Normalized solutions for Kirchhoff equations with Choquard nonlinearity: mass Super-Critical Case

1.   Introduction and main results

In this paper we are interested in the following Kirchhoff equation with Choquard nonlinearity:

under the constraint

where $a, b, c > 0$, $\mu\in\mathbb{R}$, $\lambda\in\mathbb{R}$, $\alpha\in(0, 3)$, $\frac{10}{3}\leq q < 6$, $3+\frac{\alpha}{3}\leq p < 3+\alpha$. $I_{\alpha}:\mathbb{R}^{N}\setminus\{0\}\mapsto\mathbb{R}$ is the Riesz potential defined by

The problem (1.1) is closely related to the equation

which is the stationary analog of the equation

where $f(x, u)$ is a general nonlinearity. The problem (1.4) was proposed by Kirchhoff ^[1] as an extension of the classical D'Alembert's wave equations for free vibration of elastic strings. In problem (1.4), $u$ denotes the displacement, the nonlinear term $f$ is the external force, and $a$ is the initial tension while $b$ is related to the intrinsic properties of the string. Mathematically, the problem (1.4) is often referred to be nonlocal as the appearance of the term $(\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\Delta u$, which depends not only on the pointwise value of $\Delta u$, but also on the integral of $|\nabla u|_{2}$ over the whole space. This phenomenon causes some mathematical difficulties, which make the study of Kirchhoff type equations particularly interesting.

After the pioneering work of Lions ^[2], (1.3) began to receive much attention and many researchers studied its steady-state model, see ^[3,4,5,6] for more important research progress. Some scholars have also considered generalizations of fixed-frequency solutions for Kirchhoff equations. Gao et al. ^[7] studied the nonlinear coupled Kirchhoff system with purely Sobolev critical exponent

where $N\geq3$, $a_{i}, b_{i}\geq0$, $i = 1, 2$, $\mu_{1}, \mu_{2}, \gamma > 0$, and $\alpha+\beta = 2^{\ast}$. They gave a complete classification of positive ground states for (1.5) in any dimension $3$ or $4$. Sun et al. ^[8] extended the results to the $p$-sub-Laplacians and obtained the multiple solutions. As to the case of bounded domains in $\mathbb{R}^{N}$, Cabanillas ^[9] studied the global existence theorem and its exponential decay. In addition, Yang and Tang ^[10] dealt with the nonlinear Kirchhoff problem with a sign potential

where $b > 0$ and the nonlinearity $f\in C(\mathbb{R}, \mathbb{R})$ exhibits subcritical growth. By using a more general global compactness lemma and a sign-changing Nehari manifold, they showed the existence of a least energy sign-changing solution for $b > 0$ that is sufficiently small and established the asymptotic behavior when $b\to0^{+}$.

Now there are two substantially different viewpoints in terms of the frequency $\lambda$ in (1.1). One is to regard the frequency $\lambda$ as a given constant. In this situation, solutions of (1.1) are critical points of the following functional

Although this is not the concern of our present article, we still refer the readers to ^{[11,12,13,14]}. Naturally, the other one is to regard $\lambda$ as an unknown parameter, which is exactly what our article is concerned. For this case, standing wave solutions are required to possess a priori prescribed $L^{2}$-norm, which also have attracted widespread attention during recent years. These solutions are commonly called as normalized solutions, which provide valuable insights into dynamical properties of stationary solutions, such as the stability or instability of orbits. In addition, it is natural to prescribe the value of the mass so that $\lambda$ can be interpreted as a Lagrange multiplier. For example, from a physical point of view, the normalized condition may represent the number of particles of each component in Bose-Einstein condensates or the power supply in the nonlinear optics framework. To obtain the solutions of (1.1)-(1.2), it suffices to consider critical points of the functional

on

with the parameter $\lambda\in\mathbb{R}$ appearing as a Lagrange multiplier.

In order to narrate the relevant results and state our motivation conveniently, we consider the following Kirchhoff-type equations with convolutional terms:

where $a > 0, b\geq0, c > 0$, $N\geq3$, $\mu, \gamma\in\mathbb{R}$, $2 < q < \frac{2N}{N-2}$, and $\frac{N+\alpha}{N} < p < \frac{N+\alpha}{N-2}$.

Taking $a = 1$ and $b = 0$, (1.8) reduces to the nonlinear Choquard equation with combined nonlinearities. The endpoints of $\frac{N+\alpha}{N}$ and $\frac{N+\alpha}{N-2}$ established in ^[15] are called lower- and upper-critical exponent. The upper-critical exponent plays a similar role as the Sobolev critical exponent in the local semilinear equations, for instance, Li ^[16] proved the existence and orbital stability of ground states of (1.8) when $p = \frac{N+\alpha}{N-2}$, $\gamma = 1$. As to the lower critical exponent, it seems to be a new feature for the Choquard equation, which is related to a new phenomenon of bubbling at infinity, see ^[17,18].

If $b > 0$, $N = 3$, then (1.8) is a nonlinear Kirchhoff equation. Zeng and Zhang ^[19] proved the existence and uniqueness of solutions to (1.8) with $q\in(2, 6)$, $\gamma = 0$, and $\mu = 1$ by using some simple energy estimates rather than the concentration-compactness principles. In addition, Ye ^[20] considered the existence and mass concentration of solutions of (1.8) under the case of $L^{2}$-critical exponent, namely, $q = \frac{14}{3}$. If $\gamma\ne0$, then (1.8) can be viewed as a Kirchhoff-Choquard type equation. Liu ^[21] considered the case of $\gamma = 1$, $\mu = 0$, and $\frac{N+\alpha}{N} < p < \frac{N+\alpha}{N-2}$, and he provided threshold values $c_{\ast}$ and $c_{\ast\ast}$ related to $c$ separating the existence and nonexistence of normalized solutions of it when $p$ belongs to different ranges. In addition, he also deduced that (1.8) has no nontrivial solutions in the cases of $p = \frac{N+\alpha}{N}$ or $p = \frac{N+\alpha}{N-2}$. For the cases of non-autonomous Kirchhoff equations, Qiu et al. ^[22] considered the following non-autonomous Kirchhoff equation with a perturbation:

where $1\leq N\leq3$, $a, b, c > 0$, $1\leq q < 2$, $2 < p < 2^{\ast}$, $h(x)\in\mathbb{R}$. They proved the existence of mountain pass solutions and bound-state solutions. Furthermore, Ni et al. ^[23] dealt with the following non-autonomous Kirchhoff equations with general nonlinearities:

where $a, b, c > 0$, $V$ is a nonnegative continuous function, and $f$ is a continuous function with $L^{2}$-subcritical growth. When $\varepsilon > 0$ is small enough, by using minimization techniques and the Lusternik-Schnirelmann theory, they pointed out that the number of normalized solutions was related to the topological richness of the set where the potential $V$ attained its minimum value.

Motivated by the above analysis, we consider (1.8) with $\gamma = 1$, $\mu\ne0$, and $N = 3$. Compared to the case of $\mu = 0$, at this time, (1.8) is regarded as a Kirchhoff type mixed equation with convolutional terms, which leads to a more complex geometric structure of the energy functional and makes the compactness analysis and energy estimates more difficult. In addition, we need to accurately determine the range of the parameter $p$ to ensure that the convolution term $(I_{\alpha}\ast|u|^{p})|u|^{p-2}u$ is the leading term.

In the present paper, we study the existence and asymptotic behavior of solutions to (1.1)-(1.2). We say that $u_{0}$ is a ground state to (1.1)-(1.2) if it is a solution to (1.1)-(1.2) having minimal energy among all the solutions belong to $S_{c}$

The following Gagliardo-Nirenberg inequality ^[24] is also crucial in our argument, that is, there exists a best constant $\mathcal{C}_{q}$ depending on $q$ such that

where $\delta_{q}: = \frac{3(q-2)}{2q}$.

The following inequality introduced in ^[25] is called Gagliardo-Nirenberg inequality of Hartree type:

where $\gamma_{p}: = \frac{3p-3-\alpha}{2p}$ and equality holds for $u = Q_{p}$, $Q_{p}$ is a nontrivial solution of

Now, we state our main results:

Theorem 1.1. Let $a, b, c > 0$, $3+\frac{\alpha}{3}\leq p < 3+\alpha$, $\alpha\in(0, 3)$, and $\mu > 0$.

(1) If $q = \frac{10}{3}$ with $\mu c^{q(1-\delta_{q})} < \frac{aq}{2\mathcal{C}_{q}^{q}}$ or $\frac{14}{3}\leq q < 6$, the problem (1.1)-(1.2) has a positive radial ground state solution of mountain pass type at a positive level $m$ for some $\lambda < 0$.

(2) If $\frac{10}{3} < q < \frac{14}{3}$, the problem (1.1)-(1.2) has a positive radial solution of mountain pass type at a positive level $m$ for some $\lambda < 0$.

Theorem 1.2. Let $a, b, c > 0$, $\frac{10}{3}\leq q\leq\frac{14}{3}\leq p < 3+\alpha$, and $\alpha\in(\frac{5}{3}, 3)$. If the following inequality

holds, where $\widetilde{C_{0}} = \left(\frac{ac^{p-3-\alpha}}{p\gamma_{p}}\|Q_{p}\|_{2}^{2p-2}\right)^{\frac{2}{2p\gamma_{p}-2}}$, then the problem (1.1)-(1.2) has a mountain pass type ground state $\tilde{u}$, with the following properties: $\tilde{u}$ is a radial function, and solves (1.1)-(1.2) for some $\lambda < 0$ and $E_{\mu}(\tilde{u}) > 0$.

Theorem 1.3. Let $u_{b}\in S_{c, r}$ be the solution of (1.1) obtained by Theorem 1.1. Then, up to a subsequence, we have $u_{b}\to u$ in $H_{r}^{1}(\mathbb{R}^{3})$ as $b\to0^{+}$, where $u\in S_{c, r}$ is a solution of

for some $\lambda < 0$.

Remark 1.4. Compared with $b = 0$, the case of $b > 0$ is more delicate because of the presence of the nonlocal term $(\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\Delta u$, which causes that the weak limit $u$ of a Palais-Smale sequence $\{u_{n}\}$ may not solve (1.1)-(1.2) and makes compactness analysis more complex. In addition, dealing with the convergence of the convolutional term $(I_{\alpha}\ast|u|^{p})|u|^{p-2}u$ is a challenge. Finally, motivated by the defocusing case of Schrödinger equations, which were studied by Soave ^[26] and Luo et al. ^[27], we discuss the existence of solutions to (1.1)-(1.2) under the case $\mu < 0$. Compared to ^[26] and ^[27], $b > 0$ has a significant impact on the analysis of compactness in the defocusing case.

Remark 1.5. When $p = 3+\alpha$, the term $(I_{\alpha}\ast|u|^{p})|u|^{p-2}u$ can be seen a Sobolev critical term and the lack of compactness is a challenge. Soave ^[26] and Li et al. ^[28] both used the method introduced by Brezis and Nirenberg ^[29] in Sobolev critical case; this method ensured that energy level is less than a threshold, which is an essential ingredient in compactness argument. However, in our article, the existence of $(\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\Delta u$ and $(I_{\alpha}\ast|u|^{p})|u|^{p-2}u$ makes it difficult for us to accurately estimate the energy level. So, the convergence of a Palais-Smale sequence is a very delicate problem, which at the moment we could not solve.

2.   Preliminaries

In this section, we give some preliminary results that will be used throughout the rest of the paper. To start, we introduce the following notations:

● $H^{1}(\mathbb{R}^{3})$ is the usual Sobolev space endowed with the norm $\|u\| = (\|\nabla u\|_{2}^{2}+\|u\|_{2}^{2})^{\frac{1}{2}}$.

● $H_{r}^{1}(\mathbb{R}^{3})$ denotes the subspace of functions in $H^{1}(\mathbb{R}^{3})$ which are radially symmetric with respect to 0, $S_{c, r} = H_{r}^{1}(\mathbb{R}^{3})\cap S_{c}$.

● $L^{p}(\mathbb{R}^{3})$($1\leq p < \infty$), denotes the Lebesgue space with the norm $\|u\|_{p} = (\int_{\mathbb{R}^{3}}|u|^{p}dx)^{\frac{1}{p}}$.

● $D^{1, 2}(\mathbb{R}^{3}) = \{u\in L^{2^{\ast}}(\mathbb{R}^{3}):\nabla u\in L^{2}(\mathbb{R}^{3})\}$.

● $o_{n}(1)$ denotes the vanishing quantities as $n\to\infty$.

Next, we give some lemmas that will be used throughout the rest of the paper.

Lemma 2.1. If $u\in H^{1}(\mathbb{R}^{3})$ is a weak solution of (1.1), then the Nehari-Pohozaev identity

holds.

Proof. If $u$ is a weak solution of

then we get

by [30, Corollary 2.5]. Furthermore, we can regard the term $(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)$ in (1.1) as a constant coefficient motivated by [28, Lemma 2.3]. Therefore, combining (2.1) with the conclusion of Pohozaev identity of Schrödinger equation ^[31], we see immediately that

In addition, since $u\in H^{1}(\mathbb{R}^{3})$ is a weak solution of (1.1)-(1.2), we have

Combining (2.2) with (2.3), we infer that

□

When the energy functional $E_{\mu}$ is unbounded from below on $S_{c}$, we introduce the Pohozaev set:

Lemma 2.1 implies that any critical point of $E_{\mu}|_{S_{c}}$ is contained in $\mathcal{P}_{c, \mu}$. For $t\in\mathbb{R}$ and $u\in S_{c}$, we define

Then, $t\star u\in S_{c}$. The map

see [32, Lemma 3.5]. Similar to ^[33], we define the fiber map

An easy computation shows that

So we see immediately that for $u\in S_{c}$, $t$ is a critical point of $\Psi_{\mu}^{u}(t)$ if and only if $t\star u\in\mathcal{P}_{c, \mu}$.

We need to recall the Hardy-Littlewood-Sobolev inequality.

Lemma 2.2. (Hardy-Littlewood-Sobolev inequality) ^[34] Let $N\geq1$, $p, r > 1$, and $0 < \alpha < N$ with $\frac{1}{p}+\frac{N-\alpha}{N}+\frac{1}{r} = 2$, $u\in L^{p}(\mathbb{R}^{N})$, $v\in L^{r}(\mathbb{R}^{N})$. Then, there exists a sharp constant $C(N, \alpha, p)$ independent of $u$ and $v$ such that

Lemma 2.3. [35, Lemma 2.3] Let $N\geq3$, $\alpha\in(0, N)$, and $p\in[\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2}]$. Assume that the sequence $\{ u_{n}\}\subset H^{1}(\mathbb{R}^{N})$ satisfies $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{N})$ as $n\to \infty$. Then,

Lemma 2.4. [32, Lemma 3.6] For $u\in S_{c}$ and $t\in\mathbb{R}$, the map $\varphi\mapsto t\ast\varphi$ from $T_{u}S_{c}$ to $T_{t\ast u}S_{c}$ is a linear isomorphism with inverse $\psi\mapsto(-t)\ast\psi$, where $T_{u}S_{c} = \{\varphi\in S_{c}:\int_{\mathbb{R}^{3}}u\varphi = 0\}$.

Lemma 2.5. Assume $\frac{3+\alpha}{3}\leq p\leq3+\alpha$. Then the energy functional $\phi$ is invariant under any orthogonal transformation in $\mathbb{R}^{3}$, where

Proof. We define the following group with orthogonal invariance:

and

with $A\in\mathcal{O}(3)$. We obtain

by $\left|\det\left(\frac{\partial\tilde{x}}{\partial x}\right)\right| = \left|\det\left(\frac{\partial \tilde{y}}{\partial y}\right)\right| = \left|\det(A)\right| = 1$. Thus, $\phi$ is invariant under any orthogonal transformation in $\mathbb{R}^{3}$. □

3.   Supercritical leading term with focusing perturbation

In this section, we prove the existence of mountain pass type critical points for $E_{\mu}|_{S_{c, r}}$ when $\mu > 0$ and we assume that $3+\frac{\alpha}{3}\leq p < 3+\alpha$.

We first investigate the mountain pass geometry of $E_{\mu}$ on $S_{c, r}$.

Lemma 3.1. Suppose that $\frac{10}{3} < q < 6$ or $q = \frac{10}{3}$ with $\mu c^{q(1-\delta_{q})} < \frac{aq}{2\mathcal{C}_{q}^{q}}$.

$\mathrm{(i)}$ There exist two positive numbers $k_{1} < k_{2}$ sufficiently small such that

where

$\mathrm{(ii)}$ There exists $u_{0}\in S_{c, r}\setminus A_{k_{2}}$ such that $E_{\mu}(u_{0}) < 0$.

Proof. $\mathrm{(i)}$ In view of (1.12), (1.13) and $2\leq q\delta_{q} < 2p\gamma_{p}$, we obtain

and

It is also clear that

Notice that $2\leq q\delta_{q} < 6$ and $\mu c^{q(1-\delta_{q})} < \frac{aq}{2\mathcal{C}_{q}^{q}}$ when $q\delta_{q} = 2$. Taking two small positive numbers $k_{1} < k_{2}$, we arrive at the desired result.

$\mathrm{(ii)}$ For $u\in S_{c, r}$, we have

Choosing $u_{0} = t\star u$ with $t > 0$ large enough, we deduce that $u_{0}\in S_{c, r}\setminus A_{k_{2}}$ and $E_{\mu}(u_{0}) < 0$. □

By Lemma 3.1, we define the mountain pass level of the functional $E_{\mu}$ on $S_{c, r}$ by

where

Clearly, we have

Lemma 3.2. Suppose that $\frac{10}{3} < q < 6$ or $q = \frac{10}{3}$ with $\mu c^{q(1-\delta_{q})} < \frac{aq}{2\mathcal{C}_{q}^{q}}$. Then, there exists a Palais-Smale sequence $\{u_{n}\}\subset S_{c, r}$ for $E_{\mu}|_{S_{c, r}}$ at the level $\sigma(c, \mu)$ with $P_{\mu}(u_{n})\to0$ as $n\to\infty$.

Proof. Motivated by ^[33], we define the augmented functional $\tilde{E}_{\mu}:\mathbb{R}\times H^{1}(\mathbb{R}^{3})\to\mathbb{R}$

where $\Psi_{\mu}^{u}(t)$ is defined in (2.6). Notice that $\tilde{E}_{\mu}$ is of class $C^{1}$. By Lemma 2.5, we know that $\tilde{E}_{\mu}$ is invariant under rotations applied to $u$. Therefore, [31, Theorem 1.28] indicates that a critical point for $\tilde{E}_{\mu}|_{\mathbb{R}\times S_{c, r}}$ is a critical point for $\tilde{E}_{\mu}|_{\mathbb{R}\times S_{c}}$.

Now, we denote

where $E_{\mu}^{0}: = \{u\in S_{c, r}:E_{\mu}(u)\leq0\}$. We easily see that if $\gamma\in\Gamma$, then $\tilde{\gamma}: = (0, \gamma)\in\tilde{\Gamma}$ and $\tilde{E}_{\mu}(\tilde{\gamma}(t)) = E_{\mu}(\gamma(t))$ for $t\in[0, 1]$; while if $\tilde{\gamma} = (\tilde{\gamma}_{1}, \tilde{\gamma}_{2})\in\tilde{\Gamma}$, then $\gamma(\cdot): = \tilde{\gamma}_{1}\star\tilde{\gamma}_{2}\in\Gamma$ and $\tilde{E}_{\mu}(\tilde{\gamma}(t)) = E_{\mu}(\gamma(t))$ for $t\in[0, 1]$. Therefore, we have

By the definition of $\sigma(c, \mu)$, for $\varepsilon_{n} = \frac{1}{n^{2}}$, there exists $\gamma_{n}\in\Gamma$ such that

setting $\tilde{\gamma}_{n} = (0, \gamma_{n})$, we obtain

According to Ekeland's variational principle [33, Lemma 2.3], there exists a sequence $\{(t_{n}, v_{n})\}\subset\mathbb{R}\times S_{c, r}$ such that

with the additional property that

Note that $\tilde{E}_{\mu}(t_{n}, v_{n}) = \tilde{E}_{\mu}(0, t_{n}\star v_{n})$ and

for $(t, \psi)\in\mathbb{R}\times H_{r}^{1}(\mathbb{R}^{3})$ with $\int_{\mathbb{R}^{3}}v_{n}\psi = 0$. Setting $u_{n} = t_{n}\star v_{n}\in S_{c, r}$, by (3.6), we obtain

We take $(1, 0)$ as a test function in (3.8), and it follows from (3.6) that

For $w\in H_{r}^{1}(\mathbb{R}^{3})$ with $\int_{\mathbb{R}^{3}}(t_{n}\star v_{n})w = 0$, we take $(0, (-t_{n})\star w)$ as a test function in (3.8). In view of (3.6) and (3.7), we have that $E_{\mu}'|_{S_{c, r}}(u_{n})\to0$ as $n\to\infty$. □

Lemma 3.3. If $r_{i} > 0, i = 1, 2, 3$, then the function

has a unique critical point at which $g$ achieves its maximum.

Proof. By direct computation, we have

Obviously, $\tilde{g}$ is decreasing, $\lim_{t\to-\infty}\tilde{g}(t) = +\infty$ and $\lim_{t\to+\infty}\tilde{g}(t) = -\infty$, so there exists a unique $t_{0}\in\mathbb{R}$ such that $\tilde{g}(t_{0}) = 0$, $\tilde{g}(t) < 0$ if $t > t_{0}$, and $\tilde{g}(t) > 0$ if $t < t_{0}$. Then, $t_{0}$ is the unique critical point of the function $g(t)$ and $g(t_{0}) = \max_{t\in\mathbb{R}}g(t) > 0$ since $g(-\infty) = 0^{+}$ and $g(+\infty) = -\infty$. □

Similar to Lemma 3.3, we have

Lemma 3.4. If $r_{i} > 0, i = 1, 2, 3, 4$ and $\tau\geq4$, then the function

has a unique critical point at which $g$ achieves its maximum.

Lemma 3.5. Suppose that $\frac{14}{3}\leq q < 6$ or $q = \frac{10}{3}$ with $\mu c^{q(1-\delta_{q})} < \frac{aq}{2\mathcal{C}_{q}^{q}}$. For any $u\in S_{c}$, there exists a unique $t_{u}\in\mathbb{R}$ such that $t_{u}\star u\in\mathcal{P}_{c, \mu}$, $t_{u}$ is the unique critical point of $\Psi_{\mu}^{u}$, and it is a strict maximum at a positive level. Moreover, $\Psi_{\mu}^{u}$ is strictly decreasing on $(t_{u}, +\infty)$ and $t_{u} < 0$ implies $P_{\mu}(u) < 0$. The map $u\in S_{c}\mapsto t_{u}\in\mathbb{R}$ is of class $C^{1}$.

Proof. If $q = \frac{10}{3}$, recalling $q\delta_{q} = 2$, letting $u\in S_{c}$, by (2.6) and (2.7), we have

and $(\Psi_{\mu}^{u})'(t) = 0\Leftrightarrow P_{\mu}(t\star u) = 0\Leftrightarrow t\star u\in\mathcal{P}_{c, \mu}$. In view of (1.12), we obtain

Since $\mu c^{q(1-\delta_{q})} < \frac{aq}{2\mathcal{C}_{q}^{q}}$ and $2p\gamma_{p} > 4$, in view of Lemma 3.3 and (3.9), there exists a unique $t_{u}\in\mathbb{R}$ such that $(\Psi_{\mu}^{u})'(t_{u}) = 0$ and $\Psi_{\mu}^{u}(t_{u}) = \max_{t\in\mathbb{R}}\Psi_{\mu}^{u}(t) > 0$.

Similarly, we deduce that $4\leq q\delta_{q} < 2p\gamma_{p}$ when $\frac{14}{3}\leq q < 6$, in view of Lemma 3.4 and (2.6), there exists a unique $t_{u}\in\mathbb{R}$ such that $(\Psi_{\mu}^{u})'(t_{u}) = 0$ and $\Psi_{\mu}^{u}(t_{u}) = \max_{t\in\mathbb{R}}\Psi_{\mu}^{u}(t) > 0$. Thus, we infer that

and $\Psi_{\mu}^{u}$ is strictly decreasing on $(t_{u}, +\infty)$. Since $(\Psi_{\mu}^{u})'(t) < 0$ if and only if $t > t_{u}$, we deduce that $t_{u} < 0$ implies $P_{\mu}(u) = (\Psi_{\mu}^{u})'(0) < 0$.

Define $\Phi: \mathbb{R}\times H_{r}^{1}(\mathbb{R}^{3})\to\mathbb{R}$ by $\Phi(t, u) = (\Psi_{\mu}^{u})'(t_{u})$. It is clear that $\Phi$ is of class $C^{1}$, $\Phi(t_{u}, u) = 0$ and $\partial_{t}\Phi(t_{u}, u) = (\Psi_{\mu}^{u})''(t_{u}) < 0$. Applying the implicit function theorem, we see that the map $u\in S_{c}\mapsto t_{u}\in\mathbb{R}$ is of class $C^{1}$. □

Lemma 3.6. Suppose that $\frac{14}{3}\leq q < 6$ or $q = \frac{10}{3}$ with $\mu c^{q(1-\delta_{q})} < \frac{aq}{2\mathcal{C}_{q}^{q}}$. We define

then $m(c, \mu) > 0$.

Proof. If $u\in\mathcal{P}_{c, \mu}$, combining $P_{\mu}(u) = 0$ with (1.12), (1.13), we obtain

So, in view of $4\leq q\delta_{q} < 2p\gamma_{p}$ or $q\delta_{q} = 2$ when $\mu c^{q(1-\delta_{q})} < \frac{aq}{2\mathcal{C}_{q}^{q}}$, there exists a positive constant $C_{0}$ such that

In addition, for $u\in\mathcal{P}_{c, \mu}$, we have

Then, we get $m(c, \mu) > 0$. □

Lemma 3.7. Suppose that $\frac{14}{3}\leq q < 6$ or $q = \frac{10}{3}$ with $\mu c^{q(1-\delta_{q})} < \frac{aq}{2\mathcal{C}_{q}^{q}}$. Then, $\sigma(c, \mu) = m(c, \mu)$, where $\sigma(c, \mu)$ is defined in (3.2).

Proof. Denote

For any $u\in\mathcal{P}_{c, \mu}\cap S_{c, r}$, recalling that

Then, there exist $t_{1} < 0$ and $t_{2} > 0$ such that $t_{1}\star u\in A_{k_{1}}$, $t_{2}\star u\in S_{c, r}\setminus A_{k_{2}}$ and $E_{\mu}(t_{2}\star u) < 0$ by Lemma 3.1. Setting $\gamma(t) = ((1-t)t_{1}+tt_{2})\star u$ for $t\in[0, 1]$, we easily see that $\gamma\in\Gamma$. In addition, by Lemma 3.5, we have $\max_{t\in[0, 1]}E_{\mu}(\gamma(t)) = E_{\mu}(u)$ if $(1-t)t_{1}+tt_{2} = 0$, from which it follows that $m_{r}(c, \mu)\geq\sigma(c, \mu)$.

On the other hand, we prove $\sigma(c, \mu)\geq m_{r}(c, \mu)$, it suffices to verify that $\gamma([0, 1])\cap\mathcal{P}_{c, \mu}\ne\emptyset$ for any $\gamma\in\Gamma$. If $u\in S_{c, r}$, by (3.12), we get

So, we have $P_{\mu}(\gamma(1)) < 4E_{\mu}(\gamma(1))\leq0$ for any $\gamma\in\Gamma$. Note that $P_{\mu}(\gamma(0)) > 0$. Let us consider the function

Obviously, $P_{\gamma}$ is continuous by (2.5). Hence, we infer that there exists $\tau_{\gamma}\in(0, 1)$ such that $P_{\gamma}(\tau_{\gamma}) = 0$, namely, $\gamma(\tau_{\gamma})\in\mathcal{P}_{c, \mu}$, which implies that $\gamma(\tau_{\gamma})\in\gamma([0, 1])\cap\mathcal{P}_{c, \mu}$, namely, $\gamma([0, 1])\cap\mathcal{P}_{c, \mu}\ne\emptyset$. Thus, we obtain

Finally, we prove $m_{r}(c, \mu) = m(c, \mu)$. Since $m_{r}(c, \mu)\geq m(c, \mu)$, we only prove $m(c, \mu)\geq m_{r}(c, \mu)$. Suppose by contradiction that there exists $\bar{u}\in\mathcal{P}_{c, \mu}\setminus S_{c, r}$ such that

Let $\bar{v} = |\bar{u}|^{\ast}$ be the symmetric decreasing rearrangement of $|\bar{u}|$, then by the properties of symmetric decreasing rearrangement, we have

Since $\bar{v}\in S_{c, r}$, by Lemma 3.5, there exists a unique $t_{\bar{v}}\in\mathbb{R}$ such that $t_{\bar{v}}\star\bar{v}\in\mathcal{P}_{c, \mu}$ and

Hence, in view of (3.15)-(3.17), we have

which is impossible. Therefore, $m_{r}(c, \mu) = m(c, \mu)$. Combining with (3.14), we have

□

In what follows, we will discuss the convergence of Palais-Smale sequences satisfying suitable additional conditions, following the ideas introduced by [28, Proposition 3.1].

Proposition 3.8. Suppose that $\frac{10}{3} < q < 6$ or $q = \frac{10}{3}$ with $\mu c^{q(1-\delta_{q})} < \frac{aq}{2\mathcal{C}_{q}^{q}}$, $\{u_{n}\}\subset S_{c, r}$ is a Palais-Smale sequence for $E_{\mu}|_{S_{c, r}}$ at the positive level $\sigma(c, \mu)$ with $P_{\mu}(u_{n})\to0$ as $n\to\infty$. Then, up to a subsequence, $u_{n}\to u$ in $H_{r}^{1}(\mathbb{R}^{3})$, and $u\in S_{c, r}$ is a solution to (1.1) for some $\lambda < 0$.

Proof. Step 1. Boundedness of $\{u_{n}\}$ in $H_{r}^{1}(\mathbb{R}^{3})$.

If $\frac{10}{3}\leq q < \frac{14}{3}$, then $2\leq q\delta_{q} < 4$. Using (1.12) and $P_{\mu}(u_{n}) = o_{n}(1)$ yields

from which we see that $\{u_{n}\}$ is bounded in $H_{r}^{1}(\mathbb{R}^{3})$.

If $\frac{14}{3}\leq q < 6$, then $4\leq q\delta_{q} < 6$. Using (1.12) and $P_{\mu}(u_{n}) = o_{n}(1)$ again, we obtain

Hence, $\{u_{n}\}$ is bounded in $H_{r}^{1}(\mathbb{R}^{3})$.

Step 2. We prove that there exist Lagrange multipliers $\lambda_{n}\to\lambda\in\mathbb{R}$.

Since $\{u_{n}\}$ is bounded in $H_{r}^{1}(\mathbb{R}^{3})$, by the compactness of $H_{r}^{1}(\mathbb{R}^{3})\hookrightarrow L^{q}(\mathbb{R}^{3})$($q\in (2, 6)$), up to a subsequence, there exists a function $u\in H_{r}^{1}(\mathbb{R}^{3})$, such that $u_{n}\rightharpoonup u$ in $H_{r}^{1}(\mathbb{R}^{3})$, $u_{n}\to u$ in $L^{q}(\mathbb{R}^{3})$ and $u_{n}\to u$ a.e. on $\mathbb{R}^{3}$. Notice $\{u_{n}\}$ is a bounded Palais-Smale sequence of $E_{\mu}|_{S_{c, r}}$, by the Lagrange multipliers rule, there exists $\{\lambda_{n}\}\subset\mathbb{R}$ such that

for every $\varphi\in H^{1}(\mathbb{R}^{3})$. Choosing $\varphi = u_{n}$, then

By (1.12) and (1.13), the boundedness of $\{u_{n}\}$ implies that $\{\|u_{n}\|_{q}\}$ and $\{\int_{\mathbb{R}^{3}}(I_{\alpha }\ast|u_{n}|^{p})|u_{n}|^{p}dx\}$ are bounded. Thus, $\{\lambda_{n}\}$ is bounded as well, up to a subsequence, $\lambda_{n}\to\lambda\in\mathbb{R}$.

Step 3. We show that $\lambda < 0$ and $u\not\equiv0$.

We first prove that

By Hardy-Littlewood-Sobolev inequality and Minkowski inequality, we obtain

By virtue of (1.12), $2 < \frac{6p}{3+\alpha} < 6$, and the boundedness of $\{u_{n}\}$ in $H_{r}^{1}(\mathbb{R}^{3})$, we deduce that $\{\|u_{n}\|_{\frac{6p}{3+\alpha}}\}$ is bounded. In addition, we have

by $\|u_{n}-u\|_{\frac{6p}{3+\alpha}}\to0$. Thus, (3.20) is established. Combining (3.19) with $P_{\mu}(u_{n}) = o_{n}(1)$, we get

Then, we obtain

by virtue of $\lambda_{n}\to\lambda$, $u_{n}\to u$ in $L^{q}(\mathbb{R}^{3})$ and (3.20). Since $\mu > 0$, $0 < \delta_{q} < 1$ and $p < 3+\alpha$, we deduce that $\lambda\leq0$, with $\lambda = 0$ if and only if $u\equiv0$. If $\lambda_{n}\to0$, we have

by the compactness of $H_{r}^{1}(\mathbb{R}^{3})\hookrightarrow L^{q}(\mathbb{R}^{3})$ and (3.20). Using again $P_{\mu}(u_{n}) = o_{n}(1)$, we have $E_{\mu}(u_{n})\to0$, which is a contradiction with $E_{\mu}(u_{n})\to m > 0$, and thus $\lambda_{n}\to\lambda < 0$ and $u\ne0$.

Step 4. $u_{n}\to u$ in $H_{r}^{1}(\mathbb{R}^{3})$.

Since $u_{n}\rightharpoonup u\ne0$ in $H_{r}^{1}(\mathbb{R}^{3})$ and $H_{r}^{1}(\mathbb{R}^{3})\hookrightarrow D^{1, 2}(\mathbb{R}^{3})$, we get

Then, (3.18) and Lemma 2.3 imply that

Test (3.18)-(3.24) with $\varphi = u_{n}-u$, we obtain

By using the Hölder inequality and the strong convergence of $u_{n}$ to $u$ in $L^{q}(\mathbb{R}^{3})$, we obtain

In addition, by the Hardy-Littlewood-Sobolev inequality and generalized Hölder inequality, we have

So, by the boundedness of $\{\|u_{n}\|_{\frac{6p}{3+\alpha}}\}$ and $\|u_{n}-u\|_{\frac{6p}{3+\alpha}}\to0$, we obtain

Similarly, we get

In view of (3.26), (3.27), (3.28), and (3.25), we have

which implies that $u_{n}\to u$ in $H_{r}^{1}(\mathbb{R}^{3})$ by $\lambda < 0$, and $u\in S_{c, r}$ solves (1.1) for some $\lambda < 0$. □

3.1.   Proof of Theorem 1.1

According to Lemma 3.2, there exists a sequence $\{u_{n}\}\subset S_{c, r}$ with the following properties

Then, by Propostion 3.8, $u_{n}\to u$ in $H_{r}^{1}(\mathbb{R}^{3})$, $u\in S_{c, r}$ is a solution to (1.1) for some $\lambda < 0$ with $E_{\mu}(u) = \sigma(c, \mu)$. In addition, by Lemma 3.7, we get $E_{\mu}(u) = \sigma(c, \mu) = m(c, \mu)$, namely, $u$ is a ground state solution under the cases of $q = \frac{10}{3}$ and $\frac{14}{3}\leq q < 6$. Recalling that $\beta_{n}(\tau)\geq0$ a.e. in $\mathbb{R}^{3}$ for every $\tau$, (3.7) and the convergence imply that $u$ is non-negative. The strong maximum principle implies that $u > 0$. □

4.   Supercritical leading term with defocusing perturbation

In this section, we prove the existence of a mountain pass type solution when $\mu < 0$.

Lemma 4.1. Suppose that $\frac{10}{3}\leq q\leq\frac{14}{3}\leq p < 3+\alpha$, $\alpha\in(\frac{5}{3}, 3)$ and $\mu < 0$. For any $u\in S_{c}$, there exists a unique $t_{u}\in\mathbb{R}$ such that $t_{u}\star u\in\mathcal{P}_{c, \mu}$, $t_{u}$ is the unique critical point of $\Psi_{\mu}^{u}$ and it is a strict maximum at a positive level. Moreover, $\Psi_{\mu}^{u}$ is strictly decreasing on $(t_{u}, +\infty)$ and $t_{u} < 0$ implies $P_{\mu}(u) < 0$. The map $u\in S_{c}\mapsto t_{u}\in\mathbb{R}$ is of class $C^{1}$.

Proof. In view of $\mu < 0$, $2\leq q\delta_{q}\leq 4 < 2p\gamma_{p}$ and (2.6), we have

Therefore, $\Psi_{\mu}^{u}(t)$ has a global maximum point at positive level. The rest of the proof is similar to that of Lemma 3.5. □

Lemma 4.2. Suppose that $\frac{10}{3}\leq q\leq\frac{14}{3}\leq p < 3+\alpha$, $\alpha\in(\frac{5}{3}, 3)$ and $\mu < 0$. Then, $m(c, \mu) > 0$, where $m(c, \mu)$ is defined in (3.10).

Proof. Combining (1.13) with $P_{\mu}(u) = 0$, we obtain

By virtue of $\mu < 0$ and (4.2), we have

Thus,

For any $u\in\mathcal{P}_{c, \mu}$, the energy functional can be seen

In view of $2p\gamma_{p} > 4$, $\mu < 0$ and (4.4), we obtain

and the desired result follows from the inequality above. □

Lemma 4.3. Suppose that $\frac{10}{3}\leq q\leq\frac{14}{3}\leq p < 3+\alpha$, $\alpha\in(\frac{5}{3}, 3)$ and $\mu < 0$. Then, there exists $k > 0$ sufficiently small such that

where $\overline{A_{k}}: = \{u\in S_{c}:\|\nabla u\|_{2}\leq k\}$.

Proof. In view of (1.13), $2p\gamma_{p} > 4$, and $\mu < 0$, if $u\in\overline{A_{k}}$ with $k$ small enough, then we obtain

and

If we could replace $k$ with a smaller positive number, combining (1.12) with Lemma 4.2, then it is obvious that

□

Proposition 4.4. Suppose that $\frac{10}{3}\leq q\leq\frac{14}{3}\leq p < 3+\alpha$, $\alpha\in(\frac{5}{3}, 3)$, and $\mu < 0$. If $\{u_{n}\}\subset S_{c, r}$ is a Palais-Smale sequence for $E_{\mu}|_{S_{c, r}}$ at non-zero level $\tilde{c}$ with $P_{\mu}(u_{n})\to0$ as $n\to\infty$ and (1.14) holds, then up to a subsequence, $u_{n}\to\tilde{u}$ in $H_{r}^{1}(\mathbb{R}^{3})$, and $\tilde{u}\in S_{c, r}$ is a solution to (1.1) for some $\lambda < 0$.

Proof. We divide the proof into four steps.

Step 1. Boundedness of $\{u_{n}\}$ in $H_{r}^{1}(\mathbb{R}^{3})$.

Combining $\mu < 0$ with $P_{\mu}(u_{n}) = o_{n}(1)$, we obtain

then we deduce that $\{\|\nabla u_{n}\|_{2}\}$ is bounded by $2p\gamma_{p} > 4$. Thus, $\{u_{n}\}$ is bounded in $H_{r}^{1}(\mathbb{R}^{3})$.

Step 2. We prove that there exist Lagrange multipliers $\lambda_{n}\to\lambda$.

By the compactness of $H_{r}^{1}(\mathbb{R}^{3})\hookrightarrow L^{q}(\mathbb{R}^{3})$($q\in (2, 6)$), up to a subsequence, there exists a function $\tilde{u}\in H_{r}^{1}(\mathbb{R}^{3})$, such that $u_{n}\rightharpoonup\tilde{u}$ in $H_{r}^{1}(\mathbb{R}^{3})$, $u_{n}\to\tilde{u}$ in $L^{q}(\mathbb{R}^{3})$ and $u_{n}\to\tilde{u}$ a.e. on $\mathbb{R}^{3}$. Notice $\{u_{n}\}$ is a bounded Palais-Smale sequence of $E_{\mu}|_{S_{c, r}}$, by the Lagrange multipliers rule, there exists $\{\lambda_{n}\}\subset\mathbb{R}$ such that

for every $\varphi\in H_{r}^{1}(\mathbb{R}^{3})$. Choosing $\varphi = u_{n}$, then we obtain

By (1.12) and (1.13), the boundedness of $\{u_{n}\}$ implies that $\{\|u_{n}\|_{q}\}$ and $\{\int_{\mathbb{R}^{3}}(I_{\alpha }\ast|u_{n}|^{p})|u_{n}|^{p}dx\}$ are bounded. Thus, $\{\lambda_{n}\}$ is bounded as well, up to a subsequence, $\lambda_{n}\to\lambda\in\mathbb{R}$.

Step 3. $\lambda < 0$.

Combining $\mu < 0$ with $P_{\mu}(u_{n}) = o_{n}(1)$, we get

using (1.13) again, we have

Now, we may assume that

by (4.6) and (4.7), we obtain

Combining (4.5) with $P_{\mu}(u_{n}) = o_{n}(1)$, we obtain

Since $\frac{10}{3}\leq q\leq\frac{14}{3}\leq p < 3+\alpha$, we get $1-\frac{1}{\gamma_{p}} < 0$ and $\frac{\delta_{q}}{\gamma_{p}} < 1$, in view of (4.7), (4.8), (4.9), and (1.12), we infer that

Thus, observing assumption (1.14), taking to the limit of (4.10) as $n\to\infty$, we obtain

Step 4. We show that $u_{n}\to\tilde{u}$ in $H^{1}(\mathbb{R}^{3})$.

Similar to (3.24)-(3.28), we easily obtain

which, being $\lambda < 0$, implies that $u_{n}\to \tilde{u}$ in $H^{1}(\mathbb{R}^{3})$. □

Denoting by $E_{\mu}^{d}$ the closed sublevel set $\{u\in S_{c, r}:E_{\mu}(u)\leq d\}$, we introduce the minimax class

with associated minimax level

where $A_{k}$ and $\tilde{E}_{\mu}(t, u)$ are defined in (3.1) and (3.5), respectively.

4.1.   Proof of Theorem 1.2

We split the proof into the following steps.

Step 1. We show $\hat{\sigma}(c, \mu) = m_{r}(c, \mu)$, where $m_{r}(c, \mu)$ is defined (3.13).

Let $u\in S_{c, r}$, by (4.1), there exist $t_{0}\ll-1$ and $t_{1}\gg1$ such that

is a path in $\Gamma_{0}$(the continuity from (2.5)), then $\hat{\sigma}(c, \mu)$ is a real number.

We claim that for every $\gamma\in\Gamma_{0}$, there exists $\tau_{\gamma}\in(0, 1)$ such that $\alpha(\tau_{\gamma})\star\beta(\tau_{\gamma})\in\mathcal{P}_{c, \mu}$. Indeed, let us consider the function

By Lemma 4.3, we get $\tilde{P}_{\gamma}(0) = P_{\mu}(\beta(0)) > 0$. On the other hand, by Lemma 4.1, since $(\Psi_{\mu}^{\beta(1)})'(t) > 0$ for every $t\in(-\infty, t_{\beta(1)}]$ and $\Psi_{\mu}^{\beta(1)}(0) = E_{\mu}(\beta(1))\leq0$, then $t_{\beta(1)} < 0$. Again by Lemma 4.1, we can see that $P_{\mu}(\beta(1)) < 0$. Therefore, by the continuity of $\tilde{P}_{\gamma}$(see (2.5)), we can deduce that there exists $\tau_{\gamma}\in(0, 1)$ such that $\tilde{P}_{\gamma}(\tau_{\gamma}) = 0$, so $\alpha(\tau_{\gamma})\star\beta(\tau_{\gamma})\in\mathcal{P}_{c, \mu}$. This implies that

Thus,

On the other hand, $\gamma_{u}$ is the corresponding path defined by (4.11), and if $u\in\mathcal{P}_{c, \mu}\cap S_{c, r}$, then by Lemma 4.1, we get

So, we infer that

Combining (4.12) with (4.13), we have

Then, we have

and $\hat{\sigma}(c, \mu) > 0$ by Lemma 4.2.

Step 2. We prove the existence of a Palais-Smale sequence $\{u_{n}\}$ of $E_{\mu}|_{S_{c, r}}$ at the level $\hat{\sigma}(c, \mu)$.

By Lemma 4.3, we infer that

Taking

Then, we consider that $\mathcal{F} = \{\gamma([0, 1]):\gamma\in\Gamma_{0}\}$ is a homotopy stable family of compact subsets of $\mathbb{R}\times S_{c, r}$ with extended boundary $B = (0, \overline{A_{k}})\cup(0, E_{\mu}^{0})$ and the assumptions of [36, Theorem 5.2] hold with the superlevel $\{\tilde{E}\geq\hat{\sigma}(c, \mu)\}$. Thus, taking any minimizing sequence $\{\gamma_{n} = (\alpha_{n}, \beta_{n})\}\subset\Gamma_{0}$ for $\hat{\sigma}(c, \mu)$ with the property that $\alpha_{n}\equiv0$ and $\beta_{n}(\tau)\geq0$ a.e. in $\mathbb{R}^{3}$ for every $\tau\in[0, 1]$, there exists a Palais-Smale sequence $\{(t_{n}, v_{n})\}\subset\mathbb{R}\times S_{c, r}$ for $\tilde{E}_{\mu}|_{\mathbb{R}\times S_{c, r}}$ at the level $\hat{\sigma}(c, \mu) > 0$ such that

as $n\to\infty$. Moreover,

From (4.15), we have $P_{\mu}(t_{n}\star v_{n})\to0$ as $n\to\infty$. In addition, we have

for every $\varphi\in T_{v_{n}}S_{c, r}$. Thus, (4.17) leads to

for every $\varphi\in T_{v_{n}}S_{c, r}$, with $o_{n}(1)\to0$ as $n\to\infty$, in the last equality, we used that $\{t_{n}\}$ is bounded, due to (4.16). From Lemma 2.4 and (4.18), we see that $\{u_{n}: = t_{n}\star v_{n}\}\subset S_{c, r}$ is a Pohozaev-Palais-Smale sequence for $E_{\mu}|_{S_{c, r}}$ at the level $m_{r}(c, \mu) = \hat{\sigma}(c, \mu)$.

Step 3. We notice that the assumptions of Proposition 4.4 are satisfied for the Palais-Smale sequence $\{u_{n}\}$ obtained in the previous step. Then, up to a subsequence, we have $u_{n}\to\tilde{u}$ strongly in $H_{r}^{1}(\mathbb{R}^{3})$ and $\tilde{u}$ is a radial normalized solution to (1.1)-(1.2) for some $\hat{\lambda} < 0$. We complete the proof. □

5.   Asymptotic behavior

In this section, we study the asymptotic behavior of normalized solutions obtained in Theorem 1.1 as $b\to0^{+}$ and give the proof of Theorem 1.3.

5.1.   Proof of Theorem 1.3

Let $\{u_{b}:0 < b < \bar{b}\}$ be the solutions to (1.1)-(1.2) obtained by Theorem 1.1, where $\bar{b}$ is small enough. We split the proof into three steps.

Step 1. We show that the family of radial solutions $\{u_{b}\}$ is bounded in $H_{r}^{1}(\mathbb{R}^{3})$.

Note that $P_{\mu}(u_{b}) = 0$. If $\frac{10}{3}\leq q < \frac{14}{3}$, namely, $2\leq q\delta_{q} < 4 < 2p\gamma_{p}$, then we obtain

While if $\frac{14}{3}\leq q < 6$, namely, $4\leq q\delta_{q} < 2p\gamma_{p}$, then

Hence, we deduce from the above two cases that $\{u_{b}\}$ is bounded in $H_{r}^{1}(\mathbb{R}^{3})$.

Step 2. We prove that there exist Lagrange multipliers $\lambda_{n}\to\lambda\in\mathbb{R}$.

Since $\{u_{b}\}$ is bounded in $H_{r}^{1}(\mathbb{R}^{3})$, there exists $u\in H_{r}^{1}(\mathbb{R}^{3})$ such that, up to a subsequence, $u_{b}\rightharpoonup u$ in $H_{r}^{1}(\mathbb{R}^{3})$, and $u_{b}\to u$ in $L^{q}(\mathbb{R}^{3})$, $u_{b}\to u$ a.e. on $\mathbb{R}^{3}$. We also know that

Thus, for every $\varphi\in H_{r}^{1}(\mathbb{R}^{3})$, we have

Choosing $\varphi = u_{b}$, then

By (1.12) and (1.13), the boundedness of $\{u_{b}\}$ implies that $\{\|u_{b}\|_{q}\}$ and $\{\int_{\mathbb{R}^{3}}(I_{\alpha }\ast|u_{b}|^{p})|u_{b}|^{p}dx\}$ are bounded. Thus, $\{\lambda_{n}\}$ is bounded as well, up to a subsequence, $\lambda_{n}\to\lambda\in\mathbb{R}$.

Step 3. We prove that $u$ is a weak solution of (1.15) and $u_{b}\to u\in H_{r}^{1}(\mathbb{R}^{3})$.

By (3.20) and $u_{b}\to u$ in $L^{q}(\mathbb{R}^{3})$, we obtain

from (5.2). Obviously, we deduce that $\lambda\leq0$, with "=" if and only if $u\equiv0$. Similar to Step $3$ of the proof of Proposition 3.8, we get $\lambda < 0$ and $u\not\equiv0$. In view of $u_{b}\rightharpoonup u\ne0$ in $H_{r}^{1}(\mathbb{R}^{3})$, $H_{r}^{1}(\mathbb{R}^{3})\hookrightarrow D^{1, 2}(\mathbb{R}^{3})$, Lemma 2.3 and (5.1), we get

for any $\varphi\in H_{r}^{1}(\mathbb{R}^{3})$ as $b\to0^{+}$. That is, $u$ satisfies

test (5.1)-(5.3) with $\varphi = u_{b}-u$, as $b\to0^{+}$, we have

which implies that $u_{b}\to u$ in $H_{r}^{1}(\mathbb{R}^{3})$ by $\lambda < 0$, and $u\in S_{c, r}$ solves (1.1) for some $\lambda < 0$.

By (5.3) and $H_{r}^{1}(\mathbb{R}^{3})\hookrightarrow L^{2}(\mathbb{R}^{3})$, $u$ is a weak solution of (1.15) and $\|u\|_{2} = c$, namely, $u$ is a normalized solution of (1.15). □

Author contributions

Zhi-Jie Wang: Writing-original draft, Writing-review and editing; Hong-Rui Sun: Supervision, Writing-review and editing, Methodology, Validation.

Use of AI tools declaration

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

Acknowledgments

Research of H.R. Sun was partially supported by the NSF of Gansu Province of China (24JRRA414, 21JR7RA535).

Conflict of interest

The authors declare there is no conflict of interest.