Note on normalized solutions to a kind of fractional Schrödinger equation with a critical nonlinearity

1.   Introduction and main results

In this paper, we study normalized solutions of the fractional Schrödinger equation with a critical nonlinearity of $|u|^{2^\ast_s-2}u$,

where $N\geq2$, $s\in(0, 1)$, $a > 0$, and $2 < p < 2^\ast_s\triangleq\frac{2N}{N-2s}$. The fractional Laplace operator $(-\Delta)^s$ is defined by

with a positive constant $C(N, s)$, and we normalize the factor $C(N, s)/2 = 1$ for convenience. For problem (1.1), $p = 2+\frac{4s}{N}$ is the $L^2$-critical exponent.

The operator $(-\Delta)^s$ arises in physics, chemistry, biology, and finance and can be seen as the infinitesimal generators of the Lévy stable diffusion process (see ^[1]). Moreover, $(-\Delta+m^2)^{\frac{1}{2}}$ appears in quantum mechanics, where $m$ is the mass of the particle under consideration (see ^[16]). The study of fractional Laplacian nonlinear equations has attracted much attention from many mathematicians working in different fields. Felmer et al. ^[11] studied the existence, regularity, and symmetry of positive solutions to the fractional Schrödinger equations in the whole space $\mathbb{R}^N$. Caffarelli et al. investigated a fractional Laplacian with free boundary conditions (see ^[6,7]). We also refer the interested readers to the works ^[5,9,10,19] for more details on the fractional operator and its applications.

Normalized solutions to Schrödinger equations with $L^2$-supercritical nonlinearity were first studied in the paper ^[14], where the energy functional was unbounded from below on the $L^2$-constraint. Recently, Soave in ^[21] proved several existence (or nonexistence) and stability (or instability) results for the Schrödinger equation with combined nonlinearities as follows:

where $N\geq3$, $\mu > 0$, $\lambda\in\mathbb{R}$ and $2 < q < 2^\ast\triangleq\frac{2N}{N-2}$. Wei and Wu in ^[22] extended the results in ^[21] in three aspects. Firstly, they obtained the existence of a solution of mountain-pass type for $N\geq3$ and $2 < q < 2+\frac{4}{N}$. Secondly, the existence and nonexistence of ground states for $2+\frac{4}{N}\leq q < 2^\ast$ with $\mu > 0$ large were obtained. Finally, they obtained the precisely asymptotic behaviors of ground states and mountain-pass solutions as $\mu\rightarrow 0$. Luo and Zhang in ^[17] dealt with the existence of normalized ground states for the fractional Schrödinger equation with combined nonlinearities as follows:

Under different assumptions on $q < p$ and $a > 0$, they proved the existence and nonexistence of normalized solutions in the $L^2$-subcritical case and $L^2$-supercritical case, respectively. But they only considered the Sobolev subcritical case $p, q < 2^\ast_s$. Motivated by the above papers, Zhang and Han in ^[25] considered problem (1.2) in the Sobolev critical case $q = 2^\ast_s$, i.e., problem (1.1). They obtained the following results:

${\rm (i)}$ Let $N\geq2$, $s\in(0, 1)$, $2 < p < 2+\frac{4s}{N}$, and assume that $0 < a < \min\{\alpha_1, \alpha_2\}$. Then, $m^+_a\triangleq\inf_{u\in V(a)^+}E(u) = \inf_{u\in V(a)}E(u) < 0$ and it can be attained by $u_{a, +}$, which is nonnegative and radially decreasing. Moreover, problem (1.1) has a ground state $(u_{a, +}, \lambda_{a, +})$ with $\lambda_{a, +} < 0$;

${\rm (ii)}$ Let $N\geq2$, $s\in(0, 1)$, $N^2 > 8s^2$, $p = 2+\frac{4s}{N}$, and assume that $0 < a < \alpha_3$. Then, $m^-_a\triangleq\inf_{u\in V(a)^-}E(u)\in(0, \frac{s}{N}S_s^{\frac{N}{2s}})$ and it can be attained by $u_{a, -}$ which is nonnegative and radially decreasing. Moreover, problem (1.1) has a ground state $(u_{a, -}, \lambda_{a, -})$ with $\lambda_{a, -} < 0$;

${\rm (iii)}$ Let $N\geq2$, $s\in(0, 1)$, $N^2 > 8s^2$, $2+\frac{4s}{N} < p < 2_s^\ast$, and assume that $0 < a < \alpha_4$. Then, $m^-_a\triangleq\inf_{u\in V(a)^-}E(u)\in(0, \frac{s}{N}S_s^{\frac{N}{2s}})$ and it can be attained by $u_{a, -}$ which is nonnegative and radially decreasing. Moreover, problem (1.1) has a ground state $(u_{a, -}, \lambda_{a, -})$ with $\lambda_{a, -} < 0$.

The constants $\alpha_1$, $\alpha_2$, $\alpha_3$, $\alpha_4$ that appear in (ⅰ)–(ⅲ) are

where $\gamma_{p, s}\triangleq\frac{N(p-2)}{2ps} < 1$, the constants $S_s$, $C(s, N, p)$ are defined in (1.3), (1.5) respectively and $V(a)$ is defined in (1.6).

We also refer to the works ^{[2,3,4,15,20,24,26]} for other related equations.

In order to state our main results, we denote the best constant of the embedding $D^{s, 2}(\mathbb{R}^N)\hookrightarrow L^{2^\ast_s}(\mathbb{R}^N)$ by

where $D^{s, 2}(\mathbb{R}^N)$ denotes the completion of the space $C_c^{\infty}(\mathbb{R}^N)$ with the norm $\|u\|_{D^{s, 2}(\mathbb{R}^N)} = \|(-\Delta)^\frac{s}{2}u\|_2$. Solutions to (1.1) can be obtained as the critical points of the associated energy functional

defined on the constraint manifold $S(a)\triangleq\left\{u\in H^s(\mathbb{R}^N):\|u\|_2^2 = a^2\right\}$, where

endowed with the natural norm

We recall the following fractional Gagliardo–Nirenberg–Sobolev inequality (see ^[12])

Define $H_r^s(\mathbb{R}^N)\triangleq\left\{u\in H^s(\mathbb{R}^N):u(x) = u(|x|)\right\}$. It is well known that $H_r^s(\mathbb{R}^N)$ is compactly embedded into $L^p(\mathbb{R}^N)$ for any $p\in(2, 2_s^\ast)$, and $H_r^s(\mathbb{R}^N)$ is a natural constraint (see ^[23]).

Lemma 1. (Lemma 2.1 in ^[25]). Let $(u, \lambda)\in S(a)\times \mathbb{R}$ be a weak solution to problem (1.1). Then $u$ belongs to the set

Moreover, $V(a)$ can be naturally divided into the following three parts:

In the $L^2$-subcritical perturbation case $2 < p < 2+\frac{4s}{N}$, since the functional (1.4) is unbounded below on $S(a)$ as $2_s^\ast < 2+\frac{4s}{N}$, it will be naturally expected that $E(u)\mid_{S(a)}$ has a second critical point of mountain pass type for problem (1.1). In this paper, we give a complete positive answer to the above expectation (see Theorem 1.1). Since $H_r^s(\mathbb{R}^N)$ is a natural constraint, we only need to find the critical point for the functional $E(u)$ defined on $H^s_r(\mathbb{R}^N)\cap S(a)$. Define

Theorem 1. Let $N\geq2$, $s\in(0, 1)$, $2 < p < 2+\frac{4s}{N}$, and assume that $0 < a < \min\{\alpha_1, \alpha_2\}$. Then $m_{a, r}^-\triangleq\inf_{u\in V_r(a)^-}E(u)\in(0, \frac{s}{N}S_s^{\frac{N}{2s}})$ and it can be attained by $u_{a, -}$ which is positive and radially decreasing. Moreover, problem (1.1) has a second solution $(u_{a, -}, \lambda_{a, -})$ with some $\lambda_{a, -} < 0$.

Remark 1. The method used in this paper can also be applied to the following Sobolev critical fractional Schrödinger equation with a parameter $\mu > 0$

which was considered in ^[27]. We leave the details to the interested readers.

Notations. The notation $C$ in the following context denotes some positive constant that might be changed from line to line and even in the same line. $a\sim b$ means that $Cb \leq a \leq Cb$ and $a\lesssim b$ ($a\gtrsim b$) means that $a\leq Cb$ ($a\geq Cb$) for some positive constant $C$. The notation $B_z(0)$ denotes the ball in $\mathbb{R}^N$ of center at origin and radius $z$.

2.   Proof of Theorem 1

As in ^[14], we use the fiber map preserving the $L^2$-norm $\tau\ast u = e^{\frac{N\tau}{2}}u(e^\tau x)$ for a.e. $x\in\mathbb{R}^N$. For $u\in S(a)$, define the auxiliary function

Lemma 2. (Lemma 3.3 in ^[25]). Let $N\geq2$, $2 < p < 2+\frac{4s}{N}$, and $0 < a < \alpha_1$. For every $u\in S(a)$, the function $\Psi_u(\tau)$ has exactly two critical points $s_u < t_u\in\mathbb{R}$ and two zeros $c_u < d_u\in\mathbb{R}$ with $s_u < c_u < t_u < d_u$. Moreover, we have the following statements:

${\rm (i)}$ $s_u\ast u\in V(a)^+$ and $t_u\ast u\in V(a)^-$. If $\tau\ast u\in S(a)$, then either $\tau = s_u$ or $\tau = t_u$.

${\rm (ii)}$ We have

and $\Psi_u(\tau)$ is strictly decreasing on $(t_u, +\infty)$. In particular, if $t_u < 0$, then $P(u) < 0$.

${\rm (iii)}$ The maps $u\in V(a)\mapsto s_u\in\mathbb{R}$ and $u\in V(a)\mapsto t_u\in\mathbb{R}$ are of class $C^1$.

Proof. Statements (ⅰ), (ⅲ), and the first part of (ⅱ) have already been shown in Lemma 3.3 in ^[25]. From the proof of Lemma 3.3 in ^[25], we know the functions $\Psi_u(\tau)$ and $\Psi_u''(\tau)$ have exactly two inflection points. In particular, $\Psi_u(\tau)$ is strictly decreasing and concave on $[t_u, +\infty)$. Hence, if $t_u < 0$, then $P(u) = \Psi_u'(0) < 0$. □

Lemma 3. Let $N\geq2$, $2 < p < 2+\frac{4s}{N}$, and $0 < a < \min\{\alpha_1, \alpha_2\}$. Then, we have

Proof. Applying (1.4), (1.5), and $\|u\|_{2_s^\ast}^2\leq S_s^{-1}\|(-\Delta)^\frac{s}{2}u\|_2^2$, we have

for every $u\in V_r(a)^-$. Define

Since $p\gamma_{p, s} < 2 < 2_s^\ast$, it is easy to see that $h(0^+) = 0^-$ and $h(+\infty) = -\infty$. Let $t_{\rm max}$ denote the strict maximum of the function $h(t)$, which is at a positive level (see Lemma 3.2 in ^[25]). For every $u\in V_r(a)^-$, by an easy computation, there exists $\tau_u\in \mathbb{R}$ such that $\|(-\Delta)^\frac{s}{2}(\tau_u \ast u)\|_2 = t_{\rm max}$. Moreover, by Lemma 2, we see that the value $0$ is the unique strict maximum of the function $\Psi_u(\tau)$. Therefore,

Since $u\in V_r(a)^-$ is arbitrarily chosen, we deduce that

□

Lemma 4. Let $N\geq2$, $2 < p < 2+\frac{4s}{N}$, and $0 < a < \min\{\alpha_1, \alpha_2\}$. Then $m^+_a\triangleq\inf_{u\in V(a)^+}E(u) = \inf_{u\in V(a)}E(u) < 0$ and it can be attained by $u_{a, +}$, which is positive and radially decreasing. Moreover, problem (1.1) has the ground state solution $(u_{a, +}, \lambda_{a, +})$ with $\lambda_{a, +} < 0$.

Proof. By using a similar method used in Theorem 1.1 in ^[25], we obtain

and it can be attained by $u_{a, +}$, which is nonnegative and radially decreasing. Moreover, problem (1.1) has the ground state $(u_{a, +}, \lambda_{a, +})$ with $\lambda_{a, +} < 0$. Finally, by the strong maximum principle for the fractional Laplacian (see Proposition 2.17 in ^[19]), we have that $u_{a, +}$ is positive. □

Lemma 5. Let $N\geq2$, $2 < p < 2+\frac{4s}{N}$, and $0 < a < \min\{\alpha_1, \alpha_2\}$. Then, we have

Proof. As in ^[18], the function $U_\varepsilon(x) = \varepsilon^{-\frac{N-2s}{2}}u^\ast(\frac{x}{\varepsilon})$ solves the equation $(-\Delta)^su = |u|^{2_s^\ast-2}u$ in $\mathbb{R}^N$, where $u^\ast(x) = \tilde{u}(x/S_s^{\frac{1}{2s}})/\|\tilde{u}\|_{2_s^\ast}$ and $\tilde{u}(x) = k(\mu^2+|x|^2)^{-\frac{N-2s}{2}}$, $x\in\mathbb{R}^N$, with $k > 0$ and $\mu > 0$ are fixed constants. Let $\chi(x)\in C_c^{\infty}(\mathbb{R}^N)$ be a cut-off function satisfying:

${\rm (a)}$ $0\leq\chi(x)\leq1$ for any $x\in \mathbb{R}^N$,

${\rm (b)}$ $\chi(x)\equiv1$ in $B_1(0)$,

${\rm (c)}$ $\chi(x)\equiv0$ in $\mathbb{R}^N\setminus \overline{B_2(0)}$.

Define $W_{\varepsilon} = \chi(x)U_\varepsilon(x)$. According to Propositions 21 and 22 in ^[18], we know that

and

Now, we define

Then, it is well known that

and

We choose $\xi = \left(\|\widehat{W}_{\varepsilon, t}\|_2/a\right)^{\frac{1}{s}}$, then $\overline{W}_{\varepsilon, t}\in H^s_r(\mathbb{R}^N)\cap S(a)$. By Lemma 2, there exists $\tau_{\varepsilon, t} > 0$ such that $(\overline{W}_{\varepsilon, t})_{\tau_{\varepsilon, t}}\in V_r(a)^-$, where $(\overline{W}_{\varepsilon, t})_{\tau_{\varepsilon, t}} = \tau_{\varepsilon, t}^{\frac{N}{2}}\overline{W}_{\varepsilon, t}(\tau_{\varepsilon, t}x)$. Thus,

Since $u_{a, +}\in V(a)^+$, by Lemma 2, we get $\tau_{\varepsilon, 0} > 1$. By (2.1), (2.2), and (2.6), we know that $\tau_{\varepsilon, t}\rightarrow0$ as $t\rightarrow +\infty$ uniformly for $\varepsilon > 0$ sufficiently small. Since $\tau_{\varepsilon, t}$ is unique by Lemma 2, it is standard to show that $\tau_{\varepsilon, t}$ is continuous for $t$, which implies that there exists $t_\varepsilon > 0$ such that $\tau_{\varepsilon, t_\varepsilon} = 1$. It follows that

Recall that $u_{a, +}\in H^s_r(\mathbb{R}^N)\cap S(a)$ and $W_\varepsilon$ are positive. By (2.4) and (2.5), we have

By the fact that $u_{a, +}$ is a solution to problem (1.1) for some $\lambda_{a, +} < 0$ and the Pohazaev identity satisfied by $u_{a, +}$, we have

Then, by (2.8) and (2.9), we deduce that $E(\overline{W}_{\varepsilon, t})\rightarrow m_a^+$ as $t\rightarrow0^+$ and

as $t\rightarrow +\infty$ uniformly for $\varepsilon > 0$ sufficiently small. Hence, there exists $t_0 > 0$ large enough such that $t_\varepsilon\in(\frac{1}{t_0}, t_0)$ and $E(\overline{W}_{\varepsilon, t}) < 0$ for $t < \frac{1}{t_0}$ and $t > t_0$. Now, we estimate $E(\overline{W}_{\varepsilon, t})$ for $\frac{1}{t_0}\leq t\leq t_0$. Let $u_{a, +}$ be a positive and radially decreasing ground state solution to problem (1.1) (see Lemma 4). Then, we have

From the definition of $\widehat{W}_{\varepsilon, t}$, we obtain

for $\frac{1}{t_0}\leq t\leq t_0$. Applying (2.11) and the inequality $(1+\hat{t})^\alpha\geq1+\alpha\hat{t}$ for $\hat{t}\geq0$ and $\alpha < 0$, we obtain that

Applying (2.8)–(2.10) and (2.12), we have

for $\frac{1}{t_0}\leq t\leq t_0$. By direct calculation, we have

Similar to (2.10), we have

By (2.1)–(2.3) and (2.13)–(2.15), we have

for $\frac{1}{t_0}\leq t\leq t_0$ by taking $\varepsilon > 0$ sufficiently small. By Lemma 3 and (2.16), we obtain

Moreover, by (2.17), for $t < \frac{1}{t_0}$ and $t > t_0$, we have

It follows from (2.16) and (2.18) that

Then, the conclusion follows from (2.7). □

For $0 < c < \min\{\alpha_1, \alpha_2\}$, let $u\in V(c)^{\pm}$. Then $v_b\triangleq\frac{b}{c}u\in S(b)$ for all $b > 0$. By Lemma 2, there exists $\tau_{\pm}(b) > 0$ such that

where $0 < b < \min\{\alpha_1, \alpha_2\}$. Clearly, $\tau_{\pm}(c) = 1$.

Lemma 6. Let $N\geq2$, $2 < p < 2+\frac{4s}{N}$, and $0 < c < \min\{\alpha_1, \alpha_2\}$. Then, $(\tau_{\pm}^s(c))'$ exist and

Moreover, $E((v_b)_{\tau_{\pm}(b)}) < E(u)$ for all $b > c$ such that $b < \min\{\alpha_1, \alpha_2\}$.

Proof. The proof is mainly inspired by ^[8,22]. Since $(v_b)_{\tau_{\pm}(b)}\in V(b)^{\pm}$, we have

Next, we define the function

Clearly, $\Phi(b, \tau_{\pm}^s(b)) = 0$ for $0 < b < \min\{\alpha_1, \alpha_2\}$. Applying $u\in V(c)^\pm$ and Lemma 1, we have

Applying the implicit function theorem, we have $(\tau_{\pm}^s(c))'$ exists and (2.19) holds. By $u\in V(c)^\pm$ once more, we deduce that

Since $(v_b)_{\tau_{\pm}(b)}\in V(b)^{\pm}$ and $u\in V(c)^\pm$, we obtain

Thus, we obtain

Since $0 < c < \min\{\alpha_1, \alpha_2\}$ is arbitrary and $(v_b)_{\tau_{\pm}(b)}\in V(b)^\pm$, we have $E((v_b)_{\tau_{\pm}(b)}) < E(u)$ for all $b > c$ such that $b < \min\{\alpha_1, \alpha_2\}$. □

Lemma 7. Let $N\geq2$, $2 < p < 2+\frac{4s}{N}$, and $0 < a < \min\{\alpha_1, \alpha_2\}$. Assume that $u\in V(a)$ is a critical point for $E(u)\mid_{V(a)}$, then $u$ is a critical point for $E(u)\mid_{S(a)}$.

Proof. By Lemma 3.1 in ^[25], we have that $V(a)$ is a smooth manifold of codimension 2 in $H^s(\mathbb{R}^N)$ and $V(a)^0$ is empty. If $u\in V(a)$ is a critical point for $E(u)\mid_{V(a)}$, then by the Lagrange multipliers rule there exist $\lambda, \mu\in \mathbb{R}$ such that

for every $\varphi\in H^s(\mathbb{R}^N)$. This implies

We have to prove that $\mu = 0$. By using the Pohozaev identity for the above equation, we know that

Applying $u\in V(a)$ and (2.20), we deduce that

Since $u\notin V(a)^0$, we have $2\|(-\Delta)^{\frac{s}{2}}u\|_2^2-\gamma_{p, s}^2p\|u\|_p^p-2_s^\ast\|u\|_{2_s^\ast}^{2_s^\ast}\neq0$. Thus, we have $\mu = 0$. Hence, $u$ is a critical point for $E(u)\mid_{S(a)}$, that is, $V(a)$ is a natural constraint. □

Lemma 8. Let $N\geq2$, $2 < p < 2+\frac{4s}{N}$, and $0 < a < \min\{\alpha_1, \alpha_2\}$. Assume that $m_{a, r}^- < m_a^++\frac{s}{N}S_s^{\frac{N}{2s}}$, then $m_{a, r}^-$ can be attained by some $u_{a, -}\in H^s_r(\mathbb{R}^N)$, which is positive and radially decreasing. Furthermore, problem (1.1) has a second solution $u_{a, -}$ with some $\lambda_{a, -} < 0$.

Proof. Let $\{\bar{u}_n\}\subset V_r(a)^-$ be a minimizing sequence. By Ekeland's variational principle (see ^[13]), there exists a new minimizing sequence $\{{u}_{n}\}$ satisfying

Therefore, we obtain

From the third property in (2.21), we have

by the Gagliardo–Nirenberg–Sobolev inequality (1.5). Then, using that $E(u_n)\leq m_{a, r}^-+1$ for $n$ large, we deduce that

This implies that $\{u_n\}$ is bounded in $H^s_r(\mathbb{R}^N)$. Therefore, $u_n\rightharpoonup u_0$ in $H^s_r(\mathbb{R}^N)$ up to a subsequence. By the Sobolev compact embedding theorem $H_r^s(\mathbb{R}^N)\hookrightarrow L^{p}(\mathbb{R}^N)$ for $2 < p < 2_s^\ast$, we have $u_n\rightarrow u_0$ strongly in $L^{p}(\mathbb{R}^N)$ as $n\rightarrow \infty$ up to a subsequence. Without loss of generality, we assume that $u_n\rightharpoonup u_0$ weakly in $H^s_r(\mathbb{R}^N)$ and $u_n\rightarrow u_0$ strongly in $L^{p}(\mathbb{R}^N)$ as $n\rightarrow \infty$. We claim that $u_0\neq0$. Otherwise, $u_n\rightarrow0$ strongly in $L^{p}(\mathbb{R}^N)$. By $P(u_{n})\rightarrow 0$ as $n\rightarrow \infty$, it follows that

Applying (2.23) and the embedding $D^{s, 2}(\mathbb{R}^N)\hookrightarrow L^{2^\ast_s}(\mathbb{R}^N)$, we obtain either $u_n\rightarrow0$ strongly in $D^{s, 2}(\mathbb{R}^N)$ as $n\rightarrow \infty$ or

According to (2.22), either $m_{a, r}^- = 0$ or $m_{a, r}^-\geq\frac{s}{N}S_s^{\frac{N}{2s}}$, which contradicts Lemmas 3 and 5. Therefore, we obtain $u_0\neq0$. Let $v_n\triangleq u_n-u_0$. Then, there are the following two cases:

${\rm (i)}$ $v_n\rightarrow0$ strongly in $H^s_r(\mathbb{R}^N)$ as $n\rightarrow \infty$;

${\rm (ii)}$ $\|(-\Delta)^\frac{s}{2}v_n\|_2^2+\|v_n\|_2^2\gtrsim 1$.

In the case (ⅰ), $u_0\in V_r(a)^-$ and $m_{a, r}^-$ is attained by $u_0$, which is nonnegative and radially decreasing. By Lemma 7, $u_0$ is a solution to problem (1.1) with $\lambda_0\in\mathbb{R}$, which appears as a Lagrange multiplier. By multiplying equation (1.1) with $u_0$ and integrating by parts, it follows from $u_0\in V_r(a)^-$ that

Hence, $\lambda_0 < 0$. By using the strong maximum principle for the fractional Laplacian, we can see that $u_0$ is positive. It remains to consider the case (ⅱ). Let $\|u_0\|_2^2 = t_0^2$. Then, by Fatou's lemma, we get $0 < t_0\leq a$. Next, we have the following two subcases:

${\rm (a)}$ $\|v_n\|_{2_s^\ast}\rightarrow0$ as $n\rightarrow \infty$ up to a subsequence;

${\rm (b)}$ $\|v_n\|_{2_s^\ast}^{2_s^\ast}\gtrsim 1$.

In the subcase (a), by Lemma 2, there exists $s_0 > 0$ such that $(u_0)_{s_0}\in V_r(t_0)^-$. Using Lemma 2 again, (2.21) and $u_n\rightarrow u_0$ strongly in $L^{p}(\mathbb{R}^N)\cap L^{2_s^\ast}(\mathbb{R}^N)$ as $n\rightarrow \infty$ up to a subsequence, we deduce that

By Lemma 6, we have $m_{t_0, r}^-\geq m_{a, r}^-$. Therefore, $E((u_0)_{s_0}) = m_{t_0, r}^-$ and $m_{t_0, r}^- = m_{a, r}^-$. If $t_0 < a$, we take $(u_0)_{s_0}$ as the test function in the proof of Lemma 6, and we deduce that $m_{t_0, r}^- > m_{a, r}^-$, which is a contradiction. Thus, in the case of (a), we have $t_0 = a$, and $m^-_{a, r}$ is attained by $(u_0)_{s_0}$, which is nonnegative and radially decreasing. As above, we can see that $(u_0)_{s_0}$ is positive and $(u_0)_{s_0}$ is a solution to problem (1.1) with $\lambda'_0 < 0$. Now, it remains to consider the case (b). Let

Clearly, in the case (b), $s_n\lesssim1$, and by the embedding $D^{s, 2}(\mathbb{R}^N)\hookrightarrow L^{2^\ast_s}(\mathbb{R}^N)$, we deduce that

Since $0 < t_0\leq a$, by Lemma 2, there exists $\tau_0 > 0$ such that $(u_0)_{\tau_0}\in V_r(t_0)^-$. We claim that $s_n\geq\tau_0$ up to a subsequence. If not, suppose the contrary: $s_n < \tau_0$ for all $n$. Define an auxiliary functional

Applying Lemma 2 once more, the Brezis–Lieb lemma (see Lemma 1.32 in ^[23]), Lemma 6, $u_n\rightarrow u_0$ strongly in $L^p(\mathbb{R}^N)$ as $n\rightarrow \infty$, and the boundedness of $\{s_n\}$, it follows that

which is impossible. Therefore, we have $s_n\geq\tau_0$ up to a subsequence. Without loss of generality, we assume that $s_n\geq\tau_0$ for all $n\in \mathbb{N}$. Again, by Lemma 2, the Brezis–Lieb lemma (see Lemma 1.32 in ^[23]), and the fact that $u_n\rightarrow u_0$ strongly in $L^p(\mathbb{R}^N)$ as $n\rightarrow \infty$, we obtain

According to $s_n\geq\tau_0$, by the proof of Theorem 1.4 in ^[27] (see Section 8 in ^[27]), we have

By Lemma 6, we have $t_0 = a$, and $m_{a, r}^-$ is attained by $(u_0)_{\tau_0}$, which is nonnegative and radially decreasing. By the above analysis, we prove that $(u_0)_{\tau_0}$ is positive and $(u_0)_{\tau_0}$ is a solution for problem (1.1) with some $\lambda''_0 < 0$. Thus, we have proved that $m_{a, r}^-$ can always be attained by $u_{a, -}$, which is positive and radially decreasing. Hence, problem (1.1) has a second solution $(u_{a, -}, \lambda_{a, -})$ with some $\lambda_{a, -} < 0$. □

We are ready to give the proof of Theorem 1.

Proof of Theorem 1. By Lemmas 3 and 5, we have $m_{a, r}^- < m_a^++\frac{s}{N}S_s^{\frac{N}{2s}}$. Then, Theorem 1 follows from Lemma 8. □

Author contributions

Xizheng Sun: Conceptualization, Investigation, Methodology, Supervision, Validation, Writing-original draft, Writing-review and editing; Zhiqing Han: Conceptualization, Methodology, Validation, Writing-original draft, Writing-review and editing. All authors have read and agreed to the published version of the manuscript.

Use of AI tools declaration

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

Conflict of interest

This work does not have any conflicts of interest.