Normalized solutions for pseudo-relativistic Schrödinger equations

1.   Introduction

This paper deals with the following pseudo-relativistic equation of the form:

where the frequency $\lambda$ as a real Lagrange parameter and is part of the unknowns, $2 < p < 2^\sharp.$ For $s\in (0, 1),$ the operator $(-\Delta+m^2)^s$ is defined in Fourier space as multiplication by the symbol $(|\xi|^2+m^2)^s$ see(^[1,2]) i.e., for any $u:\mathbb{R}^N\rightarrow \mathbb{R}$ belonging to the Schwartz space $S(\mathbb{R}^N)$ of rapidly decreasing functions,

where we denote by

the Fourier transform of $u.$ Aslo, we show an alternative definition of $(-\Delta+m^2)^s$ (see ^[2,3]):

where $P.V.$ is the Cauchy principal value, $K_\iota$ is the modified Bessel function of the third kind of index $\iota$ (see ^[4,5]) and

Once $m\rightarrow0,$ then $(-\Delta+m^2)^s$ reduces to the classical fractional Laplacian $(-\Delta)^s$ defined via Fourier transform by

At the same time, by singular integrals, we also get

for $s\in(0, 1).$ We observe that the most important difference between operators $(-\Delta)^s$ and $(-\Delta+m^2)^s$ is showed in scaling: the first one is homogeneous in scaling, while the second one is inhomogeneous, which is evident from the Bessel function $K_\iota$ in (1.2). There are many scholars devoted to the exploration of fractional Schrödinger equation

where $(-\Delta)^s$ as the fractional Laplacian, $f(u)$ represents the nonlinearity, the function $V(x): \mathbb{R}^N\rightarrow \mathbb{R}$ is an external potential function, and $\Omega$ is a bounded domain in $\mathbb{R}^N$ or $\Omega = \mathbb{R}^N.$ It was first introduced in the work of Laskin ^[6,7] and originated from an expansion of the Feynman path integral from Brownian-like to Lévy-like quantum mechanical paths. Note that the Feynman path integral produces the classical Schrödinger equation, however, the fractional Schrödinger equation is obtained by the path integral over Lévy trajectories.

When $s = \frac{1}{2},$ the operator $\sqrt{-\Delta+m^2}$ associates with the free Hamiltonian of a free relativistic particle of mass $m.$ It is worth noting that works of Lieb and Yau ^[8,9] on the stability of relativistic matter bring great inspiration to the exploration of $\sqrt{-\Delta+m^2}.$ There are some results for this topic, here we just quote a few, please refer to ^[10,11,12]. In particularly, it is interesting to consider results for fractional equations involving the operator $\sqrt{-\Delta u+m^2}$ with $m > 0.$ From the perspective of mathematics, many scholars focused on finding a solution to the following pseudo-relativistic equation

with $g(u) = |u |^{2^\sharp-2}u$. Now, there are two different approaches to consider problem (1.4) according to the characteristics of the frequency $\lambda:$

($i$) the frequency $\lambda$ is a fixed given constant,

($ii$) the frequency $\lambda$ is part of the unknown in problem (1.4).

In case $(i)$, we use a variant of extension method ^[13] to consider problem (1.4) due to the presence of the nonlocal operator $\sqrt{-\Delta u +m^2}u$ and we shall introduce this tool in detail in Section 2. Therefore, it can be seen that the solution of problem (1.4) is a critical point connected with the energy functional $\mathcal {I}_\lambda(v)$ defined in $H^1(\mathbb{R}_+^{N+1})$ by

In this case, we are devoted to looking for the ground state solutions because they possess many more properties, such as positivity, symmetry and stability. In particularly, the ground state solutions are regarded as minimizers of $\mathcal {I}_\lambda$ on the Nehari manifold

(see ^[14]). In addition, by building a nonempty closed subset of the sign-changing Nehari manifold, Yang and Tang ^[15] obtained the existence of least energy sign-changing solutions for Schrödinger-Poisson system involving concave-convex nonlinearities in $\mathbb{R}^3.$

Alternatively, in case $(ii)$ other papers are devoted to looking for nontrivial solutions of problem (1.4) when the frequency $\lambda$ is unknown. In this situation, $\lambda$ is regarded as a Lagrange multiplier. Moreover, this method from the perspective of physics seems particularly interesting because of the conservation of mass and the mass has a clear physical meaning. On the other hand, such solutions help us to better understand the dynamical properties, such as orbital stability or instability, where $\vartheta > 0$ represents the strength of attractive interactions between cold atoms. In general, the solutions with prescribed $L^2$-norms of solutions is called normalized solutions, i.e., the solutions satisfy $|u|_2 = c > 0$ for a priori given $c$. Here, in order to look for normalized solutions of problem (1.1), we shall take advantage of a variant of extension method ^[13] and transform problem (1.1) into a local problem in a upper half-space $\mathbb{R}_+^{N+1}$ with Neumann boundary condition. In addition, we look for the critical point of the functional on the constraint manifold $\mathcal {S}(a).$ We shall introduce $\mathcal {S}(a)$ and the upper half-space $\mathbb{R}_+^{N+1}$ in detail in Section 2.

In recent years, many scholars have paid great attention to exploration of normalization solutions to various classes of local and non-local problems, and have obtained many results, which are not only of special significance in physics, but also closely related to nonlinear optics and Bose-Einstein condensation. In addition, more and more mathematical scholars begin to explore also solutions with prescribed $L^2$-norms. This kind of problems was first proposed by Jeanjean in ^[16], who considered the existence of normalized solutions for the Schrödinger equations

where $N\geq 1, \lambda \in \mathbb{R}$ and $g$ satisfies suitable assumptions. Inspired by pioneering work of Jeanjean ^[16], with the help of variational methods, Alves et al. ^[17] considered the existence of normalized solutions to the nonlinear Schrödinger equation with critical growth both when $N\geq3$ and $N = 2.$ The author in ^[18] established existence and several properties of ground states for the following critical equation

Later, Soave ^[19] also was interested in existence and qualitative properties of normalized solutions of the nonlinear Schrödinger equation with combined power nonlinearities driven by two different Laplacian operators. With the aid of an approximation method, Deng and Wu ^[20] obtained the existence of normalized solutions for the Schrödinger equation, and the positive solution is mountain-pass type for $p = 2^\ast.$ Li and Zou ^[21] were interested in the exploration of fractional Schrödinger equation, they obtained the existence of multiple normalized solutions in both the $L^2$-subcritical and $L^2$-supercritical cases by truncation technique, concentration-compactness principle, genus theory and a fiber map. Wang et al. ^[22] explored the existence results of normalized solutions for $p$-Laplacian equations in the case $(\frac{N+2}{N}p, p^\ast)$ by a mountain-pass argument and constrained variational methods. Yao et al. ^[23] considered several nonexistence and existence results of normalized solutions for the Choquard equations involving lower critical exponent by variational methods. With the aid of a perturbation method, Jeanjean et al. ^[24] verified the existence of two solutions involving a prescribed $L^2$-norm for a quasi-linear Schrödinger equation. We point out that, in ^{[19,25,26,27]}, several applications are discussed. However, results about the pseudo-relativistic equation are relatively few, as far as we know.

Inspired by the works above, we treat existence of the multiple normalized solutions for problem (1.1). Undoubtedly, we shall encounter some difficulties in proving the existence of the normalized solutions of problem (1.1). One is that Sobolev critical exponent $2^\sharp = \frac{2N}{N-1}$ which makes the lack of compactness occur. On the other hand, since the embedding $\mathbb{X}_{rad}(\mathbb{R}_+^{N+1})\hookrightarrow L^2(\mathbb{R}^N)$ is not compact, we observe that the weak limit of $(PS)$ sequence can not be established in the constraint manifold $\mathcal {S}(a).$ Therefore, we have to prove that the Lagrange multipliers $\lambda$ are non-negative in case $2 < p < 2+2/N < 2^\sharp$, which is crucial for us to be able to obtain the compactness. Using the compactness principle, the difficulty is solved.

In the following, in case $2 < p < 2+2/N < 2^\sharp$, the energy functional $\mathcal {J}$ is unbounded from below on $\mathcal {S}(a)$, which results in the failure to get the existence of the solution to problem (1.1) via minimizing problem. In the case $2 < p < 2+\frac{2}{N},$ inspired by ^[17,28], we use a truncation technique that allows the truncation function to be bounded from below and coercive.

Finally, problem (1.1) is nonlocal, we shall encounter new difficulties and the study of this kind of equations becomes very meaningful. Therefore, by the extension method in ^[13], we transform problem (1.1) into a local problem in a upper half-space with a nonlinear Neumann boundary condition.

Our main result is stated in the following theorem:

Theorem 1.1. Let $2 < p < 2+\frac{2}{N} < 2^\sharp$ be satisfied. Then for given $k\in \mathbb{N},$ there exists $\beta > 0$ independent of $k$ and $\vartheta_k: = \vartheta(k)$ such that problem (1.1) has at least $k$ couples $(u_j, \lambda_j)\in H^{\frac{1}{2}}(\mathbb{R}^N)\times \mathbb{R}$ of weak solutions for $\vartheta\geq \vartheta_k$ and $a\in(0, (\beta/\vartheta)^{\frac{1}{1-\theta}}]$, with $\int_{\mathbb{R}^N}|u_j|^2dx = a^2, \lambda_j < 0$ for all $j\in [1, k]$ and $\theta = \frac{(p-2)(N-1)}{2}.$

The organizational structure of present paper in what follows. In Section 2, we give some necessary preliminaries and outline the variational framework. In Section 3, we are devoted to the proof of Theorem 1.1.

2.   Preliminaries

Let $H^{\frac{1}{2}}(\mathbb{R}^{N})$ be the fractional Sobolev space defined as the completion of $C^{\infty}_{c}(\mathbb{R}^{N})$ with the following norm

Therefore, $H^{\frac{1}{2}}(\mathbb{R}^{N})$ is continuously embedded in $L^{p}(\mathbb{R}^{N})$ for all $p\in [2, 2^\sharp)$ and $H^{\frac{1}{2}}(\mathbb{R}^{N})$ is compactly embedded in $L^{p}_{loc}(\mathbb{R}^{N})$ for all $p\in [1, 2^\sharp)$, please refer to ^[2,4,29,30]. Let $H^{1}(\mathbb{R}^{N+1}_{+})$ denote the completion of $C^{\infty}_{c}(\overline{\mathbb{R}^{N+1}_{+}})$ in the norm:

According to Lemma 3.1 in ^[3], for $s\in (0, 1),$ we have that the continuous imbedding $H^1(\mathbb{R}^{N+1}_{+})\hookrightarrow L^{2\gamma}(\mathbb{R}^{N+1}_{+}, y^{1-2s})$, this fact means

for some $\hat{S} > 0$, where $\gamma: = 1+\frac{2}{N-2s}$, and $L^{r}(\mathbb{R}^{N+1}_{+}, y^{1-2s})$ is the weighted Lebesgue space for $r\in (1, \infty)$, equipped with the norm

Using Lemma 3.1.2 in ^[31], it follows that $H^1(\mathbb{R}^{N+1}_{+})$ compactly embedded in $L^{2}(B_{R}^{+}, y^{1-2s})$ for all $R > 0$. In the lignt of Proposition 5 in ^[3], there exists a (unique) linear trace operator ${\rm Tr}: H^1(\mathbb{R}^{N+1}_{+})\rightarrow H^{\frac{1}{2}}(\mathbb{R}^N)$ such that

where $\sigma_{s}: = 2^{1-2s}\Gamma(1-s)/\Gamma(s)$, please refer to ^[32,33]. For the sake of simplicity, we will show ${\rm Tr}(v)$ by $v(\cdot, 0)$. It is worth noting that (2.2) implies

for all $v\in H^1(\mathbb{R}^{N+1}_{+})$, which is equivalent to

To simplify the notation, we can get rid of the constant $\sigma_{s}$ in (2.4).

In the following, we define the work space

equipped with the norm

Clearly, $\mathbb{X}\subset H^1(\mathbb{R}_+^{N+1})$ and using (2.3), we see that

Moreover, $\mathbb{X}$ is a Hilbert space equipped with the inner product

At the same time, $\mathbb{X}^\ast$ is the dual space of $\mathbb{X}.$

Now, we recall some results in the case $s\in(0, 1).$ Since $Tr(H^1(\mathbb{R}_+^{N+1}))\subset H^{\frac{1}{2}}(\mathbb{R}^N)$ and the embedding $H^{\frac{1}{2}}(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N)$ is continuous for any $q\in [2, 2_s^\ast]$ and $s\in (0, 1).$ we have the following results.

Theorem 2.1. ^[34] For any $u\in H^1(\mathbb{R}_+^{N+1}, y^{1-2s})$ and for any $q\in [2, 2_s^\ast]$

where $\kappa_s = 2^{1-2s}\frac{\Gamma(1-s)}{\Gamma(s)}$ and $u(x) = v(x, 0)$ is the trace of $v$ on $\partial \mathbb{R}_+^{N+1}.$

Theorem 2.2. ^[34] Let $H_{rad}^1 = \{u\in H^1(\mathbb{R}_+^{N+1}, y^{1-2s}): u \ \ \mathit{\mbox{is radially symmetric with respect to}} \ \ x\}.$ Then $H_{rad}^1(\mathbb{R}_+^{N+1}, y^{1-2s})\hookrightarrow\hookrightarrow L^q(\mathbb{R}^N)$ for any $q\in(2, 2_s^\ast).$

We recall the trace inequality with $s = \frac{1}{2}$(see Theorem 2.1 in ^[32]):

for all $v\in H_0^{1}(\mathbb{R}_+^{N+1}),$ where $H_0^{1}(\mathbb{R}_+^{N+1})$ as the completion of $C_c(\overline{\mathbb{R}_+^{N+1}})$ in the norm

and the best constant is given by

This constant is obtained on the family of functions $\omega_\epsilon = \mathcal {E}_{1/2}(u_\epsilon)$, where $\mathcal {E}_{{1}/{2}}$ denotes the $\frac{1}{2}$-harmonic extension ^[13], and

see ^[29,32]. Therefore,

where

as the Poisson kernel for the extension problem in $\mathbb{R}_+^{N+1}.$

We observe that $\omega_\epsilon(x, y) = \epsilon^{\frac{1-N}{2}}\omega_1(\frac{x}{\epsilon}, \frac{y}{\epsilon}).$ We are devoted to studying the existence and multiplicity of normalized solutions of problem (1.1) in present paper. To consider problem (1.1) by variational methods, we make full use of a variant of the extension method ^[13] given in ^[3,29,33]. To be more precise, the nonlocal operator $\sqrt{-\Delta+m^2}$ in $\mathbb{R}^N$ can be achieved by a local problem in $\mathbb{R}^N\times (0, \infty)$. In the following, we shall describe this construction in detail. For any function $u\in H^{\frac{1}{2}}(\mathbb{R}^N),$ there exists a unique function $v\in H^1(\mathbb{R}_+^{N+1})$ (here, $\mathbb{R}_+^{N+1} = \{(x, y)\in \mathbb{R}^N\times \mathbb{R}:y > 0\}$ such that

Set

we have the following equation

with the solution $w(x, y) = -\frac{\partial v}{\partial y}(x, y).$ By (2.6), we have

and hence $T^2 = (-\Delta_x+m^2).$ Thus, the operator $T$ that maps the Dirichlet-type data $u$ to the Neumann-type data $-\frac{\partial v}{\partial y}(x, 0)$ is actually $\sqrt{-\Delta+m^2}.$ Therefore, for problem (1.1), we shall consider the following nonlinear boundary value problem:

Furthermore, we shall look for the critical points of the energy functional $\mathcal {J}:\mathbb{X}_{rad}(\mathbb{R}_+^{N+1})\rightarrow \mathbb{R}$ associated with problem (2.7):

on the constraint

3.   Proof of Theorem 1.1

Let us start the section by recalling the definition of genus. Let $X$ be a Banach space and $D$ be a subset of $X.$ The set $D$ is called to be symmetric if $-u \in D$ for all $u\in D$. Denote by $\Sigma$ the family of closed symmetric subsets $D$ of $X$ such that $0 \notin D,$ that is

For $D\in \Sigma,$ we define

and $\Sigma_k = \{D\in \Sigma\, :\, \gamma(D)\geq k\}.$ Now, we are ready to give some lemmas that play important roles in proving Theorem 1.1.

Lemma 3.1. Let $v\in H^{1}(\mathbb{R}_+^{N+1})$ and $2 < t < 2^\sharp,$ then

where $\theta = \frac{(t-2)(N-1)}{2}.$

Proof. Since $v\in H^{1}(\mathbb{R}_+^{N+1})$ and $2 < t < 2^\sharp,$ by Hölder inequality and (2.5), we obtain

where $\theta = \frac{(t-2)(N-1)}{2}.$ Then we have completed the proof of Lemma 3.1.

We state the concentration-compactness principle for $s = \frac{1}{2}$ in what follows.

Lemma 3.2 (Proposition 3.1 in ^[35]). Let $\{v_k\}$ be a bounded tight sequence in $H^1(\mathbb{R}_+^{N+1}),$ such that $v_k$ converges weakly to $v$ in $H^1(\mathbb{R}_+^{N+1}).$ Let $\mu, \nu$ be two non-negative measures on $\mathbb{R}_+^{N+1}$ and $\mathbb{R}^N$ respectively and such that

and

in the sense of measures. Then, there exist an at most countable set $I$ and three families $\{x_i\}_{i\in I},$ $\{\mu_i\}_{i\in I},$ $\{\nu_i\}_{i\in I},$ with $\mu_i, \nu_i\geq 0$ for all $i\in I$, such that

Lemma 3.3. Let $\{v_k\}$ in be a sequence in $H^1(\mathbb{R}_+^{N+1})$ as in Lemma 3.2 and define

Then

where $\mu, \nu$ are the finite non-negative measures in Lemma 3.2.

Proof. Fix a sequence $\{v_k\}$ in $H^1(\mathbb{R}_+^{N+1}),$ as in the statement of Lemma 3.2. Let $\eta\in C^\infty_c(\overline{\mathbb{R}_+^{N+1}})$ such that $0 \leq\eta\leq 1,$ $\eta = 0$ in $B_1^+$ and $\eta = 1$ in $(B_2^c)^+.$ Take $R > 0$ and put $\eta_R(x, y) = \eta(\frac{x}{R}, \frac{y}{R}).$ We write

We first observe that

So by (3.1),

On the other hand, since $\mu$ is finite, $1-\eta_R^2$ has compact support and $\eta_R\rightarrow 0$ a.e. in $\mathbb{R}_+^{N+1},$ by the definition of $\mu$ and the Dominated convergence theorem, we have

Using (3.5)-(3.6) in (3.4), we can obtain (3.2). Arguing similarly for $\nu,$ we see that

Thus, the first part of (3.3) is proved.

In order to verify the last part of (3.3), we consider again the function $\eta_R.$ Let $K: =$supp$(\eta_R).$ By the fact that

and applying this to $\eta_Ru_k$ in $H^1(\mathbb{R}_+^{N+1}),$ we get

for all $k$. On the other hand,

By the definition of $\eta_R,$ we know

Using the Hölder inequality, the boundedness of $\{v_{k}\}_k$ in $H^1(\mathbb{R}_+^{N+1})$ and (3.10), we get

Therefore, together with (3.10) and taking $R\rightarrow \infty,$ $k\rightarrow \infty$ in (3.11), we obtain

Putting (3.10)-(3.12) into (3.8), we obtain the desired conclusion.

For $v\in \mathcal {S}(a)$, by Lemma 3.1 and (3.7), we have

where

and $\theta = \frac{(p-2)(N-1)}{2}.$ By $2 < p < 2+\frac{2}{N},$ we get that $0 < \theta < 1$ and there exists $\beta > 0$ such that $\vartheta a^{1-\theta}\leq \beta$. Thus, the function $\mathcal {K}$ has a positive local maximum. To be more precisely, there exist two numbers $0 < W_1 < W_2 < \infty$ such that $\mathcal {K} < 0$ in the intervals $(0, W_1)$ and $(W_2, \infty),$ while $\mathcal {K} > 0$ in $(W_1, W_2).$ Suppose that $\sigma\in C^\infty(\mathbb{R}^+, [0, 1])$ is a nonincreasing function such that $\sigma(t) = 1$ for $t\leq W_1$ and $\sigma(t) = 0$ for $t\geq W_2.$

We define the truncated functional by

For $v\in \mathcal {S}(a)$, by Lemma 3.1 and (3.7), we get

where

Therefore, with the help of the definition of $\sigma$, we obtain $\widetilde{\mathcal {K}} < 0$ in $(0, W_1)$ and $\widetilde{\mathcal {K}} > 0$ in $(W_2, \infty)$ when $a\in \Big(0, \Big(\beta/\vartheta\Big)^{\frac{1}{1-\theta}}\Big]$. From now on, we assume that

Without loss of generality, taking $W_1 > 0$ small enough if necessary, we also assume

Lemma 3.4. $(a)$ $\mathcal {J}_\sigma\in C^1(\mathbb{X}_{rad}(\mathbb{R}_+^{N+1}), \mathbb{R}).$

$(b)$ $\mathcal {J}_\sigma$ is coercive and bounded from below on $\mathcal {S}(a).$ Furtheremore, if $\mathcal {J}_\sigma\leq0,$ then $\|v\|\leq W_1$ and $\mathcal {J}_\sigma(v) = \mathcal {J}(v).$

$(c)$ ${\mathcal {J}_\sigma}|_{\mathcal {S}(a)}$ satisfies the $(PS)_c$ condition for all $c < 0.$

Proof. $(a)$ and $(b)$ hold true with the aid of a standard argument.

For $(a)$. As the proof of the Proposition B.10 in the book ^[36], conclusion $(a)$ is satisfied.

For $(b)$. Let $v \in \mathcal {S}(a)$, by the definition of $\sigma,$ we obtain $\sigma(\|v\|^2) = 0$ when $\|v\|\rightarrow \infty$. Thus,

since $N(p-2) < 2$ and $\theta = \frac{(p-2)(N-1)}{2},$ that is $\mathcal {J}_\sigma$ is coercive. On the other hand, it follows from the definition of $\widetilde{\mathcal {K}}(t)$ that $\widetilde{\mathcal {K}}$ has a maximum value, and then $\mathcal {J}_\sigma(v)$ is bounded from below on $\mathcal {S}(a).$ Furthermore, if $\mathcal {J}_\sigma(v)\leq 0$, so $\widetilde{\mathcal {K}} < 0$. Also, by the definition of $\widetilde{\mathcal {K}}$, we obtain $\|v\|\leq W_1$. Therefore, from the definition of $\sigma$, we get $\sigma = 1.$ This fact implies $\mathcal {J}_\sigma(v) = \mathcal {J}(v)$.

For $(c).$ Assume that $\{v_k\}_k$ is a $(PS)_c$ sequence of $\mathcal {J}_\sigma$ restricted to $\mathcal {S}(a)$ with $c < 0,$ that is,

By $(b), \|v_k\|\leq W_1$ for $k$ large enough. Therefore, $\{v_k\}_k$ is bounded in $\mathbb{X}_{rad}(\mathbb{R}_+^{N+1}).$ Then, up to subsequence, there exists $\mathbb{X}_{rad}(\mathbb{R}_+^{N+1})$ such that $v_k\rightharpoonup v$ in $\mathbb{X}_{rad}(\mathbb{R}_+^{N+1})$ and $v_k\rightarrow v$ in $L^p(\mathbb{R}^N)$ for all $p\in (2, 2^\sharp)$ and $v_k \rightarrow v$ a.e. in $\mathbb{R}^N.$ Due to the fact that $2 < p < 2+\frac{2}{N} < 2^\sharp,$ we get

Moreover, we claim $v\neq0.$ Otherwise, $\mathop{\lim}\limits_{k\rightarrow \infty}\int_{\mathbb{R}^N}|v_k|^pdx = 0.$ Combining this and (3.13), we see that

which is impossible and proves the claim.

Let

Thus, $\mathcal {S}(a) = \Psi^{-1}(\{\frac{a^2}{2}\}).$ By the Lagrange multiplier, there exists $\lambda_a\in \mathbb{R}$ such that

in $(H^{1}(\mathbb{R_+}^{N+1}))^\ast.$ Therefore, using this fact, we have

With the help of Proposition 5.12 in ^[14], there exists $\lambda_k\in \mathbb{R}$ such that

as $k\rightarrow \infty.$ Hence, for $\varphi \in \mathbb{X}_{rad}(\mathbb{R}_+^{N+1}),$

In particular,

The boundedness of $\{\|v_k\|\}_k$ implies that $\{\lambda_k\}_k$ is also bounded in $\mathbb{R}.$ Therefore, up to a subsequence, there exists $\lambda_a\in\mathbb{R}$ such that $\lambda_k\rightarrow \lambda_a$ as $k\rightarrow \infty.$ Therefore, by (3.15) and a standard argument, we obtain that $v$ satisfies problem (3.14). In fact, for any $\varphi\in \mathbb{X}_{rad}(\mathbb{R}_+^{N+1}),$ it follows from the definition of weak convergence that

as $k\rightarrow \infty.$ Since $\lambda_k\rightarrow \lambda_a$ as $k\rightarrow \infty$, we also obtain that

as $k\rightarrow \infty.$ Moreover, since $\{|v_k|^{2^\sharp-2}v_k\}_k$ is bounded in $L^{\frac{2^\sharp}{2^\sharp-1}}(\mathbb{R}^N)$ and

then

This implies that

as $k\rightarrow \infty.$ Next, we show that $\lambda_a < 0.$ Indeed, thanks to $2 < p < 2+\frac{2}{N} < 2^\sharp,$ we have

Therefore,

which shows $\lambda_a < 0.$

In the following, we shall recover the compactness with an application of the concentration-compactness principle ^[35]. Indeed, since $\|v_k\|\leq W_1$ for $k$ enough large, using the Prokhorov theorem [37, Theorem 8.6.2], there exist two positive measures $\mu, \nu\in \mathcal {M}(\mathbb{R}_+^{N+1})$ such that

Hence, Lemma 3.2-Lemma 3.3 hold. Together with Lemma 3.2, either $v_k\rightarrow v$ in $L^{2^\sharp}(\mathbb{R}^N)$ or there exists a (at most countable) set of distinct points $\{x_i\}_i\subset \mathbb{R}^N$ and positive numbers $\{\nu_i\}_i$ such that

If the latter holds, we can also verify $v_k\rightarrow v$ in $L^{2^\sharp}(\mathbb{R}^N)$. We shall verify the following three claims hold.

$\textbf{Claim 1.}$ We verify that $\mu(x_i)\leq \nu_i$ for any $i\in I$.

Assume that $x_i\in \mathbb{R}^N$ for some $i\in I.$ For any $\rho > 0,$ we define, $\varphi_\rho(x, y) = \varphi(\frac{x-x_i}{\rho}, \frac{y}{\rho}),$ where $\varphi\in C_c(\overline{\mathbb{R}_+^{N+1}})$ such that $\varphi = 1$ in $B_1^+$ and $\varphi = 0$ in $(B_2^+)^c, \varphi\in [0, 1]$ and $\|\nabla \varphi\|_{L^\infty(\mathbb{R}_+^{N+1})}\leq 2.$ We suppose that $\rho > 0$ such that supp$(\varphi_\rho(\cdot, 0))\subset \mathbb{R}_+^{N+1}.$ By the boundedness of $\{v_k\}$ in $\mathbb{X}_{rad}(\mathbb{R}_+^{N+1}),$ we know that $\{\varphi_\rho v_k\}$ is also bounded in $\mathbb{X}_{rad}(\mathbb{R}_+^{N+1}).$ Therefore,

That means

Consequently,

Together with the definition of $\varphi_\rho,$ we obtain

Moreover, (3.19) implies

In the following, we show that

In fact, by the Hölder inequality, the boundedness of $\{v_k\}_k$ in $\mathbb{X}_{rad}(\mathbb{R}_+^{N+1}),$ the fact that $\|\nabla\varphi_\rho\|_{L^\infty(\mathbb{R}_+^{N+1})}\leq\frac{C}{\rho}$ and $\mathbb{X}(\mathbb{R}_+^{N+1})$ is compactly embedded into $L^2(B_\rho^+(x_i, 0), y^{1-2s})$ with $s = \frac{1}{2},$ we obtain

By Hölder inequality with $\frac{1}{r}+\frac{r-1}{r} = 1$ and (2.1), we have

which shows that (3.24) holds. Therefore, inserting (3.21)-(3.24) into (3.20), taking $k\rightarrow \infty$ and $\rho\rightarrow 0^+$, we obtain

and the claim holds.

$\textbf{Claim 2.}$ We claim that $\mu_\infty\leq \nu_\infty.$

Let $\phi\in C_c^\infty(\overline{\mathbb{R}_+^{N+1}})$ such that $0\leq\phi\leq 1$, $\phi = 0$ in $B_1^+$ and $\phi = 1$ in $(B_2^c)^+.$ Take $R > 0$ and put $\phi_R(x) = \phi(\frac{x-x_i}{R}, \frac{y}{R})$. Again, by the boundedness of $\{v_k\}_k$ in $\mathbb{X}_{rad}({\mathbb{R}_+^{N+1}}),$ we know that $\{v_k\phi_R\}_k$ is also bounded in $\mathbb{X}_{rad}({\mathbb{R}_+^{N+1}}).$ Hence,

From the aforementioned proof, we obtain

By Hölder's inequality, $0\leq \phi_R\leq 1$ and $\{v_k\}$ is bounded in $\mathbb{X}_{rad}(\mathbb{R}_+^{N+1}),$ we have

as $R \rightarrow \infty.$ Therefore,

By the proof of Lemma 3.3 in ^[38], we obtain

and

Therefore, it follows from (3.25) that

and this proves Claim 2.

$\textbf{Claim 3.}$ We shall veify that $\nu_i = 0$ for any $i\in I$ and $\nu_\infty = 0.$

By contradiction, we suppose that there exists $i\in I$ such that $\nu_{i} > 0.$ Steps 1 implies that

It implies that $\nu_i\geq S_\ast^N.$ If this case is valid, we get

which is impossible by (3.13). If the latter holds, by the same discussion above, we get

which contradicts with (3.13), and together with Lemma 3.2 implies $v_k\rightarrow v$ in $L^{2^\sharp}_{loc}({\mathbb{R}^N})$ Moreover, combining with Lemma 3, we obtain $v_k\rightarrow v$ in $L^{2^\sharp}(\mathbb{R}^N).$ Taking into account (3.15)–(3.17), we obtain

Since $\lambda_a < 0$,

Hence,

Moreover, we obtain

By (3.26), we get

Then $v_k\rightarrow v$ in $\mathbb{X}_{rad}(\mathbb{R}_+^{N+1})$ and $|v_k(x, 0)|_2 = a.$ The proof of Lemma 3.4 is completed. □

Set

for $\varepsilon > 0.$ By the fact that $\mathcal {J}_\sigma$ is even and continuous on $\mathbb{X}_{rad}(\mathbb{R}_+^{N+1}),$ gives that $\mathcal {J}_\sigma^{-\epsilon}$ is closed and symmetric. Consequently, the following lemma is true and its proof is the same as that of Lemma 3.2 in ^[28].

Lemma 3.5. Given $k\in \mathbb{N}$, there exist $\epsilon_k: = \epsilon(k)$ and $\vartheta_k: = \vartheta(k)$ such that whenever $0 < \epsilon\leq \epsilon_k$ and $\vartheta\geq \vartheta_k, \gamma(\mathcal {J}_\sigma^{-\epsilon})\geq k.$

Set

and

for all $k\in E$ by Lemma 3.4 (b). In order to verify Theorem 1.1, we given by

Therefore, we obtain that the following result holds.

Lemma 3.6. If $c = c_k = c_{k+1} = \cdots = c_{k+m},$ then $\gamma(\mathcal {K}_c)\geq m+1.$ Especially, $\mathcal {J}_\sigma$ has at least $m+1$ nontrivial critical points.

Proof. For $\epsilon > 0,$ we know that $\mathcal {J}_\sigma^{-\epsilon}\in \Sigma.$ With the help of Lemma 3.5, for any $k\in\mathbb{N}$, there exists $\epsilon_k = \epsilon(k) > 0$ and $\vartheta_k = \vartheta(k)$ such that if $0 < \epsilon\leq \epsilon_k$ and $\vartheta\geq \vartheta_k,$ we have $\gamma(\mathcal {J}_\sigma^{-\epsilon})\geq k.$ Therefore, $\mathcal {J}_\sigma^{-\epsilon_k}\in \Sigma_k,$ and

Let $0 > c = c_k = c_{k+1} = \cdots = c_{k+m}$ are satisfied. Therefore, Lemma 3.4 (c) shows that $\mathcal {J}_\sigma$ satisfies the $(PS)_c$ condition. Consequently, $\mathcal {K}_c$ is a compact set. Theorem 2.1 in ^[39] yields that $\mathcal {J}_\sigma|_{\mathcal {S}(a)}$ has at least $m + 1$ critical points. □

Proof of Theorem 1.1. By Lemma 3.4 $(b)$ the critical points of $\mathcal {J}_\sigma$ obtained in Lemma 3.6 are the critical points of $\mathcal {J}$. Hence, we complete the proof.

Use of AI tools declaration

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

Acknowledgments

Sihua Liang is supported by the Science and Technology Development Plan Project of Jilin Province, China (Grant No. YDZJ202201ZYTS582), the Research Foundation of Department of Education of Jilin Province (Grant No. JJKH20230902KJ) and Innovation and Entrepreneurship Talent Funding Project of Jilin Province (No.2023QN21).

Conflict of interest

The authors declare there is no conflict of interest.