An Existence Result for a Class of Magnetic Problems in Exterior Domains

Abstract

In this paper we deal with the existence of solutions for the following class of magnetic semilinear Schrödinger equation

$$\begin{aligned} (P) \qquad \qquad \left\{ \begin{aligned}&(-i\nabla + A(x))^2u +u = |u|^{p-2}u,\;\;\text{ in }\;\;\Omega ,\\&u=0\;\;\text{ on }\;\;\partial \Omega , \end{aligned} \right. \end{aligned}$$

where $N \ge 3$, $\Omega \subset {\mathbb {R}}^N$ is an exterior domain, $p\in (2, 2^*)$ with $2^*=\frac{2N}{N-2}$, and $A: {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N$ is a continuous vector potential verifying $A(x) \rightarrow 0\;\;\text{ as }\;\;|x|\rightarrow \infty .$

1 Introduction

In this paper we investigate the existence of solutions for the following magnetic semilinear Schrödinger equation

$$\begin{aligned} (P) \qquad \qquad \left\{ \begin{aligned}&(-i\nabla + A(x))^2u +u= |u|^{p-2}u\;\;\text{ in }\;\;\Omega ,\\&u=0\;\;\text{ on }\;\;\partial \, \Omega , \end{aligned} \right. \end{aligned}$$

where $N \ge 3$, $p\in (2, 2^*)$, $2^*:=\frac{2N}{N-2}$ is the critical Sobolev exponent, $\Omega \subset {\mathbb {R}}^N$ is an exterior domain, i.e. $\Omega$ is an unbounded domain with smooth boundary $\partial \Omega \ne \emptyset$ such that ${\mathbb {R}}^N {\setminus } \Omega$ is bounded, and $A\in C({\mathbb {R}}^N, {\mathbb {R}}^N)$ satisfies

$$\begin{aligned} \qquad \qquad \qquad \qquad A(x) \rightarrow 0\;\;\text{ as }\;\;|x|\rightarrow \infty . \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (A) \end{aligned}$$

During the past years there has been a considerable interest in the existence of solutions for elliptic equations in exterior domains, more precisely, for problems of the type

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta u + u = f(u), \quad \text{ in } \,\,\, \Omega , \\ u \in H^1_0(\Omega ), \end{array} \right. \end{aligned}$$

(1.1)

where f is a continuous function satisfying some technical conditions. The main difficulty in dealing with (1.1) is the lack of compactness of the Sobolev embedding due to the unboundedness of the domain. In order to overcome this difficulty, in some papers, authors assumed certain type of symmetry on $\Omega$; see for instance [9, 23] and [25].

In [12], Benci and Cerami studied the existence of nontrivial solutions for the problem

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta u + \lambda u = |u|^{p-2}u, \quad \text{ in } \,\,\, \Omega , \\ u \in H^1_0(\Omega ), \end{array} \right. \end{aligned}$$

(1.2)

in an exterior domain $\Omega$ without assuming symmetry, with $2 \le p < 2^*$ and $\lambda >0$. In that article, they proved that (1.2) does not have a ground state solution and this fact yields a series of difficulties. The key idea exploited by the authors was to analyze the behavior of Palais-Smale sequences, obtaining a precise estimate of the energy levels where the Palais-Smale condition fails. The authors proved that if $p=1+\frac{8}{N}$ for $N=5, 6, 7$ or $p<\frac{2(N-1)}{N-2}$ for $N=3, 4$,

• there exists $\lambda _{*}>0$ such that, for every $\lambda \in (0, \lambda _{*})$, (1.2) has at least one positive solution,

• for every $\lambda$ there exists a $\rho =\rho (\lambda )$ such that if ${\mathbb {R}}^{N}{\setminus } \Omega \subset B(x_{0}, \rho )$, with $x_{0}\in {\mathbb {R}}^{N}{\setminus } \Omega$, (1.2) has at least one positive solution.

Later, existence results were obtained for more general problems

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta u + \lambda u = f(x,u) \quad \text{ in } \,\,\, \Omega , \\ u \in H^1_0(\Omega ), \end{array} \right. \end{aligned}$$

(1.3)

where f is a continuous function satisfying

$$\begin{aligned} \lim _{|x|\rightarrow +\infty } f(x,t)= f_{\infty }(t), \,\,\,\text{ for } \text{ all } \,\, t \in {\mathbb {R}}. \end{aligned}$$

In [9], Bahri and Lions studied (1.3) with $f(x,t)=b(x) |t|^{p-2}t$ where $b(x)\rightarrow b >0$ as $|x|\rightarrow +\infty$. Using topological arguments, they showed that (1.3) has a solution when $\Omega$ is an arbitrary exterior domain, for all $\lambda >0$. In the autonomous case, using the technique introduced in [9], Li and Zheng [28] proved that (1.3) possesses at least one positive solution, with f asymptotically linear satisfying some assumptions, in particular, a property of convexity (see also [21]). Whereas in [29], Maia and Pellacci established an existence result without the hypothesis of convexity.

In the above-mentioned papers, a key point to prove the results of existence is the uniqueness, up to a translation, of the positive solution for the “equation at infinity” associated with (1.3) given by

$$\begin{aligned} -\Delta u + \lambda u = f_{\infty }(u) \quad \text{ in } \,\,\, {\mathbb {R}}^N. \end{aligned}$$

(1.4)

However, when the exterior domain is the exterior of a ball, more precisely $\Omega ={\mathbb {R}}^{N} {\setminus } B(0, R)$ for some $R>0$, it is possible to explore some groups of rotation in order to get multiple solutions without using the uniqueness of solutions of the limit problem when R is large enough, see for example Cao and Noussair [16] and Clapp, Maia and Pellacci [22]. In [2], Alves and de Freitas studied the existence of a positive solution for a class of elliptic problems in exterior domains involving critical growth. Finally, we mention a recent paper due to Alves, Molica Bisci and Torres Ledesma [5], in which a fractional elliptic equation with Dirichlet-type condition set in an exterior domain is considered.

In the last years time-independent magnetic Schrödinger equations in bounded domains or in whole of ${\mathbb {R}}^N$ have received a special attention. A basic motivation to study these equations stems from the search of standing wave solutions for the time-dependent nonlinear Schrödinger equation of the type

$$\begin{aligned} i h \dfrac{ \partial \psi }{ \partial t } = \left( \dfrac{h}{i} \nabla - A(z) \right) ^2 \psi + U(z)\psi - f(|\psi |^2)\psi ,~~z \in {\mathcal {D}}, \, t\in {\mathbb {R}}, \end{aligned}$$

where ${\mathcal {D}} \subset {\mathbb {R}}^N$, with $N \ge 2$, is a smooth domain, the function $\psi$ takes values in ${\mathbb {C}}$, h is the Planck constant, i is the imaginary unit and $A:{\mathbb {R}}^N \rightarrow {\mathbb {R}}^N$ denotes a magnetic potential. For the interested reader in this subject, we cite the papers by Alves, Figueiredo and Furtado [3, 4], Ambrosio [6, 7], Arioli and Szulkin [8], Barile [10, 11], Chabrowski and Szulkin [17], Cingolani [18], Cingolani and Clapp [19], Cingolani, Jeanjean and Secchi [20], Ding and Liu [24], Esteban and Lions [26] and the references therein.

After a careful bibliography review, we did not find any paper concerned with magnetic semilinear Schrödinger equations in exterior domains. Motivated by this fact and the above-mentioned papers, the aim of this paper is to give a first existence result for (P). We emphasize that the main difficulty in dealing with this type of problem is related to the uniqueness, up to a translation, of the solution for the limit problem. Recently, in a very interesting paper due to Bonheure, Nys and Van Schaftingen [14], we found a partial answer for this question when the magnetic field A satisfies some technical conditions; see [14, Theorem 1].

The main result of this paper can be stated as follows:

Theorem 1.1

Suppose that (A) holds. Then, there exist $\rho _0>0$ and $\epsilon >0$ such that if ${\mathbb {R}}^N {\setminus } \Omega \subset B(0, \rho )$, $\rho < \rho _0$ and $\Vert A\Vert _\infty \le \epsilon$, then problem (P) has at least one weak solution.

The proof of Theorem 1.1 will be done done via variational methods inspired by [2] and [12]. However, with respect to [2, 12], a more careful analysis will be needed and some refined estimates will be given. The diamagnetic inequality in [26] will play a fundamental role.

We point out that Theorem 1.1 complements the study in magnetic semilinear Schrödinger equations, in the sense that we obtain an existence result for a magnetic Schrödinger equation in an exterior domain.

The paper is organized as follows. In Sect. 2, we introduce suitable function spaces and collect some useful results concerning the limit problem that we will work with. In Sect. 3, we establish a compactness result in the spirit of [12] for the energy functional associated with problem (P). In Sect. 4, we show some technical estimates that will be used in Sect. 5, where Theorem 1.1 is proved.

Notations: In this paper, we use the following notations:

• For $q \in (2, 2^*)$, we define $q'$ as the conjugate exponent of q, that is, $q' := \frac{q}{q-1}.$

• The usual norm of the Lebesgue spaces $L^t(\Omega )$ for $t \in [1,\infty ]$, will be denoted by $|\,.\,|_{t}$, and the norm of the Sobolev space $H^1_0(\Omega )$, by $\Vert \,.\,\Vert$;

• C denotes (possibly different) any positive constant.

2 Preliminary Results and the Limit Problem

In what follows, we denote by $H_A^1(\Omega , {\mathbb {C}})$ the Hilbert space obtained by the closure of $C_0^{\infty }(\Omega ,{\mathbb {C}})$ under the scalar product

$$\begin{aligned} \langle u,v \rangle :=\Re \left( \int _{\Omega }(\nabla _A u(x) \overline{\nabla _A v(x)}+u(x)\overline{v(x)})\,dx\right) , \end{aligned}$$

where $\Re (w)$ denotes the real part of $w \in {\mathbb {C}}$, ${\overline{w}}$ is its complex conjugate, $\nabla _Au:=(D_1u,D_2u,...,D_Nu)$ where $D_j:=-i\partial _j+A_j(x)$, for $j=1,2,...,N.$ The norm induced by this inner product is given by

$$\begin{aligned} \Vert u\Vert :=\left( \int _{\Omega }(|\nabla _A u(x)|^{2}+|u(x)|^2)\,dx\right) ^{\frac{1}{2}}. \end{aligned}$$

We also consider the Hilbert space $H_A^1({\mathbb {R}}^N, {\mathbb {C}})$ defined as

$$\begin{aligned} H_A^1({\mathbb {R}}^N, {\mathbb {C}}):=\{u\in L^2({\mathbb {R}}^N, {\mathbb {C}}): \nabla _A u\in L^2({\mathbb {R}}^N, {\mathbb {C}}^N)\} \end{aligned}$$

endowed with the scalar product

$$\begin{aligned} \langle u,v\rangle _A := \Re \left( \int _{{\mathbb {R}}^N} (\nabla _Au(x) \cdot {{\overline{\nabla _Av(x)}}} + u(x)\overline{v(x)})dx \right) . \end{aligned}$$

Then we can define the norm

$$\begin{aligned} \Vert u\Vert _{A} := \left( \int _{{\mathbb {R}}^N} \left( |\nabla _Au(x)|^2 + |u(x)|^2 \right) dx \right) ^{\frac{1}{2}}. \end{aligned}$$

By [26, Proposition 2.1-(i)], we know that $C_{c}^{\infty }({\mathbb {R}}^N, {\mathbb {C}})$ is dense in $H_{A}^{1}({\mathbb {R}}^N, {\mathbb {C}})$. A direct computation shows that $H_A^1(\Omega , {\mathbb {C}}) \subset H_{A}^{1}({\mathbb {R}}^N, {\mathbb {C}})$.

As proved in [26, Sect. 2], for any $u \in H_A^1({\mathbb {R}}^N, {\mathbb {C}})$, there holds the diamagnetic inequality, namely

$$\begin{aligned} |\nabla |u|| \le \left| \Re \left( \nabla _A u \;\mathrm{sign}(u)\right) \right| \le |\nabla u_A|,\;\;\text{ a.e. } \text{ in } \, {\mathbb {R}}^N, \end{aligned}$$

(2.1)

where

$$\begin{aligned} \mathrm{sign}(u)= \left\{ \begin{array}{ll} \frac{\overline{u(x)}}{|u(x)|} &{} \text{ if } u(x)\ne 0,\\ 0 &{} \text{ if } u(x)= 0. \end{array} \right. \end{aligned}$$

Hence, if $u\in H_{A}^{1}({\mathbb {R}}^N, {\mathbb {C}})$, then $|u|\in H^1({\mathbb {R}}^N,{\mathbb {R}})$. Furthermore, as a consequence of the diamagnetic inequality, we have that the embedding

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

(2.2)

is continuous for any $s\in [2, 2^*]$.

2.1 Limit Problem

In this subsection, we consider the scalar limit problem associated with (P), namely

$$\begin{aligned} (P_0)\qquad \qquad {\left\{ \begin{array}{ll} -\Delta u + u = |u|^{p-2}u\;\;\text{ in }\;\;{\mathbb {R}}^N,&{}\\ u\in H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}}),&{} \end{array}\right. } \end{aligned}$$

where $p\in (2,2^*)$. The reader is invited to see that $u=u_{1}+\imath u_{2}$, with $u_{1}$ and $u_{2}$ real valued, is a solution of $(P_0)$ if and only if $u_{1}$ and $u_{2}$ solve the following elliptic system

$$\begin{aligned} (S_0) \qquad \qquad \left\{ \begin{array}{l} -\Delta u_1+u_1=\left( \sqrt{|u_1|^2+|u_2|^2}\right) ^{p-2}u_1, \;\; \text{ in } \;\; {\mathbb {R}}^N, \\ \\ -\Delta u_2+u_2=\left( \sqrt{|u_1|^2+|u_2|^2}\right) ^{p-2}u_2, \;\; \text{ in } \;\; {\mathbb {R}}^N, \end{array} \right. \end{aligned}$$

which is a system of the gradient type.

Note that, the solutions of $(P_0)$ are critical points of the functional

$$\begin{aligned} \begin{aligned} I_0 : H^{1}({\mathbb {R}}^N, {\mathbb {C}})&\rightarrow {\mathbb {R}}\\ u&\rightarrow I_0(u):=\frac{1}{2}\int _{{\mathbb {R}}^N}(|\nabla u(x)|^2 + |u(x)|^2)dx - \frac{1}{p}\int _{{\mathbb {R}}^N}|u(x)|^p dx. \end{aligned}\nonumber \\ \end{aligned}$$

(2.3)

If $c_0$ denotes the mountain pass level of $I_{0}$ and ${\mathcal {N}}_0$ is the Nehari manifold defined as

$$\begin{aligned} {\mathcal {N}}_0 := \{u\in H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}}){\setminus } \{0\}: I'_0(u)u = 0\}, \end{aligned}$$

it is well-known (see [30]) that

$$\begin{aligned} c_0 = c^{*}_{0}:=\inf _{u\in {\mathcal {N}}_0} I_0(u), \end{aligned}$$

from where it follows that $c_0$ is the least energy of $(P_0)$. We recall that $u\in H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})$ is a least energy solution of $(P_0)$ if $I_{0}(u)=c^{*}_{0}$ and $I'_{0}(u)=0$, and $c_{0}^{*}$ is called the least energy of $(P_{0})$.

Lemma 2.1

The following fact holds: u is a least energy solution of $(P_0)$ if, and only if, $v(x):=|u(x)|\in H^1({\mathbb {R}}^N)$ is a least energy solution of

$$\begin{aligned} -\Delta v + v = |v|^{p-2}v, \text{ in } {\mathbb {R}}^{N}, \, \; v>0. {(P_\infty )} \end{aligned}$$

Moreover, $(P_0)$ and $(P_\infty )$ have the same least energy.

Proof

The proof can be done as in [24, Lemma 2.5]. $\square$

Lemma 2.2

The following facts hold:

• (1) ${c}_{0} >0$ is the least energy of $(P_\infty )$;

• (2) ${\mathcal {N}}_0 \ne \emptyset$;

• (3) ${c}_0$ is attained, and the set

  $$\begin{aligned} {\mathcal {R}}_0:=\{u\in {\mathcal {N}}_0: { I}_0(u) = {c}_0,\;\;u(0) = \Vert u\Vert _\infty \} \end{aligned}$$

  is compact in $H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})$;

• (4) There exists $C, c>0$ such that

  $$\begin{aligned} |u(x)| \le Ce^{-c|x|}\;\;\forall x\in {\mathbb {R}}^N, u\in {\mathcal {R}}_0. \end{aligned}$$

Proof

See [24, Lemma 2.6]. $\square$

Let $I_\infty :H^{1}({\mathbb {R}}^N,{\mathbb {R}}) \rightarrow {\mathbb {R}}$ be the energy functional given by

$$\begin{aligned} I_\infty (w):=\frac{1}{2}\int _{{\mathbb {R}}^N}(|\nabla w(x)|^2 + |w(x)|^2)dx - \frac{1}{p}\int _{{\mathbb {R}}^N}|w(x)|^p dx. \end{aligned}$$

Note that $I_{\infty }$ is defined on $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$ while $I_{0}$ is defined on $H^{1}({\mathbb {R}}^{N}, {\mathbb {C}})$. If ${c}_\infty$ denotes the mountain pass level of $I_{\infty }$ and ${\mathcal {N}}_\infty$ is the Nehari manifold given by

$$\begin{aligned} {\mathcal {N}}_\infty := \{w\in H^{1}({\mathbb {R}}^N,{\mathbb {R}}){\setminus } \{0\}: {I}'_{\infty }(w)w = 0\}, \end{aligned}$$

then (see [30])

$$\begin{aligned} c_{\infty } =c^{*}_{\infty }:= \inf _{w\in {\mathcal {N}}_\infty } {I}_\infty (w). \end{aligned}$$

Let $\varphi \in H^1({\mathbb {R}}^N,{\mathbb {R}})$ be a positive ground state solution of $(P_\infty )$, that is,

$$\begin{aligned} I_\infty (\varphi ) = c^{*}_\infty =c_{\infty } \quad \text{ and }\quad I'_\infty (\varphi ) = 0. \end{aligned}$$

(2.4)

The function $\varphi$ can be chosen radial and decreasing with respect to |x|; see [13].

An immediate consequence of Lemmas 2.1 and 2.2, we have the equality $c_0=c_\infty$, and so, $\varphi$ is also a ground state solution for $(P_0)$.

Before concluding this section, we state an important result that is a particular case of a result due to Bonheure, Nys and Van Schaftingen [14, Theorem 1], which will be crucial in our approach.

Theorem 2.3

There is $\kappa >0$ such that if $w \in H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})$ is a critical point of $I_0$ with $I_0(w) \le c_0 +\kappa$, then there are $a \in {\mathbb {R}}^N$ and $\theta \in {\mathbb {R}}$ such that $w(x)=e^{i \theta }\varphi (x-a)$, for all $x \in {\mathbb {R}}^N$. Hence, $I_0(w)=c_0$.

3 A Compactness Result for Energy Functional

In this section, we study some compactness property of the energy functional $I_A: H_A^1(\Omega , {\mathbb {C}}) \rightarrow {\mathbb {R}}$ associated with (P) given by

$$\begin{aligned} I_A(u) := \frac{1}{2}\int _{\Omega }(|\nabla _Au(x)|^2 + |u(x)|^2)dx - \frac{1}{p}\int _{\Omega }|u(x)|^pdx. \end{aligned}$$

In the sequel, we denote by $c_A$ the mountain pass level of I that satisfies the equality below

$$\begin{aligned} c_A = \inf _{u\in {\mathcal {N}}_A}I_A(u), \end{aligned}$$

(3.1)

where ${\mathcal {N}}_A$ is the Nehari manifold of $I_A$ given by

$$\begin{aligned} {\mathcal {N}}_A := \{u\in H_A^1(\Omega , {\mathbb {C}}) {\setminus } \{0\}: I'_A(u)u=0\}. \end{aligned}$$

Theorem 3.1

The equality $c_0 = c_A$ holds true. Hence, there is no $u\in H_A^1(\Omega , {\mathbb {C}})$ such that

$$\begin{aligned} I_A(u) = c_A\quad \text{ and }\quad I'_A(u) =0, \end{aligned}$$

and so, problem (P) has no ground state solution.

Proof

By using the diamagnetic inequality (2.1),

$$\begin{aligned} c_0 \le c_A. \end{aligned}$$

(3.2)

Recalling that $\varphi$ satisfies (2.4) and that $c_0=c_\infty$, we have that

$$\begin{aligned} I_0(\varphi ) = c_0 \quad \text{ and }\quad I'_0(\varphi )=0. \end{aligned}$$

Let $(y_n)\subset \Omega$ be a sequence such that $|y_n|\rightarrow +\infty$, and $\rho$ be the smallest positive number satisfying

$$\begin{aligned} {\mathbb {R}}^N {\setminus } \Omega \subset B(0,\rho ) = \{x\in {\mathbb {R}}^N: |x|< \rho \}. \end{aligned}$$

Furthermore, let us define $\zeta \in C^\infty ({\mathbb {R}}^N, [0,1])$ by

$$\begin{aligned} \zeta (x) :=\xi \left( \frac{|x|}{\rho } \right) , \end{aligned}$$

where $\xi :[0, +\infty )\rightarrow [0,1]$ is a smooth non-decreasing function such that

$$\begin{aligned} \xi (t) = 0\;\;\forall t\le 1\quad \text{ and }\quad \xi (t) = 1,\;\;\forall t\ge 2. \end{aligned}$$

Now, we consider the sequence

$$\begin{aligned} \psi _n(x) := \zeta (x) \varphi (x-y_n), \end{aligned}$$

and fix $t_n>0$ such that $t_n \psi _n\in {\mathcal {N}}_A$. By making the change of variable $z = x-y_n$, we deduce

$$\begin{aligned} \Vert \zeta \varphi (\cdot -y_n) - \varphi (\cdot -y_n)\Vert _{L^p({\mathbb {R}}^N)} = \left( \int _{{\mathbb {R}}^N}|(\zeta (x+y_n) - 1)\varphi (x)|^pdx\right) ^{1/p}. \end{aligned}$$

Since $|y_n|\rightarrow +\infty$ as $n\rightarrow +\infty$, it is easy to check that

$$\begin{aligned} |(\zeta (x+y_n) - 1)\varphi (x)|^p \rightarrow 0\;\; \text{ a.e. } x\in {\mathbb {R}}^N. \end{aligned}$$

As

$$\begin{aligned} |(\zeta (x+y_n) - 1)\varphi (x)|^p \le |\zeta (x+y_n) - 1|^p|\varphi (x)|^p \le 2^p|\varphi (x)|^p\in L^1({\mathbb {R}}^N), \end{aligned}$$

the dominated convergence theorem ensures that

$$\begin{aligned} \int _{{\mathbb {R}}^N} |\zeta (x+y_n)\varphi (x) - \varphi (x)|^pdx \rightarrow 0\;\;\text{ as }\;\;n\rightarrow +\infty , \end{aligned}$$

or equivalently,

$$\begin{aligned} \zeta (\cdot + y_n)\varphi \rightarrow \varphi \;\;\text{ in }\;\;L^p({\mathbb {R}}^N). \end{aligned}$$

(3.3)

Recalling that $\zeta (x) = 0$ for $x\in {\mathbb {R}}^N {\setminus } \Omega$, by the previous discussion,

$$\begin{aligned} \begin{aligned} \Vert \zeta \varphi (\cdot -y_n)\Vert _{L^p(\Omega )}&= \left( \int _{\Omega } |\zeta (x)\varphi (x-y_n)|^pdx \right) ^{1/p}\\&=\left( \int _{{\mathbb {R}}^N} |\zeta (x+y_n)\varphi (x)|^pdx\right) ^{1/p} = \left( \int _{{\mathbb {R}}^N}|\varphi (x)|^pdx\right) ^{1/p} + o_n(1). \end{aligned} \end{aligned}$$

On the other hand, for each $j \in \{1,..,N\}$,

$$\begin{aligned} \begin{aligned}&\int _{{\mathbb {R}}^N}|D_j(\zeta (x)\varphi (x-y_n))|^2dx = \int _{{\mathbb {R}}^N} |(-i\partial _j + A_j(x))(\zeta (x)\varphi (x-y_n))|^2dx\\&\quad = \int _{{\mathbb {R}}^N} \left( |(\partial _j(\zeta (x+y_n)\varphi (x))|^2+|A_j(x+y_n)\zeta (x+y_n)\varphi (x)|^2\right) dx\\&\quad = \int _{{\mathbb {R}}^N} \left( |\zeta (x+y_n)\partial _j \varphi (x) + \varphi (x)\partial _j \zeta (x+y_n)|^2+|A_j(x+y_n)\zeta (x+y_n)\varphi (x)|^2\right) dx. \end{aligned} \end{aligned}$$

Since $\zeta (x+y_n) \rightarrow 1$, $A(x + y_n) \rightarrow 0$ as $n\rightarrow +\infty$ and $\partial _j \zeta (x+y_n) \rightarrow 0$, the dominated convergence theorem implies that

$$\begin{aligned} \int _{{\mathbb {R}}^N}|D_j(\zeta (x)\varphi (x-y_n))|^2dx = \int _{{\mathbb {R}}^N} |\partial _j \varphi (x)|^2dx + o_n(1), \end{aligned}$$

and so,

$$\begin{aligned} \int _{{\mathbb {R}}^N}|\nabla _A(\zeta (x)\varphi (x-y_n))|^2dx = \int _{{\mathbb {R}}^N} |\nabla \varphi (x)|^2dx + o_n(1). \end{aligned}$$

(3.4)

By the previous analysis together with the fact that $\varphi \in {\mathcal {N}}_0$, using translation invariance it is not difficult to prove that $t_n \rightarrow 1$. Thus, by definition of $c_A$, (3.3) and (3.4), we get

$$\begin{aligned} c_A \le I_A(t_n\psi _n) = c_0+ o_n(1), \end{aligned}$$

that leads to

$$\begin{aligned} c_A \le c_0. \end{aligned}$$

(3.5)

From (3.2) and (3.5),

$$\begin{aligned} c_0= c_A. \end{aligned}$$

(3.6)

Now, suppose by contradiction that there is $v_0\in H_A^1(\Omega , {\mathbb {C}})$ such that

$$\begin{aligned} I_A(v_0) = c_A\quad \text{ and }\quad I'_A(v_0) =0. \end{aligned}$$

By (2.1), (3.6), and recalling that $c_0=c_\infty$, we deduce that the function $w=|v_0| \in H_{0}^{1}({\mathbb {R}}^N, {\mathbb {R}})$ is a ground state solution of $(P_\infty )$, that is,

$$\begin{aligned} -\Delta w + w = |w|^{p-2}w,\quad \text{ in } \quad {\mathbb {R}}^N. \end{aligned}$$

Since $w \ge 0$ in ${\mathbb {R}}^N$ and $w \not =0$, the strong maximum principle ensures that $w(x)>0$ for all $x \in {\mathbb {R}}^N$, which is impossible because $v_0 = 0$ in ${\mathbb {R}}^N {\setminus } \Omega$. $\square$

3.1 A Compactness Lemma

In this section, we prove a compactness result involving the energy functional ${I}_A$ associated with (P). In order to do this, we need to consider the energy functional ${I}_0: H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})\rightarrow {\mathbb {R}}$ associated with $(P_0)$ defined as in (2.3). With the above notations, we are able to prove the following compactness result.

Lemma 3.2

Let $(u_n)\subset H_A^1(\Omega , {\mathbb {C}})$ be a sequence such that

$$\begin{aligned} {I}_A(u_n) \rightarrow c\quad \text{ and }\quad {I}'_A(u_n) \rightarrow 0\;\;\text{ as }\;\;n\rightarrow +\infty . \end{aligned}$$

(3.7)

Then, up to a subsequence, there exists a weak solution $u^0\in H_A^1(\Omega , {\mathbb {C}})$ of (P) such that

$$\begin{aligned} u_n \rightarrow u^0 \quad \text{ in } \quad H^{1}({\mathbb {R}}^N) \end{aligned}$$

or there are k functions $(u_{n}^{j})\subset H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})$, $1\le j \le k$ such that

$$\begin{aligned} \begin{aligned}&u_{n}^{0} = u_n \rightharpoonup u^0\;\;\text{ in }\;\;H_A^1(\Omega , {\mathbb {C}}),\\&\quad u_{n}^{j} \rightharpoonup u^j\;\;\text{ in }\;\;H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})\;\;\text{ for }\;\;1\le j \le k, \end{aligned} \end{aligned}$$

where $u^j$ are nontrivial weak solutions of $(P_0)$, for every $1\le j \le k$. Furthermore

$$\begin{aligned} \Vert u_n\Vert ^2 \rightarrow \Vert u_0\Vert ^2 + \sum _{j=1}^{k}\Vert u^j\Vert _{0}^{2} \end{aligned}$$

and

$$\begin{aligned} {I}_A(u_n) \rightarrow {I}_A(u^0) + \sum _{j=1}^{k}{I}_0(u^j). \end{aligned}$$

Proof

We proceed by steps.

Step 1. The sequence $(u_n)$ is bounded in $H_A^1(\Omega , {\mathbb {C}})$.

By (3.7),

$$\begin{aligned} \langle {I}'_A(u_n), \psi \rangle= & {} \Re \int _{\Omega }( \nabla _Au_n(x) \cdot \overline{\nabla _A \psi (x)} + u_n(x) \overline{\psi (x)} - |u_n(x)|^{p-2}u_n (x)\overline{\psi (x)} )dx \nonumber \\= & {} o(1) \end{aligned}$$

(3.8)

for any $\psi \in H_A^1(\Omega , {\mathbb {C}})$ and

$$\begin{aligned} \frac{1}{2} \int _{\Omega } \left( |\nabla _Au_n(x)|^2 + |u_n(x)|^2 \right) dx - \frac{1}{p}\int _{\Omega }|u_n(x)|^{p}dx \rightarrow c. \end{aligned}$$

(3.9)

Choosing $\psi = u_n$ in (3.8), we obtain

$$\begin{aligned} \Vert u_n\Vert ^2 - \int _{\Omega }|u_n(x)|^pdx = o_n(1) \end{aligned}$$

(3.10)

which combined with (3.9) and (3.10) gives

$$\begin{aligned} \left( \frac{1}{2}-\frac{1}{p}\right) \Vert u_n\Vert ^2 = c + o_n(1). \end{aligned}$$

(3.11)

Therefore, $(u_n)$ is bounded in $H_A^1(\Omega , {\mathbb {C}})$.

Consequently, up to a subsequence, there exists $u^0\in H_A^1(\Omega , {\mathbb {C}})$ such that

$$\begin{aligned} \begin{aligned}&u_n \rightharpoonup u^0\;\;\text{ in }\;\;H_A^1(\Omega , {\mathbb {C}}),\\&u_n \rightarrow u^0\;\;\text{ in }\;\;L_{loc}^{p}(\Omega , {\mathbb {C}}) \;\;\text{ for }\;\;p\in [2, 2^*),\\&u_n(x) \rightarrow u^0(x)\;\;\text{ a.e. } \text{ in }\;\;\Omega . \end{aligned} \end{aligned}$$

(3.12)

We claim that $u^0$ is a weak solution of (P). In fact, for an arbitrary function $\psi \in H_A^1(\Omega , {\mathbb {C}})$, the limit ${I}'_A(u_n) \rightarrow 0$ in $(H_A^1(\Omega , {\mathbb {C}}))^*$ yields

$$\begin{aligned} \langle {I}'_A(u_n), \psi \rangle \rightarrow 0\;\;\text{ as }\;\;n\rightarrow \infty , \end{aligned}$$

that is,

$$\begin{aligned} \lim _{n\rightarrow \infty } \Re \left( \int _{\Omega } (\nabla _Au_n(x) \overline{\nabla _A \psi (x)} + u_n(x) \overline{\psi (x)})dx - \int _{\Omega } |u_n(x)|^{p-2}u_n(x)\overline{\psi (x)}dx \right) = 0.\nonumber \\ \end{aligned}$$

(3.13)

Since $u_n \rightharpoonup u^0$ in $H_A^1(\Omega , {\mathbb {C}})$, it follows that

$$\begin{aligned}&\lim _{n\rightarrow \infty } \Re \int _{\Omega } (\nabla _Au_n(x) \overline{\nabla _A \psi (x)} + u_n(x) \overline{\psi (x)})dx \nonumber \\&\quad = \Re \int _{\Omega } (\nabla _Au^0(x) \overline{\nabla _A\psi (x)} + u^0(x)\overline{\psi (x)})dx. \end{aligned}$$

(3.14)

From the boundedness of $(u_n)$ in $H_A^1(\Omega , {\mathbb {C}})$ and Sobolev embedding, we know that $(|u_n|^{p-2}u_n)$ is a bounded sequence in $L^{\frac{p}{p-1}}(\Omega , {\mathbb {C}})$. Moreover, by (3.12), we see that

$$\begin{aligned} |u_n|^{p-2}u_n \rightarrow |u^0|^{p-2}u^0\;\;\text{ a.e. } \text{ in }\;\;\Omega . \end{aligned}$$

Consequently, by [27, Lemma 4.8], $|u^0|^{p-2}u^0$ is the weak limit of the sequence $(|u_n|^{p-2}u_n)$ in $L^{\frac{p}{p-1}}(\Omega , {\mathbb {C}})$. Hence,

$$\begin{aligned} \lim _{n\rightarrow \infty }\Re \int _{\Omega } |u_n(x)|^{p-2}u_n(x) \overline{\psi (x)}dx = \Re \int _{\Omega }|u^0(x)|^{p-2}u^0(x)\overline{\psi (x)}dx. \end{aligned}$$

(3.15)

From (3.13)–(3.15),

$$\begin{aligned} \Re \left( \int _{\Omega } (\nabla _Au^0(x)\overline{\nabla _A\psi (x)} + u^0(x)\overline{\psi (x)})dx - \int _{\Omega }|u^0(x)|^{p-2}u^0(x)\overline{\psi (x)}dx\right) = 0, \end{aligned}$$

which means that ${I}'_A(u^0) = 0$, and so, $u^0$ is a weak solution of (P).

Now, let $\Psi _n^1$ be the function defined as

$$\begin{aligned} \Psi _{n}^{1}(x) := {\left\{ \begin{array}{ll} (u_n- u^0)(x),&{}x\in \Omega \\ 0,&{}x\in {\mathbb {R}}^N{\setminus } \Omega . \end{array}\right. } \end{aligned}$$

With the above notations, we are able to prove the following steps:

Step 2.

$$\begin{aligned} {I}_{0}(\Psi _n^1) = {I}_A(\Psi _n^1) + o_n(1) = {I}_A(u_n) - {I}_A(u^0) + o_n(1). \end{aligned}$$

(3.16)

Note that by the Brezis-Lieb lemma [15],

$$\begin{aligned} \int _{\Omega } |u_n(x)|^pdx = \int _{\Omega }|u^0(x)|^pdx + \int _{{\mathbb {R}}^N} |\Psi _{n}^{1}(x)|^pdx + o_n(1). \end{aligned}$$

(3.17)

Moreover,

$$\begin{aligned} \begin{aligned} \Vert \Psi _n^1\Vert _0^2&=\int _{{\mathbb {R}}^N}|\nabla _0\Psi _n^1(x)|^2dx =\int _{{\mathbb {R}}^N}|-i\nabla \Psi _n^1(x)|^2dx\\&= \int _{{\mathbb {R}}^N} |-i\nabla \Psi _n^1(x) + (A(x) - A(x))\Psi _n^1(x)|^2dx\\&=\int _{{\mathbb {R}}^N} |\nabla _A\Psi _n^1(x) - A(x)\Psi _n^1(x)|^2dx.\\ \end{aligned} \end{aligned}$$

Using condition (A), it is easy to prove that

$$\begin{aligned} \lim _{n\rightarrow \infty } \int _{{\mathbb {R}}^N} {{|A(x) \Psi _n^1(x)|^2}} dx = 0. \end{aligned}$$

Therefore,

$$\begin{aligned} \int _{{\mathbb {R}}^N} |\nabla _A\Psi _n^1(x) - A(x)\Psi _n^1(x)|^2dx=\int _{{\mathbb {R}}^N} |\nabla _A\Psi _n^1(x)|^2dx+o_n(1)=\Vert \Psi _n^1\Vert _A^2+o_n(1). \end{aligned}$$

Consequently,

$$\begin{aligned} \Vert \Psi _n^1\Vert _{0}^{2} = \Vert \Psi _{n}^{1}\Vert _A^{2} +o_n(1) = \Vert u_n\Vert _A^{2} - \Vert u^0\Vert _A^{2} + o_n(1).\nonumber \\ \end{aligned}$$

(3.18)

Now, (3.16) follows from (3.17) and (3.18).

Step 3.

$$\begin{aligned} {I}'_{0}(\Psi _n^1) = {I}'_A(\Psi _n^1) + o_n(1) = {I}'_A(u_n) - {I}'_A(u^0) + o_n(1) = o_n(1). \end{aligned}$$

(3.19)

For each $v\in H_A^1(\Omega , {\mathbb {C}}) \subset H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})$ with $\Vert v\Vert \le 1$, one has

$$\begin{aligned} {I}'_A(\Psi _n^1)v = \Re \left( \int _{\Omega }(\nabla _A \Psi _n^1(x)\cdot \overline{\nabla _A v(x)} + \Psi _n^1(x) \overline{v(x)})dx - \int _{\Omega } |\Psi _n^1(x)|^{p-2}\Psi _n^1(x)\overline{v(x)}dx \right) \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {I}'_{0}(\Psi _n^1)v = \Re \left( \int _{{\mathbb {R}}^N}(\nabla _0 \Psi _n^1(x)\cdot \overline{\nabla _0 v(x)} + \Psi _n^1(x)\overline{v(x)})dx - \int _{{\mathbb {R}}^N}|\Psi _n^1(x)|^{p-2}\Psi _n^1(x)\overline{v(x)}dx \right) . \end{aligned}\nonumber \\ \end{aligned}$$

Then, as in the previous step,

$$\begin{aligned} \begin{aligned} \left| \langle {I}'_A(\Psi _n^1) - {I}'_0(\Psi _n^1), v \rangle \right|&= \left| \Re \left( \int _{\Omega } (\nabla _A\Psi _n^1(x)\cdot \overline{\nabla _A v (x)} - \nabla _0\Psi _n^1(x)\cdot \overline{\nabla _0 v(x)}) dx \right) \right| = o_n(1) \end{aligned} \end{aligned}$$

for every $v\in H_A^1(\Omega , {\mathbb {C}})$ with $\Vert v\Vert \le 1$. Consequently,

$$\begin{aligned} {I}'_{0}(\Psi _n^1) = {I}'_A(\Psi _n^1) + o_n(1), \quad \text{ in } \quad H_A^1(\Omega , {\mathbb {C}})'. \end{aligned}$$

(3.20)

Now, we are going to show that

$$\begin{aligned} {I}'_A(\Psi _n^1) = {I}'_A(u_n) - {I}'_A(u^0) + o_n(1) = o_n(1). \end{aligned}$$

(3.21)

Note that

$$\begin{aligned} \begin{aligned}&\langle {I}'_A(\Psi _n^1) - {I}'_A(u_n) + {I}'_A(u^0), v\rangle \\&\quad = \Re \left( \int _{\Omega } \nabla _A\Psi _n^1(x) \overline{\nabla _A v(x) } + \Psi _n^1(x) \overline{v(x) }dx - \int _{\Omega }|\Psi _n^1(x) |^{p-2}\Psi _n^1(x) \overline{v(x) }dx \right) \\&\qquad -\Re \left( \int _{\Omega } \nabla _Au_n(x) \overline{\nabla _A v(x) } + u_n(x) \overline{v(x) } dx - \int _{\Omega } |u_n(x) |^{p-2}u_n(x) \overline{v(x) }dx \right) \\&\qquad +\Re \left( \int _{\Omega } \nabla _A u^0(x) \overline{\nabla _A v(x) } + u^0(x) \overline{v(x) }dx - \int _{\Omega }|u^0(x) |^{p-2}u^0(x) \overline{v(x) } dx\right) . \end{aligned} \end{aligned}$$

As $\Psi _n^1 = u_n-u^0$ in $\Omega$, by Hölder inequality,

$$\begin{aligned} \begin{aligned}&|\langle {I}'_A(\Psi _n^1) - {I}'_A(u_n) + {I}'_A(u^0), v \rangle | \\&\quad = \left| \Re \int _{\Omega } \left( |u_n(x)|^{p-2}u_n(x) - |u^0(x)|^{p-2}u^0(x) - |\Psi _n^1(x)|^{p-2}(x)\Psi _n^1(x) \right) {\overline{v}}dx\right| \\&\quad \le \int _{\Omega } ||u_n(x)|^{p-2}u_n(x) - |u^0(x)|^{p-2}u^0(x) - |\Psi _n^1(x)|^{p-2}\Psi _n^1(x)||v(x)|dx\\&\quad \le C \left( \int _{\Omega } ||u_n(x)|^{p-2}u_n(x) - |u^0(x)|^{p-2}u^0(x) - |\Psi _n^1(x)|^{p-2}\Psi _n^1(x)|^{\frac{p}{p-1}} dx\right) ^{\frac{p-1}{p}}. \end{aligned} \end{aligned}$$

Recalling that by [1, Lemma 3] there holds

$$\begin{aligned} \left( \int _{\Omega } ||u_n(x)|^{p-2}u_n(x) - |u^0(x)|^{p-2}u^0(x) - |\Psi _n^1(x)|^{p-2}\Psi _n^1(x)|^{\frac{p}{p-1}} dx\right) ^{\frac{p-1}{p}} = o_n(1), \end{aligned}$$

it follows that

$$\begin{aligned} {I}'_A(\Psi _n^1) = {I}'_A(u_n) - {I}'_A(u^0) + o_n(1). \end{aligned}$$

As $u^0$ is critical point of $I_A$, we have ${I}'_A(u^0) = 0$ and thus (3.19) holds.

If $\Psi _n^1 \rightarrow 0$ in $H_A^1(\Omega , {\mathbb {C}})$ the statements of the main result are verified. Thus, we can suppose that

$$\begin{aligned} \Psi _n^1 \not \rightarrow 0\;\;\text{ in }\;\;H_A^1(\Omega , {\mathbb {C}}). \end{aligned}$$

(3.22)

By using the fact that

$$\begin{aligned} {I}_0(\Psi _{n}^{1}) = \frac{1}{2}\Vert \Psi _{n}^{1}\Vert _0^2 - \frac{1}{p}\int _{{\mathbb {R}}^N}|\Psi _{n}^{1}(x)|^pdx, \end{aligned}$$

and ${I}'_{0}(\Psi _{n}^{1}) = o_n(1)$, we have

$$\begin{aligned} {I}'_{0}(\Psi _{n}^{1})\Psi _{n}^{1} = \Vert \Psi _{n}^{1}\Vert _{0}^{2} - \int _{{\mathbb {R}}^N} |\Psi _{n}^{1}(x)|^pdx = o_n(1). \end{aligned}$$

(3.23)

Therefore,

$$\begin{aligned} {I}_{0}(\psi _{n}^{1}) = \left( \frac{1}{2} - \frac{1}{p}\right) \Vert \Psi _{n}^{1}\Vert _0^{2} + o_n(1). \end{aligned}$$

By (3.22), there is $\alpha >0$ such that

$$\begin{aligned} {I}_{0}(\Psi _{n}^{1})\ge \alpha >0. \end{aligned}$$

(3.24)

Now, let us decompose ${\mathbb {R}}^N$ into N-dimensional unit hypercubes $Q_i$ whose vertices have integer coordinates and put

$$\begin{aligned} d_n := \max _{i}\Vert \Psi _{n}^{1}\Vert _{L^p(Q_i)}. \end{aligned}$$

(3.25)

Arguing as in [12, Lemma 3.1], there is $\gamma >0$ such that

$$\begin{aligned} d_n \ge \gamma >0. \end{aligned}$$

(3.26)

Denote by $(y_n^1)$ the center of a hypercube $Q_i$ in which $\Vert \Psi _{n}^{1}\Vert _{L^p(Q_i)} = d_n$. We claim that $(y_n^1)$ is unbounded sequence in ${\mathbb {R}}^N$. Arguing by contradiction, let us suppose that $(y_n^1)$ is bounded in ${\mathbb {R}}^N$. Then, there is $R>0$ such that

$$\begin{aligned} \int _{B(0,R)}|\Psi _{n}^{1}(x)|^pdx \ge \int _{Q_i(y_n^1)}|\Psi _{n}^{1}(x)|^pdx = d_{n}^{p}> \gamma ^p >0. \end{aligned}$$

(3.27)

On the other hand, since $\Psi _{n}^{1} \rightharpoonup 0$ in $H_0^1({\mathbb {R}}^N, {\mathbb {C}})$, the local compactness of the Sobolev embedding gives

$$\begin{aligned} \int _{B(0,R)}|\Psi _{n}^{1} (x)|^pdx \rightarrow 0,\;\;\text{ as }\;\;n\rightarrow +\infty , \end{aligned}$$

against (3.27). Therefore, the sequence $(y_n^1)$ is unbounded. Since

$$\begin{aligned} \Vert \Psi _{n}^{1} (\cdot + y_{n}^{1})\Vert _{0} = \Vert \Psi _{n}^{1} \Vert _0 \quad \forall n \in {\mathbb {N}}, \end{aligned}$$

we deduce that $(\Psi _{n}^{1} (\cdot + y_n^1))$ is a bounded sequence in $H_0^1({\mathbb {R}}^N, {\mathbb {C}})$. Then, there is $u^1\in H_0^1({\mathbb {R}}^N, {\mathbb {C}})$ such that

$$\begin{aligned} \Psi _{n}^{1} (\cdot + y_n^1) \rightharpoonup u^1\;\;\text{ in }\;\;H_0^1({\mathbb {R}}^N, {\mathbb {C}}) \end{aligned}$$

and

$$\begin{aligned} \Psi _{n}^{1} (\cdot + y_n^1) \rightarrow u^1\;\;\text{ in }\;\;L_{loc}^{p}({\mathbb {R}}^N, {\mathbb {C}}). \end{aligned}$$

Step 4. $u^1$ is a nontrivial weak solution of $(P_0)$.

First, by (3.27), we derive that $u^1\ne 0$, and by a straightforward computation

$$\begin{aligned} {I}'_{0}(\Psi _{n}^{1}(\cdot + y_n^1))\varphi = o_n(1),\;\;\forall \varphi \in C_{0}^{\infty }({\mathbb {R}}^N, {\mathbb {C}}). \end{aligned}$$

Thus, passing to the limit as $n \rightarrow +\infty$, we find

$$\begin{aligned}&\Re \left( \int _{{\mathbb {R}}^N} (\nabla _0 u^1(x) \cdot \overline{\nabla _d \varphi (x)} + u^1(x)\overline{\varphi (x)})dx\right) \\&\quad = \Re \left( \int _{{\mathbb {R}}^N} |u^1(x)|^{p-2}u^1(x)\overline{\varphi (x)}dx\right) ,\;\;\forall \varphi \in C_{0}^{\infty }({\mathbb {R}}^N, {\mathbb {C}}). \end{aligned}$$

Now, the density of $C_{0}^{\infty }({\mathbb {R}}^N,{\mathbb {C}})$ in $H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})$ ensures that

$$\begin{aligned}&\Re \left( \int _{{\mathbb {R}}^N}(\nabla _0u^1(x)\cdot \overline{\nabla _d w(x)} + u^1(x)\overline{w(x)})dx \right) \\&\quad = \Re \left( \int _{{\mathbb {R}}^N} |u^1(x)|^{p-2}u^1(x)\overline{w(x)}dx\right) ,\;\;\forall w\in H_0^1({\mathbb {R}}^N, {\mathbb {C}}), \end{aligned}$$

i.e., the function $u^1$ is a nontrivial weak solution of problem ($P_0$).

We can repeat this process obtaining the sequences

$$\begin{aligned} \Psi _{n}^{j}(x) = \Psi _{n}^{j-1}(x + y_n^{j-1})-u^{j-1}(x),\;\;j\ge 2, \end{aligned}$$

with

$$\begin{aligned} |y_n^j|\rightarrow +\infty ,\;\;\text{ as }\;\;n\rightarrow +\infty \end{aligned}$$

and

$$\begin{aligned} \Psi _{n}^{j-1}(x + y_n^{j-1}) \rightharpoonup u^{j-1}\;\;\text{ in }\;\;H_0^1({\mathbb {R}}^N,{\mathbb {C}}), \end{aligned}$$

(3.28)

where each function $u^j$ is a nontrivial weak solution of problem ($P_{0}$).

Now, an inductive argument ensures that

$$\begin{aligned} \begin{aligned} \Vert \Psi _{n}^{j}\Vert _{0}^{2}&= \Vert \Psi _{n}^{j-1}\Vert _{0}^{2} - \Vert u^{j-1}\Vert _{0}^{2} + o_n(1) = \Vert u_n\Vert _A^2 - \Vert u^0\Vert _A^2 - \sum _{i=1}^{j-1}\Vert u^i\Vert _{0}^{2} + o_n(1) \end{aligned}\nonumber \\ \end{aligned}$$

(3.29)

and

$$\begin{aligned} \begin{aligned} {I}_{0}(\Psi _{n}^{j})&= {I}_{0}(\Psi _{n}^{j-1}) - {I}_{0}(u^{j-1}) + o_n(1)={I}_A(u_n) - {I}_A(u^0) - \sum _{i=1}^{j-1}{I}_0(u^i) + o_n(1). \end{aligned}\nonumber \\ \end{aligned}$$

(3.30)

Since $u^j$ is a nontrivial solution of ($P_0$), it follows that

$$\begin{aligned} I_0(u^j) \ge c_0, \end{aligned}$$

(3.31)

for every $1\le j\le k$. Now, the rest of the proof follows as in [12, Lemma 3.1]. $\square$

Corollary 3.3

Let $(u_n)$ be as in Lemma 3.2 with $c< c_0$. Then $(u_n)$ admits a strongly convergent subsequence. Hence, the functional ${I}_A$ verifies the $(PS)_c$ condition, for every $c \in \left( 0,c_0 \right) .$

Proof

The argument is standard. However, we give the details for the reader’s convenience. Thanks to our hypotheses, one has

$$\begin{aligned} {I}_A(u_n)\rightarrow c\;\;\text{ and }\;\;{I}'_A(u_n)\rightarrow 0\;\;\text{ as }\;\;n\rightarrow +\infty , \end{aligned}$$

with $c<c_0$. Without loss of generality, we can suppose that $(u_n)$ is bounded in $H_A^1(\Omega , {\mathbb {C}})$. Then, up to some subsequence, there exists $u^0\in H_A^1(\Omega , {\mathbb {C}})$ such that

$$\begin{aligned} u_n \rightharpoonup u^0\;\;\text{ in }\;\;H_A^1(\Omega , {\mathbb {C}}). \end{aligned}$$

If $u \not \rightarrow u^0$ in $H_A^1(\Omega , {\mathbb {C}})$, by Lemma 3.2 we must have $k \ge 1$. Hence,

$$\begin{aligned} {I}_A(u_n)\rightarrow c \ge c_0, \end{aligned}$$

which contradicts $c<c_0$. Thereby,

$$\begin{aligned} u_n \rightharpoonup u^0\;\;\text{ and }\;\;\Vert u_n\Vert ^2 \rightarrow \Vert u^0\Vert ^2, \end{aligned}$$

and this implies that $u_n \rightarrow u^0$ in $H_A^1(\Omega , {\mathbb {C}})$. $\square$

Corollary 3.4

Assume that there exists $(u_n)$ for which all the assumptions of Lemma 3.2 hold. If

$$\begin{aligned} c_0< c < c_0+\kappa _1, \end{aligned}$$

(3.32)

where $\kappa _1 < \min \{\kappa ,c_0\}$, then $(u_n)$ admits a strongly convergent subsequence. Hence, the energy functional $I_A$ satisfies the $(PS)_c$ condition, for every $c \in \left( c_0, c_0+\kappa _1 \right) ,$ where $\kappa$ is given by Theorem 2.3.

Proof

Assume by contradiction that $u_n \not \rightarrow u^0$ in $H_A^1(\Omega , {\mathbb {C}})$. By Lemma 3.2, it follows that $k \ge 1$. As $I_0(u^j) \ge c_0$, we must have $k=1$, and so,

$$\begin{aligned} I_A(u_n) \rightarrow I_A(u^0)+I_0(u^1). \end{aligned}$$

We claim that $u^0 \not = 0$, otherwise

$$\begin{aligned} I_0(u^1)=I_A(u_n)+o_n(1)=c+o_n(1) \end{aligned}$$

and thus $u^1$ is a critical point of $I_0$ with $I_0(u^1)<c_0+\kappa _1$. Hence, by Theorem 2.3, we must have $I_0(u^1)=c_0$, which is impossible because $c>c_0$. From this, $u^0 \not = 0$ and

$$\begin{aligned} I_A(u^0)\ge c_A=c_0. \end{aligned}$$

Hence, the limit equality

$$\begin{aligned} I_A(u_n)=I_A(u^0)+I_0(u^1)+o_n(1), \end{aligned}$$

yields that

$$\begin{aligned} \lim _{n \rightarrow +\infty }I_A(u_n)=I_A(u^0)+I_0(u^1)\ge 2c_0, \end{aligned}$$

which gives an absurd. Thus, we must have $u_n \rightarrow u^0$ in $H_A^{1}(\Omega , {\mathbb {C}})$. This shows the desired result. $\square$

4 Technical Estimates

The main goal this section is to establish some technical estimates that we will use in the proof of Theorem 1.1.

We start by introducing the following operator

$$\begin{aligned} \begin{aligned} \Phi _\rho : {\mathbb {R}}^N&\rightarrow H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})\\ y&\rightarrow \Phi _\rho (y):= \phi _{y,\rho }, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \phi _{y,\rho }(x) :=\zeta (x)\varphi (x-y) = \xi \left( \frac{|x|}{\rho } \right) \varphi (x-y), \end{aligned}$$

$\varphi$ is the positive ground state of $(P_{\infty })$ satisfying (2.4) and $\zeta , \xi$ are given as in the proof of Theorem 3.1. A direct computation ensures that the functions $\phi _{y, \rho }$ belong to $H_{A}^{1}(\Omega , {\mathbb {C}})$ and $L^p(\Omega , {\mathbb {C}})$, respectively. From now on, we take $t_{y,\rho }>0$ such that $\psi _{\rho }(y) = t_{y,\rho }\phi _{y,\rho }\in {\mathcal {N}}_{A}$.

Lemma 4.1

The following relations hold:

• (i) $\displaystyle \limsup _{\rho \rightarrow 0}I_A(\psi _{\rho }(y)) \le c_0+M\Gamma \Vert \varphi \Vert ^2_{L^{2}(\Omega )}$ uniformly in y;

• (ii) $I_A(\psi _\rho (y)) \rightarrow c_0$ as $|y|\rightarrow +\infty$, for every $\rho$,

where $M:=\sum _{j=1}^{N}\Vert A_j\Vert _\infty ^2$ and $\Gamma >0$ is a constant independent of y.

Proof

(i) Note that

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N} |\phi _{y,\rho }(x) - \varphi (x-y)|^2dx&= \int _{{\mathbb {R}}^N} \left| \left( \xi \left( \frac{|x|}{\rho }\right) -1\right) \varphi (x-y) \right| ^2dx\\&= \int _{{\mathbb {R}}^N}\left| \left( \xi \left( \frac{|x|}{\rho }\right) -1\right) \varphi (x-y) \right| ^2dx\\&=\int _{B(0,2\rho )} \left| \left( \xi \left( \frac{|x|}{\rho }\right) -1\right) \varphi (x-y) \right| ^2dx \\&\le C_1\varphi ^2(0)|B(0, 2\rho )| \rightarrow 0\;\;\text{ as }\;\;\rho \rightarrow 0, \end{aligned} \end{aligned}$$

from where it follows that

$$\begin{aligned} \phi _{y,\rho } \rightarrow \varphi (\cdot - y) \quad \text{ in } \quad L^{2}({\mathbb {R}}^N) \quad \text{ as }\;\;\rho \rightarrow 0,\;\;\text{ uniformly } \text{ in }\;\;y\in {\mathbb {R}}^N. \end{aligned}$$

Consequently,

$$\begin{aligned} \Vert \phi _{y,\rho }\Vert _{L^2(\Omega )}^2\rightarrow \Vert \varphi (\cdot - y)\Vert _{L^2({\mathbb {R}}^N)}^{2} = \Vert \varphi \Vert _{L^2({\mathbb {R}}^N)}^{2}\;\;\text{ as }\;\;\rho \rightarrow 0,\;\;\text{ uniformly } \text{ in }\;\;y\in {\mathbb {R}}^N.\nonumber \\ \end{aligned}$$

(4.1)

In the same way, we can show that

$$\begin{aligned} \phi _{y,\rho } \rightarrow \varphi (\cdot - y) \quad \text{ in } \quad L^{p}({\mathbb {R}}^N) \quad \text{ as }\;\;\rho \rightarrow 0,\;\;\text{ uniformly } \text{ in }\;\;y\in {\mathbb {R}}^N \end{aligned}$$

and

$$\begin{aligned} \Vert \phi _{y,\rho }\Vert _{L^p(\Omega )}^p\rightarrow \Vert \varphi (\cdot - y)\Vert _{L^p({\mathbb {R}}^N)}^{p} = \Vert \varphi \Vert _{L^p({\mathbb {R}}^N)}^{p}\;\;\text{ as }\;\;\rho \rightarrow 0,\;\;\text{ uniformly } \text{ in }\;\;y\in {\mathbb {R}}^N.\nonumber \\ \end{aligned}$$

(4.2)

On the other hand, note that for each $j \in \{1,..,N\}$,

$$\begin{aligned} \begin{aligned} \int _{\Omega } |D_j\phi _{y,\rho }(x)|^2dx&= \int _{{\mathbb {R}}^N} |-i\partial _j\phi _{y,\rho }(x) +A_j(x)\phi _{y,\rho }(x)|^2dx\\&= \int _{{\mathbb {R}}^N} \left( |\partial _j\phi _{y,\rho }(x)|^2+|A_j(x)\phi _{y,\rho }(x)|^2\right) dx \\&\le \int _{{\mathbb {R}}^N} |\partial _j\phi _{y,\rho }(x)|^2\,dx+\Vert A_j\Vert ^2_\infty \Vert \varphi \Vert ^{2}_{L^{2}(\Omega )}. \end{aligned} \end{aligned}$$

We claim that

$$\begin{aligned} \int _{{\mathbb {R}}^N} |\partial _j\phi _{y,\rho }(x)|^2\,dx\rightarrow \int _{{\mathbb {R}}^N} |\partial _j\varphi (x)|^2\,dx\;\;\text{ as }\;\;\rho \rightarrow 0. \end{aligned}$$

(4.3)

Indeed, note that

$$\begin{aligned} \int _{{\mathbb {R}}^N}\left| \varphi (x-y)\partial _j \xi \left( \frac{|x|}{\rho } \right) \right| ^2dx \le \Vert \varphi \Vert _\infty ^2 |B(0, \rho )|, \end{aligned}$$

which yields

$$\begin{aligned} \lim _{\rho \rightarrow 0}\int _{{\mathbb {R}}^N}\left| \varphi (x-y)\partial _j \xi \left( \frac{|x|}{\rho } \right) \right| ^2dx=0, \quad \text{ uniformly } \text{ in } \;\; y \in {\mathbb {R}}^N. \end{aligned}$$

Since

$$\begin{aligned} \lim _{\rho \rightarrow 0}\int _{{\mathbb {R}}^N}\left| \xi \left( \frac{|x|}{\rho } \right) \partial _j \varphi (x-y)\right| ^2dx=\int _{{\mathbb {R}}^N}\left| \partial _j \varphi (x-y)\right| ^2dx, \quad \text{ uniformly } \text{ in } \;\; y\in {\mathbb {R}}^N, \end{aligned}$$

we can deduce that (4.3) holds.

As $\psi _{\rho }(y) = t_{y,\rho }\phi _{y,\rho } \in {\mathcal {N}}_A$, it follows that

$$\begin{aligned} t_{y,\rho }\Vert \phi _{y,\rho }\Vert _A^2 = t_{y,\rho }^{p-1}\int _{\Omega }|\phi _{y,\rho }(x)|^pdx. \end{aligned}$$

This combined with diamagnetic inequality (2.1) leads to

$$\begin{aligned} t_{y,\rho }\Vert \phi _{y,\rho }\Vert _{H^{1}({\mathbb {R}}^N)}^2 \le t_{y,\rho }^{p-1} \int _{{\mathbb {R}}^N}|\phi _{y,\rho }(x)|^pdx. \end{aligned}$$

Now, recalling that

$$\begin{aligned} \Vert \phi _{y,\rho }\Vert _{H^{1}({\mathbb {R}}^N)}^2 \rightarrow \Vert \varphi \Vert _{H^{1}({\mathbb {R}}^N)}^2 \quad \text{ and } \quad \Vert \phi _{y,\rho }\Vert _{L^{p}({\mathbb {R}}^N)}^2 \rightarrow \Vert \varphi \Vert _{L^{p}({\mathbb {R}}^N)}^2 \end{aligned}$$

as $\rho \rightarrow 0$, uniformly in y, we can infer that

$$\begin{aligned} \liminf _{\rho \rightarrow 0} t_{y,\rho } \ge 1 \;\;\text{ uniformly } \text{ in }\;\;y\in {\mathbb {R}}^N. \end{aligned}$$

On the other hand, we also know that the limits below

$$\begin{aligned}&\limsup _{\rho \rightarrow 0}\Vert \phi _{y,\rho }\Vert _A^2 \le \Vert \varphi \Vert _{H^1({\mathbb {R}}^N)}^2 +M \Vert \varphi \Vert ^{2}_{L^{2}(\Omega )} \quad \text{ and }\\&\quad \limsup _{\rho \rightarrow 0}\int _{\Omega } |\phi _{y,\rho }(x)|^pdx = \int _{{\mathbb {R}}^N}|\varphi (x)|^pdx, \end{aligned}$$

are uniform in $y\in {\mathbb {R}}^N$. These facts imply that there exists $C>0$ that depends on M, which is independent of $y\in {\mathbb {R}}^N$, such that

$$\begin{aligned} \limsup _{\rho \rightarrow 0} t_{y,\rho } \le C \;\;\text{ uniformly } \text{ in }\;\;y\in {\mathbb {R}}^N. \end{aligned}$$

Indeed,

$$\begin{aligned} \limsup _{\rho \rightarrow 0} t_{y, \rho }= \limsup _{\rho \rightarrow 0} \left( \frac{\Vert \phi _{y, \rho }\Vert _{A}^{2}}{\Vert \phi _{y, \rho }\Vert ^{p}_{L^{p}({\mathbb {R}}^{N})}}\right) ^{\frac{1}{p-2}}\le \left( \frac{\Vert \varphi \Vert ^{2}_{H^{1}({\mathbb {R}}^{N})}+M\Vert \varphi \Vert ^{2}_{L^{2}(\Omega )}}{\Vert \varphi \Vert _{L^{p}({\mathbb {R}}^{N})}^{p}}\right) ^{\frac{1}{p-2}}=:C. \end{aligned}$$

Therefore, using the fact that $\varphi$ satisfies (2.4) and that $c_0=c_\infty$, we obtain

$$\begin{aligned} \limsup _{\rho \rightarrow 0} I_A(\psi _\rho (y)) = \limsup _{\rho \rightarrow 0} I_A(t_{y, \rho }\phi _{y,\rho }) \le {{C^{2} I_\infty (\varphi )}}+M\Gamma \Vert \varphi \Vert ^{2}_{L^{2}(\Omega )} = c_0+M\Gamma \Vert \varphi \Vert ^{2}_{L^{2}(\Omega )}, \end{aligned}$$

where $\Gamma :=\frac{C^2}{2}$. We point out that, from the above calculations, $\Gamma$ is bounded when $M \rightarrow 0$. This information will be useful in the next section.

(ii) For each fixed $\rho$, let us consider an arbitrary sequence $(y_n)\subset {\mathbb {R}}^N$ with $|y_n|\rightarrow \infty$ as $n\rightarrow +\infty$ and let $t_{y_n, \rho }>0$ such that $t_{y_n, \rho }\phi _{y_n,\rho }\in {\mathcal {N}}_{A}$. As in the proof of Theorem 3.1,

$$\begin{aligned} \Vert \psi _{y_n, \rho }\Vert _A^2 \rightarrow \Vert \varphi (\cdot -y_n)\Vert _0^2, \quad \ \int _{\Omega }|\phi _{y_n, \rho }(x)|^pdx \rightarrow \int _{{\mathbb {R}}^N}|\varphi (x-y_n)|^pdx \quad \text{ and } \quad t_{y_n, \rho } \rightarrow 1. \end{aligned}$$

From this,

$$\begin{aligned} I_A(\psi _\rho (y_n)) = I_A(t_{y_n, \rho }\phi _{y_n, \rho })= I_0(\varphi ) +o_n(1)=I_\infty (\varphi ) +o_n(1)=c_\infty +o_n(1)=c_0+o_n(1). \end{aligned}$$

$\square$

In light of the previous lemma, we can prove the corollary below.

Corollary 4.2

There is $\rho _0>0$ and $\epsilon >0$ such that if $\Vert A\Vert _\infty <\epsilon$, then

$$\begin{aligned} \sup _{y\in {\mathbb {R}}^N} I_A(\psi _\rho (y)) < c_0 + \kappa _1, \quad \forall \rho \le \rho _0. \end{aligned}$$

(4.4)

Proof

By Lemma 4.1—Part (i), one has

$$\begin{aligned} \limsup _{\rho \rightarrow 0} I_A(\psi _\rho (y)) \le c_0+M\Gamma \Vert \varphi \Vert _{L^{2}({\mathbb {R}}^N)}^{2}, \end{aligned}$$

where $M=\sum _{j=1}^{N}\Vert A_j\Vert _\infty ^2$, for every $y\in {\mathbb {R}}^N$. So, there is $\rho _0>0$ small enough such that

$$\begin{aligned} \begin{aligned} \sup _{y\in {\mathbb {R}}^N} I_A(\psi _\rho (y)) \le c_0+2M\Gamma \Vert \varphi \Vert _{L^{2}({\mathbb {R}}^N)}^{2}, \quad \forall \rho <\rho _0. \end{aligned} \end{aligned}$$

Fixing $\epsilon >0$ such that if $\Vert A\Vert _\infty <\epsilon$ then $2M\Gamma \Vert \varphi \Vert _{L^{2}(\Omega )}^{2}<\kappa _1$, we must have that

$$\begin{aligned} \begin{aligned} \sup _{y\in {\mathbb {R}}^N} I_A(\psi _\rho (y)) \le c_0+ 2M\Gamma \Vert \varphi \Vert _{L^{2}(\Omega )}^{2}< c_0+\kappa _1. \end{aligned} \end{aligned}$$

This ends the proof of the corollary. $\square$

Hereafter, let us fix $\rho \in (0, \rho _0)$, such that

$$\begin{aligned} {\mathbb {R}}^N {\setminus } \Omega \subset B(0, \rho ). \end{aligned}$$

Furthermore, we consider the barycenter function given by

$$\begin{aligned} \begin{aligned} \tau : H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})&\rightarrow {\mathbb {R}}^N\\ u&\rightarrow \tau (u):= \int _{{\mathbb {R}}^N}|u(x)|^2\chi (|x|)xdx, \end{aligned} \end{aligned}$$

where $\chi \in C({\mathbb {R}}^+, {\mathbb {R}})$ is a non-increasing real function such that

$$\begin{aligned} \chi (t) = {\left\{ \begin{array}{ll} 1,&{}t\in (0,R]\\ \frac{R}{t},&{}t>R, \end{array}\right. } \end{aligned}$$

for some $R>0$ such that ${\mathbb {R}}^N {\setminus } \Omega \subset B(0, R)$. By definition of $\chi$,

$$\begin{aligned} \chi (|x|)|x|\le R,\;\;\forall x\in {\mathbb {R}}^N. \end{aligned}$$

(4.5)

Set

$$\begin{aligned} {\mathcal {T}}_0 := \{u\in {\mathcal {N}}_A: \tau (u) = 0\} \subset {\mathcal {N}}_A\subset H_A^1(\Omega , {\mathbb {C}}). \end{aligned}$$

Lemma 4.3

If

$$\begin{aligned} \lambda _0:= \inf _{u\in {\mathcal {T}}_0} I_A(u), \end{aligned}$$

then

$$\begin{aligned} c_0 < \lambda _0 \end{aligned}$$

(4.6)

and there exists $R_0>0$, with $R_0>\rho$ such that:

• (i) If $y\in {\mathbb {R}}^N$ with $|y|\ge R_0$, then

  $$\begin{aligned} I_A(\psi _\rho (y)) \in \Big (c_0, \frac{c_0+\lambda _0}{2} \Big ). \end{aligned}$$

• (ii) If $y\in {\mathbb {R}}^N$ with $|y| = R_0$, then

  $$\begin{aligned} \langle \tau (\psi _\rho (y)), y\rangle >0. \end{aligned}$$

Proof

Since ${\mathcal {T}}_0 \subset {\mathcal {N}}_A$ and

$$\begin{aligned} c_0 = c_A = \inf _{u\in {\mathcal {N}}_A}I_A(u), \end{aligned}$$

we have

$$\begin{aligned} c_0 \le \lambda _0. \end{aligned}$$

Now we are going to show that $c_0\ne \lambda _0$. Suppose by contradiction that $c_0 = \lambda _0$. Then, there exists a minimizing sequence $(u_n)\subset {\mathcal {T}}_0 \subset H_A^1(\Omega , {\mathbb {C}})$ such that

$$\begin{aligned} I_A(u_n)\rightarrow c_0\quad \text{ and }\quad \tau (u_n) = 0\;\;\forall n\in {\mathbb {N}}. \end{aligned}$$

By the Ekeland variational principle [30, Theorem 2.4], we can find a sequence $(w_n)\subset {\mathcal {N}}_A$ such that

$$\begin{aligned} I_A(w_n)\rightarrow c_0,\quad \nabla _{{\mathcal {N}}_A} I_A(w_n)\rightarrow 0\quad \text{ and }\quad \Vert w_n - u_n\Vert _A\rightarrow 0. \end{aligned}$$

A well-known computation shows that $(u_n)$ and $(w_n)$ are bounded. Moreover, there exists a sequence $(\lambda _n)\subset {\mathbb {R}}$ such that

$$\begin{aligned} I'_A(w_n) - \lambda _n J'_A(w_n) = o_n(1), \end{aligned}$$

(4.7)

where $J_A(u) = \langle I'_A(u), u\rangle$. Using standard arguments, we have that $\lambda _n \rightarrow 0$, and then (4.7) yields

$$\begin{aligned} I'_A(w_n)\rightarrow 0. \end{aligned}$$

Assuming that $w_n \rightharpoonup w^0$ in $H_A^1(\Omega , {\mathbb {C}})$, we have that $I'_A(w^0)=0$, and so, $w^0$ is a solution for (P). Hence, by Theorem 3.1, we cannot have $w_n \rightarrow w^0$ in $H_A^1(\Omega , {\mathbb {C}})$, because this convergence would imply in $I_A(w^0)=c_0=c_A$. Thereby, by Lemma 3.2,

$$\begin{aligned} c_0 = \lim _{n\rightarrow +\infty } I_A(w_n) = I_A(w^0) + \sum _{j=1}^{k}I_0(u^j) \ge I_A(w^0) + kc_0. \end{aligned}$$

As $I_A(w^0)\ge 0$, then $k=1$, $w^0=0$ and $u^1$ is a nontrivial solution of $(P_0)$ with $I_0(u^1) = c_0$. From Theorem 2.3, there are $\theta >0$ and $a \in {\mathbb {R}}^N$ such that

$$\begin{aligned} u^1(x)=e^{i\theta }\varphi (x-a), \quad \forall x \in {\mathbb {R}}^N. \end{aligned}$$

(4.8)

Since $w_n \rightharpoonup w^0=0$, we get

$$\begin{aligned} \Psi _n(x+y_n^1) = w_n(x+y_n^1) \rightharpoonup u^1(x) \end{aligned}$$

and

$$\begin{aligned} \Vert \Psi _n(\cdot + y_n^1)\Vert _A^2 = \Vert w_n(\cdot + y_n^1)\Vert _A^2 \rightarrow \Vert u^1\Vert _0^2, \end{aligned}$$

where $(y_n^1)$ must be a sequence satisfying $|y_n^1|\rightarrow \infty$. Therefore,

$$\begin{aligned} w_n(\cdot +y_n^1) \rightarrow u^1\;\;\text{ in }\;\;H_A^1({\mathbb {R}}^N, {\mathbb {C}}). \end{aligned}$$

Setting

$$\begin{aligned} u= u^1,\;\; y_n = y_n^1\quad \text{ and }\quad v_n(x+y_n^1) = w_n(x+y_n^1) - u^1(x), \end{aligned}$$

we have

$$\begin{aligned} v_n(x) = w_n(x) - u(x-y_n)\quad \text{ and }\quad \Vert v_n\Vert _A^2 = \Vert w_n(\cdot + y_n) - u\Vert _A^2. \end{aligned}$$

Therefore, the strong convergence of $(w_n(\cdot + y_n))$ to $u^1$ yields $v_n \rightarrow 0$ in $H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})$.

Next, we consider the following sets

$$\begin{aligned} ({\mathbb {R}}^N)_{n}^{+} :=\{x\in {\mathbb {R}}^N: \langle x, y_n+a\rangle >0\}\quad \text{ and }\quad ({\mathbb {R}}^N)_{n}^{-} = {\mathbb {R}}^N {\setminus } ({\mathbb {R}}^N)_{n}^{+}, \end{aligned}$$

where the vector a is given in (4.8). Using the fact that $|y_n|\rightarrow +\infty$ as $n\rightarrow +\infty$, we claim that there is a ball

$$\begin{aligned} B(y_n+a, r_*) = \{x\in {\mathbb {R}}^N: |x-y_n-a| < r_*\} \subset ({\mathbb {R}}^N)_n^+ \end{aligned}$$

such that

$$\begin{aligned} \varphi (x-y_n-a) \ge \frac{1}{2}\varphi (0)>0,\;\;\forall x\in B(y_n+a, r_*). \end{aligned}$$

(4.9)

It is easy to see that (4.9) holds for $r^*>0$ small enough, because $\varphi$ is positive, radial, strictly decreasing with respect to |x| and

$$\begin{aligned} \varphi (0)=\max _{z \in {\mathbb {R}}^N}\varphi (z). \end{aligned}$$

On the other hand, for each $r_*>0$ fixed, there is $n_0$ such that

$$\begin{aligned} \langle x, y_n+a\rangle> \frac{|x|^2 + |y_n+a|^2 - r_{*}^{2}}{2} \ge \frac{|y_n+a|^2 - r_{*}^{2}}{2}>0,\;\;\forall n\ge n_0,\;\;\forall x\in B(y_n+a, r_*), \end{aligned}$$

showing that

$$\begin{aligned} B(y_n+a, r_*) \subset ({\mathbb {R}}^N)_{n}^{+},\;\;\text{ for } \text{ n } \text{ large } \text{ enough }. \end{aligned}$$

Hence, for n large enough,

$$\begin{aligned} |\varphi (x-y_n-a)|^2, \chi (|x|), \langle x, y_n\rangle >0\quad \forall x\in ({\mathbb {R}}^N)_{n}^{+},\quad \quad B(y_n+a, r_*)\subset ({\mathbb {R}}^N)_{n}^{*}, \end{aligned}$$

and $|x|>R$ for every $x\in B(y_n+a, r_*)$. Using this information, we find

$$\begin{aligned} \begin{aligned} \int _{({\mathbb {R}}^N)_{n}^{+}}|u(x-y_n)|^2\chi (|x|)\langle x, y_n+a\rangle dx&=\int _{({\mathbb {R}}^N)_{n}^{+}} |\varphi (x-y_n-a)|^2\chi (|x|) \langle x, y_n+a\rangle dx\\&\ge \frac{|\varphi (0)|^2}{4}\int _{B(y_n+a, r_*)} \chi (|x|) \langle x, y_n+a\rangle dx\\&= \frac{|\varphi (0)|^2}{8}R|B(y_n+a, r_*)||y_n+a|>0. \end{aligned} \end{aligned}$$

Recalling that for each $x\in ({\mathbb {R}}^N)_n^-$,

$$\begin{aligned} |x-y_n-a| \ge |x|, \end{aligned}$$

and using again the fact that $\varphi$ is radial with relation to the origin and decreasing, it follows that

$$\begin{aligned} |u(x-y_n)|^2\chi (|x|)|x| = |\varphi (x-y_n-a)|^2\chi (|x|)|x| \le R |\varphi (|x|)|^2\in L^1({\mathbb {R}}^N). \end{aligned}$$

This fact, combined with the limit

$$\begin{aligned} \varphi (\cdot -y_n-a) \rightarrow 0\;\;\text{ as }\;\;|y_n| \rightarrow +\infty \end{aligned}$$

ensures that

$$\begin{aligned} \int _{({\mathbb {R}}^N)_n^-} |u(x-y_n)|^2\chi (|x|) |x|dx = o_n(1). \end{aligned}$$

(4.10)

Therefore, by the Cauchy-Schwarz inequality and (4.10),

$$\begin{aligned} \begin{aligned}&\langle \tau (u(x-y_n)), \frac{y_n+a}{|y_n+a|}\rangle = \int _{({\mathbb {R}}^N)_{n}^{+}} |\varphi (x-y_n-a)|^2 \chi (|x|)\langle x, \frac{y_n+a}{|y_n+a|}\rangle dx \\&\qquad + \int _{({\mathbb {R}}^N)_{n}^{-}} |\varphi (x-y_n-a)|^2\chi (|x|)\langle x, \frac{y_n+a}{|y_n+a|}\rangle dx\\&\quad \ge \frac{R|\varphi (0)|^2}{8} |B(y_n+a, r_*)| - \int _{({\mathbb {R}}^N)_{n}^{-}} |\varphi (x-y_n-a)|^2\chi (|x|)|x|dx\\&\quad \ge \frac{R|\varphi (0)|^2}{8}|B(y_n+a, r_*)| - o_n(1) >0. \end{aligned}\nonumber \\ \end{aligned}$$

(4.11)

Now, using the fact that $w_n \rightarrow u^1$ in $H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})$ together with the limit $\tau (w_n) = o_n(1)$, we find that

$$\begin{aligned} \tau (u(x-y_n)) = o_n(1), \end{aligned}$$

(4.12)

which contradicts (4.11), and so,

$$\begin{aligned} c_0 < \lambda _0. \end{aligned}$$

Now we are ready to prove the assertion (i) of Lemma 4.3. As $\psi _\rho (y) = t_{y, \rho }\phi _{y,\rho }\in {\mathcal {N}}_0$, by Theorem 3.1,

$$\begin{aligned} I_A(\psi _\rho (y)) > c_A = c_0,\;\;\forall y\in {\mathbb {R}}^N. \end{aligned}$$

By Lemma 4.1-part (ii), for each $\rho$ fixed

$$\begin{aligned} I_A(\psi _\rho (y)) \rightarrow c_0\;\;\text{ as }\;\;|y|\rightarrow \infty . \end{aligned}$$

(4.13)

Thereby, for a given $\epsilon _1 \in (0, \frac{\lambda _0-c_0}{2})$, there is $R_0>0$ such that

$$\begin{aligned} \Big | I_A(\psi _\rho (y)) - c_0\Big | < \epsilon _1\;\;\text{ whenever }\;\;|y|\ge R_0. \end{aligned}$$

From this

$$\begin{aligned} I_A(\psi _\rho (y)) \in \left( c_0, \frac{\lambda _0+c_0}{2} \right) ,\;\;\forall y\in {\mathbb {R}}^N\;\;\text{ such } \text{ that }\;\;|y|\ge R_0. \end{aligned}$$

Finally, let us show assertion (ii) of Lemma 4.3, by definition of $\psi _\rho (y)$ and arguing as above with |y| large enough, we derive

$$\begin{aligned} \begin{aligned} \langle \tau (\psi _\rho (y)), \frac{y}{|y|}\rangle&= t_{y,\rho }^{2}\int _{({\mathbb {R}}^N)_n^+}|\phi _{y,\rho }(x)|^2\chi (|x|)\langle x,\frac{y}{|y|}\rangle dx\\&\qquad + t_{y,\rho }^{2}\int _{({\mathbb {R}}^N)_n^-}|\phi _{y,\rho }(x)|^2\chi (|x|)\langle x, \frac{y}{|y|}\rangle dx\\&\ge t_{y,\rho }^{2}\int _{B(y, r_*)}|\phi _{y,\rho }(x)|^2\chi (|x|)\langle x,\frac{y}{|y|}\rangle dx\\&\qquad + t_{y,\rho }^{2}\int _{({\mathbb {R}}^N)_n^-}|\phi _{y,\rho }(x)|^2\chi (|x|)\langle x, \frac{y}{|y|}\rangle dx\\&\ge t_{y,\rho }^{2}\frac{|\varphi (0)|^2}{4}\int _{B(y, r_*)}\left| \xi \left( \frac{|x|}{\rho }\right) \right| ^2\chi (|x|)\langle x, \frac{y}{|y|}\rangle dx - o(1). \end{aligned} \end{aligned}$$

As $t_{y, \rho } \rightarrow 1$ as $|y|\rightarrow +\infty$, we have for $|y| = R_0$ large,

$$\begin{aligned} \left\langle \tau (\psi _\rho (y)), \frac{y}{|y|}\right\rangle >0. \end{aligned}$$

(4.14)

$\square$

5 Proof of Theorem 1.1

In the sequel, we consider the sets

$$\begin{aligned} \Sigma:= & {} \{\psi _\rho (y): |y|\le R_0\} \subset H_A^1(\Omega , {\mathbb {C}}), \\ {\mathcal {H}}:= & {} \left\{ h\in C({\mathcal {N}}_A, {\mathcal {N}}_A): h(u) = u,\;\;\forall u\in {\mathcal {N}}_A\;\;\text{ with }\;\;I_A(u) < \frac{\lambda _0+c_0}{2} \right\} \end{aligned}$$

and

$$\begin{aligned} \Upsilon := \{B\in {\mathcal {N}}_A:\;\;B = h(\Sigma ),\;\;h\in {\mathcal {H}}\}. \end{aligned}$$

Lemma 5.1

If $B\in \Upsilon$, then $B\cap {\mathcal {T}}_0 \ne \emptyset$.

Proof

We are going to show that, for every $B\in \Upsilon$, there exists $u\in B$ such that $\tau (u) = 0$. Equivalently, we prove that: for every $h\in {\mathcal {H}}$, there exists ${\tilde{y}}\in {\mathbb {R}}^N$ with $|{\tilde{y}}|\le R_0$ such that

$$\begin{aligned} (\tau \circ h \circ \psi _\rho )({\tilde{y}}) = 0. \end{aligned}$$

(5.1)

For any $h\in {\mathcal {H}}$, we set the functions

$$\begin{aligned} {\mathcal {J}}:=\tau \circ h \circ \psi _\rho : {\mathbb {R}}^N \rightarrow {\mathbb {R}}^N \end{aligned}$$

and ${\mathcal {F}}:[0,1]\times {\overline{B}}(0, R_0)\rightarrow {\mathbb {R}}^N$ given by

$$\begin{aligned} {\mathcal {F}} (t, z) := t{\mathcal {J}}(z) + (1-t)z. \end{aligned}$$

We claim that $0\not \in {\mathcal {F}}(t, \partial B(0, R_0))$. Indeed, for $|y| = R_0$, by Lemma 4.3—Part (i) we have

$$\begin{aligned} I_A(\psi _\rho (y)) < \frac{\lambda _0+c_0}{2}. \end{aligned}$$

Hence,

$$\begin{aligned} {\mathcal {F}}(t,y)=t(\tau \circ \psi _\rho )(y) + (1-t)y \end{aligned}$$

and

$$\begin{aligned} \langle {\mathcal {F}}(t,y), y\rangle = t \langle \tau (\psi _\rho (y)), y\rangle + (1-t)\langle y,y\rangle . \end{aligned}$$

(5.2)

Now

$\circ$:

If $t=0$, then $\langle {\mathcal {F}}(0,y), y\rangle = |y|^2 = R_{0}^{2}>0$;

$\circ$:

If $t=1$, then by Lemma 4.3—Part (ii) we have $\langle {\mathcal {F}}(1,y), y\rangle = \langle \tau (\psi _\rho (y)), y \rangle >0$;

$\circ$:

If $t\in (0,1)$, then $\langle {\mathcal {F}}(t,y), y\rangle >0$, since the terms $t, 1-t, \langle \tau (\psi _\rho (y)), y\rangle$ and $|y|^2$ are positives.

Then, by using the homotopy-invariance of the Brouwer degree, one has

$$\begin{aligned} d({\mathcal {F}}(t, \cdot ), B(0, R_0), 0) = \text{ constant },\;\;\forall t\in [0,1]. \end{aligned}$$

Recalling that

$$\begin{aligned} d({\mathcal {J}}, B(0, R_0), 0) = 1\ne 0, \end{aligned}$$

there exists ${\tilde{y}}\in B(0, R_0)$ such that ${\mathcal {J}}({\tilde{y}}) = 0$, that is,

$$\begin{aligned} {\mathcal {J}}({\tilde{y}}) = (\tau \circ h \circ \psi _\rho )({\tilde{y}}) = 0. \end{aligned}$$

This completes the proof of Lemma 5.1. $\square$

Now, let us define

$$\begin{aligned} c:=\inf _{B\in \Upsilon } \sup _{u\in B} I_A(u), \end{aligned}$$

(5.3)

$$\begin{aligned} {\mathcal {K}}_c := \{u\in {\mathcal {N}}_A: I_A(u) = c\;\;\text{ and }\;\;\nabla I_{A}\Big |_{{\mathcal {N}}_A}(u) = 0\} \end{aligned}$$

and

$$\begin{aligned} L_\gamma := \{u\in {\mathcal {N}}_A: I_A(u) \le \gamma \}, \end{aligned}$$

for every $\gamma \in {\mathbb {R}}$.

Proof of Theorem 1.1. We choose $\rho = \rho _0$, where $\rho _0$ is given in Corollary 4.2. We claim that the constant c defined in (5.3) is a critical value for $I_A$, that is, ${\mathcal {K}}_c \ne \emptyset$. We start our analysis by noting that

$$\begin{aligned} c_0< c < c_0+\kappa _1. \end{aligned}$$

(5.4)

First of all, we recall that by Lemma 5.1, $B\cap {\mathcal {T}}_0 \ne \emptyset$ for every $B\in \Upsilon$. Then, for each $B\in \Upsilon$, there is ${\tilde{u}}\in B \cap {\mathcal {T}}_0$ such that

$$\begin{aligned} \inf _{u\in {\mathcal {T}}_0}I_A(u) \le \inf _{u\in B\cap {\mathcal {T}}_0} I_A(u) \le I_A({\tilde{u}}) \le \sup _{u\in B\cap {\mathcal {T}}_0} I_A(u)\le \sup _{u\in B}I_A(u). \end{aligned}$$

(5.5)

By Lemma 4.3 and (5.5),

$$\begin{aligned} c_0 < \lambda _0 = \inf _{u\in {\mathcal {T}}_0}I_A(u) \le \sup _{u\in B}I_A(u), \quad \forall B\in \Upsilon . \end{aligned}$$

Thus

$$\begin{aligned} c_0 < \lambda _0\le \inf _{B\in \Upsilon } \sup _{u\in B}I_A(u) = c. \end{aligned}$$

(5.6)

Since

$$\begin{aligned} c\le \sup _{u\in B}I_A(u), \;\;\forall B\in \Upsilon , \end{aligned}$$

(5.7)

it follows that

$$\begin{aligned} c\le \sup _{|y|\le R_0} I_A(h(\psi _\rho (y))),\;\;\forall h\in {\mathcal {H}}. \end{aligned}$$

Now, taking $h \equiv I$, we find

$$\begin{aligned} c\le \sup _{|y|\le R_0}I_A(\psi _\rho (z))\le \sup _{y\in {\mathbb {R}}^N}I_A(\psi _\rho (y)). \end{aligned}$$

The last inequality, together with Corollary 4.2 and (5.6) leads to

$$\begin{aligned} c_0< c < c_0+\kappa _1, \end{aligned}$$

(5.8)

which proves (5.4).

Suppose by contradiction that ${\mathcal {K}}_c = \emptyset$. Recall that

$$\begin{aligned} \frac{\lambda _0+c_0}{2} \le \frac{c+c_0}{2}< c < c_0+\kappa _1. \end{aligned}$$

By Corollary 3.4 and the deformation lemma [30], there is a continuous map

$$\begin{aligned} \eta :[0,1]\times {\mathcal {N}}_A\rightarrow {\mathcal {N}}_A \end{aligned}$$

and a positive number $\varepsilon _0$ such that

• (a) $L_{c+\varepsilon _0}{\setminus } L_{c-\varepsilon _0} \subset \subset L_{c_0 + \kappa _1} {\setminus } L_{\frac{\lambda _0+c_0}{2}}$,

• (b) $\eta (t,u) = u,\;\;\forall u\in L_{c-\varepsilon _0} \cup \{{\mathcal {N}}_A {\setminus } L_{c+\varepsilon _0}\}\;\;\text{ and }\;\;\forall t\in [0,1]$,

• (c) $\eta (1, L_{c + \frac{\varepsilon _0}{2}}) \subset L_{c - \frac{\varepsilon _0}{2}}$.

Fix ${\tilde{B}}\in \Upsilon$ such that

$$\begin{aligned} c\le \sup _{u\in {\tilde{B}}} I_A(u) < c + \frac{\varepsilon _0}{2}. \end{aligned}$$

Since

$$\begin{aligned} I_A(u) < c + \frac{\varepsilon _0}{2},\;\;\forall u\in {\tilde{B}}, \end{aligned}$$

it follows that

$$\begin{aligned} {\tilde{B}} \subset L_{c+\frac{\varepsilon _0}{2}}. \end{aligned}$$

Now, by (c), one has

$$\begin{aligned} I_A(u) < c-\frac{\varepsilon _0}{2},\;\;\forall u\in \eta (1, {\tilde{B}}), \end{aligned}$$

that is,

$$\begin{aligned} \sup _{u\in \eta (1, {\tilde{B}})}I_A(u) < c-\frac{\varepsilon _0}{2}. \end{aligned}$$

(5.9)

On the other hand, we notice that $\eta (1, \cdot ) \in C({\mathcal {N}}_A, {\mathcal {N}}_A)$. Moreover, since ${\tilde{B}} \in \Upsilon$, there exists $h\in {\mathcal {H}}$ such that ${\tilde{B}} = h(\Sigma )$. Consequently,

$$\begin{aligned} {\tilde{h}} = \eta (1, \cdot ) \circ h\in C({\mathcal {N}}_A, {\mathcal {N}}_A). \end{aligned}$$

Since $h\in {\mathcal {H}}$, it follows that

$$\begin{aligned} h(u) = u,\;\;\forall u\in {\mathcal {N}}_A\;\;\text{ with }\;\;I_A(u) < \frac{\lambda _0 + c_0}{2} \end{aligned}$$

and

$$\begin{aligned} {\tilde{h}}(u) = \eta (1,u)\;\;\forall u\in {\mathcal {N}}_A\;\;\text{ with }\;\;I_A(u) < \frac{\lambda _0 + c_0}{2}. \end{aligned}$$

Taking into account that

$$\begin{aligned} \frac{\lambda _0+c_0}{2} < c-\epsilon _0, \end{aligned}$$

by item (b), we easily have

$$\begin{aligned} {\tilde{h}}(u) = \eta (1,u) = u,\;\;\forall u\in {\mathcal {N}}_A\;\;\text{ with }\;\;I_A(u)< \frac{\lambda _0 + c_0}{2} < c-\epsilon _0. \end{aligned}$$

Then ${\tilde{h}} \in {\mathcal {H}}$. Moreover

$$\begin{aligned} \eta (1,{\tilde{A}}) \in \Gamma , \end{aligned}$$

owing to $\eta (1, {\tilde{B}}) = {\tilde{h}}(\Sigma )$. Therefore, exploiting the definition of c, we have

$$\begin{aligned} c\le \sup _{u\in \eta (1, {\tilde{B}})} I_A(u), \end{aligned}$$

which contradicts (5.9). Thereby, ${\mathcal {K}}_c \ne \emptyset$ and c is a critical value of $I_A$ on ${\mathcal {N}}_A$, namely there is at least one nontrivial weak solution of (P). Hence, Theorem 1.1 is proved.