Normalized solutions for the mixed dispersion nonlinear Schrödinger equations with four types of potentials and mass subcritical growth

1.   Introduction

Over the past several decades, the mixed dispersion nonlinear Schrödinger equation

has been studied by many researchers. Biharmonic Schrödinger equations have played an important role in considering the small biharmonic dispersion terms in the transmission of intense laser beams in a bulk medium with Kerr nonlinearity; see ^[1,2]. Biharmonic Schrödinger equations are also important in depicting the motion of a vortex filament in an incompressible fluid; see ^[3]. Since then, biharmonic Schrödinger equations have received attention due to whose applications in physics.

An interesting topic is to study the standing waves of Eq (1.1). By applying the ansatz $\psi(t, x) = e^{i\lambda t}u(x)$, Eq (1.1) yields the following equation:

where $\gamma > 0$, $\lambda\in \mathbb{R}$, $\beta\in \mathbb{R}$, and $u:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a function which does not rely on time. Moreover, if $u(x)$ is a solution to (1.2), we can obtain that $\psi (t, x) = e^{i\lambda t}u(x)$ is a solution to (1.1).

Above all, when $\gamma = 0$, $\beta = 1$ and $V(x) \equiv 0$, we consider the existence of the $L^2$-constraint variational problem

When $f(u) = \vert u\vert^{2\sigma} u \, (0 < \sigma < \frac{2}{N})$, by assuming $H^1$-precompactness of any minimizing sequences, Cazenave and Lions ^[4] obtained the existence of the $L^2$-constraint minimization problem. To this end, the subadditivity assumption

is very crucial. Due to the assumption (1.3), we can eliminate the dichotomy of minimizing sequences.

If only $V(x)\equiv0$, many papers are dedicated to this equation:

Bonheure et al. considered a mixed dispersion nonlinear Schrödinger equation in ^[5]. More precisely, they studied the existence of the ground states and positive solutions. They also studied the multiplicity of radial solutions and the standing waves of the related dispersive equation. Recently, Goubet and Manoubi ^[6] studied semilinear Schrödinger equations with a non-standard dispersion that is discontinuous at $x = 0$. They obtained both the existence and the uniqueness of standing waves for these equations. Then, they discussed the orbital stability of these standing waves in a subspace of the energy space, by using some classical methods such as the concentration-compactness method of Lions. In ^[7], Khiddi and Essafi obtained the existence of infinitely many solutions for a class of quasilinear Schrödinger equations without assuming the 4-superlinear at infinity on the nonlinearity. The approach is based on the fountain theorem, and the involved potential term is continuous and satisfies suitable regularities. In ^[8], Alotaibi et al. studied both the existence and nonexistence of global weak solutions to a class of inhomogeneous nonlinear Schrödinger equations. The main problem is related to gradient, which requires certain specific estimates to develop the precise proofs of results. The approach is based on rescaled test function arguments derived from the Mitidieri and Pokhozhaev method, and it also involves the Fujita critical exponent. In ^[9], Bonheure et al. studied two related constraint minimization problems: One is related to a constraint on the $L^2$-norm, and another one is related to a constraint on the $L^2\sigma+2$-norm. They also studied the attainability and the qualitative properties of minimizers, namely, their sign, symmetry, decay and so on. In ^[10], Fernández et al. established non-homogeneous Gagliardo-Nirenberg-type inequalities depending on the Tomas-Stein inequality. They proved the attainability of minimizers in the mass-subcritical and mass-critical cases. For more research about the biharmonic Schrödinger equations, see ^{[11,12,13,14,15]} and the references therein.

Usually, if $V(x) \equiv 0$, the scaling $u(sx)$ is useful, and we can show (1.3). However, when $V(x)\not\equiv 0$ the scaling $u(sx)$ does not work generally, and it is harder to show the subadditivity condition. Therefore, the $L^2$-constraint minimization problem is hard. Just because of this, the solutions to the problem would not be enough. For the biharmonic Schrödinger equations with a potential, see ^[16] and the references therein.

Although the biharmonic nonlinear Schrödinger equations are related to physics, they are far from being properly understood. The nonlinear Schrödinger equations have been studied in ^{[17,18,19,20,21]}, but the fourth order Schrödinger equations have been studied very little. Apart from some papers already mentioned, there are actually few papers dealing with biharmonic nonlinear Schrödinger equations.

Inspired by the past work of ^[22,23,24], in this paper, we consider the attainability of minimizers of the $L^2$-constraint variational problem:

where

$\gamma > 0$, $a > 0$, $\sigma\in(0, \frac{2}{N})$ with $N\ge 2$ and the continuous bounded function $V:\mathbb{R}^{N}\rightarrow [0, +\infty)$. Here, we consider four functions:

$(V_1)$ $V$ is a function that is 1-periodic in $x_1, x_2, \cdots, x_N$.

$(V_2)$ $V$ is an asymptotically periodic function. Namely there exists a function $V_q : \mathbb{R}^{N}\rightarrow \mathbb{R}$ which is 1-periodic in $x_1, x_2, \cdots, x_N$, and $V$ satisfies the following conditions:

$(V_3)$ $V\in L^{\infty}(\mathbb{R}^{N})$, and

$(V_4)$ Suppose that $\mu W(x) = V(x)$ and a constant $M_1 > 0$ such that

Moreover, $\Omega = \text{int}(W^{-1}(0)) \neq \emptyset$.

Next, we describe the first result of this paper.

Theorem 1.1. Let $\gamma > 0$, $\sigma\in(0, \frac{2}{N})$ and assume that $(V_1)$ holds or $(V_2)$ holds. There exists a constant $\delta(a) > 0$ for any $a > 0$, and if $\vert V\vert_\infty < \delta$ when V satisfies $(V_1)$, or $\vert V_q\vert_\infty < \delta$ when V satisfies $(V_2)$, $m_{\gamma, a} < 0$ is achieved.

Our second result is combined with the $L^2$-constraint variational problem:

where

$\gamma > 0$, $a > 0$, $\varepsilon > 0$ are real numbers, and $\sigma\in(0, \frac{2}{N})$ with $N\ge 2$.

The second result is as follows.

Theorem 1.2. Let $\gamma > 0$, $\sigma\in(0, \frac{2}{N})$ and assume that $(V_3)$ holds. Then, there exist two constants $\delta(a)$, $\varepsilon_0 > 0$ for any $a > 0$, and if $\vert V\vert_{\infty} < \delta$, $m_{\gamma, a, \varepsilon} < 0$ is achieved for any $\varepsilon\in(0, \varepsilon_0)$.

In $(V_4)$, we choose $r > 0$ such that $B_r(x_1)\subset\Omega$. We study a constrained variational problem:

where

and $\gamma > 0$, $a > 0$, $\sigma\in(0, \frac{2}{N})$ with $N\ge 2$. Finally, we describe the third main result.

Theorem 1.3. Let $\gamma > 0, \sigma\in(0, \frac{2}{N})$ and assume that $V$ satisfies $(V_4)$. Then, there exist two constants $r_0(a)$, $\mu_0(a) > 0$ for any $a > 0$ such that $m_{\gamma, a, \mu} < 0$ is achieved for any $\mu\ge \mu_0$, $r\ge r_0$.

Notation

● $C, C_1, C_2, \ldots$ represent positive constants, and they are independent of each other.

● $B_{r}(y)$ represents an open ball centered at $y\in\mathbb{R}^N$ with radius $r > 0$, $B^{c}_{r}(y)$ represents its complement in $\mathbb{R}^{N}$.

● $\Vert \cdot \Vert$ represents the common norm of the Sobolev space $H^{2}(\mathbb{R}^{N})$, and $\vert\cdot \vert_{p}$ represents the common norm of the Lebesgue space $L^{p}(\mathbb{R}^{N})$, for $p\in[1, \infty]$.

● $o_n(1)$ represents a real number sequence with $o_n(1)\rightarrow 0$ as $n\rightarrow +\infty$.

2.   Variational framework and some preliminaries

In the following, we study the functional $J_{\gamma}:E\rightarrow \mathbb{R}$, namely,

constrained on the sphere in $L^2(\mathbb{R}^{N})$ given by

where $\gamma > 0$, and the continuous function $V:\mathbb{R}^{N}\rightarrow [0, +\infty)$. $E$ is described as the space

and the norm of $E$ is given by

We can infer $E = H^2(\mathbb{R}^{N})$ if $V\in L^{\infty}(\mathbb{R}^{N})$.

According to the definition of $E$, it is obvious that the embedding $E\hookrightarrow H^2(\mathbb{R}^{N})$ is continuous. The following embeddings

and

are continuous, too.

In addition, we introduce two Gagliardo-Nirenberg interpolation inequalities; see ^[25,26,27]. When the function $u\in H^2(\mathbb{R}^{N})$, we have

where

and

where

the constants $B = B_N(\sigma) > 0$ and $C = C_N(\sigma) > 0$. Therefore, we have

and

Since $\sigma\in(0, \frac{2}{N})$, we know that $\sigma N < 2$. Hence, $J_{\gamma}$ is bounded from below on $S(a)$ for any $a > 0$, $\gamma > 0$. Relying on the above arguments, we infer that

is well-defined.

Lemma 2.1. Let $V\in L^{\infty}(\mathbb{R}^{N})$ and $\gamma > 0$. There exists a constant $\delta(a) > 0$ for each $a > 0$ such that $m_{\gamma, a} < 0$ when $\vert V\vert_{\infty} < {\delta}$.

Proof. We choose $u_1\in S(a)$ for every $a > 0$, and set

By calculation, we have

and

Therefore, we infer that

As $\sigma\in(0, \frac{2}{N})$, there exists a constant $k < 0$ such that

Now, we choose fixed $\delta = \frac{-D_k}{a^2}$ and consider $\vert V\vert_\infty < \delta$, and we have

which shows $m_{\gamma, a} < 0$.

Lemma 2.2. Let $\gamma > 0$, and there are $x_1\in\mathbb{R}^{N}$, $r > 0$ and

Then, there exists a constant $r_0 > 0$ that does not rely on $\mu$ in $(V_4)$ and such that $m_{\gamma, a} < 0$ for any $r\ge r_0$.

Proof. We choose $u_1\in S(a)\cap C_0^{\infty}(\mathbb{R}^{N})$, $x_1\in\mathbb{R}^{N}$ with $V(x) = 0$ for any $x\in B_r (x_1)$ and set

By calculation we have

and

which lead to

As $\sigma\in(0, \frac{2}{N})$, there exists a constant $k < 0$ such that

Now, we can choose $M_1 = \text{sup} \{\vert x\vert \, : x\in\text{supp}(u_1) \},$ $r_0 = e^{-k} M_1$ for each $r\ge r_0 > 0$. Then, it is easy to deduce that

Hence,

which shows $m_{\gamma, a} < 0.$

Lemma 2.3. Let the conditions of Lemma 2.1 hold or Lemma 2.2 hold. When $0 < a_2 < a_1$, then $\frac{a_2^2}{a_1^2} m_{\gamma, a_1} < m_{\gamma, a_2} < 0$.

Proof. We set $\omega > 1$ such that $a_1 = \omega a_2$ and choose a minimizing sequence $(u_n)\subset S(a_2)$ with respect to $m_{\gamma, a_2}$, namely,

Let $\tilde{u}_n = \omega u_n$, and it is easy to see $\tilde{u}_n \in S(a_1)$. Then

Claim 2.4. There exist two constants $C_1 > 0$ and $n_1 \in \mathbb{N}$ such that $\int_{\mathbb{R}^{N}} \vert u_n \vert^{2\sigma+2} dx \ge C_1$ for any $n\ge n_1$.

If not, we can infer that

and if necessary we can choose a subsequence. Let us recall that

which is a contradiction. Hence, the proof of Claim 2.4 is completed.

Applying Claim 2.4, $\omega^2 - \omega^{2\sigma+2} < 0$, we can get that for $n\in \mathbb{N}$ big enough

Taking the limit $n\rightarrow + \infty$, we have

namely,

and the proof of the Lemma 2.3 is completed.

Lemma 2.5. Suppose that there exists a minimizing sequence $(u_n)\subset S(a)$ with respect to $m_{\gamma, a}$ such that $u\neq 0$, $u_n(x)\rightarrow u(x)$ a.e. in $\mathbb{R}^{N}$, and $u_n\rightharpoonup u$ in $H^2 (\mathbb{R}^{N})$. Then, $u_n\rightarrow u$ in $H^2 (\mathbb{R}^{N})$, $J_{\gamma}(u) = m_{\gamma, a}$ and $u\in S(a)$.

Proof. Actually, if not, we can obtain that $\vert u\vert_2 = c \neq a$. Relying on $u\neq 0$ and Fatou's lemma, it is easy to infer that $c\in(0, a)$. Applying Brezis-Lieb lemma (see[28,Lemma 1.32]), we can obtain

and

Let $\tilde{u}_n = u_n -u$, $b_n = \vert \tilde{u}_n\vert_2$, and assume $\vert \tilde{u}_n\vert_2 \rightarrow b$. We infer $a^2 = c^2 + b^2$ and $b_n \in (0, a)$ for $n$ big enough. Moreover, by using Lemma 2.3, we have

Taking the limit $n\rightarrow +\infty$, we obtain that

As $c\in(0, a)$, we can apply the Lemma 2.3 in (2.8), and it is easy to obtain this inequality:

We get a contradiction, which implies $\vert u\vert_2 = a$, namely, $u\in S(a)$.

Since $\vert u_n \vert_2 = \vert u\vert_2 = a$, $u_n \rightharpoonup u$ in $L^2(\mathbb{R}^{N})$,

Using the interpolation theorem in the Lebesgue space, it is easy to obtain that

Moreover, as $\int_{\mathbb{R}^{N}} \gamma\vert \Delta u \vert^2 + \vert \nabla u \vert^2 + V(x)\vert u \vert^{2} dx$ is convex and continuous in $H^2(\mathbb{R}^{N})$, this functional is weakly lower semicontinuous, namely,

The above limit together with $m_{\gamma, a} = \lim\limits_{n\rightarrow +\infty} J_{\gamma}(u_n)$ shows that

Since $u\in S(a)$, $J_{\gamma}(u) = m_{\gamma, a}$, and $J_{\gamma}(u_n)\rightarrow J_{\gamma}(u)$. Since $u_n\rightarrow u$ in $L^{2\sigma+2}(\mathbb{R}^{N})$, we infer $u_n \rightarrow u$ in $H^2(\mathbb{R}^{N})$.

3.   Periodic case

We suppose that $V$ satisfies $V_1$ and set a minimizing sequence $u_n \subset S(a)$ with respect to $m_{\gamma, a}$, namely,

As $\sigma\in(0, \frac{2}{N})$, the above limit combined with (2.3) and (2.4) ensures $(\vert\Delta u_n\vert_2)$ and $(\vert\nabla u_n\vert_2)$ are bounded sequences. Hence, $(u_n)$ is a bounded sequence in $H^2(\mathbb{R}^{N})$. Furthermore, there exist a subsequence of $(u_n)$, still represented by $(u_n)$, and $u\in H^2(\mathbb{R}^{N})$ such that

and

As the discussion of Claim 2.4, there is a constant $C_1 > 0$, and the following inequality is established

Lemma 3.1. If $(u_n)\subset S(a)$ is a minimizing sequence, then $(u_n)$ can be chosen to be a new minimizing sequence $\tilde{u}_n\subset S(a)$ such that $\tilde{u}_n \rightharpoonup \tilde{u}$ and $\tilde{u} \neq 0$.

Proof. Above all, there exist $\beta > 0$, $R > 0$ and $y_n \in \mathbb{R}^{N}$ such that

If not, we can infer that $u_n \rightarrow 0$ in $L^{2\sigma+2}(\mathbb{R}^{N})$, which is a contradiction. A brief discussion implies that we can suppose $y_n \in \mathbb{R}^{N}$ and $R > 0$ big enough in (3.2). Then, setting $\tilde{u}_n(x) = u(x+y_n)$, we infer $(\tilde{u}_n)\subset S(a)$ and $(\tilde{u}_n)$ is also a minimizing sequence with respect to $m_{\gamma, a}$. Furthermore, there exists $\tilde{u}\in H^2(\mathbb{R}^{N})\backslash\{0\}$ such that

and

The proof is completed.

3.1.   Proof of Theorem 1.1 (Part I)

Proof. Using Lemma 3.1, we can get a bounded minimizing sequence $(u_n)\subset S(a)$ with respect to $m_{\gamma, a}$ and its weak limit $u \neq 0$. Now, by Lemma 2.5, it is easy to infer $u\in S(a)$, $J_{\gamma}(u) = m_{\gamma, a}$ and $u_n\rightarrow u$ in $H^2(\mathbb{R}^{N}).$ Hence, by the Lagrange multiplier, there exists a constant $\lambda(a)\in \mathbb{R}$ such that

where $\Psi:H^2(\mathbb{R}^{N}) \rightarrow \mathbb{R}$, and

According to (3.3),

Therefore, the proof is completed when $V$ satisfies $(V_1)$.

4.   Asymptotically periodic case

In this section, let $V\not\equiv V_q$. Therefore, there exists a measurable set $\Omega\subset\mathbb{R}^{N}$ with $\vert \Omega\vert > 0$ and

Then, let us represent by $J_{\gamma, q} :H^2(\mathbb{R}^{N})\rightarrow \mathbb{R}$ the functional

and the constant

Depending on the conditions of Theorem 1.1, as the discussion in Section 3, there exists $u_p \in S(a)$ such that $J_{\gamma, p}(u_p) = m_{\gamma, a, p}$. Furthermore, by (4.1), we can infer

Now, we choose a minimizing sequence $(u_n)\subset S(a)$ with respect to $m_{\gamma, a}$, namely,

Since $\sigma\in(0, \frac{2}{N})$, as the discussion in Section 2, sequence $(u_n)$ is bounded in $H^2(\mathbb{R}^{N})$. Therefore, there exist $u\in H^2(\mathbb{R}^{N})$ and a subsequence of $(u_n)$, still represented by $(u_n)$, such that

and

Lemma 4.1. If $u$ is the weak limit of $(u_n)\subset S(a)$, then $u \neq 0$.

Proof. If not, we have that $u_n\rightharpoonup 0$ in $H^2(\mathbb{R}^{N})$. Then,

which leads to

Since $u_n\rightarrow 0$ in $L^2_{loc}(\mathbb{R}^{N})$, the condition (1.6) shows

Let $n\rightarrow +\infty$ in (4.3), and applying the above limit, we get that

We get a conclusion that contradicts (4.2). Hence, $u$ is nontrivial.

4.1.   Proof of Theorem 1.1 (Part II)

Relying on Lemma 4.1, first of all, we have a minimizing sequence $(u_n)\subset S(a)$ with respect to $m_{\gamma, a}$ and its weak limit $u\neq 0$. By using Lemma 2.5, we can obtain $u_n\rightarrow u$ in $H^2(\mathbb{R}^{N})$, $u\in S(a)$, $J_{\gamma}(u) = m_{\gamma, a}$. The rest is similar to Theorem 1.1 when $V$ satisfies $(V_1)$, and we do not repeat it.

5.   Proof of Theorem 1.2

In this section, let us represent by $J_{\gamma, \infty}, J_{\gamma, 0}, :H^2(\mathbb{R}^{N})\rightarrow \mathbb{R}$ these functionals:

and

Furthermore, let us represent by $m_{\gamma, a, \infty}$, $m_{\gamma, a, 0}$ these constants:

and

According to the conditions of Theorem 1.2, we can get a constant $\delta(a) > 0$ for any $a > 0$, and $\vert V\vert_\infty < \delta$. Hence, by $(V_3)$, we have $V_0 < \delta$ and $V_{\infty} < \delta$. Depending on Section 3, we can get two functions $u_0, u_{\infty}\in S(a)$ and $J_{\gamma, \infty}(u_{\infty}) = m_{\gamma, a, \infty}$, $J_{\gamma, 0}(u_0) = m_{\gamma, a, 0}$. Moreover, from Lemma 2.1 and (1.7),

Lemma 5.1. $m_{\gamma, a, 0}\ge \limsup\limits_{\varepsilon\rightarrow 0^+}m_{\gamma, a, \varepsilon}$.

Proof. In the following, let $x_1\in \mathbb{R}^{N}$ such that

and $v_{\varepsilon}(x) = u_1(x-\frac{x_1}{\varepsilon}).$ So, $v_{\varepsilon}\in S(a)$, and

Taking the limit $\varepsilon\rightarrow 0^+$, we have

From Lemma 5.1 and (5.1), there exists $\varepsilon_0 > 0$ such that

We choose a minimizing sequence $(u_n)\subset S (a)$ with respect to $m_{\gamma, a, \varepsilon}$, namely,

As $\sigma\in(0, \frac{2}{N})$, the above limit combined with (2.3) and (2.4) ensures $(\vert\Delta u_n\vert_2)$ and $(\vert\nabla u_n\vert_2)$ are bounded sequences, from where we infer that $(u_n)$ is bounded in $H^2(\mathbb{R}^{N})$. Therefore, there exist a function $u\in H^2(\mathbb{R}^{N})$ and a subsequence of $(u_n)$, still represented by $(u_n)$, such that

and

Lemma 5.2. When $\varepsilon\in(0, \varepsilon_0)$, $u\neq 0$, where $u$ is the weak limit of $(u_n)$.

Proof. If not, we have $u = 0$. Then,

Relying on $(V_3)$, for each given $\eta > 0$, there exist a constant $R > 0$ and

Therefore,

We recall that $(u_n)$ is bounded in $H^2(\mathbb{R}^{N})$, $u_n\rightarrow 0$ in $L^2(B_{R/\varepsilon} (0))$. Hence, can infer that

where $C_1 > 0$. As $\eta > 0$ is arbitrary, we have

and we get a conclusion that contradicts (5.2). Hence, $u\neq 0$ when $\varepsilon\in(0, \varepsilon_0)$.

5.1.   Proof of Theorem 1.2

Depending on Lemma 5.2, we can choose $(u_n)\subset S(a)$ which is a minimizing sequence with respect to $m_{\gamma, a, \varepsilon}$ such that $u_n\rightharpoonup u$ and $u\neq 0$. Moreover, using Lemma 2.5, it follows that $u_n\rightarrow u$ in $H^2(\mathbb{R}^{N})$, $u\in S(a)$, and $J_{\gamma, \varepsilon}(u) = m_{\gamma, a, \varepsilon}$.

6.   Proof of Theorem 1.3

The following part is dedicated to studying the attainability of minimizers when $V$ satisfies $(V_4)$. We can choose a minimizing sequence $(u_n)\subset S(a)$ with respect to $m_{\gamma, a, \mu}$, namely,

Since $\sigma\in(0, \frac{2}{N})$, arguing as the discussion in Section 2, we infer that $(u_n)$ is bounded in $E$. Therefore, we can get $u\in E$ and a subsequence of $(u_n)$, still represented by $(u_n)$, such that

and

Lemma 6.1. There exist two constants $C_1$, $r_0 > 0$, which do not rely on $\mu > 0$, and

Proof. Set $u_1 \in S(a)\cap C_0^{\infty} (\mathbb{R}^{N})$ with supp$(u_1)\subset\Omega$, where $\Omega = \text{int}(W^{-1}(0)) \neq \emptyset$, $x_1\in\Omega$, and

By calculation, we have

and

Therefore,

As $\sigma\in(0, \frac{2}{N}),$ there exists a constant $k < 0$ such that

Now, we choose $M_1 = \text{sup} \{ \vert x\vert :x\in\text{supp}(u_1)\}$ and $r_0 = e^{-k}M_1$. If $r\ge r_0 > 0$, and then

Hence,

It shows there is a constant $C_1 > 0$ that does not rely on $\mu$ and $m_{\gamma, a, \mu} < -C_1$ when $\mu > 0$. Let us recall

Then, we have

and

where $C_1$ is independent of $\mu$.

Lemma 6.2. There exist two constants $\mu_0 > 0$, $R > 0$, and when $\mu \ge \mu_0 > 0$, this inequality is established:

where $C_1 > 0$ is given in Lemma 6.1.

Proof. Actually, as the discussion in [29,Lemma 2.5], for each $\varepsilon > 0$, there exist $R > 0$ and $\mu_0 > 0$ such that

Now, apply that the result $(u_n)$ is bounded in $L^{2^*}(\mathbb{R}^{N})$ by a constant, and the constant is independent of $\mu$. In the following, we can use interpolation theorem of Lebesgue spaces and the proper $\varepsilon > 0$ to get the expected result.

Lemma 6.3. When $\mu\ge \mu_0 > 0$, $u\neq 0$, where $u$ is the weak limit of $(u_n)$.

Proof. If not, we have that $u = 0$ for some $\mu\ge \mu_0$. As $u_n\rightarrow 0$ in $L^{2\sigma+2}(B_R(0))$ for each $R > 0$, according to Lemmas 6.1 and 6.2 we have

We get a contradiction, which implies that there exists a constant $\mu_0 > 0$ such that $u\neq0$ when $\mu\ge \mu_0$.

6.1.   Proof of Theorem 1.3

According to Lemma 6.3, we can get $(u_n)\subset S(a)$ which is a minimizing sequence with respect to $m_{\gamma, a, \mu}$ such that $u_n\rightharpoonup u$ and $u\neq0$. Moreover, we can use Lemma 2.5, and then, $u_n\rightarrow u$ in $E$, $u\in S(a)$ and $J_{\gamma, \mu}(u) = m_{\gamma, a, \mu}$.

Acknowledgments

The author would like to thank the anonymous referee for many valuable comments to improve the manuscript.

Conflict of interest

The author declares there is no conflict of interest.