therefore, we note that

Then, applying a similar argument to the proof of [19, Proposition 4.4], and passing to a subsequence if necessary, $Q_{n}\to Q$ in $C_{loc}^{2, \alpha}(\mathbb{R}^{N})$, for some $\alpha\in (0, 1)$. It is easy to see that $Q$ satisfies, in weak sense,

According to [6, Theorem 1.5], we derive that $Q = 0$, which contradicts $Q(0) = 1$. □

Define

By direct calculation, we see that $\tilde{u}_{n}(0) = \|\tilde{u}_{n}\|_{L^{\infty}(\mathbb{R}^{N})} = 1$ and

Lemma 5.5. Let $1\leqslant N < 4s$, $s\in (\frac{1}{2}, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{5})$. Let $(\omega_{c}, u_{c})$ be the solution given by Theorem 2. Then,

Proof. We assume by contradiction that there exists a sequence $c_{n}\to+\infty$ such that

Letting $x = 0$ in (5.7), one sees that

which is a contradiction. □

Lemma 5.6. Let $1\leqslant N < 4s$, $s\in (\frac{1}{2}, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{5})$. Let $(\omega_{c}, u_{c})$ be the solution given by Theorem 2. Then, $u_{c}(0) = \|u_{c}\|_{L^{\infty}(\mathbb{R}^{N})} \to 0$, as $c\to+\infty$.

Proof. Recalling that $\omega_{c}\to 0^{+}$ and $\|u_{n}\|_{D^{s, 2}(\mathbb{R}^{N})}\to 0$ as $c\to+\infty$, assume by contradiction that $\liminf_{n\to+\infty}u_{n}(0) > 0$. According to $f(u_{n})-\omega_{n}u_{n}\in L^{\infty}(\mathbb{R}^{N})$, applying a similar argument to the proof of [19, Proposition 4.4], and passing to a subsequence if necessary, we assume that $u_{n}\to u_{c}$ in $C_{loc}^{2, \alpha}(\mathbb{R}^{N})$, for some $\alpha\in (0, 1)$ with $u_{c}(0) = \max_{x\in \mathbb{R}^{N}}u_{c}(x) > 0$, and $u_{c}$ is a nonnegative bounded radial solution of

Then, by [6, Theorem 1.1], we derive that $u_{c}\equiv0$, which contradicts $u_{c}(0) > 0$. □

Lemma 5.7. Let $1\leqslant N < 4s$, $s\in (\frac{1}{2}, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{5})$. Let $(\omega_{c}, u_{c})$ be the solution given by Theorem 2. Then,

Proof. Assume by contradiction that there exists a sequence $c_{n}\to+\infty$ such that

Thus,

Set

A direct calculation shows that $\hat{u}_{n}(0) = \|\hat{u}_{n}\|_{L^{\infty}(\mathbb{R}^{N})} = 1$ and

From $(H_{2})$, Lemma 5.4 and (5.8), we derive that the right side of (5.9) is of $L^{\infty}(\mathbb{R}^{N})$. Therefore, applying a similar argument to the proof of [19, Proposition 4.4] and passing to a subsequence if necessary, we assume that $\hat{u}_{n}\to \hat{u}_{c}$ in $C_{loc}^{2, \alpha}(\mathbb{R}^{N})$, for some $\alpha\in (0, 1)$. Combining Lemma 5.6 and $(H_{4})$, noting that $e_{n}\to 0$, we deduce that $\hat{u}_{c}$ is a nonnegative bounded radial solution of

Thanks to [6, Theorem 1.5], we derive that $\hat{u}_{c} = 0$, which contradicts $\hat{u}_{c}(0) = 1$. □

Lemma 5.8. Let $1\leqslant N < 4s$, $s\in (\frac{1}{2}, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{5})$. Let $(\omega_{c}, u_{c})$ be the solution given by Theorem 2. Then, $\tilde{u}_{n}\to 0$ as $|x|\to+\infty$ uniformly for large $n\in \mathbb{N}$.

Proof. (5.7) can be rewritten as

where $g_n(x) = -\frac{\tilde{u}_{n}(x)}{(a+be_{n})} +\frac{f(u_{n}(0)\tilde{u}_{n}(x))}{\omega_{n}u_{n}(0)(a+be_{n})}$. By Lemma 5.4, it is clear that $g_n\in L^\infty(\mathbb{R}^N)$. Moreover, it is uniformly bounded for sufficiently large $n$. Due to the fact $\{u_n\}$ converges strongly in $H^s(\mathbb{R}^N)$, the interpolation inequality yields that there exists $g\in L^r(\mathbb{R}^N)$ such that $g_n\rightarrow g$ in $L^r(\mathbb{R}^N)$ for $r\in [2, +\infty)$. Thus, by ^[3], we observe that

where $K$ is a Bessel potential and it satisfies the following properties:

$(D_1)$ $K$ is positive, radially symmetric, and smooth in $\mathbb{R}^N\setminus\{0\}$.

$(D_2)$ There exists $C > 0$ such that $K(x)\leq \frac{C}{|x|^{N+2s}}$ for $x\in \mathbb{R}^N\setminus\{0\}$.

$(D_3)$ $K\in L^{r}(\mathbb{R}^N)$ for $r\in [1, \frac{1}{1-s})$. For any $\sigma > 0$, we see that

It follows from $(D_2)$ that

By Hölder's inequality and $(D_3)$, we obtain

This yields that there exists $R_1 > 0$ independent of $\sigma > 0$ such that

Here, we use the fact $2 < \frac{1}{1-s}$ and $\left(\int_{\{|x-y| < \frac{1}{\sigma}\}} |g|^2 \mathrm{d}y\right)^{\frac{1}{2}} \rightarrow 0$ as $|x|\rightarrow +\infty$. Combining (5.10) and (5.11), we obtain

and, thus, $\tilde{u}_{n}(x)\rightarrow 0$ as $|x|\rightarrow +\infty$ uniformly for large $n$. □

Proof of Theorem 3. For each sequence $\{c_{n}\}\to+\infty$, we define

According to Lemma 5.7, we observe that

Together with Lemmas 5.5, 5.7, and 5.8, we deserve that $\lim_{|x|\to+\infty} v_{n}(x) = 0$ uniformly for large $n$. Moreover, $v_{n}$ is the solution of

Hence, a similar argument to the proof of [19, Proposition 4.4] implies that $v_{n}\to Q$ in $C_{loc}^{2, \alpha}(\mathbb{R}^{N})$ for some $\alpha\in (0, 1)$. Recalling that $\omega_{n}\to 0^{+}$ and $\|u_{n}\|_{D^{s, 2}(\mathbb{R}^{N})}\to 0$ as $n\to+\infty$, from $(H_{4})$, we obtain that $Q$ is nontrivial nonnegative solution of

From ^[4], we know that $Q$ is a radial, positive, and strictly decreasing in $x$. □

Proof of Theorem 4. From (5.13), we have

Since (5.12) and $\omega_{n}\to 0^{+}$, by $(H_{4})$, we observe that

as $n\to+\infty$. Together with $\|u_{n}\|_{D^{s, 2}(\mathbb{R}^{N})}\to 0$ as $n\to+\infty$, it follows from (5.14) that

Noting that $\lim_{|x|\to+\infty}v_{n}(x) = 0$ uniformly in $n\in\mathbb{N}$, there exist $R > 0$ large enough and $N_{0}\in \mathbb{N}$ such that

for $|x| > R$ and $n\geqslant N_{0}$. Therefore,

Arguing as in the proof of [19, Lemma 5.6], we see that

Thus, $v_{n}\to Q$ in $L^{2}(\mathbb{R}^{N})$. By direct calculation, we see that

Combining with Remark 3 and recalling that $\omega_{n}\to 0$, we deduce that $\|v_{n}\|_{L^{2}(\mathbb{R}^{N})}^{2}$ and $\|v_{n}\|_{D^{s, 2}(\mathbb{R}^{N})}^{2}$ are comparable. According to the fact $v_{n}\to Q$ in $L^{2}(\mathbb{R}^{N})$, there exist $C_{3}, C_{4} > 0$ such that

for all $n\in \mathbb{N}$. Therefore, there exist $C_{5}, C_{6} > 0$ such that

Hence, $\{v_{n}\}$ is a bounded sequence in $H^{s}(\mathbb{R}^{N})$. Up to a subsequence, $v_{n}\rightharpoonup v$ in $H^{s}(\mathbb{R}^{N})$. Moreover, from Lemma 1.1 and the fact $v_{n}\to Q$ in $L^{2}(\mathbb{R}^{N})$, one gets that $v_{n}\to Q$ in $L^{q}(\mathbb{R}^{N}), q\in [2, 2_{s}^{*})$. Applying (5.15), we observe that $\|v_{n}\|_{D^{s, 2}(\mathbb{R}^{N})}^{2} \to \|Q\|_{D^{s, 2}(\mathbb{R}^{N})}^{2}$, which yields $v_{n}\to Q$ in $H^{s}(\mathbb{R}^{N})$. □

5.3.   The case of $c\to 0^{+}$

Let $c_{n}\to 0^{+}$. From Lemma 5.3 and (5.4), we obtain that $\omega_{n}\to +\infty$ and $e_{n}\to +\infty$ as $n\to+\infty$.

Lemma 5.9. Let $1\leqslant N < 4s$, $s\in (\frac{1}{2}, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{5})$. Let $(\omega_{c}, u_{c})$ be the solution given by Theorem 2. Then,

and

Proof. For each sequence $\{c_{n}\}\to 0^{+}$, set

A direct calculation shows that $\tilde{v}_{n}(0) = \|\tilde{v}_{n}\|_{L^{\infty}(\mathbb{R}^{N})} = 1$ and

Letting $x = 0$ in (5.16), and applying $(H_{2})$, there exists $C > 0$ such that

which yields that $u_{n}(0)\to+\infty$, since $\omega_{n}\to+\infty, \gamma > \lambda > 2$. Moreover, by $\lambda\leqslant \gamma$, we deserve that

By the arbitrary of $c_{n}$, we complete the proof. □

Lemma 5.10. Let $1\leqslant N < 4s$, $s\in (\frac{1}{2}, 1)$, and suppose that $f$ satisfies $(H_{1})$–$(H_{5})$. Let $(\omega_{c}, u_{c})$ be the solution given by Theorem 2. Then,

Proof. We assume by contradiction that there exists a sequence $c_{n}\to 0^{+}$ such that

which implies that $\omega_{n} = o([u_{n}(0)]^{\gamma-2})$. Set

Then, $\hat{v}_{n}(0) = \|\hat{v}_{n}\|_{L^{\infty}(\mathbb{R}^{N})} = 1$ and

Note that

Together with $u_{n}(0)\to +\infty$ and $\lambda\leqslant \gamma$, we conclude that

This indicates that $\frac{1}{u^{\gamma-1}_{n}(0)} f(u_{n}(0)\hat{v}_{n})$ is of $L^{\infty}(\mathbb{R}^{N})$. Therefore, applying a similar argument to the proof of [19, Proposition 4.4], and passing to a subsequence if necessary, we assume that $\hat{v}_{n}\to \hat{v}$ in $C_{loc}^{2, \alpha}(\mathbb{R}^{N})$, for some $\alpha\in (0, 1)$. Combining Lemma 5.6 and $(H_{4})$, noting that $e_{n}\to +\infty$, we deduce that $\hat{v}$ is a nonnegative bounded radial solution of

Thanks to [6, Theorem 1.5], we derive that $\hat{v} = 0$, which contradicts $\hat{v}(0) = 1$. □

Proof of Theorem 5. For each sequence $\{c_{n}\}\to 0^{+}$, set

Furthermore, $\bar{v}_{n}$ is the solution of

From $(H_{2})$, we obtain that

Combining $\omega_{n}\to +\infty$ and $\lambda\leqslant \gamma$, we deduce that $\frac{f(\omega_{n}^{\frac{1}{\gamma-2}} \bar{v}_{n})} {\omega_{n}^{\frac{\gamma-1}{\gamma-2}}}\in L^{\infty}(\mathbb{R}^{N})$. Hence, a similar discussion to the proof of Lemma 5.8 and Theorem 3 implies that $\bar{v}_{n}\to U$ in $C_{loc}^{2, \alpha}(\mathbb{R}^{N})$, for some $\alpha\in (0, 1)$. Recalling that $\omega_{n}\to +\infty$ and $e_{n}\to +\infty$ as $n\to+\infty$, from $(H_{4})$, we obtain that $U$ is nontrivial nonnegative solution of

From ^[4], we know that $U$ is radial, positive, and strictly decreasing in $x$. □

Proof of Theorem 6. Letting $c_{n}\to 0^{+}$ and recalling (5.18), we derive that

By the proof of Theorem 5, we also obtain that $\lim_{|x|\to+\infty}\bar{v}_{n}(x) = 0$ uniformly in $n\in\mathbb{N}$. It follows from $(H_{2})$ and $\omega_{n}\to +\infty$ as $n\to +\infty$ that

as $|x|\to +\infty$ uniformly in $n\in\mathbb{N}$. Hence, there exist $R > 0$ large enough and $N_{1}\in \mathbb{N}$ such that

for $|x| > R$ and $n\geqslant N_{1}$. Arguing similarly as in the proof of Theorem 4, we can show that $\bar{v}_{n}\to U$ in $L^{2}(\mathbb{R}^{N})$. By direct calculation, we see that

Combining with Remark 3 and recalling that $e_{n}\to +\infty$, we deduce that $\|\bar{v}_{n}\|_{L^{2}(\mathbb{R}^{N})}^{2}$ and $\|\bar{v}_{n}\|_{D^{s, 2}(\mathbb{R}^{N})}^{2}$ are comparable. Similar as the proof of Theorem 4, we observe that $\bar{v}_{n}\to U$ in $H^{s}(\mathbb{R}^{N})$. □

Author contributions

Min Shu, Haibo Chen and Jie Yang: Conceptualization, Methodology, Validation, Writing-original draft, Writing-review & editing. All authors contributed equally to this work.

Use of Generative-AI tools declaration

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

Acknowledgments

The authors thank the anonymous referees for their valuable comments and nice suggestions to improve the results.

J. Yang is supported by Natural Science Foundation of Hunan Province of China (Nos. 2023JJ30482 and 2022JJ30463), Research Foundation of Education Bureau of Hunan Province (Nos. 23A0558 and 22A0540), and Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province.

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.