Ground state solution for the Schrödinger equation with Hardy potential and critical Sobolev exponent

Abstract

In this paper, we consider the following Schrödinger equation $\begin{matrix} (0.1) & \{\begin{matrix} - Δ u - μ \frac{u}{| x |^{2}} + V (x) u = K (x) | u |^{2^{*} - 2} u + f (x, u), x \in R^{N}, \\ u \in H^{1} (R^{N}), \end{matrix} \end{matrix}$ where $N ⩾ 4$ , $0 ⩽ μ < \overline{μ}$ , $\overline{μ} = \frac{{(N - 2)}^{2}}{4}$ , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

Keywords

Schrödinger equation generalized Nehari manifold asymptotically periodic

1. Introduction and main result

In this paper, we consider the following Schrödinger equation $\begin{matrix} (1.1) & \{\begin{matrix} - Δ u - μ \frac{u}{| x |^{2}} + V (x) u = K (x) | u |^{2^{*} - 2} u + f (x, u), x \in R^{N}, \\ u \in H^{1} (R^{N}), \end{matrix} \end{matrix}$ where $N ⩾ 4$ , $0 ⩽ μ < \overline{μ}$ , $\overline{μ} = \frac{{(N - 2)}^{2}}{4}$ , V is periodic in x and 0 lies in a spectral gap of $- Δ - \frac{μ}{| x |^{2}} + V$ , K and f are asymptotically periodic in x.

The existence of solutions of the problems related to (1.1) has been studied extensively. The Hardy potential is critical in nonrelativistic quantum mechanics, as it represent an intermediate threshold between regular potentials and singular potentials, for more details see [11]. The ground state solution is a nontrivial solution with least energy, which has a certain stability, and then aroused a lot of authors’ interest. When $μ = 0$ , $K (x) \equiv 0$ , there are a lot of works on the existence of nontrivial solutions and ground state solutions for problem (1.1) with periodic potential, see [2, 3, 7, 15]. When $μ = 0$ , $K (x) > 0$ , [22] proved that (1.1) has a ground state solution depending on the spectrum of the operator $- Δ + V$ in $L^{2} (R^{N})$ and asymptotically potential. When $μ \neq 0$ , $f (x, u) \equiv 0$ , the ground state solution of (1.1) is obtained in an open set of $R^{N}$ containing the origin in [14]. When $V (x) = λ$ , $K (x) = 1$ , $f (x, u) \equiv 0$ and in a smooth open bounded domain $Ω \in R^{N}$ , Cao and Yan [5] considered problem (1.1) in the case of $N ⩾ 7$ and $μ \in (0, \frac{{(N - 2)}^{2}}{4} - 4)$ and proved that problem (1.1) possessed infinitely many solutions. Chen and Zou [9] obtained a ground state solution for each fixed $λ > 0$ . Moreover, if $N ⩾ 7$ and $μ \in (0, \frac{{(N - 2)}^{2}}{4} - 4)$ , this problem had infinitely many sign-changing solutions for each fixed $λ > 0$ . When $K (x) = 1$ , $V (x)$ and $f (x, u)$ are under suitable assumptions, Deng, Jin and Peng [10] first gave a representation to the Palais–Smale sequence related to (1.1) and then obtained an existence result of positive solutions of (1.1). When $K (x) = 0$ , [12] considered semilinear (1.1) with asymptotically linear $f (x, u)$ and proved the existence and asymptotical behavior of ground state solutions of Nehari–Pankov type. Some other latest researches on solutions of equations with Hardy potential can be consulted in [1, 18, 20], positive classical solution and sign-changing solutions are certified by the authors.

We are inspired by the above articles and utilizing the method in [22], in which the author discussed semilinear Schrödinger equation with indefinite linear part, therefore we consider a new problem which is different from the past and expand the existing achievements. Let operator $A = - Δ - \frac{μ}{| x |^{2}} + V$ in $L^{2} (R^{N})$ . We only consider that 0 lies in a gap of $σ (A)$ because the else case is too complicated. Let $F$ be the class of functions $h \in L^{\infty} (R^{N})$ such that, for each $κ > 0$ the set ${x \in R^{N} : | h (x) | ⩾ κ}$ has finite Lebesgue measure. For $V, K \in C (R^{N})$ , the following assumptions are also needed on them: $(V)$

V is 1-periodic in each of $x_{1}, x_{2}, \dots, x_{N}$ , $0 \notin σ (- Δ - \frac{μ}{| x |^{2}} + V)$ and $σ (- Δ - \frac{μ}{| x |^{2}} + V) \cap (- \infty, 0) \neq \emptyset$ ;

(K)

There exists $K_{p} \in C (R^{N})$ , 1-periodic in each of $x_{1}, x_{2}, \dots, x_{N}$ , and a point $x_{0} \in R^{N}$ such that $K - K_{p} \in F$ and

$K (x) ⩾ K_{p} (x) > 0$ for all $x \in R^{N}$ ;

$K (x_{0}) = {max}_{R^{N}} K (x)$ , $K (x) - K (x_{0}) = o (| x - x_{0} |^{2})$ as $x \to x_{0}$ , and $V (x_{0}) < 0$ .

Let

{E (λ) : - \infty < λ < + \infty}

be the spectral family of

A

, then for a fixed ν,

E (ν) L^{2} (R^{N})

is the subspace of

L^{2} (R^{N})

corresponding to

λ ⩽ ν

(see [7]). Let

\begin{matrix} X = D (| A |^{\frac{1}{2}}), X^{-} = E (0) E, X^{+} = (id - E (0)) E . \end{matrix}

σ (A) \subset (0, + \infty)

, then

dim X^{-} = 0

, otherwise

X^{-}

is infinite-dimensional.

For any $u \in E$ , we can easily see that $u = u^{+} + u^{-}$ , where $\begin{matrix} u^{-} : = E (0) u \in X^{-}, u^{+} = (id - E (0)) u \in X^{+} . \end{matrix}$ Define an inner product $\begin{matrix} ⟨ u, v ⟩ = {⟨ | A |^{\frac{1}{2}} u, | A |^{\frac{1}{2}} v ⟩}_{L^{2}}, u, v \in X \end{matrix}$ and the corresponding norm $\begin{matrix} ‖ u ‖ = {| | A |^{\frac{1}{2}} u |}_{2}, u \in X, \end{matrix}$ where ${⟨ \cdot, \cdot ⟩}_{L^{2}}$ denotes the inner product of $L^{2} (R^{N})$ and $| \cdot |_{p}$ denotes the norm of $L^{p} (R^{N})$ . By Hardy inequality, $\begin{matrix} \int_{R^{N}} \frac{u^{2}}{| x |^{2}} d x ⩽ \frac{1}{\overline{μ}} \int_{R^{N}} | \nabla u |^{2} d x, \forall u \in X, \end{matrix}$ and note that $0 ⩽ μ < \overline{μ}$ , it is easy to see that $‖ u ‖_{μ} = {(\int_{R^{N}} (| \nabla u |^{2} - μ \frac{u^{2}}{| x |^{2}}) d x)}^{1 / 2}$ is equivalent to the usual norm $‖ u ‖ = {(\int_{R^{N}} | \nabla u |^{2})}^{1 / 2}$ in X. Owing to condition $(V)$ that $X = H^{1} (R^{N})$ with equivalent norm. Therefore, X embeds continuously in $L^{p} (R^{N})$ for all $2 ⩽ p ⩽ 2^{*}$ . In addition, there exists the decomposition $X = X^{+} \oplus X^{-}$ orthogonal with respect to both ${⟨ \cdot, \cdot ⟩}_{L^{2}}$ and $⟨ \cdot, \cdot ⟩$ .

It is well known that the solutions of equation (1.1) are critical points of the functional $\begin{matrix} I (u) = \frac{1}{2} \int_{R^{N}} (| \nabla u |^{2} - μ \frac{u^{2}}{| x |^{2}} + V (x) u^{2}) d x - \frac{1}{2^{*}} \int_{R^{N}} K (x) | u |^{2^{*}} d x - \int_{R^{N}} F (x, u) d x . \end{matrix}$ For $\forall u, v \in X$ , there is $\begin{matrix} (1.2) & \begin{matrix} ⟨ I^{'} (u), v ⟩ & = \int_{R^{N}} (\nabla u \nabla v - μ \frac{u v}{| x |^{2}} + V (x) u v) d x - \int_{R^{N}} K (x) | u |^{2^{*} - 2} u v d x \\ - \int_{R^{N}} f (x, u) v d x . \end{matrix} \end{matrix}$ Furthermore, $\begin{matrix} (1.3) & I (u) = \frac{1}{2} ({‖ u^{+} ‖}^{2} - {‖ u^{-} ‖}^{2}) - \frac{1}{2^{*}} \int_{R^{N}} K (x) | u |^{2^{*}} d x - \int_{R^{N}} F (x, u) d x, \end{matrix}$ and $\begin{matrix} ⟨ I^{'} (u), u ⟩ = {‖ u^{+} ‖}^{2} - {‖ u^{-} ‖}^{2} - \int_{R^{N}} K (x) | u |^{2^{*}} d x - \int_{R^{N}} f (x, u) u d x, \end{matrix}$ for $\forall u = u^{+} + u^{-} \in X$ .

The following assumptions should be made on $f (x, u) \in C (R^{N} \times R, R)$ :

(f 1)

$| f (x, u) | ⩽ a (1 + | u |^{q - 1})$ for some $a > 0$ and $2 < q < 2^{*}$ ;

(f 2)

$f (x, u) = o (u)$ uniformly in x as $u \to 0$ ;

(f 3)

$u \mapsto \frac{f (x, u)}{| u |}$ is nondecreasing on $(- \infty, 0)$ and $(0, \infty)$ ;

(f 4)

there exists a function $f_{p} \in C (R^{N} \times R, R)$ , 1-periodic in each of $x_{1}, x_{2}, \dots, x_{N}$ , such that

$| f (x, u) | ⩾ | f_{p} (x, u) |$ for $\forall (x, u) \in R^{N} \times R$ ;

$| f (x, u) - f_{p} (x, u) | ⩽ | h (x) | (| u | + | u |^{q - 1})$ for $\forall (x, u) \in R^{N} \times R$ , $h \in F$ ;

$u \mapsto \frac{f_{p} (x, u)}{| u |}$ is nondecreasing on $(- \infty, 0)$ and $(0, \infty)$ .

Theorem 1.1.
Suppose that $β > 1$ , $(V)$ , $(K)$ , $(f 1)$ – $(f 4)$ hold, then problem ( 1.1 ) has a ground state solution.

The proof of Theorem 1.1 is mainly based on the generalized Nehari manifold $\begin{matrix} M = {u \in X ∖ X^{-} : ⟨ I^{'} (u), u ⟩ = ⟨ I^{'} (u), v ⟩ = 0, \forall v \in X^{-}}, \end{matrix}$ which is a subset of the Nehari manifold $\begin{matrix} N = {u \in X ∖ {0} : ⟨ I^{'} (u), u ⟩ = 0} . \end{matrix}$ In order to overcome the difficulty of finding a ground state solution, we take advantage of the method similar to [15, 22], which is transforming our problem into that seeking for a minimizer of the functional on generalized Nehari manifold.

The paper is organized as follows. In Section 2, preliminaries are given. In Section 3, the variational framework is on exhibition which is important for the proof. In Section 4, we deal with the non-periodicity of nonlinearity, the main result is obtained.
2. Preliminaries

For $X = H^{1} (R^{N})$ is a is real Hilbert space with the orthogonal decomposition $X = X^{+} \oplus X^{-}$ with the inner product $\begin{array}{l} (u, u) = \int_{R^{N}} (| \nabla u |^{2} - μ \frac{u^{2}}{| x |^{2}} + V (x) u^{2}) d x, \forall u \in X^{+}, \\ - (u, u) = \int_{R^{N}} (| \nabla u |^{2} - μ \frac{u^{2}}{| x |^{2}} + V (x) u^{2}) d x, \forall u \in X^{-} . \end{array}$ A solution $\tilde{u} \in X$ of (1.1) is called a ground state solution if $\begin{matrix} I (\tilde{u}) = min {I (u) : u \in X ∖ {0}, I^{'} (u) = 0} . \end{matrix}$ The limit equation of (1.1) (the corresponding periodic equation of (1.1)) plays a essential role in the process of finding ground state solution for (1.1). The limit equation is defined as follows $\begin{matrix} (2.1) & \{\begin{matrix} - Δ u - μ \frac{u}{| x |^{2}} + V (x) u = K_{p} (x) | u |^{2^{*} - 2} u + f_{p} (x, u), x \in R^{N}, \\ u \in X, \end{matrix} \end{matrix}$ and the corresponding functional is $\begin{matrix} I_{p} (u) = \frac{1}{2} ({‖ u^{+} ‖}^{2} - {‖ u^{-} ‖}^{2}) - \frac{1}{2^{*}} \int_{R^{N}} K_{p} (x) | u |^{2^{*}} d x - \int_{R^{N}} F_{p} (x, u) d x, \end{matrix}$ where $F_{p} (x, u) = \int_{0}^{u} f_{p} (x, s) d s$ .

Set $\begin{array}{l} g (x, u) = K (x) | u |^{2^{*} - 2} u + f (x, u), g_{p} (x, u) = K_{p} (x) | u |^{2^{*} - 2} u + f_{p} (x, u), \\ G (x, u) = \int_{0}^{u} g (x, s) d s, G_{p} (x, u) = \int_{0}^{u} g_{p} (x, s) d s . \end{array}$ The following three lemmas can be seen in [22], which are important to our result.

Lemma 2.1.
Conditions $(f 1)$ and $(f 2)$ imply that for all $ε > 0$ there exists a $a_{ε} > 0$ such that $\begin{matrix} (2.2) & | f (x, u) | ⩽ ε | u | + a_{ε} | u |^{q - 1}, \forall u \in R, \end{matrix}$ for $2 < q < 2^{}$ .

Conditions* $(f 2)$ and $(f 3)$ imply $\begin{matrix} (2.3) & 0 ⩽ 2 F (x, u) ⩽ f (x, u) u, \forall u \in R . \end{matrix}$ Conditions K- $(i)$ , $(f 2)$ and $(f 3)$ imply $\begin{matrix} (2.4) & 0 < 2 G (x, u) < g (x, u) u, \forall u \neq 0 . \end{matrix}$ Conditions $(f 2)$ and $(f 4)$ - $(i)$ imply $\begin{matrix} (2.5) & f_{p} (x, u) = o (u) uniformly in x as u \to 0 . \end{matrix}$ Then together with $(f 4)$ - $(iii)$ yield $\begin{matrix} (2.6) & 0 ⩽ 2 F_{p} (x, u) ⩽ f_{p} (x, u) u, \forall u \in R . \end{matrix}$ Conditions K- $(i)$ , $(f 2)$ and $(f 4)$ yield $\begin{matrix} (2.7) & 0 < 2 G_{p} (x, u) < g_{p} (x, u) u, \forall u \neq 0 . \end{matrix}$ Equations ( 2.3 ), ( 2.6 ) and $(f 4)$ - $(i)$ yield $\begin{matrix} (2.8) & F (x, u) ⩾ F_{p} (x, u), \forall u \in R . \end{matrix}$

For convenience, defining $\begin{matrix} (2.9) & \tilde{G} (x, u) = \frac{1}{2} g (x, u) u - G (x, u), {\tilde{G}}_{p} (x, u) = \frac{1}{2} g_{p} (x, u) u - G_{p} (x, u) . \end{matrix}$ According to ( 2.4 ), ( 2.7 ), there are $\begin{matrix} (2.10) & \tilde{G} (x, u) > 0, {\tilde{G}}_{p} (x, u) > 0, \forall u \neq 0 . \end{matrix}$
Lemma 2.2.
Let $u, v \in R$ , $s \in [0, + \infty)$ and $x \in R^{N}$ . Suppose that K- $(i)$ , $(f 2)$ and $(f 4)$ hold, then $\begin{matrix} g_{p} (x, u) (\frac{s^{2} - 1}{2} u + s v) + G_{p} (x, u) - G_{p} (x, s u + v) ⩽ 0 . \end{matrix}$ For the derivative of the function I, the next lemma is obtained directly.
Lemma 2.3.
Suppose that $(V)$ , $(K)$ , $(f 1)$ – $(f 2)$ hold, then
$I^{'}$ maps bounded sets in X into bounded sets in $X^{}$ ;

$I^{'}$ is weakly sequentially continuous. Namely, if* $u_{n} ⇀ u$ in X, then $I^{'} (u_{n}) ⇀ I^{'} (u)$ in $X^{}$ .

The next remark is easily gained similar to Lemma 2.3.
Remark 2.4.
Because of $(f 1)$ , $(f 2)$ and $(f 4)$ - $(i)$ , it is easily seen that $(f 1)$ , $(f 2)$ also hold with $f_{p}$ instead of f. Therefore, if $(V)$ , $(K)$ , $(f 1)$ , $(f 2)$ and $(f 4)$ - $(i)$ are satisfied, then $I_{p}^{'}$ is weakly sequentially continuous.

The next lemma can be seen in [17]. Lemma 2.5.
Suppose that* $h (x, t)$ is nondecreasing in $t \in R$ and $h (x, 0) = 0$ for any $x \in R^{N}$ , then for $\forall θ ⩾ 0$ , $τ, σ \in R$ , $\begin{matrix} (\frac{1 - θ^{2}}{2} τ - θ σ) h (x, τ) | τ | ⩾ \int_{θ τ + σ}^{τ} h (x, s) | s | d s . \end{matrix}$

3. Variational setting

In this section we depict the variational framework for the research of our problem. Because of our strongly indefinite problem, we will put to use the generalized Nehari manifold to find a ground state solution, which can be explicit seen in [15, 16].

The generalized Nehari manifold M corresponding to I is defined by $\begin{matrix} M = {u \in X ∖ X^{-} : ⟨ I^{'} (u), u ⟩ = ⟨ I^{'} (u), v ⟩ = 0, \forall v \in X^{-}} \end{matrix}$ and the least energy on M is defined by $c = {inf}_{M} I$ .

For all $u \in X ∖ X^{-}$ , define $\begin{matrix} (3.1) & \hat{X} (u) : = X^{-} \oplus R^{+} u = X^{-} \oplus R^{+} u^{+} = \hat{X} (u^{+}), \end{matrix}$ where $R^{+} = [0, + \infty)$ .

Lemma 3.1.
Suppose that $(f 2)$ , $(f 3)$ hold, if $u \in M$ , then for $\forall θ ⩾ 0$ , $w \in X^{-}$ and $w \neq 0$ , $\begin{matrix} I (u) > I (θ u + w), \end{matrix}$ that is u is the unique global maximum of $I |_{\hat{X} (u)}$ .
Proof.
Because of $u \in M$ , $\forall θ ⩾ 0$ , $w \in X^{-}$ and $w \neq 0$ , so that $θ u + w \in \hat{X} (u)$ . For any $x \in R^{N}$ , by direct calculation, $\begin{array}{l} I (u) - I (θ u + w) \\ = \frac{1}{2} \int_{R^{N}} (| \nabla u |^{2} - μ \frac{u^{2}}{| x |^{2}} + V (x) u^{2}) d x - \frac{1}{2^{}} \int_{R^{N}} K (x) | u |^{2^{}} d x - \int_{R^{N}} F (x, u) d x \\ - \frac{1}{2} \int_{R^{N}} ({| \nabla (θ u + w) |}^{2} - μ \frac{{(θ u + w)}^{2}}{| x |^{2}} + V (x) {(θ u + w)}^{2}) d x \\ + \frac{1}{2^{}} \int_{R^{N}} K (x) | θ u + w |^{2^{}} d x + \int_{R^{N}} F (x, θ u + w) d x \\ = \frac{1 - θ^{2}}{2} ⟨ I^{'} (u), u ⟩ + \frac{1}{2} ‖ w ‖^{2} - θ ⟨ I^{'} (u), w ⟩ \\ + \int_{R^{N}} [\frac{1 - θ^{2}}{2} f (x, u) u - θ f (x, u) w - \int_{θ u + w}^{u} f (x, s) d s] \\ + \int_{R^{N}} [\frac{1 - θ^{2}}{2} K (x) | u |^{2^{}} - θ K (x) | u |^{2^{} - 2} u w - \int_{θ u + w}^{u} K (x) | s |^{2^{} - 2} s d s] . \end{array}$ According to $(f 2)$ – $(f 3)$ and Lemma 2.5 that for $\forall θ ⩾ 0$ , $τ, σ \in R$ , $\begin{matrix} (\frac{1 - θ^{2}}{2} τ - θ σ) f (x, τ) ⩾ \int_{θ τ + σ}^{τ} f (x, s) d s, \end{matrix}$ and $\begin{matrix} (\frac{1 - θ^{2}}{2} τ - θ σ) K (x) | τ |^{2^{} - 2} τ ⩾ \int_{θ τ + σ}^{τ} K (x) | s |^{2^{} - 2} s d s . \end{matrix}$ Therefore, for $u \in M$ , $w \neq 0$ , so that $I (u) > I (θ u + w)$ , that means u is the unique global maximum of $I |_{\hat{X} (u)}$ . □
Lemma 3.2.
If $(f 2)$ , $(f 3)$ are satisfied, then there exists* $α > 0$ such that ${inf}_{S_{α}^{+}} I > 0$ , where $S_{α}^{+} : = {u \in X^{+} : ‖ u ‖ = α}$ . Moreover, $c ⩾ {inf}_{S_{α}^{+}} I > 0$ .
Proof.
For equation (2.2), we have $\int_{R^{N}} F (x, u) = o (‖ u ‖^{2})$ as $u \to 0$ . For $u \in X^{+}$ that $I (u) = \frac{1}{2} ‖ u ‖^{2} - \frac{1}{2^{}} \int_{R^{N}} K (x) | u |^{2^{}} d x - \int_{R^{N}} F (x, u) d x$ , if we want ${inf}_{S_{α}^{+}} I > 0$ , we only need a sufficiently small $α > 0$ . According to Lemma 3.1 that for every $u \in M$ , there is $θ ⩾ 0$ such that $θ u \in \hat{X} (u) \cap S_{α}^{+}$ , and then $c ⩾ {inf}_{S_{α}^{+}} I$ . □
Lemma 3.3.
If $U \subset X^{+} ∖ {0}$ is a compact subset, then for every $u \in U$ , there exists $R > 0$ such that $\begin{matrix} I (u) ⩽ 0, \end{matrix}$ on $\hat{X} (u) ∖ B_{R} (0)$ .
Proof.
It is sufficiently to show that $I (t u + v) ⩽ 0$ for $t ⩾ 0$ , $v \in X^{-}$ and $‖ t u + v ‖ ⩾ R$ for large enough $R > 0$ . Arguing indirectly, assume that $e \in X^{+} ∖ {0}$ with $‖ e ‖ = 1$ , then for $u_{n} \in U$ that $u_{n} = t_{n} e$ for $t_{n} ⩾ 0$ . Moreover assume $v_{n} \in X^{-}$ , then $t_{n} e + v_{n} \in \hat{X} (u)$ and $‖ t_{n} e + v_{n} ‖ \to + \infty$ such that $I (t_{n} e + v_{n}) ⩾ 0$ as $n \to + \infty$ . Set $\begin{matrix} w_{n} = \frac{t_{n} e + v_{n}}{‖ t_{n} e + v_{n} ‖} = w_{n}^{-} + τ_{n} e, \end{matrix}$ so that $‖ w_{n}^{-} + τ_{n} e ‖ = 1$ . Passing to subsequence, we may assume that $w_{n} ⇀ w$ in X, $w_{n} \to w$ a.e. on $R^{N}$ , $w_{n}^{-} ⇀ w^{-}$ in $X^{-}$ , $τ_{n} \to τ$ and $\begin{matrix} (3.2) & 0 ⩽ \frac{I (t_{n} e + v_{n})}{‖ t_{n} e + v_{n} ‖^{2}} = \frac{τ_{n}^{2}}{2} - \frac{‖ v_{n}^{-} ‖^{2}}{2} - \frac{1}{2^{}} \int_{R^{N}} K (x) \frac{| t_{n} e + v_{n} |^{2^{}}}{‖ t_{n} e + v_{n} ‖^{2}} d x - \int_{R^{N}} \frac{F (x, t_{n} e + v_{n})}{‖ t_{n} e + v_{n} ‖^{2}} d x . \end{matrix}$ Since $‖ w_{n}^{-} ‖^{2} ⩽ τ_{n}^{2} = 1 - ‖ w_{n}^{-} ‖^{2}$ and therefore $\frac{1}{\sqrt{2}} ⩽ τ_{n} ⩽ 1$ . So for a subsequence, $τ_{n} \to τ > 0$ . Since $w = τ e + w^{-} \neq 0$ and $| t_{n} e + v_{n} | \to \infty$ if $w \neq 0$ , then by the Fatou lemma that $\begin{matrix} \int_{R^{N}} K (x) \frac{| t_{n} e + v_{n} |^{2^{}}}{‖ t_{n} e + v_{n} ‖^{2}} d x = \int_{R^{N}} K (x) | t_{n} e + v_{n} |^{2^{} - 2} w_{n}^{2} d x \to + \infty, \end{matrix}$ which contradicts to (3.2). □
Lemma 3.4.
If $(f 2)$ , $(f 3)$ are satisfied, then
$M \cap \hat{X} (u) \neq \emptyset$ ;

For each $u \in X ∖ X^{-}$ , the set $M \cap \hat{X} (u)$ consist of precisely one point $\hat{m} (u)$ which is the unique global maximum of $I |_{\hat{X} (u)}$ .

Proof.
Since $\hat{X} (u) = \hat{X} (u^{+})$ , we may assume that $u \in X^{+}$ , $‖ u ‖ = 1$ . By Lemma 3.3, there exists $R > 0$ such that $I (u) ⩽ 0$ on $\hat{X} (u) ∖ B_{R} (0)$ . By Lemma 3.2, $I (t u) > 0$ for small $t > 0$ , and since $I ⩽ 0$ on $\hat{X} (u) ∖ B_{R} (0)$ , then $0 < {sup}_{\hat{X} (u)} I < + \infty$ . It is easy to see that I is weakly upper semicontinuous on $\hat{X} (u)$ , therefore $I (u_{0}) = {sup}_{\hat{X} (u)} I$ for some $u_{0} \in \hat{X} (u) ∖ {0}$ . This means $u_{0}$ is a critical point of $I |_{\hat{X} (u)}$ , so $⟨ I^{'} (u_{0}), u_{0} ⟩ = ⟨ I^{'} (u_{0}), v ⟩$ for all $v \in \hat{X} (u)$ . Consequently, $u_{0} \in M \cap \hat{X} (u)$ , as required. □

From Lemma 3.4, we have $\forall v \in \hat{X} (w)$ then $\hat{m} (v) = \hat{m} (w)$ . Define $m : = \hat{m} |_{S_{1}^{+}} : S_{1}^{+} \to M$ , where $S_{1}^{+}$ is given in Lemma 3.2. Then m is a bijection from $S_{1}^{+}$ to M. Thus, we introduce the functional $φ : S_{1}^{+} \to R$ by $φ (u) = I (m (u))$ . The result below shows that the critical points of φ on $S_{1}^{+}$ and those of I on M are corresponded by the mapping m. Thus, looking for critical points of I is equivalent to looking for those of φ on $S_{1}^{+}$ .
Lemma 3.5.
If $(V)$ , $(K)$ , $(f 1)$ – $(f 3)$ are satisfied, then the following result hold:
If ${v_{n}}$ is a PS sequence for φ, then ${m (v_{n})}$ is a PS sequence for I;

${inf}_{S_{1}^{+}} φ = {inf}_{M} I$ . Moreover, if v is a critical point of φ, then $m (v)$ is a nontrivial critical point of I.

A minimizer of $I |_{M}$ is a ground state solution of equation ( 1.1 ).

Proof.
The proof can be seen in [22]. □

From Lemma 3.5(iii), we realize that the issue about to looking for a ground state solution for (1.1) can be transformed into that of finding a minimizer of $I |_{M}$ . In the process of seeking for the minimizer, we mainly demand to conquer the troubles emerged by the critical growth condition and the non-periodicity of the nonlinearity. Below we firstly deal with the former difficulty. As previously mentioned, we shall prove that when the functional level lies in a suitable interval, one can restore the compactness. Letting $S_{μ}$ be the best constant for the Sobolev embedding $D^{1, 2} (R^{N}) ↪ L^{2^{}} (R^{N})$ given by $\begin{matrix} S_{μ} = inf_{u \in D^{1, 2} (R^{N}) ∖ {0}} \frac{\int_{R^{N}} (| \nabla u |^{2} - μ \frac{u^{2}}{| x |^{2}}) d x}{| u |_{2^{}}^{2}} . \end{matrix}$
Lemma 3.6.
If $(V)$ , $(K)$ , $(f 1)$ – $(f 3)$ are satisfied, $c \in (0, c^{})$ , where* $c^{} = \frac{1}{N} | K |_{\infty}^{- \frac{N - 2}{2}} S_{μ}^{\frac{N}{2}}$ , then the* ${(PS)}_{c}$ sequence cannot be vanishing.
Proof.
The proof is similar to [7]. We argue by contradiction. Suppose that ${u_{n}}$ is a ${(PS)}_{c}$ sequence, i.e. satisfying $\begin{matrix} (3.3) & I (u_{n}) \to c, I^{'} (u_{n}) \to 0, \end{matrix}$ and ${u_{n}}$ is vanishing. Firstly, we claim that the ${(PS)}_{c}$ sequence is bounded. By $(f 1)$ and $(f 2)$ that for each $ε > 0$ there exists $C_{ε}$ such that $| f (x, u) | ⩽ ε | u | + C_{ε} | u |^{2^{* - 1}}$ [7]. Note that by (2.3), $\begin{matrix} c + 1 + ‖ u_{n} ‖ ⩾ I (u_{n}) - \frac{1}{2} ⟨ I^{'} (u_{n}), u_{n} ⟩ ⩾ \frac{1}{N} \int_{R^{N}} K (x) | u_{n} |^{2^{}} d x . \end{matrix}$ Since $K (x)$ is bounded below by a positive constant, then $\begin{matrix} (3.4) & | u_{n} |_{2^{}}^{2^{}} ⩽ C ‖ u_{n} ‖ . \end{matrix}$ Using the Hölder and Sobolev inequalities, we obtain for large enough n, $\begin{matrix} {‖ u_{n}^{+} ‖}^{2} & = ⟨ I^{'} (u_{n}), u_{n}^{+} ⟩ + \int_{R^{N}} K (x) | u_{n} |^{2^{} - 2} u_{n} u_{n}^{+} d x + \int_{R^{N}} f (x, u_{n}) u_{n}^{+} d x \\ ⩽ ‖ u_{n}^{+} ‖ + C | u_{n} |_{2^{}}^{2^{} - 1} ‖ u_{n}^{+} ‖ + C (ε ‖ u_{n} ‖ + C_{ε} | u |_{2^{}}^{2^{} - 1}) ‖ u_{n}^{+} ‖ . \end{matrix}$ Hence by (3.4), $\begin{matrix} ‖ u_{n}^{+} ‖ ⩽ C_{1} (ε ‖ u_{n} ‖ + C_{ε} ‖ u_{n} ‖^{\frac{2^{} - 1}{2^{}}}), \end{matrix}$ and a similar inequality holds for $‖ u_{n}^{-} ‖$ . Choosing ε sufficiently small, we see that ${u_{n}}$ must be bounded. For ${u_{n}}$ satisfying (3.3) which is vanishing, namely $\begin{matrix} (3.5) & lim_{n \to \infty} sup_{y \in R^{N}} \int_{B_{1} (y)} u_{n}^{2} d x = 0, \end{matrix}$ then by Lion’s concentration compactness lemma ([19], Lemma 1.21) that $\begin{matrix} (3.6) & u_{n} \to 0 in L^{q} (R^{N}) for 2 < q < 2^{} . \end{matrix}$ Let ${z_{n}}$ be a bounded sequence in X. Since for each $ε > 0$ there is $c_{1} (ε)$ such that $| f (x, u) | ⩽ ε | u | + c_{1} (ε) | u |^{q - 1}$ , then $\begin{matrix} \int_{R^{N}} | f (x, u_{n}) | | z_{n} | d x ⩽ c_{2} (ε) ‖ u_{n} ‖ ‖ z_{n} ‖ + c_{3} (ε) | u_{n} |_{q}^{q - 1} ‖ z_{n} ‖ . \end{matrix}$ Using this and a similar argument for F and we have $\begin{matrix} (3.7) & \int_{R^{N}} f (x, u_{n}) z_{n} d x \to 0, \int_{R^{N}} F (x, u_{n}) d x \to 0 . \end{matrix}$ Therefore, by (3.3) we get $\begin{matrix} (3.8) & c = I (u_{n}) - ⟨ I^{'} (u_{n}), u_{n} ⟩ = \frac{1}{N} \int_{R^{N}} K (x) | u_{n} |^{2^{}} d x + o_{n} (1) . \end{matrix}$ Recall $(E (λ))$ is the spectral family of $A$ in $L^{2} (R^{N})$ . Let $u_{n} = u_{n}^{+} + u_{n}^{-} \in X^{+} \oplus X^{-}$ and $u_{n}^{+} = w_{n} + z_{n}$ , where $w_{n} \in E (ν) L^{2}$ , $z_{n} \in (id - E (ν)) L^{2}$ , $ν > 0$ large enough (to be determined).

By ([7], Proposition 2.4), $w_{n} \in X$ , hence also $z_{n} \in X$ , moreover,there exists $C > 0$ such that $| u_{n}^{-} |_{s} ⩽ C | u_{n}^{-} |_{2}$ , where $s = \frac{2 N}{N - 4}$ if $n > 4$ and s may be taken arbitrarily large if $N = 4$ . Then there exists $\tilde{C} > 0$ such that $| u_{n}^{-} |_{s} ⩽ \tilde{C} ‖ u_{n}^{-} ‖$ . In the same way that $| w_{n}^{-} |_{s} ⩽ C | w_{n}^{-} |_{2} ⩽ \tilde{C} ‖ w_{n}^{-} ‖$ . Also by the boundedness of ${u_{n}}$ so that $| u_{n}^{-} |_{s}$ is still bounded. Let r be such that $\frac{2^{} - 1}{r} + \frac{1}{s} = 1$ , then $2 < r < 2^{}$ (for $N = 4$ , s needs to be larger than 4). Since $u_{n} \to 0$ in $L^{r} (R^{N})$ by (3.6), then similar to (3.7) we have that $\int_{R^{N}} f (x, u_{n}) u_{n}^{-} d x = o_{n} (1)$ . Therefore, according to (3.3) that $\begin{matrix} {‖ u_{n}^{-} ‖}^{2} & = - ⟨ I^{'} (u_{n}), u_{n}^{-} ⟩ - \int_{R^{N}} K (x) | u_{n} |^{2^{} - 2} u_{n} u_{n}^{-} d x - \int_{R^{N}} f (x, u_{n}) u_{n}^{-} d x \\ ⩽ | K |_{\infty} | u_{n} |_{r}^{2^{} - 1} {| u_{n}^{-} |}_{s} + o_{n} (1) \to 0 . \end{matrix}$ Similarly, $\begin{matrix} ‖ w_{n} ‖^{2} = \int_{R^{N}} K (x) | u_{n} |^{2^{} - 2} u_{n} w_{n} d x + o_{n} (1) \to 0 . \end{matrix}$ Hence, $\begin{matrix} (3.9) & u_{n} - z_{n} = w_{n} + u_{n}^{-} \to 0, \end{matrix}$ and therefore $\begin{matrix} (3.10) & \begin{matrix} ‖ z_{n} ‖^{2} & = \int_{R^{N}} (| \nabla z_{n} |^{2} - μ \frac{z_{n}^{2}}{| x |^{2}} + V (x) z_{n}^{2}) d x = \int_{R^{N}} K (x) | u_{n} |^{2^{} - 2} u_{n} z_{n} d x + o_{n} (1) \\ = \int_{R^{N}} K (x) | u_{n} |^{2^{}} d x + o_{n} (1) . \end{matrix} \end{matrix}$ Since $z_{n} \in (id - E (ν)) L^{2} \cap X$ , we have $\int_{R^{N}} (| \nabla z_{n} |^{2} - μ \frac{z_{n}^{2}}{| x |^{2}} + V (x) z_{n}^{2}) d x ⩾ ν | z_{n} |_{2}^{2}$ and for $δ > 0$ $\begin{matrix} δ \int_{R^{N}} (| \nabla z_{n} |^{2} - μ \frac{z_{n}^{2}}{| x |^{2}}) d x ⩾ δ (ν - | V |_{\infty}) | z_{n} |_{2}^{2} ⩾ - \int_{R^{N}} V (x) z_{n}^{2} d x \end{matrix}$ whenever ν is large enough. Therefore, for each $δ > 0$ , we many find sufficiently large $ν > 0$ such that $\begin{matrix} (3.11) & (1 - δ) \int_{R^{N}} (| \nabla z_{n} |^{2} - μ \frac{z_{n}^{2}}{| x |^{2}}) d x ⩽ \int_{R^{N}} (| \nabla z_{n} |^{2} - μ \frac{z_{n}^{2}}{| x |^{2}} + V (x) z_{n}^{2}) d x . \end{matrix}$ Combining (3.9), (3.10) and (3.11) gives $\begin{array}{l} (1 - δ) S_{μ} | K |_{\infty}^{- \frac{2}{2^{}}} {(\int_{R^{N}} K (x) | u_{n} |^{2^{}} d x)}^{\frac{2}{2^{}}} \\ ⩽ (1 - δ) S_{μ} | u_{n} |_{2^{}}^{2} \\ = (1 - δ) S_{μ} | z_{n} |_{2^{}}^{2} + o_{n} (1) ⩽ (1 - δ) \int_{R^{N}} (| \nabla z_{n} |^{2} - μ \frac{z_{n}^{2}}{| x |^{2}}) d x \\ ⩽ \int_{R^{N}} (| \nabla z_{n} |^{2} - μ \frac{z_{n}^{2}}{| x |^{2}} + V (x) z_{n}^{2}) d x + o_{n} (1) = \int_{R^{N}} K (x) | u_{n} |^{2^{}} d x + o_{n} (1) . \end{array}$ Passing to the limit and using (3.8) we obtain that $\begin{matrix} (1 - δ) S_{μ} | K |_{\infty}^{- \frac{2}{2^{}}} {(N c)}^{\frac{2}{2^{}}} ⩽ c N, \end{matrix}$ hence either $c = 0$ or ${(1 - δ)}^{\frac{N}{2}} c^{} ⩽ c < c^{}$ which are both impossible because δ may be chosen arbitrarily small. □

In order to restore the compactness of our problem (1.1), we need to estimate the least energy c in the interval $(0, \frac{1}{N} | K |_{\infty}^{- \frac{N - 2}{2}} S_{μ}^{\frac{N}{2}})$ given in Lemma 3.6. By Lemma 3.2, it suffices to prove that $\begin{matrix} c < c^{} = \frac{1}{N} | K |_{\infty}^{- \frac{N - 2}{2}} S_{μ}^{\frac{N}{2}} . \end{matrix}$

For $0 < μ < \overline{μ}$ , setting $β = \sqrt{\overline{μ} - μ}$ , Catrina and Wang [6] proved that all positive solutions which achieve $S_{μ}$ are of the form $u_{ε} (x) = ε^{\frac{2 - N}{2}} u (\frac{x}{ε})$ , $ε > 0$ , where $\begin{matrix} u (x) = \frac{C}{| x |^{\frac{N - 2}{2} - β} {(1 + | x |^{\frac{4 β}{N - 2}})}^{\frac{N - 2}{2}}} \end{matrix}$ for an appropriate constant $C > 0$ . Let $η \in C_{0}^{\infty} (R^{N}, [0, 1])$ , with $| \nabla η | ⩽ 2$ and $\begin{matrix} η (x) = \{\begin{matrix} 1, & | x | ⩽ \frac{r}{2}, \\ 0, & | x | ⩾ r . \end{matrix} \end{matrix}$ Set $\begin{matrix} U_{ε} (x) = η (x) u_{ε} (x), V_{ε} (x) = \frac{U_{ε} (x)}{| U_{ε} (x) |_{2^{}}}, x \in R^{N}, \end{matrix}$ where $u_{ε} (x)$ and $η (x)$ defined as before. Then we have the following estimate (see [4, 8]and [10]): $\begin{array}{l} \int_{R^{N}} (| \nabla V_{ε} |^{2} - μ \frac{V_{ε}^{2}}{| x |^{2}}) d x = S_{μ} + O (ε^{2 β}); \\ | \nabla V_{ε} |_{1} = O (ε^{β}), | V_{ε} |_{1} = O (ε^{β}); \\ \int_{R^{N}} | V_{ε} |^{2} d x = \{\begin{matrix} O (ε^{2}), & β > 1, \\ O (ε^{2 β} | ln ε |), & β = 1, \\ O (ε^{2 β}), & β < 1, \end{matrix} \\ \int_{R^{N}} | V_{ε} |^{2^{} - 1} d x = \{\begin{matrix} O (ε^{\frac{N - 2}{2}}), & β > \frac{{(N - 2)}^{2}}{2 (N + 2)}, \\ O (ε^{\frac{β (N + 2)}{N - 2}} | ln ε |), & β = \frac{{(N - 2)}^{2}}{2 (N + 2)}, \\ O (ε^{\frac{β (N + 2)}{N - 2}}), & β < \frac{{(N - 2)}^{2}}{2 (N + 2)} . \end{matrix} \end{array}$ The next lemma can be seen in [7].
Lemma 3.7.
If $V \in L^{\infty} (R^{N})$ satisfies $(V)$ , then $| u |_{1, \infty} ⩽ C | u |_{2}$ for some constant $C > 0$ and all $u \in E (0) L^{2}$ .

Let $\begin{matrix} Z_{ε} = X^{-} \oplus R V_{ε} \equiv X^{-} \oplus R V_{ε}^{+} . \end{matrix}$ We may assume without loss of generality that $K (0) = | K |_{\infty}$ and $V (0) < 0$ . Moreover, r in the definition of $V_{ε}$ may be chosen such that $V (x) ⩽ - α$ for some $α > 0$ and all x with $| x | ⩽ r$ .
Lemma 3.8.
If $ε > 0$ is small enough, then ${sup}_{Z_{ε}} I < c^{} = \frac{1}{N} | K |_{\infty}^{- \frac{N - 2}{2}} S_{μ}^{\frac{N}{2}}$ . Moreover,* $c ⩽ {sup}_{M} I < c^{} = \frac{1}{N} | K |_{\infty}^{- \frac{N - 2}{2}} S_{μ}^{\frac{N}{2}}$ .
Proof.
Let $\begin{matrix} J (u) : = \frac{1}{2} ({‖ u^{+} ‖}^{2} - {‖ u^{-} ‖}^{2}) - \frac{1}{2^{}} \int_{R^{N}} K (x) | u |^{2^{}} d x . \end{matrix}$ Since $J (u) ⩾ I (u)$ for all u, it suffices to show that ${sup}_{Z_{ε}} J < \frac{1}{N} | K |_{\infty}^{- \frac{N - 2}{2}} S_{μ}^{\frac{N}{2}}$ .

In what follows we adapt the argument in [7] and on pp. 52–53 in [19]. If $u \neq 0$ , then $\begin{matrix} (3.12) & max_{t ⩾ 0} J (t u) = \frac{1}{N} \frac{{(\int_{R^{N}} (| \nabla u |^{2} - μ \frac{u^{2}}{| x |^{2}} + V (x) u^{2}) d x)}^{\frac{N}{2}}}{{(\int_{R^{N}} K (x) | u |^{2^{}} d x)}^{\frac{N - 2}{2}}}, \end{matrix}$ whenever the integral in the numerator above is positive, and the maximum is 0 otherwise. Let $| u |_{2^{}, K}^{2^{}} : = \int_{R^{N}} K (x) | u |^{2^{}} d x$ . It is easy to see from (3.12) that if $\begin{matrix} (3.13) & m_{ε} : = sup_{u \in Z_{ε}, | u |_{2^{}, K} = 1} \int_{R^{N}} (| \nabla u |^{2} - μ \frac{u^{2}}{| x |^{2}} + V (x) u^{2}) d x < \frac{S}{| K |_{\infty}^{\frac{N - 2}{N}}}, \end{matrix}$ then ${sup}_{Z_{ε}} I < {sup}_{Z_{ε}} J < \frac{1}{N} | K |_{\infty}^{- \frac{N - 2}{2}} S_{μ}^{\frac{N}{2}}$ . So it remains to show (3.13) is satisfying for all small $ε > 0$ .

Now, we need to apply the estimate about $V_{ε}$ . Since $\int_{R^{N}} (| \nabla V_{ε}^{-} |^{2} - μ \frac{{(V_{ε}^{-})}^{2}}{| x |^{2}} + V (x) {(V_{ε}^{-})}^{2}) d x ⩽ 0$ , $\int_{R^{N}} (| \nabla V_{ε}^{-} |^{2} - μ \frac{{(V_{ε}^{-})}^{2}}{| x |^{2}}) d x ⩽ C | V_{ε}^{-} |_{2}^{2} ⩽ C ‖ V_{ε}^{-} ‖^{2} \to 0$ as $ε \to 0$ , therefore $| V_{ε}^{-} |_{2^{}} ⩽ C ‖ V_{ε}^{-} ‖ \to 0$ and $| V_{ε}^{+} |_{2^{}} = 1$ . Suppose that $| u |_{2^{}, K} = 1$ and write $u = u^{-} + s V_{ε} = u^{-} + s V_{ε}^{-} + s V_{ε}^{+}$ , then by the above argument we have $| u^{-} |_{2^{}} ⩽ C$ and $| s | ⩽ C$ for some constant $C > 0$ independent of ε. By Lemma 3.7 and convexity of $| \cdot |_{2^{}, K}$ we obtain $\begin{matrix} (3.14) & \begin{matrix} 1 & = | u |_{2^{}, K}^{2^{}} ⩾ | s V_{ε} |_{2^{}, K}^{2^{}} + 2^{} \int_{R^{N}} {(s V_{ε})}^{2^{} - 1} u^{-} d x \\ ⩾ | s V_{ε} |_{2^{}, K}^{2^{}} - C | V_{ε} |_{2^{} - 1}^{2^{} - 1} {| u^{-} |}_{2} . \end{matrix} \end{matrix}$ Moreover, by Lemma 3.7 again that $\begin{matrix} (3.15) & \begin{matrix} \int_{R^{N}} (\nabla V_{ε} \nabla u^{-} - μ \frac{V_{ε} u^{-}}{| x |^{2}} + V (x) V_{ε} u^{-}) d x & ⩽ C (| \nabla V_{ε} |_{1} + | V_{ε} |_{1}) {| u^{-} |}_{2} \\ = O (ε^{β}) {| u^{-} |}_{2} . \end{matrix} \end{matrix}$ Since $V (x) ⩽ - α < 0$ for $x \in supp V_{ε}$ and $K (x) - K (0) = o (| x |^{2})$ as $x \to 0$ , then $\begin{matrix} (3.16) & \int_{R^{N}} V (x) | V_{ε} |^{2} d x ⩽ \{\begin{matrix} - O (ε^{2}), & β > 1, \\ - O (ε^{2 β} | ln ε |), & β = 1, \\ - O (ε^{2 β}), & β < 1, \end{matrix} \end{matrix}$ and $\begin{matrix} (3.17) & \begin{matrix} | V_{ε} |_{2^{}, K}^{2^{}} & = | K |_{\infty} \int_{R^{N}} V_{ε}^{2^{}} d x + \int_{R^{N}} (K (x) - K (0)) V_{ε}^{2^{}} d x \\ = | K |_{\infty} + o (ε^{2}) . \end{matrix} \end{matrix}$ Let $β > 1$ , applying (3.14), (3.15), (3.16) and (3.17), also the fact $\begin{matrix} - {| u^{-} |}_{2}^{2} + O (ε^{β}) {| u^{-} |}_{2} ⩽ O (ε^{2 β}), \end{matrix}$ so that we obtain $\begin{matrix} m_{ε} ⩽ & - {‖ u^{-} ‖}^{2} + \frac{\int_{R^{N}} (| \nabla V_{ε} |^{2} - μ \frac{V_{ε}^{2}}{| x |^{2}} + V (x) V_{ε}^{2}) d x}{| V_{ε} |_{2^{}, K}^{2}} | s V_{ε} |_{2^{}, K}^{2} + O (ε^{β}) {| u^{-} |}_{2} \\ ⩽ & - C {| u^{-} |}_{2}^{2} + \frac{\int_{R^{N}} (| \nabla V_{ε} |^{2} - μ \frac{V_{ε}^{2}}{| x |^{2}} + V (x) V_{ε}^{2}) d x}{| K |_{\infty}^{\frac{N - 2}{N}} + o (ε^{2})} {(1 + C | V_{ε} |_{2^{} - 1}^{2^{} - 1} {| u^{-} |}_{2})}^{\frac{2}{2^{}}} \\ + O (ε^{β}) {| u^{-} |}_{2} \\ = & - C {| u^{-} |}_{2}^{2} + \frac{S_{μ} + O (ε^{2 β}) - O (ε^{2})}{| K |_{\infty}^{\frac{N - 2}{N}} + o (ε^{2})} + O (ε^{β}) {| u^{-} |}_{2} \\ ⩽ & \frac{S_{μ}}{| K |_{\infty}^{\frac{N - 2}{N}}} - O (ε^{2}) + o (ε^{2}) . \end{matrix}$ Hence, (3.13) holds for all sufficient small ε. Then by Lemma 3.4 (ii) we obtain $c < c^{}$ . □

Now we get through the difficulty caused by the non-periodicity of equation (1.1). Nevertheless, we can not use the invariance of the functional under translation to look for a minimizer as shown in [7] because of the non-periodicity of (1.1). However, the limit equation of (1.1) (i.e. equation (2.1)) is periodic, we may make use of the periodicity of (2.1), the relationship of the functionals and derivatives between equation (1.1) and (2.1) to find the minimizer.
Lemma 3.9.
If $(V)$ , $(K)$ , $(f 1)$ – $(f 4)$ are satisfied, then* $I (u) ⩽ I_{p} (u)$ , for all $u \in X$ .
Proof.
By $(f 2)$ – $(f 4)$ , then (2.8) holds, then the conclusion is easily obtained. □

As in the proof of [13], Lemma 5.1 and [21], Lemma 4.1,we have the following lemmas respectively: Lemma 3.10.
If $(K)$ is satisfied, assume that ${u_{n}} \subset X$ is bounded and $φ_{n} (x) = φ (x - x_{n})$ , where $φ \in X$ and $x_{n} \in R^{N}$ . If $| x_{n} | \to + \infty$ , then $\begin{matrix} \int_{R^{N}} (K (x) - K_{p} (x)) | u_{n} |^{2^{} - 2} u_{n} φ_{n} d x \to 0 . \end{matrix}$
Lemma 3.11.
If $(f 4) - (ii)$ is satisfied, assume that* ${u_{n}} \subset X$ satisfies $u_{n} ⇀ 0$ and $φ_{n} \in X$ is bounded, then $\begin{matrix} \int_{R^{N}} (f (x, u_{n}) - f_{p} (x, u_{n})) φ_{n} d x \to 0 . \end{matrix}$
Remark 3.12.
If $(f 4) - (ii)$ is satisfied, assume that ${u_{n}} \subset X$ satisfies $u_{n} ⇀ 0$ . Note that $\begin{matrix} \int_{R^{N}} (F (x, u_{n}) - F_{p} (x, u_{n})) d x = \int_{R^{N}} \int_{0}^{1} (f (x, t u_{n}) u_{n} - f_{p} (x, t u_{n}) u_{n}) d t d x, \end{matrix}$ then similar to the proof of Lemma 3.11, we have $\begin{matrix} \int_{R^{N}} (F (x, u_{n}) - F_{p} (x, u_{n})) d x \to 0 . \end{matrix}$

4. Proof of Theorem 1.1

We first prove a lemma for the functional $I_{p}$ . Then, by using some techniques for asymptotically periodic equations in [13], we prove Theorem 1.1.

Lemma 4.1.

If $(V)$ , $(K)$ , $(f 1)$ – $(f 4)$ are satisfied, $u \in X ∖ X^{-}$ satisfies that $I_{p}^{'} (u) = 0$ , then u is the global maximum of $I_{p} |_{\hat{X} (u)}$ .

Proof.

Set $z \in \hat{X} (u)$ , then there exists $s ⩾ 0$ and $v \in X^{-}$ such that $z = s u + v$ . Let $\begin{matrix} B (w_{1}, w_{2}) = \int_{R^{N}} (\nabla w_{1} \nabla w_{2} - μ \frac{w_{1} w_{2}}{| x |^{2}} + V (x) w_{1} w_{2}) d x, w_{1}, w_{2} \in X . \end{matrix}$ Then $\begin{matrix} (4.1) & \begin{matrix} I_{p} (z) - I_{p} (u) \\ = \frac{1}{2} [B (s u + v, s u + v) - B (u, u)] + \int_{R^{N}} [G_{p} (x, u) - G_{p} (x, z)] d x \\ = \int_{R^{N}} ({| \nabla (s u + v) |}^{2} - μ \frac{{(s u + v)}^{2}}{| x |^{2}} + V (x) {(s u + v)}^{2}) d x \\ - \int_{R^{N}} (| \nabla u |^{2} - μ \frac{u^{2}}{| x |^{2}} + V (x) u^{2}) d x + \int_{R^{N}} [G_{p} (x, u) - G_{p} (x, z)] d x \\ = - \frac{‖ v ‖^{2}}{2} + B (u, \frac{s^{2} - 1}{2} u + s v) + \int_{R^{N}} [G_{p} (x, u) - G_{p} (x, z)] d x \\ = - \frac{‖ v ‖^{2}}{2} + \int_{R^{N}} g_{p} (x, u) (\frac{s^{2} - 1}{2} u + s v) d x + \int_{R^{N}} [G_{p} (x, u) - G_{p} (x, z)] d x, \end{matrix} \end{matrix}$ here the last equality in the above follows from the fact that $I_{p}^{'} (u) = 0$ and therefore $\begin{matrix} B (u, \frac{s^{2} - 1}{2} u + s v) = \int_{R^{N}} g_{p} (x, u) (\frac{s^{2} - 1}{2} u + s v) d x . \end{matrix}$ According to Lemma 2.2 it follows that $\begin{matrix} \int_{R^{N}} g_{p} (x, u) (\frac{s^{2} - 1}{2} u + s v) d x + \int_{R^{N}} [G_{p} (x, u) - G_{p} (x, z)] d x ⩽ 0 . \end{matrix}$ Then by (4.1) we get $I_{p} (z) ⩽ I_{p} (u)$ . Namely, $I_{p} (u) = {max}_{z \in \hat{X} (u)} I_{p} (z)$ . □

Proof of Theorem 1.1.

By the argument in Section 3, we only need to find a minimizer of I on M.

Assume that $w_{n} \in S_{1}^{+}$ is a minimizing sequence such that $φ (w_{n}) \to {inf}_{S_{1}^{+}} φ$ . By the Ekeland variational principle, we suppose $φ^{'} (w_{n}) \to 0$ . Lemma 3.5(i) implies that $I^{'} (u_{n}) \to 0$ , where $u_{n} = m (w_{n}) \in M$ . Moreover, by Lemma 3.5(ii), we have $I (u_{n}) = φ (w_{n}) \to c$ . By the proof in Lemma 3.6, we get that ${u_{n}}$ is bounded in X. Up to a subsequence, we assume that $u_{n} ⇀ \tilde{u}$ in X, $u_{n} \to \tilde{u}$ in $L_{loc}^{2} (R^{N})$ and $u_{n} (x) \to \tilde{u} (x)$ a.e. on $R^{N}$ . Using Lemma 2.3 (ii), we have $I^{'} (\tilde{u}) = 0$ .

Below we shall prove that if $\tilde{u} \neq 0$ , it is just a minimizer. Otherwise, if $\tilde{u} = 0$ , by concentration compactness lemma and the periodicity of (2.1), we can still find a minimizer. Namely, we distinguish two cases that $\tilde{u} \neq 0$ and $\tilde{u} = 0$ .

Case 1: $\tilde{u} \neq 0$ . Then by (2.10) and the fact that $\tilde{u} \neq 0$ , we have $\begin{matrix} (4.2) & I (\tilde{u}) = I (\tilde{u}) - \frac{1}{2} ⟨ I^{'} (\tilde{u}), \tilde{u} ⟩ = \int_{R^{N}} \tilde{G} (x, \tilde{u}) > 0 . \end{matrix}$ However, $I ⩽ 0$ on $X^{-}$ . Then $\tilde{u} \notin X^{-}$ . So $\tilde{u} \in M$ and $I (\tilde{u}) ⩾ c$ . Note that $\tilde{G} (x, u_{n}) ⩾ 0$ by (2.10). By the Fatou lemma we have $\int_{R^{N}} \tilde{G} (x, \tilde{u}) d x ⩽ lim_{_} \int_{R^{N}} \tilde{G} (x, u_{n}) d x$ . Moreover, $\begin{matrix} (4.3) & c + o_{n} (1) = I (u_{n}) - \frac{1}{2} ⟨ I (u_{n}), u_{n} ⟩ = \int_{R^{N}} \tilde{G} (x, u_{n}) d x \end{matrix}$ and together with (4.2) imply that $I (\tilde{u}) ⩽ c$ . Therefore, $I (\tilde{u}) = c$ .

Case 2: $\tilde{u} = 0$ . This case is more complicated. According to Lemma 3.2, Lemma 3.6 and Lemma 3.8, we first infer that ${u_{n}}$ is non-vanishing, then we can follow the similar idea in [13] and [22] to construct a minimiser.

For the fact that ${u_{n}}$ is non-vanishing, so that there exists $x_{n} \in R^{N}$ and $δ_{0} > 0$ such that $\begin{matrix} (4.4) & \int_{B_{1} (x_{n})} u_{n}^{2} (x) d x > δ_{0} . \end{matrix}$ Without loss of generality, we assume that $x_{n} \in Z^{N}$ . Since $u_{n} \to \tilde{u}$ in $L_{loc}^{2} (R^{N})$ and $\tilde{u} = 0$ , we may assume that $| x_{n} | \to \infty$ up to a subsequence. Denote ${\overline{u}}_{n}$ by ${\overline{u}}_{n} (\cdot) = u_{n} (\cdot + x_{n})$ . Passing to a subsequence, we suppose that ${\overline{u}}_{n} ⇀ \overline{u}$ in X, ${\overline{u}}_{n} \to \overline{u}$ in $L_{loc}^{2} (R^{N})$ and ${\overline{u}}_{n} (x) \to \overline{u} (x)$ a.e. on $R^{N}$ . By (4.4) we have $\begin{matrix} \int_{B_{1} (0)} {\overline{u}}_{n}^{2} (x) d x > δ_{0} . \end{matrix}$ So $\overline{u} \neq 0$ .

Claim I: $\begin{matrix} (4.5) & I_{p}^{'} (\overline{u}) = 0 . \end{matrix}$ Indeed, for all $ψ \in X$ , set $ψ_{n} : = ψ (\cdot + x_{n})$ . From Lemma 3.10, replacing $φ_{n}$ by $ψ_{n}$ it follows that $\begin{matrix} \int_{R^{N}} (K (x) - K_{p} (x)) | u_{n} |^{2^{*} - 2} u_{n} ψ_{n} d x \to 0 . \end{matrix}$ Moreover, replacing $φ_{n}$ by $ψ_{n}$ in Lemma 3.11 implies $\begin{matrix} \int_{R^{N}} (f (x, u_{n}) - f_{p} (x, u_{n})) ψ_{n} d x \to 0 . \end{matrix}$ Consequently, $\begin{matrix} ⟨ I^{'} (u_{n}), ψ_{n} ⟩ - ⟨ I_{p}^{'} (u_{n}), ψ_{n} ⟩ \to 0 . \end{matrix}$ Noting that $I^{'} (u_{n}) \to 0$ and $‖ ψ_{n} ‖ = ‖ ψ ‖$ , we have $⟨ I^{'} (u_{n}), ψ_{n} ⟩ \to 0$ , then $⟨ I_{p}^{'} (u_{n}), ψ_{n} ⟩ \to 0$ . Moreover, by the periodicity of $K_{p}$ and $f_{p}$ in the variable x and $x_{n} \in Z^{N}$ , we get $\begin{matrix} ⟨ I_{p}^{'} (u_{n}), ψ_{n} ⟩ = ⟨ I_{p}^{'} ({\overline{u}}_{n}), ψ ⟩ . \end{matrix}$ Then $⟨ I_{p}^{'} ({\overline{u}}_{n}), ψ ⟩ \to 0$ . For the arbitrary of ψ, $I_{p}^{'} ({\overline{u}}_{n}) ⇀ 0$ in $X^{*}$ . Since $I_{p}^{'}$ is weakly sequentially continuous (Remark 2.4), so that (4.5) holds.

Claim II: $\begin{matrix} (4.6) & I_{p} (\overline{u}) ⩽ c . \end{matrix}$ Replacing $φ_{n}$ by $u_{n}$ , Lemma 3.11 yields $\begin{matrix} (4.7) & \int_{R^{N}} (f (x, u_{n}) - f_{p} (x, u_{n})) u_{n} d x \to 0 . \end{matrix}$ It follows from Remark 3.12 that $\begin{matrix} (4.8) & \int_{R^{N}} (F (x, u_{n}) - F_{p} (x, u_{n})) d x \to 0 . \end{matrix}$ By the condition $K ⩾ K_{p}$ , (4.7) and (4.8), we get $\begin{matrix} \int_{R^{N}} {\tilde{G}}_{p} (x, u_{n}) d x & = \frac{1}{N} \int_{R^{N}} K_{p} | u_{n} |^{2^{*}} d x + \int_{R^{N}} (\frac{1}{2} f_{p} (x, u_{n}) u_{n} - F_{p} (x, u_{n})) d x \\ ⩽ \frac{1}{N} \int_{R^{N}} K | u_{n} |^{2^{*}} d x + \int_{R^{N}} (\frac{1}{2} f_{p} (x, u_{n}) u_{n} - F_{p} (x, u_{n})) d x \\ = \frac{1}{N} \int_{R^{N}} K | u_{n} |^{2^{*}} d x + \int_{R^{N}} (\frac{1}{2} f (x, u_{n}) u_{n} - F (x, u_{n})) d x + o_{n} (1) \\ = \int_{R^{N}} \tilde{G} (x, u_{n}) d x + o_{n} (1), \end{matrix}$ where ${\tilde{G}}_{p}$ and $\tilde{G}$ are given in (2.9). Noting that ${\tilde{G}}_{p}$ is 1-periodic in x, we have $\begin{matrix} \int_{R^{N}} {\tilde{G}}_{p} (x, {\overline{u}}_{n}) d x = \int_{R^{N}} {\tilde{G}}_{p} (x, u_{n}) d x . \end{matrix}$ Therefore, $\begin{matrix} \int_{R^{N}} {\tilde{G}}_{p} (x, {\overline{u}}_{n}) d x ⩽ \int_{R^{N}} \tilde{G} (x, u_{n}) d x + o_{n} (1) . \end{matrix}$ Note that ${\overline{u}}_{n} (x) \to \overline{u} (x)$ a.e. on $R^{N}$ , from (2.10) and the Fatou Lemma it follows that $\begin{matrix} \int_{R^{N}} {\tilde{G}}_{p} (x, \overline{u}) d x + o_{n} (1) ⩽ \int_{R^{N}} {\tilde{G}}_{p} (x, {\overline{u}}_{n}) d x . \end{matrix}$ So $\begin{matrix} \int_{R^{N}} {\tilde{G}}_{p} (x, \overline{u}) d x ⩽ \int_{R^{N}} \tilde{G} (x, u_{n}) d x + o_{n} (1) . \end{matrix}$ Combining with (4.3) and (4.5) we obtain that $\begin{matrix} c + o_{n} (1) & = \int_{R^{N}} \tilde{G} (x, u_{n}) d x ⩾ \int_{R^{N}} {\tilde{G}}_{p} (x, \overline{u}) d x + o_{n} (1) \\ = I_{p} (\overline{u}) - \frac{1}{2} ⟨ I_{p}^{'} (\overline{u}), \overline{u} ⟩ + o_{n} (1) = I_{p} (\overline{u}) + o_{n} (1) . \end{matrix}$ Then (4.6) is obtained.

Claim III: $\overline{u} \notin X^{-}$ . In fact, by (2.10) $\begin{matrix} I_{p} (\overline{u}) = I_{p} (\overline{u}) - \frac{1}{2} ⟨ I_{p}^{'} (\overline{u}), \overline{u} ⟩ = \int_{R^{N}} {\tilde{G}}_{p} (x, \overline{u}) d x > 0 . \end{matrix}$ However, on $X^{-}$ , $I_{p} (\overline{u}) ⩽ 0$ , hence $\overline{u} \notin X^{-}$ .

Using Lemma 3.5, there exists $\hat{m} (u) \in M$ such that $I (\hat{m} (u)) = {max}_{v \in \hat{X} (u)} I (v)$ . Then $I (\hat{m} (u)) ⩾ c$ . Below we show that $I (\hat{m} (u)) ⩽ c$ . Indeed, by Lemma 4.1, we have ${max}_{v \in \hat{X} (u)} I_{p} (v) = I_{p} (\overline{u})$ since (4.5). Using (4.6) we get ${max}_{v \in \hat{X} (u)} I_{p} (v) ⩽ c$ . Lemma 3.9 implies that $I (\hat{m} (u)) ⩽ I_{p} (\hat{m} (u))$ . Then $I (\hat{m} (u)) ⩽ c$ . So $I (\hat{m} (u)) = c$ . Therefore $\hat{m} (u)$ is a minimizer of I on M. This ends the proof. □

Footnotes

Acknowledgements

Jing Zhang is supported by The Natural Science Foundation of Inner Mongolia Autonomous Region (No. 2019MS01004, 2020LH01009) and the National Natural Science Foundation of China (No. 11962025), Lin Li is supported by Research Fund of National Natural Science Foundation of China (No. 11861046), China Postdoctoral Science Foundation (No. 2019M662796), Chongqing Municipal Education Commission (No. KJQN20190081).

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