Abstract
In this paper, using variational methods, we establish the existence and multiplicity of multi-bump solutions for the following nonlinear magnetic Schrödinger equation
where \(\lambda >0\), f(t) is a continuous function with exponential critical growth, the magnetic potential \(A:{\mathbb {R}}^2\rightarrow {\mathbb {R}}^2\) is in \(L^{2}_{loc}({\mathbb {R}}^2)\) and the potentials V, \(Z:{\mathbb {R}}^{2}\rightarrow {\mathbb {R}}\) are continuous functions verifying some natural conditions. We show that if the zero set of the potential V has several isolated connected components \(\Omega _{1}, \ldots , \Omega _{k}\) such that the interior of \(\Omega _{j}\) is non-empty and \(\partial \Omega _{j}\) is smooth, then for \(\lambda >0\) large enough, the equation has at least \(2^{k}-1\) multi-bump solutions.
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1 Introduction and main results
This paper is devoted to the qualitative analysis of solutions for the nonlinear magnetic Schrödinger equation in \({{\mathbb {R}}}^2\). We are concerned with the existence and multiplicity of multi-bump solutions if the reaction has an exponential critical behavior. In the first part of this section, we recall some significant historical moments related to the development of the Schrödinger theory. The main result and an associated multiplicity property are described in the second part of the present section.
1.1 Historical comments
The Schrödinger equation is central in quantum mechanics and it plays the role of Newton’s laws and conservation of energy in classical mechanics, that is, it predicts the future behaviour of a dynamical system. It is striking to point out that talking about his celebrating equation, Erwin Schrödinger said: “I don’t like it, and I’m sorry I ever had anything to do with it". The linear Schrödinger equation is a central tool of quantum mechanics, which provides a thorough description of a particle in a non-relativistic setting. Schrödinger’s linear equation is
where u is the Schrödinger wave function, m is the mass of the particle, \(\hbar \) denotes Planck’s renormalized constant, E is the energy, and V stands for the potential energy.
Schrödinger also established the classical derivation of his equation, based upon the analogy between mechanics and optics, and closer to de Broglie’s ideas. He developed a perturbation method, inspired by the work of Lord Rayleigh in acoustics, proved the equivalence between his wave mechanics and Heisenberg’s matrix, and introduced the time dependent Schrödinger’s equation
where \(p<2N/(N-2)\) if \(N\ge 3\) and \(p<+\infty \) if \(N=2\).
In physical problems, a cubic nonlinearity corresponding to \(p = 3\) in Eq. (1.1) is common; in this case problem (1.1) is called the Gross–Pitaevskii equation. In the study of Eq. (1.1), Floer and Weinstein [24] and Oh [37] supposed that the potential V is bounded and possesses a non-degenerate critical point at \(x=0\). More precisely, it is assumed that V belongs to the class (\(V_a\)) (for some real number a) introduced in Kato [30]. Taking \(\gamma >0\) and \(\hbar >0\) sufficiently small and using a Lyapunov–Schmidt type reduction, Oh [37] proved the existence of bound state solutions of problem (1.1), that is, a solution of the form
Using the Ansatz (1.2), we reduce the nonlinear Schrödinger equation (1.1) to the semilinear elliptic equation
The change of variable \(y=\hbar ^{-1}x\) (and replacing y by x) yields
where \(V_\hbar (x)=V(\hbar x)\).
Let us also recall that in his 1928 pioneering paper, Gamow [25] proved the tunneling effect, which lead to the construction of the electronic microscope and the correct study of the alpha radioactivity. The notion of “solution" used by him was not explicitly mentioned in the paper but it is coherent with the notion of weak solution introduced several years later by other authors such as J. Leray, L. Sobolev and L. Schwartz. Most of the study developed by Gamow was concerned with the bound states u(x, t) defined in (1.2), where u solves the stationary equation
for a given potential V(x). Gamow was particularly interested in the Coulomb potential but he also proposed to replace the resulting potential by a simple potential that keeps the main properties of the original one. In this way, if \(\Omega \) is a subdomain of \({{\mathbb {R}}}^N\), Gamow proposed to use the finite well potential
It seems that the first reference dealing with the limit case, the so-called infinite well potential,
was the book by the 1977 Nobel Prize Mott [36]. The more singular case in which \(V_0\) is the Dirac mass \(\delta _0\) is related with the so-called Quantum Dots, see Joglekar [29]. In contrast with classical mechanics, in quantum mechanics the incertitude appears (the Heisenberg principle). For instance, for a free particle (i.e. with \(V(x)\equiv 0\)), in nonrelativistic quantum mechanics, if the wave function \( u (\cdot ,t)\) at time \(t=0\) vanishes outside some compact region \({{\overline{\Omega }}}\) then at an arbitrarily short time later the wave function is nonzero arbitrarily far away from the original region \({{\overline{\Omega }}}\). Thus, the wave function instantaneously spreads to infinity and the probability of finding the particle arbitrarily far away from the initial region is nonzero for all \(t>0\). We refer to Díaz [20] for more details. Finally, we point out that sublinear Schrödinger equations with lack of compactness and indefinite potentials have been studied by Bahrouni, Ounaies and Rădulescu [10, 11].
1.2 Main results
Consider the following nonlinear Schrödinger equation
where \(\lambda >0\) is a parameter, V, Z, f are continuous functions verifying some assumptions, has been studied by many researchers. In [21], Ding and Tanaka studied problem (1.4) assuming \(f(t)=\vert t\vert ^{q-1}t\). In this mentioned paper, the authors established the existence of multi-bump positive solutions for the problem
where \(2<p<2N/(N-2)\) for \(N\ge 3\) and \(2<p<\infty \) for \(N=1, 2\). The authors showed that problem (1.5) has at least \(2^{k}-1\) multi-bump solutions for \(\lambda \) large enough. These solutions have the following properties: for each non-empty subset \(\Gamma \subset \{1, 2, \ldots , k\}\) and \(\epsilon >0\) fixed, there is \(\lambda ^{*}>0\) such that problem (1.5) possesses a solution \(u_{\lambda }\) for all \(\lambda \ge \lambda ^{*}=\lambda ^{*}(\epsilon )\), satisfying:
and
where \(\Omega _{\Gamma }=\underset{j\in \Gamma }{\bigcup }\Omega _{j}\) and \(c_{j}\) is the minimax level of the energy functional related to the problem
In [2], using variational methods, Alves et al. considered the existence of multi-bump positive solutions for the following problem with critical growth
where \(\lambda , \beta >0\), \(p\in (1, 2^{*}-1)\), \(2^{*}=2N/N-2, N\ge 3\). In [2], due to the critical growth of the nonlinearity in \({\mathbb {R}}^{N}\), the method applied in [21] does not hold. In [8], using a new approach, Alves et al. established the same results for the following equation
where f is continuous with exponential critical growth. Due to the exponential critical growth of the nonlinearity in \({\mathbb {R}}^{2}\), some estimates in [8] are completely different from the case \(N\ge 3\). For the further research about the nonlinear Schrödinger equation with the deepening potential well, we refer to [1, 4,5,6,7, 9, 12, 27, 34] and their references.
In recent years, the nonlinear magnetic Schrödinger equation
has also received considerable attention. This class of problems has some relevant physical applications, such as nonlinear optics and plasma physics. The function u(x, t) takes on complex values, \(\hbar \) is the Planck constant, i is the imaginary unit, \(A:{\mathbb {R}}^2\rightarrow {\mathbb {R}}^2\) is the magnetic potential.
When one looks for standing wave solutions \( u (x, t):=e^{-iEt/\hbar }u(x)\), with \(E\in {\mathbb {R}}\), of Eq. (1.8), the problem can be reduced by
As far as we know, the first result seems to be established in [23], where the existence of standing waves to problem (1.9) has been obtained for \(\hbar >0\) fixed and for special classes of magnetic fields. In this way, the authors obtained the existence of solutions by solving an appropriate minimization problem for the corresponding energy functional in the cases \(N=2\) and \(N=3\). After that, Kurata [31] proved that the problem has a least energy solution for any \(\epsilon >0\) when a technical condition relating V(x) and A(x) is assumed. Under this technical condition, Kurata proved that the associated energy functional satisfies the Palais–Smale condition at any level. In [3], by combining a local assumption on V, the penalization techniques of del Pino and Felmer [19] and the Ljusternik–Schnirelmann theory, Alves et al. obtained the multiple solutions. We would like to refer to [16,17,18, 22, 28, 35] for other results related with the problem (1.9).
Recently, there are many works concerning the following magnetic Schrödinger equation with deepening potential well
In particular, Tang [39] considered multi-bump solutions of problem (1.10) with critical frequency in which \(Z(x)\equiv 0\) and f satisfies subcritical growth. Liang and Shi [33] considered multi-bump solutions of problem (1.10) with critical nonlinearity for the case \(N\ge 3\). It is quite natural to consider multi-bump solutions for the problem when the nonlinearity satisfies the exponential critical growth in \(N=2\). To the best of our knowledge, this problem has not been considered. Motivated by [3, 8, 33], the main goal of the present paper is to prove the existence of multi-bump solutions for problem (1.10), considering a class of nonlinearity with exponential critical growth in \({\mathbb {R}}^{2}\). Because the nonlinearity has exponential critical growth in \({\mathbb {R}}^{2}\), some properties that are valid for \(N\ge 3\), do not necessarily hold for the class of problems studied in this paper. Therefore, we need to take different approaches in some estimates. On the other hand, as we will see later, due to the presence of the magnetic field A(x), problem (1.10) cannot be changed into a pure real-valued problem, hence we should deal with a complex-valued directly, which causes several new difficulties in employing the methods in dealing with our problem. Our problem is more complicated than the pattern studied in [8] and we need additional technical estimates.
We now present the general assumptions used in the statement of the main result of this paper.
- (A):
-
\(A:{\mathbb {R}}^2\rightarrow {\mathbb {R}}^2\) is in \(L^{2}_{loc}({\mathbb {R}}^2)\).
- \((V_{_{1}})\):
-
\(V(x)\in C({\mathbb {R}}^{2}, {\mathbb {R}})\) with \(V(x)\ge 0\).
- \((V_{_{2}})\):
-
The potential well \(\Omega =\text {int}\, V^{-1}(0)\) is a non-empty bounded open set with smooth boundary \(\partial \Omega \) and \({\overline{\Omega }}=V^{-1}(0)\), \(\Omega \) can be decomposed in k connected components \(\Omega _{1}, \ldots , \Omega _{k}\) with \(\text {dist}(\Omega _{i}, \Omega _{j})>0\), \(i\ne j\).
- \((V_{_{3}})\):
-
There exist two positive constants \(M_{0}\) and \(M_{1}\) such that
$$\begin{aligned} \lambda V(x)+Z(x)\ge M_{0}, \quad \forall x\,\in {\mathbb {R}}^{2},\, \lambda \ge 1, \end{aligned}$$and
$$\begin{aligned} \vert Z(x)\vert \le M_{1}, \quad \forall x\,\in {\mathbb {R}}^{2}. \end{aligned}$$We assume that the reaction f is a continuous function satisfying the following conditions.
- \((f_{_{1}})\):
-
\(f(t)=0\) if \(t\le 0\).
- \((f_{_{2}})\):
-
We have
$$\begin{aligned} \lim _{t\rightarrow +\infty } \frac{f(t^{2})t}{e^{\alpha t^{2}}} = \left\{ \begin{array}{ll} 0, &{} \hbox {for } \alpha > 4\pi ,\\ +\infty ,&{} \hbox {for } 0<\alpha <4\pi \,. \end{array}\right. \end{aligned}$$ - \((f_{_{3}})\):
-
There is a positive constant \(\theta >2\) such that
$$\begin{aligned} 0<\frac{\theta }{2} F(t)\le tf(t), \quad \quad \forall t>0, \end{aligned}$$where \(F(t)=\int _{0}^{t}f(s)ds\).
- \((f_{_{4}})\):
-
There exist constants \(p>2\) and \(C_{p}>0\) such that
$$\begin{aligned} f(t)\ge C_{p}t^{(p-2)/2}\quad \text {for all}\, t>0, \end{aligned}$$where
$$\begin{aligned} C_{p}&>\Big (\frac{k\theta (p-2)}{M^{*}p(\theta -2)}\Big )^{(p-2)\diagup 2}S_{p}^{p},\quad \quad M_{*}=\min \{1, M_{0}\}, \\ S_{p}&=\max \left\{ \underset{\varphi \in H_{A}^{0,1}(\Omega _{j})\backslash \{0\}}{\inf }\frac{\Big (\int _{\Omega _{j}}(\vert \nabla _{A}\varphi \vert ^{2}+Z(x)\vert \varphi \vert ^{2})dx\Big )^{1/2}}{\Big (\int _{\Omega _{j}}\vert \varphi \vert ^{p}dx\Big )^{1/p}}, \quad j=1, \ldots , k \right\} . \end{aligned}$$ - \((f_{_{5}})\):
-
f(t) is an increasing function in \([0,\infty )\).
The main result in this paper is stated below.
Theorem 1.1
Assume that (A), \((V_{1})\)–\((V_{3})\) and \((f_{1})\)–\((f_{5})\) hold. Then, for any non-empty subset \(\Gamma \) of \(\{1, 2, \ldots , k\}\), there exists \(\lambda ^{*}\) such that for all \(\lambda \ge \lambda ^{*}\), problem (1.10) has a nontrivial solution \(u_{\lambda }\). Moreover, the family \(\{u_{\lambda }\}_{\lambda \ge \lambda ^{*}}\) has the following properties: for any sequence \(\lambda _{n}\rightarrow \infty \), we can extract a subsequence \(\lambda _{n_{i}}\) such that \(u_{\lambda _{n_{i}}}\) converges strongly in \(H_{A}^{1}({\mathbb {R}}^{2}, {\mathbb {C}})\) to a function u which satisfies \(u(x)=0\) for \(x\not \in \Omega _{\Gamma }\) and the restriction \(u|_{\Omega _{j}}\) is a least energy solution of
where \(\Omega _{\Gamma }=\underset{j\in \Gamma }{\bigcup }\Omega _{j}\).
Corollary 1.2
Under the assumptions of Theorem 1.1, there exists \(\lambda _{*}>0\) such that for all \(\lambda \ge \lambda _{*}\), problem (1.10) has at least \(2^{k}-1\) nontrivial solutions.
The paper is organized as follows. In Sect. 2 we introduce the functional setting and we give some preliminary results. In Sect. 3, we study the modified problem. We prove the Palais–Smale condition for the modified energy functional for \(\lambda \) large and study \(L^{\infty }\)-estimates for the solution and the behavior of \((PS)_{\infty }\) sequences. In Sect. 4, we adapt the deformation flow method in order to establish the existence of a special critical point, which is crucial for showing the existence of multi-bump solutions for \(\lambda \) large enough and hence to prove Theorem 1.1. We refer to the recent monograph by Papageorgiou, Rădulescu and Repovš [38] for some of the abstract methods used in this paper.
1.3 Notation
-
\(C, C_1, C_2, \ldots \) denote positive constants whose exact values are inessential and can change from line to line;
-
\(B_{R}(y)\) denotes the open disk centered at \(y\in {\mathbb {R}}^2\) with radius \(R>0\) and \(B^{c}_{R}(y)\) denotes the complement of \(B_{R}(y)\) in \({\mathbb {R}}^{2}\);
-
\(\Vert \cdot \Vert \), \(\Vert \cdot \Vert _{q}\), and \(\Vert \cdot \Vert _{L^{\infty }(\Omega )}\) denote the usual norms of the spaces \(H^{1}({\mathbb {R}}^{2}, {\mathbb {R}})\), \(L^{q}({\mathbb {R}}^{2}, {\mathbb {R}})\), and \(L^{\infty }(\Omega , {\mathbb {R}})\), respectively, where \(\Omega \subset {\mathbb {R}}^{2}\);
-
\(o_{n}(1)\) denotes a real sequence with \(o_{n}(1)\rightarrow 0\) as \(n \rightarrow +\infty \).
2 Abstract setting and preliminary results
In this section, we outline the variational framework for problem (1.10) and give some auxiliary properties.
For \(u: {\mathbb {R}}^{2}\rightarrow {\mathbb {C}}\), let us denote by
and
The space \(H_{A}^{1}({\mathbb {R}}^{2}, {\mathbb {C}})\) is a Hilbert space endowed with the scalar product
where \({\text {Re}}\) and the bar denote the real part of a complex number and the complex conjugation, respectively. Moreover, we denote by \(\Vert u\Vert _{A}\) the norm induced by this inner product. The spaces \(H_{A}^{1}({\mathbb {R}}^{2}, {\mathbb {C}})\) and \(H^{1}({\mathbb {R}}^{2}, {\mathbb {R}})\) are not comparable, more precisely, in general \(H_{A}^{1}({\mathbb {R}}^{2}, {\mathbb {C}})\not \subset H^{1}({\mathbb {R}}^{2}, {\mathbb {R}})\) and \(H^{1}({\mathbb {R}}^{2}, {\mathbb {R}})\not \subset H_{A}^{1}({\mathbb {R}}^{2}, {\mathbb {C}})\).
By hypothesis (A), on the space \(H_{A}^{1}({\mathbb {R}}^{2}, {\mathbb {C}})\) we have the following diamagnetic inequality (see e.g. [32, Theorem 7.21]):
Let
with the norm
For \(\lambda \ge 1\), it is easy to see that \((E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}}), \Vert \cdot \Vert _{\lambda })\) is a Hilbert space and \(E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\subset H_{A}^{1}({\mathbb {R}}^{2}, {\mathbb {C}})\).
Let \(K\subset {\mathbb {R}}^{2}\) be an open set. We define
Let \(H_{A}^{0, 1}(K, {\mathbb {C}})\) be the Hilbert space defined by the closure of \(C_{0}^{\infty }(K, {\mathbb {C}})\) under the norm \(\Vert u\Vert _{H_{A}^{1}(K)}\).
The diamagnetic inequality (2.1) implies that if \(u\in E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\), then \(|u|\in H^{1}({\mathbb {R}}^{2}, {\mathbb {R}})\) and \(\Vert u\Vert \le C\Vert u\Vert _{\lambda }\). Therefore, the embedding \(E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\hookrightarrow L^{r}({\mathbb {R}}^{2}, {\mathbb {C}})\) is continuous for \(r\ge 2\) and the embedding \(E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\hookrightarrow L^{r}_{\text {loc}}({\mathbb {R}}^{2}, {\mathbb {C}})\) is compact for \(r\ge 1\).
Remark 2.1
Since the embedding \(H_{A}^{1}({\mathbb {R}}^{2}, {\mathbb {C}})\hookrightarrow L^{r}_{\text {loc}}({\mathbb {R}}^{2}, {\mathbb {C}})\) is compact for \(r\ge 1\), a standard argument shows that the following infimum in \((f_{4})\) is achieved and
for \(j=1, \ldots , k\).
We recall that \(u\in E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\) is a weak solution of (1.10), if
for all \(\phi \in E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\).
The weak solutions of problem (1.10) are the critical points of \(I_{\lambda }: E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\rightarrow {\mathbb {R}}\) given by
where \(F(t)=\int _{0}^{t}f(s)ds\). It is easy to prove that \(I_{\lambda }\in C^{1}(E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}}), {\mathbb {R}})\).
In view of \((V_{3})\), for any open set \(K \subset {\mathbb {R}}^{2}\), it is easy to see that
where \(\Vert u\Vert _{2, K}^{2}=\int _{K}\vert u\vert ^{2}dx\). The following property is an immediate consequence of the above consideration.
Lemma 2.2
There exist \(\delta _{0}, \nu _{0}>0\) with \(\delta _{0}\thicksim 1\) and \(\nu _{0}\thicksim 0\) such that for any open set \(K \subset {\mathbb {R}}^{N}\)
The below estimates involving f are the key points in this paper. By \((f_{1})\) and \((f_{2})\), fixed \(q>2\), for any \(\zeta >0\) and \(\alpha >4\pi \), there exists a constant \(C>0\) depending on q, \(\alpha \), \(\zeta \), such that
and, using \((f_{3})\), we have
and
Now we recall a version of the Trudinger-Moser inequality in the whole space \({\mathbb {R}}^{2}\) due to Cao [15] (see also [13], Lemma 2.3) for functions belonging to \(H^{1}({\mathbb {R}}^{2}, {\mathbb {R}})\).
Lemma 2.3
If \(\alpha >0\) and \(u\in H^{1}({\mathbb {R}}^{2}, {\mathbb {R}})\), then
Moreover, if \(\Vert \nabla u\Vert _{2}^{2}\le 1\), \(\Vert u\Vert _{2}\le M<+\infty \), and \(0<\alpha < 4\pi \), then there exists a positive constant \(C(M, \alpha )\), which depends only on M and \(\alpha \), such that
To finish this section, in what follows, for each \(j\in \{1, 2, \ldots , k\}\), we fix a bounded open subset \(\Omega '_{j}\) with smooth boundary such that
- (i):
-
\(\overline{\Omega _{j}}\subset \Omega '_{j}\);
- (ii):
-
\(\overline{\Omega '_{i}}\bigcap \overline{\Omega '_{j}}= \emptyset \) for all \(i\ne j\).
3 An auxiliary problem
Since \({\mathbb {R}}^{2}\) is unbounded, we know that the Sobolev embeddings are not compact, as so \(I_{\lambda }\) cannot verify the Palais–Smale condition. In order to overcome this difficulty, we adapt an argument of the penalization method introduced by del Pino and Felmer [19] and Ding and Tanaka [21].
Let \(\nu _{0}>0\) be a constant given in Lemma 2.2, \(\kappa>\frac{\theta }{\theta -2}>1\) and \(a>0\) verifying \(f(a)=\frac{\nu _{0}}{\kappa }\) and \({\tilde{f}}, {\widetilde{F}}:{\mathbb {R}}\rightarrow {\mathbb {R}}\) given by
and
Note that
From now on, we fix a non-empty subset \(\Gamma \subset \{1, \ldots , k\}\) and
the function
It follows from (3.1) that g satisfies the following inequality
Let \(\Phi _{\lambda }:E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\rightarrow {\mathbb {R}}\) be the functional defined by
Standard arguments show that \(\Phi _{\lambda }\in C^{1}(E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}}), {\mathbb {R}})\) and its critical points are weak solutions of
Our aim is to obtain nontrivial solutions of (3.5) which are solutions of the original problem (1.10). More precisely, if \(u_{\lambda }\) is a nontrivial solution of (3.5) verifying \(\vert u_{\lambda }(x)\vert ^{2}\le a\) in \({\mathbb {R}}^{2}\backslash \Omega '_{\Gamma }\), then it is a nontrivial solution to (1.10).
3.1 The Palais–Smale condition and consequences
We start this subsection studying the boundedness of the Palais–Smale sequence related to \(\Phi _{\lambda }\), that is, a sequence \((u_{n})\subset E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\) verifying
for some \(c\in {\mathbb {R}}\)(shortly \((u_{n})\) is a \((PS)_{c}\) sequence).
Lemma 3.1
If \((u_{n})\) is a \((PS)_{c}\) sequence to \(\Phi _{\lambda }\), it follows that
where \(\delta _{0}\) is given in Lemma 2.2.
Proof
From the definition of Palais–Smale sequence, we have
On the other hand, from (3.6), (3.2), \(\kappa >\theta /(\theta -2)\), \((f_{3})\) and Lemma 2.2, we obtain
Therefore,
This shows that \((u_{n})\) is bounded and
which completes the proof. \(\square \)
For each fixed \(j\in \Gamma \), let us denote by \(c_{j}\) the minimax level of the functional \(I_{j}:H_{A}^{0, 1}(\Omega _{j})\rightarrow {\mathbb {R}}\) given by
and
where
It is well known that the critical points of \(I_{j}\) are weak solutions of the following problem
In the next lemma, we denote by S the following real number
Lemma 3.2
If \((f_{1})-(f_{5})\) hold, then \(0<S<M^{*}\delta _{0}(\frac{1}{2}-\frac{1}{\theta })\).
Proof
For each \(j\in \{1,\ldots , k\}\), we may choose a function \(\varphi _{j}\in H_{A}^{0,1}(\Omega _{j}, {\mathbb {C}})\) such that
Notice that
Hence
On the other hand, by \((f_{4})\) we have
Since \(\delta _{0}\) may be chosen close to 1, the last inequality implies that
This completes the proof of the lemma. \(\square \)
Proposition 3.3
For any \(\lambda \ge 1\), the functional \(\Phi _{\lambda }\) satisfies the \((PS)_{c}\) condition for all \(c\in (0, S]\), that is, if \(c\in (0, S]\), any \((PS)_{c}\)-sequence \((u_{n})\subset E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\) of \(\Phi _{\lambda }\) has a strongly convergent subsequence in \(E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\).
Proof
Let \((u_{n})\subset E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\) be a \((PS)_{c}\)-sequence for \(\Phi _{\lambda }\) at the level \(c\in (0, S]\). By Lemmas 3.1 and 3.2 we obtain
Thus, up to a subsequence, \(u_{n}\rightharpoonup u\) in \(E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\) and \(u_{n}\rightarrow u\) in \(L^{q}_{\mathrm{loc}}({\mathbb {R}}^{2}, {\mathbb {C}})\) for all \(q\ge 1\) as \(n\rightarrow +\infty \). Moreover, by (3.6) and (2.2), fixed \(q>2\), for any \(\zeta >0\) and \(\alpha >4\pi \), there exists a constant \(C>0\), which depends on q, \(\alpha \), \(\zeta \), such that for any \(\phi \in E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\),
Arguing as in [18, Lemma 2.5], we have
Thus, u is a critical point of \(\Phi _{\lambda }\).
Now, we take \(R>0\) such that \(\Omega '_{\Gamma }\subset B_{\frac{R}{2}}(0)\). Let \(\phi _{R}\in C^{\infty }({\mathbb {R}}^{2}, {\mathbb {R}})\) be a cut-off function such that
where \(C>0\) is a constant independent of R. By a direct computation, one has
and
Therefore,
Notice that
Using the Hölder inequality and (3.7) we obtain
Moreover, we have
which implies that for any \(\zeta >0\), there exists \(R^{*}>0\) large, if \(R>R^{*}\), one has
Similarity, by (3.6) and (2.2), fixed \(q>2\), for any \(\zeta >0\) and \(\alpha >4\pi \), there exists a constant \(C>0\), which depends on q, \(\alpha \), \(\zeta \), such that
Since \(u_{n}\rightarrow u\) in \(L_{\mathrm{loc}}^{r}({\mathbb {R}}^{2}, {\mathbb {C}})\), for all \(r\ge 1\), up to a subsequence, we have that
Then
Moreover, \(\vert u_{n}\vert \rightarrow \vert u\vert \) in \(L_{\mathrm{loc}}^{r}({\mathbb {R}}^{2}, {\mathbb {R}})\) for all \(r\ge 1\).
Let
where \(\alpha >4\pi \) with \(\alpha \Vert \vert u_{n}\vert \Vert <4\pi \) for n large. Using (3.6) and \((f_{2})\), it is easy to see that
and by Lemma 2.3,
Then [14, Theorem A.I] implies
Moreover, by (3.8) and the definition of g, we have
for any \(\zeta >0\).
Hence
Finally, since \(\Phi _{\lambda }'(u)=0\), we have
Thus, the sequence \((u_{n})\) strong converges to u in \(E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\). \(\square \)
Our next step is to study the behavior of a \((PS)_{\infty , c}\) sequence, that is, a sequence \((u_{n})\subset H_{A}^{1}({\mathbb {R}}^{2}, {\mathbb {C}})\) satisfying
Proposition 3.4
Let \((u_{n})\) be a \((PS)_{\infty , c}\) sequence with \(c\in (0, S]\). Then, for some subsequence, still denoted by \((u_{n})\), there exists \(u\in H_{A}^{1}({\mathbb {R}}^{2}, {\mathbb {C}})\) such that
Moreover,
- (i):
-
\(u\equiv 0\) in \({\mathbb {R}}^{2}\backslash \Omega _{\Gamma }\) and \(u|_{\Omega _{j}}\) is a solution of (3.6), for \(\forall \,j\in \Gamma \);
- (ii):
-
\(\Vert u_{n}-u\Vert _{\lambda _{n}}\rightarrow 0\);
- (iii):
-
\(u_{n}\) also satisfies
$$\begin{aligned}&\lambda _{n}\int _{{\mathbb {R}}^{2}}V(x)\vert u_{n}\vert ^{2}dx\rightarrow 0,\\&\Vert u_{n}-u\Vert ^{2}_{\lambda _{n}, {\mathbb {R}}^{2}\backslash \Omega _{\Gamma }}\rightarrow 0,\\&\Vert u_{n}\Vert ^{2}_{\lambda _{n}, {\mathbb {R}}^{2}\backslash \Omega '_{j}}\rightarrow \int _{\Omega _{j}}(\vert \nabla _{A}u\vert ^{2}+Z(x)\vert u\vert ^{2})dx. \end{aligned}$$
Proof
As in the proof of Proposition 3.3, it is easy to check that
Thus \((u_{n})\) is bounded in \(H_{A}^{1}({\mathbb {R}}^{2}, {\mathbb {C}})\) and we may assume that for some \(u\in H_{A}^{1}({\mathbb {R}}^{2}, {\mathbb {C}})\), up to a subsequence, if necessary
To show (i), we fix the set \(C_{m}=\{x\in {\mathbb {R}}^{2}: V(x)\ge \frac{1}{m}\}\). Then, for n large
The last inequality together with Fatou’s lemma imply
Therefore, \(u(x)=0\) on \(\bigcup _{m=1}^{+\infty }C_{m}={\mathbb {R}}^{2}\backslash \Omega \), from which we can assert that \(u|_{\Omega _{j}} \in H_{A}^{0, 1}(\Omega _{j}, {\mathbb {C}})\) for any \(j\in \{1, 2, \ldots , k\}\).
Since \(\Phi '_{\lambda _{n}}(u_{n})\varphi \rightarrow 0\) as \(n\rightarrow \infty \), for each \(\varphi \in C_{0}^{\infty }(\Omega _{j}, {\mathbb {C}})\)(and hence for each \(\varphi \in H_{A}^{0, 1}(\Omega _{j}, {\mathbb {C}})\)), from (3.9) and the similar arguments in Proposition 3.3, we have
showing that \(u|_{\Omega _{j}}\) is a solution of problem (3.6) for each \( j\in \{1, \ldots , k\}\).
For each \(j\in \{1, \ldots , k\}\backslash \Gamma \), setting \(\varphi =u|_{\Omega _{j}}\) in (3.11), we have
By Lemma 2.2 and the fact that \({\tilde{f}}(t^{2})t^{2}\le \frac{\nu _{0}}{\kappa }t^{2}\) for all \(t\in {\mathbb {R}}\), it yields
Thus \(u=0\) in \(\Omega _{j}\) for \(j\in \{1, 2, \ldots , k\}\backslash \Gamma \), it means that (i) holds.
For (ii), using the similar arguments in the proof of Proposition 3.3, for each \(\zeta >0\), there exists \(R>0\) such that
Using the same arguments as in the proof of Proposition 3.3 and (i), the above inequality implies that
Now, by (i) again, we have
Thus, the sequence \((u_{n})\) strong converges to u in \(E_{\lambda _{n}}({\mathbb {R}}^{2}, {\mathbb {C}})\) and (ii) holds.
To prove (iii), notice that from (i) and (ii),
Moreover, from (i) and (ii), it is also easy to obtain that as \(n\rightarrow \infty \)
Therefore, the proof is complete. \(\square \)
Proposition 3.5
For each \(\lambda \ge 1\), let \(u_{\lambda }\) be a nontrivial solution of problem (3.5) with \(\Vert u_{\lambda }\Vert ^{2}_{\lambda }<1\). Then, there exists \(K, \lambda ^{*}>0\) such that
Proof
Let \((\lambda _{n})\) be a sequence with \(\lambda _{n}\rightarrow \infty \) and define \(u_{n}(x)=u_{\lambda _{n}}(x)\). For any \(R>0\) and \(0<r\le R/2\), let \(\eta \in C^{\infty }({\mathbb {R}}^{2})\), \(0\le \eta \le 1\) with \(\eta (x)=1\) if \(\vert x\vert \ge R\) and \(\eta (x)=0\) if \(\vert x\vert \le R-r\) and \(\vert \nabla \eta \vert \le 2/r\).
For each \(n\in N\) and \(L>0\), we consider the functions
where \(\beta >1\) will be determined later.
By straightforward computations, we have
and
Taking the real part of \(\nabla _{A}u_{n}\overline{\nabla _{A}z_{L, n}}\) and using the diamagnetic inequality (2.1), we obtain
Taking \(z_{L, n}\) as the test function, we have
By (3.12), the Young inequality (with \(\tau >0\)), (3.6), (2.4), for \(\alpha >4\pi \) and for a fixed \(q>2\), given \(0<\zeta <M_{0}\), there exists \(C>0\) such that
Hence, choosing \(\tau >0\) sufficiently small, we get
Moreover, arguing similarly to (3.13), we can conclude that
On the other hand, using the Sobolev embedding, (3.14), (3.15), the Hölder inequality with \(t,\sigma ,\tau >1\), \(1/\sigma +1/\tau =1/t\), \(\sigma (q-2)\ge 2\), and the inequality \((e^{t}-1)^{s}\le e^{ts}-1\), for \(s>1\) and \(t\ge 0\), we have
Since \((u_{n})\subset H_{A}^{1}({\mathbb {R}}^{2}, {\mathbb {C}})\) is a a \((PS)_{\infty , c}\) sequence, up to a subsequence, by Proposition 3.4, we have \(u_{n}\rightarrow u\) in \(H_{A}^{1}({\mathbb {R}}^{2}, {\mathbb {C}})\). By (3.16), it follows that
and, applying the Fatou’s Lemma as \(L\rightarrow +\infty \), we obtain
Next, if we take \(\zeta :=\frac{q(t-1)}{2t}\), \(\beta :=\zeta ^{m}\), with \(m\in {\mathbb {N}}^*\), and \(s:=\frac{2t}{t-1}\), we obtain
for every \(m\in {\mathbb {N}}^*\). Then, for \(r=r_{m}:=R/2^m\), \(m\in {\mathbb {N}}^*\), using also that \(2/t<2\), we get
Hence, passing to the limit as \(m\rightarrow +\infty \) in the last inequality, we obtain
For \(x_{0}\in {\mathbb {R}}^{2}\), we can use the same argument taking \(\eta \in C_{0}^{\infty }({\mathbb {R}}^{2}, [0, 1])\) with \(\eta (x)=1\) if \(\vert x-x_{0}\vert \le {\tilde{\rho }}\), \(\eta (x)=0\) if \(\vert x-x_{0}\vert >2\rho \), with \({\tilde{\rho }} < \rho \), and \(\vert \nabla \eta \vert \le 2/{\tilde{\rho }}\), to prove that
Thus, by (3.17), (3.18), and using a standard covering argument and the boundedness of \((|u_{\lambda }|)\) in \(L^q({\mathbb {R}}^2,{\mathbb {R}})\), it follows that there exists \(K>0\) such that
Hence the proof is complete. \(\square \)
Proposition 3.6
Let \((u_{\lambda })\) be a family of nontrivial solutions of problem (3.3) with \(\Vert u_{\lambda }\Vert ^{2}_{\lambda }<1\) and \(\lambda \ge 1\). Then, there exists \(\lambda ^{*}>0\) such that
Proof
We use notation \(B_{r}(x)=\{y\in {\mathbb {R}}^{2}:\vert x-y\vert < r\}\). Since \(u_{\lambda }\in E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\) is a critical point of \(\Phi _{\lambda }(u)\), that is, \(u_{\lambda }\) satisfies the following equation
By Kato’s inequality
there holds
since \(\vert u_{\lambda }\vert \ge 0\) and \((\lambda V(x)+Z(x))\ge M_{0}>0\) if \(\lambda \ge 1\), we have
Using Proposition 3.5 and the subsolution estimate (see [26] Theorem 8.17), there exists a constant C(r) such that
By Proposition 3.4, for any sequence \(\lambda _{n}\rightarrow \infty \), we can extract a subsequence \(\lambda _{n_{i}}\) such that
In particular,
Since \(\lambda _{n}\rightarrow \infty \) is arbitrary, we have
Thus, choosing \(r\in (0, \text {dist}(\Omega _{\Gamma }, {\mathbb {R}}^{2}\backslash \Omega '_{\Gamma }))\), we have uniformly in \(x\in {\mathbb {R}}^{2}\backslash \Omega '_{\Gamma }\) that
The proof is now complete. \(\square \)
4 The existence of multi-bump positive solutions
In this section, for each \(j\in \Gamma \), we denote by \(\Phi _{\lambda , j}:H_{A}^{ 1}(\Omega '_{j}, {\mathbb {C}})\rightarrow {\mathbb {R}}\) the functional given by
It is easy to check that the functional \(\Phi _{\lambda , j}\) satisfies the mountain pass geometry. In what follows, we denote by \(c_{\lambda , j}\) the minimax level related to the above functional defined by
where
Therefore, there exist \((u_{n})\subset H_{A}^{0, 1}(\Omega _{j}, {\mathbb {C}})\) and \((u_{\lambda , n})\subset H_{A}^{1}(\Omega '_{j}, {\mathbb {C}})\) verifying
and
From \((f_{1})\) and \((f_{3})-(f_{5})\), it is easy to prove that
and these inequalities imply that \(I_{j}\) and \(\Phi _{\lambda , j}\) satisfy the \((PS)_{c_{j}}\) and \((PS)_{c_{\lambda , j}}\) conditions, respectively. Therefore, it is easy to prove that there exist two nontrivial functions \(w_{j}\in H_{A}^{0, 1}(\Omega _{j})\) and \(w_{\lambda , j}\in H_{A}^{1}(\Omega '_{j})\) verifying
and
Moreover, we have the following lemma.
Lemma 4.1
The following assertions hold:
- (i):
-
\(0<c_{\lambda , j}\le c_{j}\) for \(\lambda \ge 1\) and \(j\in \Gamma \).
- (ii):
-
\( c_{j}\)(\(c_{\lambda , j}\) respectively) is a least energy level for \(I_{j}(u)\)(\(\Phi _{\lambda , j}(u)\) respectively), that is
$$\begin{aligned} c_{j}=\inf \Big \{I_{j}(u): u\in H_{A}^{0, 1}(\Omega _{j})\backslash \{0\}, I'_{j}(u)u=0\Big \}, \end{aligned}$$and
$$\begin{aligned} c_{\lambda , j}=\inf \Big \{\Phi _{\lambda , j}(u): u\in H_{A}^{1}(\Omega '_{j})\backslash \{0\}, \Phi '_{\lambda , j}(u)u=0\Big \}. \end{aligned}$$ - (iii):
-
\(c_{j}=\max _{t>0}I_{j}(tw_{j})\), \(c_{\lambda , j}=\max _{t>0}\Phi _{\lambda , j}(tw_{\lambda , j})\).
- (iv):
-
\(c_{\lambda , j}\rightarrow c_{j}\) as \(\lambda \rightarrow \infty \) for any \(j\in \Gamma \).
Proof
From \((f_{4})\), it is easy to prove that \(c_{\lambda , j}>0\) and \(c_{j}>0\) for any \(j\in \Gamma \) and \(\lambda \ge 1\).
Now for any \(u\in H_{A}^{0, 1}(\Omega _{j})\), we may extend u to \({\tilde{u}}\in H_{A}^{1}(\Omega '_{j})\) by
and \(H_{A}^{0, 1}(\Omega _{j})\subset H_{A}^{1}(\Omega '_{j})\). Thus, we have \(\Lambda _{j}\subset \Lambda _{\lambda , j}\) and
Thus (i) holds. The proof of (ii) and (iii) is standard by using the monotonicity of the term f(t) with respect to t for \(t>0\).
Now we prove (iv). Using Proposition 3.4, we may extract a subsequence \(\lambda _{n}\rightarrow \infty \) such that
where \(u_{0}\in H_{A}^{0, 1}(\Omega _{j})\) is a solution of (3.6) and
By the definition of \(c_{j}\), we have
Together with (i), we get (iv). \(\square \)
4.1 A special critical value of \(\Phi _{\lambda }\)
In what follows, let us fix \(R>1\) such that
and
From the definition of \(c_{j}\), it is easy to check that
We consider \(\Gamma =\{1, 2, \ldots , l\}(l\le k)\), and the maps
and
We remark that \(\gamma _{0}\in \Lambda _{*}\), so \(\Lambda _{*}\ne \emptyset \) and \(b_{\lambda , \Lambda }\) is well defined.
Lemma 4.2
For any \(\gamma \in \Lambda _{*}\), there exists \((t_{1}, t_{2}, \ldots , t_{l})\in \left[ 1/R^{2}, 1\right] ^{l}\) such that
Proof
For a given \(\gamma \in \Lambda _{*}\), let us consider the map \({\tilde{\gamma }}:\left[ 1/R^{2}, 1\right] ^{l}\rightarrow {\mathbb {R}}^{l}\) defined by
where
For any \((s_{1}, s_{2}, \ldots , s_{l})\in \partial ([1/R^{2}, 1]^{l})\), it follows that
and
Thus,
Using this fact, it follows from the topological degree
Hence, there exists \((t_{1}, t_{2}, \ldots , t_{l})\in \left( 1/R^{2}, 1\right) ^{l}\) satisfying
The proof is thus completed. \(\square \)
In the sequel, let us denote by \(c_{\Gamma }=\sum _{j=1}^{l}c_{j}\). From \((f_{4})\), we know that
Proposition 4.3
The following facts hold
- (i):
-
\(\sum _{j=1}^{l}c_{\lambda , j}\le b_{\lambda , \Gamma }\le c_{\Gamma }\) for all \(\lambda \ge 1\).
- (ii):
-
\(\Phi _{\lambda }(\gamma (s_{1}, s_{2}, \ldots , s_{l}))< c_{\Gamma }\) for all \(\lambda \ge 1\), \(\gamma \in \Lambda _{*}\) and \((s_{1}, s_{2}, \ldots , s_{l})\in \partial ([1/R^{2}, 1]^{l})\).
Proof
Since \(\gamma _{0}\) defined in (4.3) belongs to \(\Lambda _{*}\), we have
Fixing \((t_{1}, t_{2}, \ldots , t_{l})\in [1/R^{2}, 1]^{l}\) given in Lemma 4.2 and recalling that \(c_{\lambda , j}\) can be characterized by
It follows that
On the other hand, recalling that \(\Phi _{\lambda , {\mathbb {R}}^{2}\backslash \Omega '_{\Gamma }}(u)\ge 0\) for all \(u\in H_{A}^{1}({\mathbb {R}}^{2}\backslash \Omega '_{\Gamma })\), we have
Thus
From the definition of \(b_{\lambda , \Gamma }\), we can obtain
This completes the proof of (i).
Since \(\gamma (s_{1}, s_{2}, \ldots , s_{l})=\gamma _{0}(s_{1}, s_{2}, \ldots , s_{l})\) on \(\partial ([1/R^{2}, 1]^{l})\), we have
Moreover, \(I_{j}( s_{j}Rw_{j})\le c_{j}\) for all \(j\in \Gamma \) and for some \(j_{0}\in \Gamma \), \(s_{j_{0}}\in \{1/R^{2}, 1\}\) and \(I_{j_{0}}( s_{j_{0}}Rw_{j_{0}})\le \frac{c_{j_{0}}}{2}\). Therefore,
for some \(\epsilon >0\). This completes the proof of (ii). \(\square \)
Corollary 4.4
The following claims hold:
- (i):
-
\(b_{\lambda , \Gamma }\rightarrow c_{\Gamma }\) as \(\lambda \rightarrow \infty \).
- (ii):
-
\(b_{\lambda , \Gamma }\) is a critical value of \(\Phi _{\lambda }\) for large \(\lambda \).
Proof
(i) For all \(\lambda \ge 1\) and for each j, we have \(0<c_{\lambda , j}\le c_{j}\). Using the same arguments in the proof of Proposition 3.4, we can prove that \(c_{\lambda , j}\rightarrow c_{j}\) as \(\lambda \rightarrow \infty \) and thus, from Proposition 4.3, \(b_{\lambda , \Gamma }\rightarrow c_{\Gamma }\) as \(\lambda \rightarrow \infty \).
(ii) Using the fact that \(\Phi _{\lambda }\) verifies that Palais–Smale condition, we can use well known arguments involving deformation lemma [40] to conclude that \(b_{\lambda , \Gamma }\) is a critical level to \(\Phi _{\lambda }\) for large \(\lambda \). \(\square \)
4.2 Proof of the main result
To prove Theorem 1.1, we need to find a nontrivial solution \(u_{\lambda }\) for the large \(\lambda \) which approaches a least energy solution in each \(\Omega _{j}(j\in \Gamma )\) and to 0 elsewhere as \(\lambda \rightarrow \infty \). To this end, we will show two propositions which imply together with the estimates made in the previous section that Theorem 1.1 holds.
Let
For small \(\mu >0\), we define
W also use the notation:
and remark that \(w=\sum _{j=1}^{l}w_{j}\in A_{\mu }^{\lambda }\cap \Phi _{\lambda }^{c_{\Gamma }}\), this shows that \(A_{\mu }^{\lambda }\cap \Phi _{\lambda }^{c_{\Gamma }}\ne \emptyset \). Fixing
We have the following uniform estimate of \(\Vert \Phi '_{\lambda }(u)\Vert _{\lambda }\) on the annulus \((A_{2\mu }^{\lambda }\backslash A_{\mu }^{\lambda })\cap \Phi _{\lambda }^{c_{\Gamma }}\).
Proposition 4.5
Let \(\mu >0\) satisfies (4.3). Then there exist \(\sigma _{0}>0\) and \(\lambda ^{*}\ge 1\) independent of \(\lambda \) such that
Proof
Arguing by contradiction, we assume that there exist \(\lambda _{n}\rightarrow \infty \) and \(u_{n}\in (A_{2\mu }^{\lambda _{n}}\backslash A_{\mu }^{\lambda _{n}})\cap \Phi _{\lambda _{n}}^{c_{\Gamma }}\) such that \(\Vert \Phi '_{\lambda }(u_{n})\Vert _{\lambda _{n}}\rightarrow 0\).
Since \(u_{n}\in A_{2\mu }^{\lambda _{n}}\) and \(\{\Vert u_{n}\Vert _{\lambda _{n}}\}\) is a bounded sequence, this shows that \(\{\Phi _{\lambda _{n}}(u_{n})\}\) is also bounded. Thus, we may assume that
up to a subsequence.
Applying Proposition 3.4, we can extract a subsequence \(u_{n}\rightarrow u\) in \(H_{A}^{1}({\mathbb {R}}^{2})\) where \(u\in H_{A}^{0, 1}(\Omega _{\Gamma })\) is a solution of (4.1) with
Since \(c_{j}\) is the least energy level for \(I_{j}\), we have two possibilities:
- (i):
-
\(I_{j}(u\mid _{\Omega _{j}} )=c_{j}\) for all \(j\in \Gamma \).
- (ii):
-
\(I_{j_{0}}(u\mid _{\Omega _{j_{0}}})=0\), that is \(u\mid _{ \Omega _{j_{0}}}\equiv 0\) for some \(j_{0}\in \Gamma \).
If (i) occurs, we have
From (4.6)–(4.8), we have \(u_{n}\in A_{\mu }^{\lambda _{n}}\) for large n, which is a contradiction to the assumption \(u_{n}\in (A_{2\mu }^{\lambda _{n}}\backslash A_{\mu }^{\lambda _{n}})\).
If (ii) occurs, from (4.7) and \(u_{n}\rightarrow u\) in \(H_{A}^{1}({\mathbb {R}}^{2})\), it follows that
which is a contradiction with the fact that \(u_{n}\in (A_{2\mu }^{\lambda _{n}}\backslash A_{\mu }^{\lambda _{n}})\). Thus neither (i) nor (ii) can hold, and the proof is completed. \(\square \)
Proposition 4.6
Let \(\mu >0\) satisfies (4.3) and \(\lambda ^{*}\ge 1\) be a constant given by in Proposition 4.5. Then, for \(\lambda \ge \lambda ^{*}\), there exists a nontrivial solution \(u_{\lambda }\) of (3.5) satisfying \(u_{\lambda }\in A_{\mu }^{\lambda }\cap \Phi _{\lambda }^{c_{\Gamma }}\).
Proof
Assuming by contradiction that there are no critical points in \(A_{\mu }^{\lambda }\cap \Phi _{\lambda }^{c_{\Gamma }}\), since the Palais–Smale condition holds for \(\Phi _{\lambda }\) in the energy level (0, S], there exists a constant \(d_{\lambda }>0\) such that
From hypothesis and Proposition 4.5, we also have
where \(\sigma _{0}>0\) is independent of \(\lambda \). In what follows, \(\Psi :E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}})\rightarrow {\mathbb {R}}\) be a continuous functional that verify
and \(H:\Phi _{\lambda }^{c_{\Gamma }}\rightarrow {\mathbb {R}}\) be a continuous functional verify
Thus, we have the inequality
Considering the deformation flow \(\eta : [0, \infty )\times \Phi _{\lambda }^{c_{\Gamma }}\rightarrow \Phi _{\lambda }^{c_{\Gamma }}\) defined by
Thus \(\eta \) has the following properties
where \(K^{*}>0\) be a constant.
Now let \(\gamma _{0}(s_{1}, s_{2}, \ldots , s_{l})\in \Lambda _{*}\) be a path defined in (4.4) and we consider \(\eta (t, \gamma _{0}(s_{1}, s_{2}, \ldots , s_{l}))\) for large t. Since for all \((s_{1}, s_{2}, \ldots , s_{l})\in \partial ([1/R^{2}, 1]^{l})\), \(\gamma _{0}(s_{1}, s_{2}, \ldots , s_{l})\not \in A_{2\mu }^{\lambda }\), thus we have by (4.10) that
and \(\eta (t, \gamma _{0}(s_{1}, s_{2}, \ldots , s_{l}))\in \Lambda _{*}\) for all \(t\ge 0\).
Since \(\text {supp} \gamma _{0}(s_{1}, s_{2}, \ldots , s_{l})(x)\subset {\overline{\Omega }}_{\Gamma }\) for all \((s_{1}, s_{2}, \ldots , s_{l})\in \partial ([1/R^{2}, 1]^{l})\), then \(\Phi _{\lambda }(\gamma _{0}(s_{1}, s_{2}, \ldots , s_{l}))\) and \(\Vert \gamma _{0}(s_{1}, s_{2}, \ldots , s_{l})\Vert _{\lambda , j}\) etc. do not depend on \(\lambda \ge 0\). On the other hand,
and \(\Phi _{\lambda }(\gamma _{0}(s_{1}, s_{2}, \ldots , s_{l}))= c_{\Gamma }\) if and only if \(s_{j}=\frac{1}{R}\), that is \(\gamma _{0}(s_{1}, s_{2}, \ldots , s_{l})(x)\mid _{\Omega _{j}}=w_{j}\) for \(j\in \Gamma \). Thus, we have that
is independent of \(\lambda \) and \(m_{0}< c_{\Gamma }\). Since \(\Vert \frac{d\eta }{dt}\Vert _{\lambda }\le 1\) for all t, u, it is easy to see that for any \(t>0\)
Since \(\Phi _{\lambda , j}(u)\in C^{1}(E_{\lambda }({\mathbb {R}}^{2}, {\mathbb {C}}), {\mathbb {R}})\) for all \(j=1, 2, \ldots , l\), and from the assumptions \((f_{1})-(f_{5})\), it is easy to see that for a large number \(T>0\), there exists a positive number \(\rho _{0}>0\) which is independent of \(\lambda \) such that for all \(j=1, 2, \ldots , l\) and \(t\in [0, T]\),
We claim that for large T,
where \(m_{0}\) is given in (4.12), \(\tau _{0}=\max \{\sigma _{0}, \sigma _{0}/\rho _{0}\}\).
In fact, if \(\gamma _{0}(s_{1}, s_{2}, \ldots , s_{l})(x)\not \in A_{\mu }^{\lambda }\), then by (4.13), we have \(\Phi _{\lambda }(\eta (T, \gamma _{0}(s_{1}, s_{2}, \ldots , s_{l})(x)))\le m_{0}\) and thus (4.14) holds. If \(\gamma _{0}(s_{1}, s_{2}, \ldots , s_{l})(x)\in A_{\mu }^{\lambda }\), we need to study the behavior of \({\tilde{\eta }}(t)=\eta (t, \gamma _{0}(s_{1}, s_{2}, \ldots , s_{l})\). Setting \({\tilde{d}}_{\lambda }:=\min \{d_{\lambda }, \sigma _{0}\}\) and \(T=\sigma _{0}\mu /(2d_{\lambda })\). Now we distinguish two cases:
-
(1)
\({\tilde{\eta }}(t)\in A_{3\mu /2}^{\lambda }\) for all \(t\in [0, T]\).
-
(2)
\({\tilde{\eta }}(t_{0})\in \partial A_{3\mu /2}^{\lambda }\) for some \(t_{0}\in [0, T]\).
If (1) holds, we have \(\Psi ({\tilde{\eta }}(t))\equiv 1\) and \(\Vert \Phi '_{\lambda }({\tilde{\eta }}(t))\Vert _{\lambda }\ge {\tilde{d}}_{\lambda }\) for all \(t\in [0, T]\). Thus, by (4.9), we have
If (2) holds, there exists \(0\le t_{1}\le t_{1}\le T\) such that
It follows from (4.17)
or
for some \(j_{0}\in \Gamma \).
Now we consider the latter case, the former case can be obtained in a similar way. By (4.16),
thus, we obtain
On the other hand, by the mean value theorem, there exists \(t_{3}\in (t_{1}, t_{2})\) such that
Moreover, from (4.10) and (4.14), we have
Thus, one has
and so (4.15) is proved. Now we recall that \({\tilde{\eta }}(T)=\eta (T, \gamma _{0}(s_{1}, s_{2}, \ldots , s_{l}))\in \Lambda _{*}\), thus
But by Corollary 4.4, we know \(b_{\lambda , \Gamma }\rightarrow c_{\Gamma }\) as \(\lambda \rightarrow \infty \), this is a contradiction to (4.18), it shows that \(\Phi _{\lambda }(u)\) has a critical point \(u\in A_{u}^{\lambda }\) for large \(\lambda \) and we have completed the proof of the proposition. \(\square \)
Proof of Theorem 1.1
From Proposition 4.6, there exists a family of nontrivial solutions \((u_{\lambda })\) to problem (3.5) verifying the following properties.
(i) For fixed \(\mu >0\), there exists \(\lambda ^{*}\) such that
Thus, from proof of Proposition 3.6, \(\mu \) fixed sufficiently small, we can conclude that
which shows that \(u_{\lambda }\) is a nontrivial solution to problem (1.10).
(ii) Fixing \(\lambda _{n}\rightarrow \infty \) and \(\mu _{n}\rightarrow 0\), the sequence \(\{u_{\lambda _{n}}\}\) verifies
Thus, from proposition 3.2, we have
from which the proof of Theorem 1.1 follows. \(\square \)
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Acknowledgements
C. Ji was supported by Shanghai Natural Science Foundation (18ZR1409100).
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Ji, C., Rădulescu, V.D. Multi-bump solutions for the nonlinear magnetic Schrödinger equation with exponential critical growth in \({\mathbb {R}}^{2}\). manuscripta math. 164, 509–542 (2021). https://doi.org/10.1007/s00229-020-01195-1
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DOI: https://doi.org/10.1007/s00229-020-01195-1
Keywords
- Nonlinear Schrödinger equation
- Magnetic field
- Exponential critical nonlinearity
- Multi-bump solution
- Variational methods