Nonlinear Perturbations of a periodic magnetic Choquard equation with Hardy-Littlewood-Sobolev critical exponent

TL;DR

This paper investigates the existence of ground state solutions for a nonlinear magnetic Choquard equation with critical Hardy-Littlewood-Sobolev exponent, considering periodic potentials and various nonlinearities, using variational methods.

Contribution

It establishes new existence results for ground states of a magnetic Choquard equation with critical exponent and periodic potentials, extending previous work to more general nonlinearities.

Findings

Existence of at least one ground state solution under certain conditions.

Results depend on the nonlinearity exponent p and parameters N, λ.

Applicable to a range of nonlinearities including power-type and critical cases.

Abstract

In this paper, we consider the following magnetic nonlinear Choquard equation \[-(\nabla+iA(x))^2u+ V(x)u = \left(\frac{1}{|x|^{\alpha}}*|u|^{2_{\alpha}^*}\right) |u|^{2_{\alpha}^*-2} u + \lambda f(u)\ \textrm{ in }\ \R^N,\] where $2_{\alpha}^{*}=\frac{2N-\alpha}{N-2}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, $\lambda>0$, $N\geq 3$, $0<\alpha< N$, $A: \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is an $C^1$, $\mathbb{Z}^N$-periodic vector potential and $V$ is a continuous scalar potential given as a perturbation of a periodic potential. Under suitable assumptions on different types of nonlinearities $f$, namely, $f(x,u)=\left(\frac{1}{|x|^{\alpha}}*|u|^{p}\right)|u|^{p-2} u$ for $(2N-\alpha)/N<p<2^{*}_{\alpha}$, then $f(u)=|u|^{p-1} u$ for $1<p<2^*-1$ and $f(u)=|u|^{2^* - 2}u$ (where $2^*=2N/(N-2)$), we prove the existence of at least one ground state solution for this equation by variational methods if $p$ belongs to some intervals depending on $N$ and $\lambda$.

Equations

$$-(\nabla +iA(x){)}^{2}u+V(x)u=\left(\frac{1}{|x{|}^{\alpha }}\ast |u{|}^{2_{\alpha }^{\ast }}\right)|u{|}^{2_{\alpha }^{\ast }-2}u+\lambda f(u)\text{ in }{\mathbb{R}}^N,$$

$$A(x+y)=A(x),\forall x\in {\mathbb{R}}^N,\forall y\in {\mathbb{Z}}^N.$$

$$-(\nabla +iA(x){)}^{2}u+\mu g(x)u=\lambda u+\left(\frac{1}{|x{|}^{\alpha }}\ast |u{|}^{2_{\alpha }^{\ast }}\right)|u{|}^{2_{\alpha }^{\ast }-2}u\text{ in }{\mathbb{R}}^N$$

$$\bigg{(}\frac{1}{|x|^{\alpha}}*|u|^{p})\bigg{)}|u|^{p-2}{u},$$

$$-(\nabla +iA(x){)}^{2}u+V(x)u=\left(\frac{1}{|x{|}^{\alpha }}\ast |u{|}^{2_{\alpha }^{\ast }})\right)|u{|}^{2_{\alpha }^{\ast }-2}u+\lambda \left(\frac{1}{|x{|}^{\alpha }}\ast |u{|}^p)\right)|u{|}^{p-2}u\text{ in }{\mathbb{R}}^N$$

$$-(\nabla+iA(x))^{2}u+V(x)u=\ \bigg{(}\frac{1}{|x|^{\alpha}}*|u|^{2_{\alpha}^{*}}\bigg{)}|u|^{2_{\alpha}^{*}-2}u+\lambda|u|^{p-1}u\ \textrm{ in }\ \mathbb{R}^{N}.$$

$$-(\nabla +iA(x){)}^{2}u+V(x)u=\lambda \left(\frac{1}{|x{|}^{\alpha }}\ast |u{|}^p\right)|u{|}^{p-2}u+|u{|}^{2^{\ast }-2}u\text{ in }{\mathbb{R}}^N,$$

$$-(\nabla +iA(x){)}^{2}u+V_{\mathcal{P}}(x)u=\left(\frac{1}{|x{|}^{\alpha }}\ast |u{|}^{2_{\alpha }^{\ast }})\right)|u{|}^{2_{\alpha }^{\ast }-2}u+\lambda f(u)\text{ in }{\mathbb{R}}^N$$

$$-\Delta u+V(x)u=\lambda u^q+u^p\text{ in }{\mathbb{R}}^N,$$

$$-\Delta u+a(x)u=\lambda |u{|}^{q-1}+|u{|}^{p-1}u\text{ in }{\mathbb{R}}^N,$$

$${\nabla }_{A}u=\nabla u+iA(x)u.$$

$$H_{A,V}^1({\mathbb{R}}^N,\mathbb{C})=\left\{u\in L^2({\mathbb{R}}^N,\mathbb{C}):{\nabla }_{A}u\in L^2({\mathbb{R}}^N,\mathbb{C}),{\int }_{{\mathbb{R}}^N}V(x)|u(x){|}^2\text{d}x<\infty \right\}$$

$$\|u\|_{A,V}=\bigg{(}\int_{\mathbb{R}^{N}}(|\nabla_{A}u|^{2}+V(x)|u|^{2})\;\textup{d}x\bigg{)}^{\frac{1}{2}}.$$

$$|\nabla |u|(x)|\le |\nabla u(x)+iA(x)u(x)|,\text{a.e. }x\in {\mathbb{R}}^N.$$

$$H_{A,V}^1({\mathbb{R}}^N,\mathbb{C})\hookrightarrow L^s({\mathbb{R}}^N,\mathbb{C})$$

$$S_{H,L}:$$

$$=\underset{u\in D_A^{1,2}({\mathbb{R}}^N)\setminus \{0\}}{\inf }\frac{{\int }_{{\mathbb{R}}^N}|{\nabla }_{A}u{|}^2\text{d}x}{{\left({\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{|u(x{)}^{2_{\alpha }^{\ast }}||u(y){|}^{2_{\alpha }^{\ast }}}{|x-y{|}^{\alpha }}\text{d}x\text{d}y\right)}^{\frac{N-2}{2N-\alpha }}}=:S_A,$$

$$u=C{\left(\frac{b}{b^2+|x-a{|}^2}\right)}^{\frac{N-2}{2}},$$

$$S_{H,L}=\frac{S}{C(N,\alpha {)}^{\frac{N-2}{2N-\alpha }}},$$

$$\bar{U}(x)=S^{\frac{(N-\alpha )(2-\alpha )}{4(N+2-\alpha )}}C(N,\alpha {)}^{\frac{2-N}{2(N+2-\alpha )}}\frac{[N(N-2){]}^{\frac{N-2}{4}}}{(|1+|x{|}^2|{)}^{\frac{N-2}{2}}}$$

$$-△u=\left({\int }_{{\mathbb{R}}^N}\frac{|u{|}^{2_{\alpha }^{\ast }}}{|x-y{|}^{\alpha }}\text{d}y\right)|u{|}^{2_{\alpha }^{\ast }-2}u\text{em}{\mathbb{R}}^N$$

$${\int }_{{\mathbb{R}}^N}|\nabla \bar{U}{|}^2\text{d}x={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{|\bar{U}(x){|}^{2_{\alpha }^{\ast }}|\bar{U}(y){|}^{2_{\alpha }^{\ast }}}{|x-y{|}^{\alpha }}\text{d}x\text{d}y=S_{H,L}^{\frac{2N-\alpha }{N+2-\alpha }}.$$

$${\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{f(x)h(y)}{|x-y{|}^{\alpha }}\text{d}x\text{d}y\le C(t,N,\alpha ,r)\Vert f{\Vert }_t\Vert h{\Vert }_r.$$

$$C(t,N,\alpha ,r)=C(N,\alpha )={\pi }^{\frac{\alpha }{2}}\frac{\Gamma (\frac{N}{2}-\frac{\alpha }{2})}{\Gamma (N-\frac{\alpha }{2})}{\left\{\frac{\Gamma (\frac{N}{2})}{\Gamma (N)}\right\}}^{-1+\frac{\alpha }{N}}.$$

$$f(x)=A({\gamma }^2+|x-a{|}^2{)}^{-(2N-\alpha )/2}$$

$$-(\nabla +iA(x){)}^{2}u+V_{\mathcal{P}}(x)u=\left(\frac{1}{|x{|}^{\alpha }}\ast |u{|}^{2_{\alpha }^{\ast }})\right)|u{|}^{2_{\alpha }^{\ast }-2}u+\lambda \left(\frac{1}{|x{|}^{\alpha }}\ast |u{|}^p)\right)|u{|}^{p-2}u,$$

$$H_{A,V_{\mathcal{P}}}^1({\mathbb{R}}^N,\mathbb{C})=\left\{u\in L^2({\mathbb{R}}^N,\mathbb{C}):{\nabla }_{A}u\in L^2({\mathbb{R}}^N,\mathbb{C})\right\}$$

$$⟨u,v{⟩}_{A,V_{\mathcal{P}}}=\mathfrak{Re}{\int }_{{\mathbb{R}}^N}\left({\nabla }_{A}u\cdot \overline{{\nabla }_{A}v}+V_{\mathcal{P}}(x)u\bar{v}\right)\text{d}x$$

$$\Vert u{\Vert }_{A,V_{\mathcal{P}}}^2={\int }_{{\mathbb{R}}^N}\left(|{\nabla }_{A}u{|}^2+V_{\mathcal{P}}|u{|}^2\right)\text{d}x.$$

$$J_{A,V_{\mathcal{P}}}(u):=\frac{1}{2}\Vert u{\Vert }_{A,V_{\mathcal{P}}}^2-\frac{1}{2\cdot 2_{\alpha }^{\ast }}D(u)-\frac{\lambda }{2p}B(u),$$

Full text

[TABLE]

where $2_{\alpha}^{*}=\frac{2N-\alpha}{N-2}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, $\lambda>0$, $N\geq 3$, $0<\alpha<N$, $A:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$ is an $C^{1}$, $\mathbb{Z}^{N}$-periodic vector potential and $V$ is a continuous scalar potential given as a perturbation of a periodic potential. Under suitable assumptions on different types of nonlinearities $f$, namely, $f(x,u)=\left(\frac{1}{|x|^{\alpha}}*|u|^{p}\right)|u|^{p-2}u$ for $(2N-\alpha)/N<p<2^{*}_{\alpha}$, then $f(u)=|u|^{p-1}u$ for $1<p<2^{*}-1$ and $f(u)=|u|^{2^{*}-2}u$ (where $2^{*}=2N/(N-2)$), we prove the existence of at least one ground state solution for this equation by variational methods if $p$ belongs to some intervals depending on $N$ and $\lambda$.

Keywords: Variational methods, magnetic Choquard equation, Hardy-Littlewood-Sobolev critical exponent

MSC[2010]: 35Q55, 35Q40, 35J20

1. Introduction

In this article we consider the problem

[TABLE]

where $\nabla+iA(x)$ is the covariant derivative with respect to the $C^{1}$, $\mathbb{Z}^{N}$-periodic vector potential $A\colon\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$, i.e,

[TABLE]

The exponent $2_{\alpha}^{*}=\frac{2N-\alpha}{N-2}$ is critical, in the sense of the Hardy-Littlewood-Sobolev inequality, $\lambda>0$, $N\geq 3,$ $0<\alpha<N,$ $V:\mathbb{R}^{N}\to\mathbb{R}$ is a continuous scalar potential and $f$ stands for different types of nonlinearities. Namely, we first consider $f(x,u)=\left(\frac{1}{|x|^{\alpha}}*|u|^{p}\right)|u|^{p-2}u$ for $(2N-\alpha)/N<p<2^{*}_{\alpha}$, then $f(u)=|u|^{p-1}u$ for $1<p<2^{*}-1$, where $2^{*}$ is the critical exponent of immersion $D^{1,2}(\mathbb{R}^{N})\hookrightarrow L^{2^{*}}(\mathbb{R}^{N})$, and finally we examine $f(u)=|u|^{2^{*}-2}u$.

Inspired by the seminal work of Coti Zelati and Rabinowitz [14], but also by Alves, Carrião and Miyagaki [1] and by Alves and Figueiredo [2], we assume that there is a continuous, $\mathbb{Z}^{N}$-periodic potential $V_{\mathcal{P}}:\mathbb{R}^{N}\to\mathbb{R}$, constants $V_{0},W_{0}>0$ and $W\in L^{\frac{N}{2}}(\mathbb{R}^{N})$ with $W(x)\geq 0$ such that

• $(V_{1})$

$V_{\mathcal{P}}(x)\geq V_{0},\quad\forall\;x\in\mathbb{R}^{N}$;

• $(V_{2})$

$V(x)=V_{\mathcal{P}}(x)-W(x)\geq W_{0},\quad\forall\;x\in\mathbb{R}^{N}$,

where the last inequality is strict on a subset of positive measure in $\mathbb{R}^{N}$.

Since the problem is considered in the whole $\mathbb{R}^{N}$ and has a critical nonlinearity in the Hardy-Littlewood-Sobolev sense, the verification of any compactness condition is not easy.

Our paper is motivated by Gao and Yang in [20], where a classical Choquard equation is considered in a bounded domain, i.e., the case $A\equiv 0$ and $V\equiv 0$ is studied in a bounded domain $\Omega$. There is a huge literature about the Choquard equation and we cite only Moroz and Van Schaftingen [27] for a good review of results on this important subject. In [20], Gao and Yang proved the existence of a ground state solution under restriction on $N$ and $\lambda$. Other recent advances in the study of the Choquard equation can be found, e.g., in [4, 5, 6, 17, 18, 22, 26, 30].

In Mukherjee and Sreenadh [28], the magnetic problem

[TABLE]

was examined. In this equation $\mu>0$ is also a parameter that interacts with the linear term in the right-hand side of the equation. Under suitable hypotheses on $g$, the existence of a ground state solution was proved. The concentration of solutions as $\mu\to\infty$ was also studied.

Changing the right-hand side of (1) to

[TABLE]

the problem was studied by Cingolani, Clapp and Secchi in [13]. In that paper the authors proved existence and multiplicity of solutions. In [12], the right-hand side (2) was generalized and a ground state solution was obtained, but the multiplicity result depend on more restrictive hypotheses than in [13].

Recent years have witnessed a growth of interest in the study of magnetic equations. The progress in this research can be found in a series of articles, e.g., [3, 7, 8, 9, 15, 16].

The main results of this paper are the following theorems.

Theorem 1**.**

For $\frac{2N-\alpha}{N}<p<2_{\alpha}^{*}$, under the hypotheses already stated on $A$, $V$ and $\alpha$, problem

[TABLE]

has at least one ground state solution if either

1. $(i)$

$\frac{N+2-\alpha}{N-2}<p<2_{\alpha}^{*}$, $N=3,4$ and $\lambda>0$; 2. $(ii)$

$\frac{2N-\alpha}{N}<p\leq\frac{N+2-\alpha}{N-2}$, $N=3,4$ and $\lambda$ sufficiently large; 3. $(iii)$

$\frac{2N-2-\alpha}{N-2}<p<2_{\alpha}^{*}$, $N\geq 5$ and $\lambda>0$; 4. $(iv)$

$\frac{2N-\alpha}{N}<p\leq\frac{2N-2-\alpha}{N-2}$, $N\geq 5$ and $\lambda$ sufficiently large.

Theorem 2**.**

For $1<p<2^{*}-1$, under the hypotheses already stated on $A$, $V$ and $\alpha$, problem

[TABLE]

has at least one ground state solution if either

1. $(i)$

$3<p<5$, $N=3$ and $\lambda>0;$ 2. $(ii)$

$p>1$, $N\geq 4$ and $\lambda>0$; 3. $(iii)$

$1<p\leq 3$, $N=3$ and $\lambda$ sufficiently large.

Theorem 3**.**

Under the hypotheses already stated on $A$, $V$ and $\alpha$, the problem

[TABLE]

has at least one ground state solution in the intervals already described in Theorem 1.

Initially, we are going to prove the existence of a ground state solution for problem (1) considering the potential $V=V_{\mathcal{P}}$, that is, we consider the problem

[TABLE]

and $f$ as in Theorems 1, 2 and 3, where we maintain the notation introduced before and suppose that $(V_{1})$ is valid.

As in Gao and Yang in [20], the key step to proof the existence of a ground state solution of problem (6) is the use of cut-off techniques on the extreme function that attains the best constant $S_{H,L}$ defined in the sequence. This allows us to estimate the mountain pass value $c_{\lambda}$ associated with the energy functional $J_{A,V_{\mathcal{P}}}$ related with (6) in terms of the level where the PS condition holds. In a demanding proof, this lead us to establish intervals for $p$ (depending on $N$ and $\lambda$) where the PS condition is satisfied, as in the seminal work of Brézis and Nirenberg [11]. After that, the proof is completed by showing the mountain pass geometry, introducing the Nehari manifold associated with (6) and applying concentration-compactness arguments. In the sequel, we consider (1) for the different nonlinearities $f$ and prove that each problem has at least one ground state solution.

We observe that the conclusion of Theorem 2 is similar to that of Theorem 1.1 in Alves, Carrião and Miyagaki [1] and Theorem 1.1 in Miyagaki [25]. Being more precise, in [1] the authors have discussed the existence of a positive solution to the semilinear elliptic problem involving critical exponents

[TABLE]

where $\lambda>0$ is a parameter, $1<q<p=2^{*}-1$ and $V:\mathbb{R}^{N}\to\mathbb{R}$ is a positive continuous function. On its turn, Miyagaki [25] has studied the existence of nontrivial solution for the following class of semilinear elliptic equation in $\mathbb{R}^{N}$ ($N\geq 3$) involving critical Sobolev exponents

[TABLE]

where $1<q<p\leq 2^{*}-1=\frac{N+2}{N-2}$ and $\lambda>0$ are constants and $a:\mathbb{R}^{N}\to\mathbb{R}$ is a continuous function such that $a(x)\geq a_{0}$ for all $x\in\mathbb{R}^{N}$, where $a_{0}>0$ is a constant.

Problems (6) and (1) are then related by showing that the minimax value $d_{\lambda}$ of the latter satisfies $d_{\lambda}<c_{\lambda}$. Once more, concentration-compactness arguments are applied to show the existence of a ground state solution.

This paper is organized as follows. In Section 2 some preliminary results will be established. Section 3, 4 and 5 are then devoted to the proofs of Theorems 1, 2 and 3, respectively.

2. Preliminary results

We denote

[TABLE]

We handle problem (1) in the space

[TABLE]

endowed with the norm

[TABLE]

Observe that the norm generated by this scalar product is equivalent to the norm obtained by considering $V\equiv 1$, see [24, Definition 7.20].

If $u\in H^{1}_{A,V}(\mathbb{R}^{N},\mathbb{C})$, then $|u|\in H^{1}(\mathbb{R}^{N})$ and the diamagnetic inequality is valid (see [13] or [24, Theorem 7.21])

[TABLE]

As a consequence of the diamagnetic inequality, we have the continuous immersion

[TABLE]

for any $s\in[2,\frac{2N}{N-2}]$. We denote $2^{*}=\frac{2N}{N-2}$ and $\|\cdot\|_{s}$ the norm in $L^{s}(\mathbb{R}^{N},\mathbb{C})$.

It is well-known that $C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{C})$ is dense in $H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$, see [24, Theorem 7.22].

Following Gao and Yang [21], we denote by $S_{H,L}$

[TABLE]

where $D^{1,2}_{A}(\mathbb{R}^{N})=\{u\in L^{2^{*}}(\mathbb{R}^{N},\mathbb{C})\})\,:\,\nabla_{A}u\in L^{2}(\mathbb{R}^{N},\mathbb{C})\}$. The equality between $S_{H,L}$ and $S_{A}$ was proved in Mukherjee and Sreenadh [28]. We remark that $S_{A}$ is attained if and only if $\textup{rot}\,A=0$ [28, Theorem 4.1]. See also [10, Theorem 1.1].

We state a result proved in [21].

Proposition 4** (Gao and Yang [21]).**

The constant $S_{H,L}$ defined in (8) is achieved if and only if

[TABLE]

where $C>0$ is a fixed constant, $a\in\mathbb{R}^{N}$ and $b\in(0,\infty)$ are parameters. Furthermore,

[TABLE]

where $S$ is the best Sobolev constant of the immersion $D^{1,2}(\mathbb{R}^{N})\hookrightarrow L^{2^{*}}(\mathbb{R}^{N})$ and $C(N,\alpha)$ depends on $N$ and $\alpha$.

If we consider the minimizer for $S$ given by $U(x):=\frac{[N(N-2)]^{\frac{N-2}{4}}}{(|1+|x|^{2}|)^{\frac{N-2}{2}}}$ (see [31, Theorem 1.42]), then

[TABLE]

is the unique minimizer for $S_{H,L}$ that satisfies

[TABLE]

with

[TABLE]

Proposition 5** (Hardy-Littlewood-Sobolev inequality, see [24]).**

Suppose that $f\in L^{t}(\mathbb{R}^{N})$ and $h\in L^{r}(\mathbb{R}^{N})$ for $t,r>1$ and $0<\alpha<N$ satisfying $\frac{1}{t}+\frac{\alpha}{N}+\frac{1}{r}=2$. Then, there exists a sharp constant $C(t,N,\alpha,r)$, independent of $f$ and $h$, such that

[TABLE]

If $t=r=\frac{2N}{2N-\alpha}$, then

[TABLE]

In this case there is equality in $\eqref{desig}$ if and only if $h=cf$ for a constant $c$ and

[TABLE]

for some $A\in\mathbb{C}$, $0\neq\gamma\in\mathbb{R}$ and $a\in\mathbb{R}^{N}$.

Lemma 6**.**

Let $U\subseteqq\mathbb{R}^{N}$ be any open set. For $1<p<\infty$, let $(f_{n})$ be a bounded sequence in $L^{s}(U,\mathbb{C})$ such that $f_{n}(x)\to f(x)$ a.e. Then $f_{n}\rightharpoonup f$ in $L^{s}(U,\mathbb{C})$.

The proof of Lemma 6 only adapts the arguments given for the real case, as in [23, Lemme 4.8, Chapitre 1].

3. The case $f(x,u)=\left(\frac{1}{|x|^{\alpha}}*|u|^{p}\right)|u|^{p-2}u$

3.1. The periodic problem

In this subsection we deal with problem (6) for $f(x,u)$ as above, that is,

[TABLE]

where $\frac{2N-\alpha}{N}<p<2_{\alpha}^{*}$.

We consider the space

[TABLE]

endowed with scalar product

[TABLE]

and, therefore

[TABLE]

We observe that the energy functional $J_{A,V_{\mathcal{P}}}$ on $H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ associated with (10) is given by

[TABLE]

where

[TABLE]

and

[TABLE]

Remark 3.1**.**

Notice that, by the Hardy-Littlewood-Sobolev inequality, the integral

[TABLE]

is well defined if

[TABLE]

Here, as also in [6], $\frac{2N-\alpha}{N}$ is called the lower critical exponent and $2_{\alpha}^{*}=\frac{2N-\alpha}{N-2}$ the upper critical exponent. This lead us to say that (1) is a critical nonlocal elliptic equation.

The Hardy-Littlewood-Sobolev inequality implies that

[TABLE]

and

[TABLE]

for constants $C_{1}(N,\alpha)$ and $C_{2}(N,\alpha)$. For any $u\in H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ the immersions (7) imply that $B$ and $D$ are well-defined. Consequently, $J_{A,V_{\mathcal{P}}}$ is well-defined.

Observe that

[TABLE]

Definition 3.1**.**

A function $u\in H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ is a weak solution of (10) if

[TABLE]

for all $\psi\in H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$.

Since the derivative of the energy functional $J_{A,V_{\mathcal{P}}}$ is given by

[TABLE]

we see that critical points of $J_{A,V_{\mathcal{P}}}$ are weak solutions of (10).

Note that, if $\psi=u$ we obtain

[TABLE]

Lemma 7**.**

The functional $J_{A,V_{\mathcal{P}}}$ satisfies the mountain pass geometry. Precisely,

1. $(i)$

there exist $\rho,\delta>0$ such that J_{A,V_{\mathcal{P}}}\big{|}_{S}\geq\delta>0 for any $u\in\mathcal{S}$, where

[TABLE] 2. $(ii)$

for any $u_{0}\in H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})\setminus\{0\}$ there exists $\tau\in(0,\infty)$ such that $\|\tau u_{0}\|_{V_{\mathcal{P}}}>\rho$ and $J_{A,V_{\mathcal{P}}}(\tau u_{0})<0$.

*Proof. * Inequalities (11) and (12) yields

[TABLE]

thus implying ($i$) if we take $\|u\|_{A,V_{\mathcal{P}}}=\rho>0$ sufficiently small.

In order to prove ($ii$), fix $u_{0}\in H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})\setminus\{0\}$ and consider the function $g_{u_{0}}\colon(0,\infty)\to\mathbb{R}$ given by

[TABLE]

We have

[TABLE]

and

[TABLE]

Thus,

[TABLE]

Since $1<p<2^{*}_{\alpha}$, we have

[TABLE]

completing the proof of ($ii$). $\hfill\Box$

The mountain pass theorem without the PS condition (see [31, Theorem 1.15]) yields a Palais-Smale sequence $(u_{n})\subset H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ such that

[TABLE]

where

[TABLE]

and $\Gamma=\left\{\gamma\in C^{1}\left([0,1],H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})\right)\,:\,\gamma(0)=0,\,J_{A,V_{\mathcal{P}}}(\gamma(1))<0\right\}$.

Lemma 8**.**

Suppose that $u_{n}\rightharpoonup u_{0}$ in $H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$. Then

[TABLE]

for all $\frac{2N-\alpha}{N}\leq s\leq 2_{\alpha}^{*}$.

*Proof. * In this proof we adapt some ideas of [7]. We can suppose that $|u_{n}(x)|^{s}\to|u_{0}(x)|^{s}$ a.e. in $\mathbb{R}^{N}$ and, as consequence of the immersion (7), $|u_{n}|^{s}$ is bounded in $L^{\frac{2N}{2N-\alpha}}(\mathbb{R}^{N})$. Thus, Lemma 6 allows us to conclude that

[TABLE]

The Hardy-Littlewood-Sobolev inequality allows us to conclude that

[TABLE]

for all $w\in L^{\frac{2N}{2N-\alpha}}(\mathbb{R}^{N})$. So, we have a continuous linear map from $L^{\frac{2N}{2N-\alpha}}(\mathbb{R}^{N})$ to $L^{\frac{2N}{\alpha}}(\mathbb{R}^{N})$. A new application of Lemma 6 yields (15). $\hfill\Box$

Corollary 9**.**

Suppose that $u_{n}\rightharpoonup u_{0}$ and consider

[TABLE]

and

[TABLE]

for $\psi\in C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{C})$. Then $B^{\prime}(u_{n})\cdot\psi\to B^{\prime}(u_{0})\cdot\psi$ and $D^{\prime}(u_{n})\cdot\psi\to D^{\prime}(u_{0})\cdot\psi$.

*Proof. * The immersion (7) guarantees that $|u_{n}|^{p-2}u_{n}$ is bounded in $L^{\frac{2N}{N+2-\alpha}}(\mathbb{R}^{N},\mathbb{C})$. Since we can suppose that $|u_{n}(x)|^{p}\to|u_{0}(x)|^{p}$ a.e. in $\mathbb{R}^{N}$, by applying Lemma 6, we conclude that

[TABLE]

for all $\frac{2N-\alpha}{N}\leq p\leq 2_{\alpha}^{*}$, as $n\to+\infty$.

Combining (15) with (16) yields

[TABLE]

as $n\to+\infty$, for all $\frac{2N-\alpha}{N}\leq p\leq 2_{\alpha}^{*}$. Consequently, for $\psi\in C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{C})$, it follows that

[TABLE]

and

[TABLE]

that is,

[TABLE]

$\hfill\Box$

Lemma 10**.**

If $(u_{n})\subset H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ is a $(PS)_{\lambda}$ sequence for $J_{A,V_{\mathcal{P}}}$, then $(u_{n})$ is bounded. In addition, if $u_{n}\rightharpoonup u$ weakly in $H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ as $n\to\infty$, then $u$ is ground state solution for problem (10).

*Proof. * Standard arguments prove that $(u_{n})$ is bounded in $H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$. Then, up to a subsequence, we have $u_{n}\rightharpoonup u$ weakly in $H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ as $n\to\infty$.

From Corollary 9 it follows that, for all $\psi\in H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$, we have

[TABLE]

where $s=p$ or $s=2_{\alpha}^{*}$.

Thus, since for all $\psi\in C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{C})$ we have $J^{\prime}_{A,V_{\mathcal{P}}}(u_{n})\cdot\psi=o_{n}(1)$, we obtain

[TABLE]

that is, $u$ is a ground state solution for (10). $\hfill\Box$

We now consider the Nehari manifold associated with the $J_{A,V_{\mathcal{P}}}$.

[TABLE]

Lemma 11**.**

There exists a unique $t_{u}=t_{u}(u)>0$ such that $t_{u}u\in\mathcal{M}_{A,V_{\mathcal{P}}}$ for all $u\in H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})\setminus\{0\}$ and $J_{A,V_{\mathcal{P}}}(t_{u}u)=\displaystyle\max_{t\geq 0}J_{A,V_{\mathcal{P}}}(tu)$. Moreover $c_{\lambda}=c^{*}_{\lambda}=c^{**}_{\lambda}$, where

[TABLE]

*Proof. * Let $u\in H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})\setminus\{0\}$ and $g_{u}$ defined on $(0,+\infty)$ given by

[TABLE]

By the mountain pass geometry (Lemma 7), there exists $t_{u}>0$ such that

[TABLE]

Hence

[TABLE]

implying that $t_{u}u\in\mathcal{M}_{A,V_{\mathcal{P}}}$, as consequence of (14). We now show that $t_{u}$ is unique. To this end, we suppose that there exists $s_{u}>0$ such that $s_{u}u\in\mathcal{M}_{A,V_{\mathcal{P}}}$. Thus, we have both

[TABLE]

Hence

[TABLE]

Since both terms in parentheses have the same sign if $t_{u}\neq s_{u}$ and we also have $B(u)>0$, $D(u)>0$ and $\lambda>0$, it follows that $t_{u}=s_{u}$.

Now, the rest of the proof follows arguments similar to that found in [1, 19, 29, 31]. $\hfill\Box$

The following result controls the level ${c_{\lambda}}$ of a Palais-Smale sequence of $J_{A,V_{\mathcal{P}}}$.

Lemma 12**.**

Let $(u_{n})\subset H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ a $(PS)_{c_{\lambda}}$ sequence for $J_{A,V_{\mathcal{P}}}$ such that

[TABLE]

with

[TABLE]

Then the sequence $(u_{n})$ verifies either

1. $(i)$

$u_{n}\to 0$* strongly in $H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C}),$ as $n\to\infty,$*

or

1. $(ii)$

There exists a sequence $(y_{n})\subset\mathbb{R}^{N}$ and constants $r,\theta>0$ such that

[TABLE]

where $B_{r}(y)$ denotes the ball in $\mathbb{R}^{N}$ of center at $y$ and radius $r>0$.

*Proof. * Suppose that ($ii$) does not hold. Applying a result by Lions [31, Lemma 1.21], it follows from inequality (11) that

[TABLE]

Since $J^{\prime}_{A,V_{\mathcal{P}}}(u_{n})u_{n}=o_{n}(1)$ as $n\to\infty$, we obtain

[TABLE]

Let us suppose that

[TABLE]

Thus, as consequence of (18), we have

[TABLE]

Since

[TABLE]

making $n\to\infty$ yields

[TABLE]

On the other hand, it follows from (13) that

[TABLE]

Thus,

[TABLE]

and from (19) and (20) we conclude that $c_{\lambda}\geq\frac{N+2-\alpha}{2(2N-\alpha)}S_{A}^{\frac{2N-\alpha}{N+2-\alpha}}$, which is a contradiction. Therefore, ($i$) is valid and the proof is complete.

We now state our result about the periodic problem (10).

Theorem 13**.**

Under the hypotheses already stated on $A$ and $\alpha$, suppose that $(V_{1})$ is valid. Then problem (10) has at least one ground state solution if either

1. $(i)$

$\frac{N+2-\alpha}{N-2}<p<2_{\alpha}^{*}$, $N=3,4$ and $\lambda>0$; 2. $(ii)$

$\frac{2N-\alpha}{N}<p\leq\frac{N+2-\alpha}{N-2}$, $N=3,4$ and $\lambda$ sufficiently large; 3. $(iii)$

$\frac{2N-\alpha-2}{N-2}<p<2_{\alpha}^{*}$, $N\geq 5$ and $\lambda>0$; 4. $(iv)$

$\frac{2N-\alpha}{N}<p\leq\frac{2N-\alpha-2}{N-2}$, $N\geq 5$ and $\lambda$ sufficiently large.

*Proof. * Let $c_{\lambda}$ be the mountain pass level and consider a sequence $(u_{n})\subset H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ such that

[TABLE]

Claim. We affirm that $c_{\lambda}<\frac{N+2-\alpha}{2(2N-\alpha)}(S_{A})^{\frac{2N-\alpha}{N+2-\alpha}}$, a result that will be shown after completing our proof, since it is very technical.

Lemma 10 guarantees that $(u_{n})$ is bounded. So, passing to a subsequence if necessary, there is $u\in H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ such that

[TABLE]

If $u\neq 0$ we are done. If $u=0$, it follows from Lemma 12 the existence of $\theta>0$ and $(y_{n})\subset\mathbb{R}^{N}$ such that

[TABLE]

A direct computation shows that we can assume that $(y_{n})\subset\mathbb{Z}^{N}$. Let

[TABLE]

Since both $V_{\mathcal{P}}$ and $A$ are $\mathbb{Z}^{N}$-periodic, we have

[TABLE]

Therefore there exists $v\in H^{1}_{A,V_{\mathcal{P}}}$ such that $v_{n}\rightharpoonup v$ weakly in $H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ and $v_{n}\to v$ in $L^{2}_{loc}(\mathbb{R}^{N},\mathbb{C})$.

We claim that $v\neq 0$. In fact, it follows from (21)

[TABLE]

Since $v_{n}\rightarrow v$ in $L^{2}_{loc}(\mathbb{R}^{N})$, we have $\|v_{n}-v\|_{L^{2}(B_{r}(0))}\to 0$ as $n\to\infty$, proving our claim.

But Corollary 9 guarantees that $J^{\prime}_{A,V_{\mathcal{P}}}(v_{n})\cdot\psi\to J^{\prime}_{A,V_{\mathcal{P}}}(v_{n})\cdot\psi$ and it follows that $J^{\prime}_{A,V_{\mathcal{P}}}(v)\cdot\psi=0$. Consequently, $v$ is a ground state solution of problem (10). $\hfill\Box$

We now prove the postponed Claim, that is, we show that $c_{\lambda}<\frac{N+2-\alpha}{2(2N-\alpha)}(S_{A})^{\frac{2N-\alpha}{N+2-\alpha}}$. Observe that, once proved the existence of $u_{\epsilon}$ as in our next result, then

[TABLE]

Lemma 14**.**

There exists $u_{\varepsilon}$ such that

[TABLE]

provided that either

1. $(i)$

$\frac{N+2-\alpha}{N-2}<p<2_{\alpha}^{*}$, $N=3,4$ and $\lambda>0$; 2. $(ii)$

$\frac{2N-\alpha}{N}<p\leq\frac{N+2-\alpha}{N-2}$, $N=3,4$ and $\lambda$ sufficiently large; 3. $(iii)$

$\frac{2N-2-\alpha}{N-2}<p<2_{\alpha}^{*}$, $N\geq 5$ and $\lambda>0$; 4. $(iv)$

$\frac{2N-\alpha}{N}<p\leq\frac{2N-2-\alpha}{N-2}$, $N\geq 5$ and $\lambda$ sufficiently large.

The arguments of this proof were adapted from the articles [20, 25]. Observe that the conditions stated in this result are exactly the same of Theorem 1 and Theorem 13.

*Proof. * We know that $U(x)=\frac{[N(N-2)]^{\frac{N-2}{4}}}{(1+|x|^{2})^{\frac{N-2}{2}}}$ is a minimizer for $S$, the best Sobolev constant of the immersion $D^{1,2}(\mathbb{R}^{N})\hookrightarrow L^{2^{*}}(\mathbb{R}^{N})$ (see [31, Theorem 1.42] or [10, Section 3]) and also a minimizer for $S_{H,L}$, according to Proposition 4.

If $B_{r}$ denotes the ball in $\mathbb{R}^{N}$ of center at origin and radius $r$, consider the balls $B_{\delta}$ and $B_{2\delta}$ and take $\psi\in C_{0}^{\infty}(\mathbb{R}^{N})$ such that, for a constant $C>0$,

[TABLE]

We define, for $\varepsilon>0,$

[TABLE]

In the proof we apply the estimates

[TABLE]

and

[TABLE]

which were obtained by Gao and Yang [21].

Case 1. $\frac{N+2-\alpha}{N-2}<p<2_{\alpha}^{*}$ and $N=3,4$ or $\frac{2N-2-\alpha}{N-2}<p<2_{\alpha}^{*}$ and $N\geq 5$.

Proof of Case 1. Consider the function $f:[0,+\infty)\to\mathbb{R}$ defined by

[TABLE]

The mountain pass geometry (Lemma 7) implies the existence of $t_{\varepsilon}>0$ such that $\displaystyle\sup_{t\geq 0}J_{A,V_{\mathcal{P}}}(tu_{\varepsilon})=J_{A,V_{\mathcal{P}}}(t_{\varepsilon}u_{\varepsilon})$. Since $t_{\varepsilon}>0$, $B(u_{\varepsilon})>0$ and $f^{\prime}(t_{\varepsilon})=0$, we obtain

[TABLE]

thus implying

[TABLE]

Now define $g:[0,S_{A}(\varepsilon)]\to\mathbb{R}$ by

[TABLE]

So,

[TABLE]

Since $t>0$ and $D(u_{\varepsilon})>0$, it follows that $g^{\prime}(t)>0$, and, consequently, $g$ is increasing in this interval. Thus,

[TABLE]

We conclude that

[TABLE]

and therefore

[TABLE]

Since $J_{A,V_{\mathcal{P}}}(tu_{\varepsilon})=g(t)-\frac{\lambda}{2p}t^{2p}B(u_{\varepsilon})$, we have

[TABLE]

But $\|u_{\varepsilon}\|_{A,V_{\mathcal{P}}}^{2}=\int_{\mathbb{R}^{N}}|\nabla u_{\varepsilon}|^{2}\textup{d}x+\int_{\mathbb{R}^{N}}(|A(x)|^{2}+V_{\mathcal{P}}(x)|u_{\varepsilon}|^{2})\textup{d}x$ implies

[TABLE]

Therefore, we conclude that

[TABLE]

Since, for all $\beta\geq 1$ and any $a,b>0$ we have $(a+b)^{\beta}\leq a^{\beta}+\beta(a+b)^{\beta-1}b$, considering

[TABLE]

it follows

[TABLE]

Taking into account (24) and (25), we conclude that

[TABLE]

We also have

[TABLE]

and

[TABLE]

We observe that, for $\varepsilon>0$ sufficiently small, it holds

[TABLE]

So,

[TABLE]

Therefore, we conclude that, for any $\varepsilon>0$ sufficiently small, we have

[TABLE]

Combining (27) with (29), for $\varepsilon$ sufficiently small, we have

[TABLE]

We claim that there is a positive constant $C_{0}$ such that, for all $\varepsilon>0$

[TABLE]

In fact, suppose that there is a sequence $(\varepsilon_{n})\subset\mathbb{R}$, $\varepsilon_{n}\to 0$ as $n\to\infty$, such that $t_{\varepsilon_{n}}\to 0$ as $n\to\infty$. Thus,

[TABLE]

Since $u_{\varepsilon_{n}}\in H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ is bounded and $t_{\varepsilon_{n}}\to 0$, as $n\to\infty$, we have $t_{\varepsilon_{n}}u_{\varepsilon_{n}}\to 0$ as $n\to\infty$, em $H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$.

The continuity of $J_{A,V_{\mathcal{P}}}$ implies that $J_{A,V_{\mathcal{P}}}(t_{\varepsilon_{n}}u_{\varepsilon_{n}})\to J_{A,V_{\mathcal{P}}}(0)=0$. Therefore,

[TABLE]

a contradiction that proves the claim.

From (26), (30) and (31) we conclude that, for some constant $C_{0}>0$ and $\varepsilon>0$ sufficiently small we have

[TABLE]

Thus,

[TABLE]

where $C_{1}=\frac{S_{A}(\varepsilon)^{2}}{2}$, $a(x)=|A(x)|^{2}+V_{p}(x)$ and $\eta=\min\{N-2,\frac{2N-\alpha}{2}\}$.

By direct computation we know that, for $\varepsilon<1$,

[TABLE]

Therefore,

[TABLE]

Since $a(x)$ is bounded, (33) and the last inequality imply that

[TABLE]

We are going to show that

[TABLE]

In order to do that, it suffices to show that

[TABLE]

and

[TABLE]

Assuming (36) and (37), let us proceed with our proof. Since

[TABLE]

from (35) follows

[TABLE]

for $\varepsilon>0$ sufficiently small.

Thus, (34) and (38) imply

[TABLE]

for $\varepsilon>0$ sufficiently small and fixed. Once (36) and (37) are verified, the proof of Case 1 is complete. $\hfill\Box$

We now prove (36).

Lemma 15**.**

If $\frac{N+2-\alpha}{N-2}<p<2_{\alpha}^{*}$ and $N=3,4$ or $\frac{2N-2-\alpha}{N-2}<p<2_{\alpha}^{*}$ and $N\geq 5$ it follows that

[TABLE]

*Proof. * This limit is evaluated considering the cases $N=3,$ $N=4$ and $N\geq 5$ as follows. We initially observe that direct computation allows us to conclude that

[TABLE]

where $\omega_{N}$ denotes the volume of the unit ball in $\mathbb{R}^{N}$.

Now, define

[TABLE]

the second equality being a consequence of (39).

$\bullet$ The case $\mathbf{N=3}$. In this case we have $5-\alpha<p<2^{*}_{\alpha}$ and therefore $5-\alpha-p<0$. We also observe that $0<\alpha<N$ implies $\min\{N-2,\frac{2N-\alpha}{2}\}=N-2=1$.

It is easy to show that

[TABLE]

Thus,

[TABLE]

Our claim follows.

$\bullet$ The case $\mathbf{N=4}$. In this case, $\frac{6-\alpha}{2}<p<2_{\alpha}^{*}$ implies $6-\alpha-2p<0$ and $\min\{N-2,\frac{2N-\alpha}{2}\}=N-2=2$, since $0<\alpha<4$.

Changing variables, we obtain

[TABLE]

So,

[TABLE]

Our claim follows by applying L’Hospital rule.

$\bullet$ The case $\mathbf{N\geq 5}$. We have

[TABLE]

It is easy to show that, if $N\geq 5$, then the integral

[TABLE]

converges.

There are two cases to be considered:

• •

$0<\alpha<4$ and $N\geq 5$;

• •

$\alpha\geq 4$ and $N\geq 5$.

Let us suppose $0<\alpha<4$ and $N\geq 5$. Since $0<\alpha<4$ we have

[TABLE]

Also $\frac{2N-\alpha-2}{N-2}<p<\frac{2N-\alpha}{N-2}$ implies $2N-\alpha-(N-2)p-2<0$. Therefore, $I_{\varepsilon}\to-\infty$ as $\varepsilon\to 0$.

Now we consider the case $\alpha\geq 4$ and $N\geq 5$. We have $N-2\geq\frac{2N-\alpha}{2}$ and therefore

[TABLE]

Since

[TABLE]

we conclude that $I_{\varepsilon}\to-\infty$. We are done. $\hfill\Box$

We now prove (37).

Lemma 16**.**

It holds

[TABLE]

*Proof. *Fix $\delta>0$ sufficiently large so that $U^{2}_{\varepsilon}(x)\leq\varepsilon^{1+\eta}$ if $|x|\geq\delta$. Since

[TABLE]

our proof is complete. $\hfill\Box$

Case 2. For $\lambda$ sufficiently large, $\frac{2N-\alpha}{N}<p\leq\frac{N+2-\alpha}{N-2}$ and $N=3,4$ or $\frac{2N-\alpha}{N}<p\leq\frac{2N-2-\alpha}{N-2}$ and $N\geq 5.$

Proof of Case 2. Define $g_{\lambda}:[0,+\infty)\to\mathbb{R}$ by

[TABLE]

We already know that $\displaystyle\lim_{t\to+\infty}g_{\lambda}(t)=-\infty$ as $t\to+\infty$ and $\displaystyle\max_{t\geq 0}g_{\lambda}(t)$ is attained at some $t_{\lambda}>0$ satisfying

[TABLE]

that is,

[TABLE]

since $g^{\prime}_{\lambda}(t_{\lambda})=0$. Thus $t_{\lambda}\to 0$ as $\lambda\to+\infty$ and

[TABLE]

Since $t_{\lambda}\to 0$ as $\lambda\to+\infty$ and $\frac{N+2-\alpha}{2(N-\alpha)}(S_{A})^{\frac{2N-\alpha}{N+2-\alpha}}>0$, we conclude that

[TABLE]

for $\lambda>0$ sufficiently large.

Therefore,

[TABLE]

for $\lambda>0$ sufficiently large. $\hfill\Box$

3.2. The proof of Theorem 1

Some arguments of this proof were adapted from the articles [2, 25].

Maintaining the notation introduced in subsection 3.1, consider the energy functional $I_{A,V}:H^{1}_{A,V}(\mathbb{R}^{N},\mathbb{C})\to\mathbb{R}$ given by

[TABLE]

We denote by $\mathcal{N}_{A,V}$ the Nehari Manifold related to $I_{A,V}$, that is,

[TABLE]

which is non-empty as a consequence of Theorem 13. As before, the functional $I_{A,V}$ satisfies the mountain pass geometry. Thus, there exists a sequence $(u_{n})\subset H^{1}_{A,V}(\mathbb{R}^{N},\mathbb{C})$ such that

[TABLE]

where $d_{\lambda}$ is the minimax level, also characterized by

[TABLE]

We stress that, as a consequence of ($V_{2}$), we have $I_{A,V}(u)<J_{A,V_{\mathcal{P}}}(u)$ for all $u\in H^{1}_{A,V}(\mathbb{R}^{N},\mathbb{C})$.

The next lemma compares the levels $d_{\lambda}$ and $c_{\lambda}$.

Lemma 17**.**

The levels $d_{\lambda}$ and $c_{\lambda}$ verify the inequality

[TABLE]

for all $\lambda>0$.

*Proof. * Let $u$ be the ground state solution of problem (10) and consider $\bar{t}_{u}>0$ such that $\bar{t}_{u}u\in\mathcal{N}_{A,V}$, that is

[TABLE]

It follows from $(V_{2})$ that

[TABLE]

Therefore,

[TABLE]

The second inequality was already known. $\hfill\Box$

Proof of Theorem 1. Let $(u_{n})$ be a $(PS)_{d_{\lambda}}$ sequence for $I_{A,V}$. As before, $(u_{n})$ is bounded in $H^{1}_{A,V}(\mathbb{R}^{N},\mathbb{C})$. Thus, there exists $u\in H^{1}_{A,V}(\mathbb{R}^{N},\mathbb{C})$ such that

[TABLE]

By the same arguments given in the proof of Theorem 13, $u$ is a ground state solution of problem (3), if $u\neq 0$.

Following close [2], we will show that $u=0$ cannot occur. Indeed, Lemma 6 yields

[TABLE]

since $W\in L^{\frac{N}{2}}(\mathbb{R}^{N},\mathbb{C})$ and $u_{n}\rightharpoonup 0$ in $H^{1}_{A,V}(\mathbb{R}^{N},\mathbb{C})$. So,

[TABLE]

showing that

[TABLE]

But, for $\varphi\in H^{1}_{A,V}(\mathbb{R}^{N},\mathbb{C})$ such that $\|\varphi\|_{A,V}\leq 1$, we have

[TABLE]

Thus,

[TABLE]

Let $t_{n}>0$ such that $t_{n}u_{n}\in\mathcal{M}_{A,V_{\mathcal{P}}}$. Mimicking the argument found in [1, 19, 29, 31], it follows that $t_{n}\to 1$ as $n\to\infty$. Therefore,

[TABLE]

Letting $n\to+\infty$, we get

[TABLE]

obtaining a contradiction with Lemma 17. This completes the proof of Theorem 1. $\hfill\Box$

4. The case $f(u)=|u|^{p-1}u$

4.1. The periodic problem

In this subsection we deal with problem (6) for $f(u)$ as above, that is,

[TABLE]

where $1<p<2^{*}-1$.

We observe that in this case the energy functional $J_{A,V_{\mathcal{P}}}$ is given by

[TABLE]

where, as before

[TABLE]

By the Sobolev immersion (7) and the Hardy-Littlewood-Sobolev inequality, we have that $J_{A,V_{\mathcal{P}}}$ is well defined.

Definition 4.1**.**

A function $u\in H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ is a weak solution of (42) if

[TABLE]

for all $\psi\in H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$.

As before, we see that critical points of $J_{A,V_{\mathcal{P}}}$ are weak solutions of (42) and

[TABLE]

We obtain that $J_{A,V_{\mathcal{P}}}$ satisfies the geometry of the mountain pass (see the proof of Lemma 7).

As in Section 3, the mountain pass theorem without the PS condition yields a sequence $(u_{n})\subset H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ such that

[TABLE]

where $c_{\lambda}=\inf_{\alpha\in\Gamma}\max_{t\in[0,1]}J_{A,V_{\mathcal{P}}}(\gamma(t))$ and $\Gamma=\left\{\gamma\in C^{1}\left([0,1],H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})\right)\,:\,\gamma(0)=0,\,J_{A,V_{\mathcal{P}}}(\gamma(1))<0\right\}$.

Considering the Nehari manifold $J_{A,V_{\mathcal{P}}}$

[TABLE]

by proceeding as in the proof of Lemma 11 we obtain

Lemma 18**.**

There exists a unique $t_{u}=t_{u}(u)>0$ such that $t_{u}u\in\mathcal{M}_{A,V_{\mathcal{P}}}$ for all $u\in H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})\setminus\{0\}$ and $J_{A,V_{\mathcal{P}}}(t_{u}u)=\displaystyle\max_{t\geq 0}J_{A,V_{\mathcal{P}}}(tu)$. Moreover $c_{\lambda}=c^{*}_{\lambda}=c^{**}_{\lambda}$, where

[TABLE]

Lemma 19**.**

Suppose that $u_{n}\rightharpoonup u_{0}$ and consider

[TABLE]

and

[TABLE]

for $\psi\in C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{C})$. Then $B^{\prime}(u_{n})\cdot\psi\to B^{\prime}(u_{0})\cdot\psi$ and $D^{\prime}(u_{n})\cdot\psi\to D^{\prime}(u_{0})\cdot\psi$ as $n\to\infty.$

Lemma 20**.**

If $(u_{n})\subset H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ is a $(PS)_{\lambda}$ sequence for $J_{A,V_{\mathcal{P}}}$, then $(u_{n})$ is bounded. In addition, if $u_{n}\rightharpoonup u$ weakly in $H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$, as $n\to\infty,$ then $u$ is ground state solution for problem (42).

Lemma 21**.**

If $(u_{n})\subset H^{1}_{A,V_{\mathcal{P}}}(\mathbb{R}^{N},\mathbb{C})$ is a sequence $(PS)_{c_{\lambda}}$ for $J_{A,V_{\mathcal{P}}}$ such that

[TABLE]

with

[TABLE]

then there exists a sequence $(y_{n})\in\mathbb{R}^{N}$ and constants $R,\theta>0$ such that

[TABLE]

where $B_{r}(y)$ denotes the ball in $\mathbb{R}^{N}$ of center at $y$ and radius $r>0$.

The proof of Lemmas 19, 20 and 21 is similar to that of Corollary 9 Lemmas 10 and 12, respectively.

Lemma 22**.**

Let $1<p<2^{*}-1$ and $u_{\varepsilon}$ as defined in (23). Then, there exists $\varepsilon$ such that

[TABLE]

provided that either

1. $(i)$

$3<p<5$, $N=3$ and $\lambda>0;$ 2. $(ii)$

$p>1$, $N\geq 4$ and $\lambda>0$; 3. $(iii)$

$1<p\leq 3$, $N=3$ and $\lambda$ sufficiently large.

*Proof. * Consider, for the cases ($i$) and ($ii$) the function $f:[0,+\infty)\to\mathbb{R}$ defined by

[TABLE]

and proceed as in the proof of Case 1, Lemma 14.

In the case of $1<p\leq 3$, $N=3$ and $\lambda$ sufficiently large, consider $g_{\lambda}:[0,+\infty)\to\mathbb{R}$ defined by

[TABLE]

and proceed as in the proof of Case 2, Lemma 14. $\hfill\Box$

Similar to the proof of Theorem 13, we now state our result about the periodic problem (42).

Theorem 23**.**

Under the hypotheses already stated on $A$ and $\alpha$, suppose that $(V_{1})$ is valid. Then problem (42) has at least one ground state solution if either

1. $(i)$

$3<p<5$, $N=3$ and $\lambda>0;$ 2. $(ii)$

$p>1$, $N\geq 4$ and $\lambda>0$; 3. $(iii)$

$1<p\leq 3$, $N=3$ and $\lambda$ sufficiently large.

4.2. Proof of Theorem 2

Some arguments of this proof were adapted from the proof of Theorem 1 below , that in turn were adapted from articles [2, 25].

Maintaining the notation already introduced, consider the functional $I_{A,V}:H^{1}_{A,V}(\mathbb{R}^{N},\mathbb{C})\to\mathbb{R}$ defined by

[TABLE]

for all $u\in H^{1}_{A,V}(\mathbb{R}^{N},\mathbb{C})$.

We denote by $\mathcal{N}_{A,V}$ the Nehari Manifold related to $I_{A,V}$, that is,

[TABLE]

which is non-empty as a consequence of Theorem 23. As before, the functional $I_{A,V}$ satisfies the mountain pass geometry. Thus, there exists a $(PS)_{d_{\lambda}}$ sequence $(u_{n})\subset H^{1}_{A,V}(\mathbb{R}^{N},\mathbb{C})$, that is, a sequence satisfying

[TABLE]

where $d_{\lambda}$ is the minimax level, also characterized by

[TABLE]

As in the Section 3, we have $I_{A,V}(u)<J_{A,V_{\mathcal{P}}}(u)$ for all $u\in H^{1}_{A,V}(\mathbb{R}^{N},\mathbb{C})$ as a consequence of ($V_{2}$).

Similar to the proof of Lemma 17 we have the following conclusion that shows as important inequality involving the levels $d_{\lambda}$ and $c_{\lambda}$, what completes the proof of Theorem 2.

Lemma 24**.**

The levels $d_{\lambda}$ and $c_{\lambda}$ verify the inequality

[TABLE]

for all $\lambda>0$.

5. The case $f(u)=|u|^{2^{*}-2}u$

5.1. Proof of Theorem 3

As observed by Gao and Yang [20], the proof of Theorem 3 is analogous to the proof of Theorem 1. The principal distinction is that the $(PS)_{c_{\lambda}}$ condition holds true below the level $\frac{1}{N}S^{\frac{N}{2}}$. It follows from [31, Lemma 1.46] that

[TABLE]

and

[TABLE]

So, we have

[TABLE]

since

[TABLE]

Observe that the last result is a consequence of

[TABLE]

and

[TABLE]

The rest of the proof is omitted here. $\hfill\Box$

Acknowledgements. The authors thank Prof. G. M. Figueiredo for many useful conversations and suggestions.