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Existence of positive solutions for a semilinear Schrödinger equation in \(\mathbb{R}^{N}\)
Boundary Value Problems volume 2015, Article number: 9 (2015)
Abstract
In this paper, we study the existence of multi-bump solutions for the semilinear Schrödinger equation \(-\Delta u+(1+\lambda a(x))u=(1-\lambda b(x))|u|^{p-2}u\), \(\forall u\in H^{1}(\mathbb{R}^{N})\), where \(N\geq1\), \(2< p<2N/(N-2)\) if \(N\geq3\), \(p>2\) if \(N=2\) or \(N=1\), \(a(x)\in C(\mathbb{R}^{N})\) and \(a(x)>0\), \(b(x)\in C(\mathbb{R}^{N})\) and \(b(x)>0\). For any \(n\in\mathbb{N}\), we prove that there exists \(\lambda(n)>0\) such that, for \(0<\lambda<\lambda(n)\), the equation has an n-bump positive solution. Moreover, the equation has more and more multi-bump positive solutions as \(\lambda\rightarrow0\).
1 Introduction and main results
In this paper we study the following time independent semilinear Schrödinger equation:
where \(N\geq1\), \(2< p<2^{*}\), 2∗ is the critical Sobolev exponent defined by \(2^{*}=\frac{2N}{N-2}\) if \(N\geq3\) and \(2^{*}=\infty\) if \(N=2\) or \(N=1\), and \(\lambda>0\) is a parameter.
This kind of equation arises in many fields of physics. For the following nonlinear Schrödinger equation:
where i is the imaginary unit, Δ is the Laplacian operator, and \(\hbar>0\) is the Planck constant. A standing wave solution of (1.1) is a solution of the form
Thus, \(\varphi(x,t)\) solves (1.1) if and only if \(u(x)\) solves the equation
where \(V(x)=N(x)-E\) and \(g(x,u)=f(x,|u|)u\). The function V is called the potential of (1.2). If \(g(x,u)=(1-\lambda b(x))|u|^{p-2}u\), then (1.2) can be written as
If \(\hbar=1\) and \(V(x)=1+\lambda a(x)\), then (1.3) is reduced to (\(\mathcal{S}_{\lambda}\)).
The nonlinear Schrödinger equation (\(\mathcal{S}_{\lambda}\)) models some phenomena in physics, for example, in nonlinear optics, in plasma physics, and in condensed matter physics, and the nonlinear term simulates the interaction effect, called the Kerr effect in nonlinear optics, among a large number of particles; see, for example, [1, 2]. The case of \(p=4\) and \(N=3\) is of particular physical interest, and in this case the equation is called the Gross-Pitaevskii equation; see [3].
The limiting equation of (\(\mathcal{S}_{\lambda}\)) is
as \(\lambda\rightarrow0\). It is well known that (1.4) has a unique positive radial solution z, which decays exponentially at ∞. This z will serve as a building block to construct multi-bump solutions of (\(\mathcal{S}_{\lambda}\)). For \(n\in\mathbb{N}\), let \(y_{1},\ldots,y_{n}\in\mathbb{R}^{N}\) be the sufficiently separated points. The profile of the function \(\sum_{i=1}^{n}z(x-y_{i})\) resembles n bumps and accordingly a solution of (\(\mathcal{S}_{\lambda}\)) which is close to \(\sum_{i=1}^{n}z(x-y_{i})\) in \(H^{1}(\mathbb{R}^{N})\) is called an n-bump solution.
As we know, multi-bump solutions arise as solutions of (1.2) as \(\hbar\rightarrow0\), under the assumption that V has several critical points; see for example [4–7]. Particularly, in the interesting paper [5], the authors proved that the solutions of (1.2) have several peaks near the point of a maximum of V. These peaks converge to the maximum of V as \(\hbar\rightarrow0\). Actually, there have been enormous studies on the solutions of (1.2) as \(\hbar\rightarrow0\), which exhibit a concentration phenomenon and are called semiclassical states. In the early results, most of the researchers focused on the case \(\inf_{x\in\mathbb{R}^{N}}V(x)>0\) and g is subcritical. Here and in the sequel, we say g is subcritical if \(g(x,u)\leq C|u|^{p-1}\) for \(2\leq p<2^{*}\) with \(2^{*}:=2N/(N-2)\) (\(N\geq3\)), and g is critical or supercritical if \(c_{1}|u|^{2^{*}-1}\leq g(x,u)\leq c_{2}|u|^{2^{*}-1}\) or only \(c_{1}|u|^{2^{*}-1}\leq g(x,u)\) for all large \(|u|\). In the case of \(\inf_{x\in\mathbb{R}^{N}}V(x)>0\), Floer and Weinstein in [8] first considered \(N=1\), \(g(u)=u^{3}\). Using the Lyapunov-Schmidt reduction argument, they proved that the system (1.2) has spike solutions, which concentrate near a nondegenerate critical point of the potential V. This result was extended to the high dimension case with \(N\geq2\) and for \(g(u)=|u|^{p-2}u\) by Oh [7, 9]. If the potential V has a nondegenerate critical point, Rabinowitz [10] obtained the existence result for (1.2) with ħ small, provided that \(0<\inf_{x\in\mathbb{R}^{N}}V(x)<\liminf_{|x|\rightarrow\infty}V(x)\). Using a global variational argument, Del Pino and Felmer [11, 12] established the existence of multi-peak solutions having exactly k maximum points provided that there are k disjoint open bounded sets \(\Omega_{i}\) such that \(\inf_{x\in\partial\Omega_{i}}V(x)>\inf_{x\in\Omega_{i}}V(x)\), each \(\Omega_{i}\) having one peak concentrating at its bottom. For the subcritical case, Refs. [1, 6, 13–15] also proved that the solutions of (1.2) are concentrated at critical points of V. There have also been recent results on the existence of solutions concentrating on manifolds; for instance, see [16–18] and the references therein.
If g is subcritical, Refs. [19, 20] first obtained the semiclassical solutions of (1.2) with critical frequency, i.e., \(\inf_{\in\mathbb{R}^{N}}V(x)=0\). They exhibited new concentration phenomena for bound states and their results were extended and generalized in [3, 21, 22]. Later, if \(\inf_{\in\mathbb{R}^{N}}V(x)=0\), Ding and Lin [23] obtained semiclassical states of (1.2) when the nonlinearity g is of the critical case. Recently, if the potentials V change sign, that is, \(\inf_{x\in\mathbb{R}^{N}}V(x)<0\), Refs. [24, 25] proved that the system (1.2) has semiclassical states.
Some researchers had also obtained multi-bump solutions for the equation
where V and f are \(T_{i}\) periodic in \(x_{i}\). Coti Zelati and Rabinowitz [26] first constructed multi-bump solutions for the Schrödinger equation (1.5). The building blocks are one-bump solutions at the mountain pass level and the existence of such solutions as well as multi-bump solutions is guaranteed by a nondegeneracy assumption of the solutions near the mountain pass level. Later, under the same nondegeneracy assumption, Coti Zelati and Rabinowitz in [27] constructed multi-bump solutions for periodic Hamiltonian systems. Multi-bump solutions have also been obtained for asymptotically periodic Schrödinger equations by Alama and Li [28]. For subsequent studies in this direction, for example, see [29–35] and the references therein. Recently, Refs. [36–38] also proved the existence of multi-bump solutions in other elliptic equations.
In this paper, we are interested in constructing multi-bump solutions of (\(\mathcal{S}_{\lambda}\)) with λ small enough. Similar results have been obtained in [39, 40] for the equations
and
To state the main result for (\(\mathcal{S}_{\lambda}\)), we need the following conditions on the functions a and b:
- (\(\mathcal{R}_{1}\)):
-
\(a(x)>0\) and \(a(x) \in C(\mathbb{R}^{N})\), \(b(x)>0\) and \(b(x) \in C(\mathbb{R}^{N})\), and
$$ \lim_{|x|\rightarrow\infty}a(x)=\lim_{|x|\rightarrow\infty}b(x)=0. $$ - (\(\mathcal{R}_{2}\)):
-
One of the following holds: (i) \(\lim_{|x|\rightarrow\infty}\frac{\ln(a(x))}{|x|}=0\); (ii) \(\lim_{|x|\rightarrow\infty}\frac{\ln(b(x))}{|x|}=0\).
Theorem 1.1
Suppose that the assumptions (\(\mathcal{R}_{1}\)) and (\(\mathcal {R}_{2}\)) hold. Then for any positive integer n there exists \(\lambda(n)>0\) such that, for \(0<\lambda<\lambda(n)\), the system (\(\mathcal{S}_{\lambda}\)) has an n-bump positive solution. As a consequence, for any positive integer n, there exists \(\lambda_{1}(n)>0\) such that, for \(0<\lambda<\lambda_{1}(n)\), the system (\(\mathcal{S}_{\lambda}\)) has at least n positive solutions.
Similar to [39, 40], the solutions in Theorem 1.1 do not concentrate near any point in the space. Instead, the bumps of the solutions we obtain are separated far apart and the distance between any pair of bumps goes to infinity as \(\lambda\rightarrow0\). The size of each bump does not shrink and is fixed as \(\lambda\rightarrow0\). This is in sharp contrast to the concentration phenomenon described above. This phenomenon has been observed by D’Aprile and Wei in [41] for a Maxwell-Schrödinger system.
We shall use the variational reduction method to prove the main results. Our argument is partially inspired by [39–42]. This paper is organized as follows. In Section 2, preliminary results are revisited. We prove Theorem 1.1 in Section 3.
2 Some preliminary works
2.1 Variational framework
In this section, we shall establish a variational framework for the system (\(\mathcal{S}_{\lambda}\)). For convenience of notation, let C and \(C_{i}\) denote various positive constants which may be variant even in the same line. In the Hilbert space \(H^{1}(\mathbb{R}^{N})\), we shall use the usual inner product,
and the induced norm \(\|u\|=\langle u,u\rangle^{\frac{1}{2}}\). Let \(|\cdot|_{q}\) denote the usual \(L^{q}(\mathbb{R}^{N})\)-norm and \((\cdot, \cdot)_{2}\) be the usual \(L^{2}(\mathbb{R}^{N})\)-inner product. Let \(n\in N\). We shall use \(\sum_{i< j}\) and \(\sum_{i\neq j}\) to represent summation over all subscripts i and j satisfying \(1\leq i< j\leq n\) and \(1\leq i\neq j\leq n\), respectively. Let us first introduce some basic inequalities which will be used later.
The following four lemmas are taken from [39, 40].
Lemma 2.1
For \(q>1\), there exists \(C>0\) such that, for any real numbers a and b,
Lemma 2.2
For \(q\geq2\), there exists \(C>0\) such that, for any \(a>0\) and \(b\in\mathbb{R}\),
Lemma 2.3
For \(q\geq2\), \(n\in N\), and \(a_{i}\geq0\), \(i=1,\ldots,n\),
and
Lemma 2.4
For \(q\geq2\), there exists \(C>0\) such that, for any \(a_{i}\geq0\), \(i = 1,\ldots,n\),
Recall that, for \(2< p<2^{*}\), the unique positive solution of the equation
has the following properties; see, for example, [40, 43–45].
Lemma 2.5
If \(2< p<2^{*}\), then every positive solution of (2.1) has the form \(z_{y}:=z(\cdot-y)\) for some \(y\in\mathbb{R}^{N}\), where \(z\in C^{\infty}(\mathbb{R}^{N})\) is the unique positive radial solution of (2.1) which satisfies, for some \(c>0\),
Furthermore, if \(\beta_{1}\leq\cdots\leq\beta_{n}\leq\cdot\cdot\cdot\) are the eigenvalues of the problem
then \(\beta_{1}=1\), \(\beta_{2}=p-1\), and the eigenspaces corresponding to \(\beta_{1}\) and \(\beta_{2}\) are spanned by z and \(\{\partial z/\partial x_{\alpha}\mid \alpha=1,\ldots,N\}\), respectively.
We shall use \(z_{y}\) as building blocks to construct multi-bump solutions of (\(\mathcal{S}_{\lambda}\)). For \(y_{i}, y_{j}\in\mathbb{R}^{N}\), the identity
will be frequently used in the sequel. The following lemma is a consequence of Lemma 2.4 in [46] (see also Lemma II.2 of [47]).
Lemma 2.6
There exists a positive constant \(c>0\) such that, as \(|y_{i}-y_{j}|\rightarrow\infty\),
For \(h>0\), \(n\geq2\), and \(n\in\mathbb{N}\), define
For convenience, we make the convention
For \(y=(y_{1},\ldots,y_{n})\in\mathcal{D}_{h}\), denote
and
Then \(H^{1}(\mathbb{R}^{N})=\mathcal{T}_{y}\oplus\mathcal{W}_{y}\). Set \(P_{\lambda}(x)=1-\lambda b(x)\), \(V_{\lambda}(x)=1+\lambda a(x)\), \(\mathcal{N}_{\lambda}=(p-1)(-\Delta+V_{\lambda})^{-1}\), and \(\mathcal{N}_{0}=\mathcal{N}\). For \(y\in\mathcal{D}_{h}\) and \(\varphi\in H^{1}(\mathbb{R}^{N})\), define
where
Noting that \(\mathcal{K}_{y}|_{\mathcal{W}_{y}}: \mathcal {W}_{y}\rightarrow\mathcal{W}_{y}\) has the form identity-compact.
Lemma 2.7
(See Lemma 2.3 of [40])
If \(h\rightarrow\infty\), then
in \(L^{p/(p-2)}(\mathbb{R}^{N})\) uniformly in \(y\in\mathcal {D}_{h}\).
Lemma 2.8
(See Lemma 2.4 of [40])
Let \(u, v\in H^{1}(\mathbb{R}^{N})\). If \(v\rightarrow0\), then
in \(L^{p/(p-2)}(\mathbb{R}^{N})\) uniformly in u in any bounded set.
Lemma 2.9
(See Lemma 2.5 of [40])
There exist \(h_{0}>0\) and \(\eta_{0}>0\) such that, for \(h>h_{0}\) and \(y\in\mathcal {D}_{h}\), \(\mathcal{K}_{y}|_{\mathcal{W}_{y}}: \mathcal {W}_{y}\rightarrow\mathcal{W}_{y}\) is invertible and
Lemma 2.10
Let \(v\in H^{1}(\mathbb{R}^{N})\). If \(\lambda\rightarrow0\), \(v\rightarrow0\), and \(h\rightarrow\infty\), then
and
Proof
By the definition of \(\mathcal{K}_{y}\), one has
Obviously, \(\mathcal{N}_{\lambda}\rightarrow\mathcal{N}\) in \(\mathcal{L}(L^{\frac{p}{p-1}}(\mathbb{R}^{N}), H^{1}(\mathbb{R}^{N}))\) as \(\lambda\rightarrow0\). Therefore, if \(\lambda\rightarrow0\), \(v\in H^{1}(\mathbb{R}^{N})\) with \(v\rightarrow0\), and \(h\rightarrow\infty\), then for \(\psi,\varphi \in H^{1}(\mathbb{R}^{N})\), and uniformly in \(y\in\mathcal{D}_{h}\),
as a consequence of Lemmas 2.7 and 2.8. Moreover, by Lemma 2.6, for \(|y_{i}-y_{j}|\rightarrow\infty\) (\(i\neq j\)), one sees that
For \(\psi,\varphi\in H^{1}(\mathbb{R}^{N})\),
as \(\lambda\rightarrow0\). We infer from (2.3)-(2.6) that, if \(\lambda\rightarrow0\), \(v\in H^{1}(\mathbb{R}^{N})\) with \(v\rightarrow0\), and \(h\rightarrow\infty\),
Similar to above arguments, one can easily acquire the second conclusion of this lemma. □
Clearly, the energy functional corresponding to the system (\(\mathcal{S}_{\lambda}\)) is defined by
where \(V_{\lambda}=(1+\lambda a(x))\) and \(P_{\lambda}=(1-\lambda b(x))\). It is easy to see that the critical points of \(\Phi_{\lambda}\) are solutions of (\(\mathcal{S}_{\lambda}\)). In the following, we shall use a Lyapunov-Schmidt reduction argument to find critical points of \(\Phi_{\lambda}\). The first procedure is to convert the problem of finding critical points of \(\Phi_{\lambda}\) to a finite dimensional problem, which consists of the following two lemmas.
Lemma 2.11
There exist \(\lambda_{0}>0\) and \(H_{0}>0\) such that, for \(0<\lambda<\lambda_{0}\) and \(h>H_{0}\), there exists a \(C^{1}\)-map
depending on h and λ, such that
-
(i)
for any \(y\in\mathcal{D}_{h}\), \(v_{h,\lambda}\in \mathcal {W}_{y}\);
-
(ii)
for any \(y\in\mathcal{D}_{h}\), \(\mathcal {P}_{y}\nabla\Phi_{\lambda}(u_{y}+v_{h,\lambda})=0\), where \(\mathcal {P}_{y}:H^{1}(\mathbb{R}^{N})\rightarrow\mathcal{W}_{y}\) is the orthogonal projection onto \(\mathcal{W}_{y}\);
-
(iii)
\(\lim_{\lambda\rightarrow0,h\rightarrow\infty}\| v_{h,\lambda,y}\|=0\) uniformly in \(y\in\mathcal{D}_{h}\); \(\lim_{|y|\rightarrow\infty}\|v_{h,\lambda,y}\|=0\) if \(n=1\).
Decreasing \(\lambda_{0}\) and increasing \(H_{0}\) if necessary, we have the following result.
Lemma 2.12
For \(0<\lambda<\lambda_{0}\) and \(h>H_{0}\), if \(y^{0}=(y_{1}^{0},\ldots,y_{n}^{0})\) is a critical point of \(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\), then \(u_{y^{0}}+v_{h,\lambda,y^{0}}\) is a critical point of \(\Phi_{\lambda}\).
Using Lemmas 2.9 and 2.10, repeating the arguments of Lemmas 2.6 and 2.7 in [40], one can easily prove Lemmas 2.11 and 2.12.
2.2 Estimates on \(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\) and \(v_{h,\lambda,y}\)
In order to prove Theorem 1.1 in the next section. We need first to estimate \(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\) and \(v_{h,\lambda,y}\). Denote \(c_{0}=\Phi_{0}(z)\), where \(\Phi_{0}\) is the functional \(\Phi_{\lambda}\) with \(\lambda=0\). Then
In the following, we first estimate \(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\). Note that
A direct computation shows that
By Lemma 2.11, we may assume that \(\|v_{h,\lambda,y}\|\leq1\). Taking \(a=u_{y}\) and \(b=v_{h,\lambda,y}\) in Lemma 2.2, we have
and
Here and in what follows, \(O(\|v_{h,\lambda,y}\|^{2})\) satisfies
for some positive constant C independent of h, λ, y. Therefore, substituting (2.9) and (2.10) into (2.8), it follows that
Denote
Then
Thus, in order to estimate the functional \(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\), it suffices to get the estimations for \(\mathcal{K}_{y}\). Since
\(\mathcal{K}_{y}\) can be rewritten as
Moreover, by the Hölder inequality one has
and
Therefore, we have
Lemma 2.13
There exist \(h_{0}>0\), \(\lambda_{0}>0\), and \(C_{i}>0\) (\(i=1,2,3\)) such that, if \(0<\lambda\leq\lambda_{0}\), \(h\geq h_{0}\), and \(y\in\mathcal{D}_{h}\), then \(\mathcal{K}_{y}\) satisfies
Proof
From Lemmas 2.4 and 2.6, one sees that
Moreover, by Lemma 2.3, we have
and by Lemma 2.1, one has
Here the fact
has been used. Substituting (2.13)-(2.15) into (2.12), one can easily get the desired conclusion. □
Next, we are in a position to estimate \(\|v_{h,\lambda,y}\|\).
Lemma 2.14
\(\|v_{h,\lambda,y}\|\) satisfies
Proof
By Lemma 2.11, for \(v\in\mathcal{W}_{y}\), one has
There exists \(\theta\in(0,1)\) such that
Substituting (2.17) into (2.16) yields
Using the operator \(\mathcal{N}\) and \(\mathcal{P}_{y}\) defined in Section 2.1, we have
By Lemma 2.4, one has
Therefore, choosing \(v=v_{h,\lambda,y}-\mathcal{P}_{y}\mathcal {N}(P_{\lambda}|u_{y}+\theta v_{h,\lambda,y}|^{p-2}v_{h,\lambda,y})\in\mathcal{W}_{y}\) in (2.18) and using Lemmas 2.9 and 2.10, we obtain, for some \(\eta>0\),
which implies, for \(\lambda>0\) sufficiently small,
Thus, we obtain the result. □
3 Proof of Theorem 1.1
The main purpose of this section is to prove Theorem 1.1. For this, we shall prove that, for \(\lambda>0\) small enough, we can choose \(\mu(\lambda)\) large enough such that the function \(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\) defined in Section 2.1 reaches its maximum in \(\mathcal{D}_{\mu}\) at some point \(y^{0}=(y_{1}^{0},\ldots,y_{n}^{0})\). Then \(u_{y^{0}}+v_{h,\lambda,y^{0}}\) is a solution of (\(\mathcal {S}_{\lambda}\)) by Lemma 2.12.
We shall mainly consider the case \(n\geq2\) since the case \(n=1\) is much easier. Define
By Lemmas 2.1 and 2.2, there exist \(\lambda'_{0}>0\), \(h'_{0}>\), and \(C'_{i}>0\) (\(i=1,2,3\)) such that, if \(0<\lambda\leq\lambda'_{0}\), \(h\geq h'_{0}\), and \(y\in\mathcal {D}_{h}\), then \(\mathcal{K}_{y}\) satisfies
Here and in the sequel, \(C_{i}\), \(C'_{i}\), and C are various positive constants independent of λ. We choose a number k such that \(k>\max\{1,12\gamma/C'_{1}\}\). Then, for any λ satisfying
there exists \(\mu^{*}=\mu^{*}(\lambda)>\mu=\mu(\lambda)>0\) such that, for \(w\in\mathbb{R}^{N}\) with \(|w|\in[\mu^{*},\mu]\),
Set
To obtain an n-bump solution of (\(\mathcal{S}_{\lambda}\)), it suffices to prove that \(\Gamma_{\lambda}\) is achieved in the interior of \(\mathcal{D}_{\mu}\).
Lemma 3.1
Assume \(n\geq2\). Then there exists \(\lambda_{1}\in(0,\lambda')\) such that, for \(0<\lambda<\lambda_{1}\),
Proof
Note that \(\mu(\lambda)\rightarrow\infty\) as \(\lambda\rightarrow0\). By Lemma 2.2 and (3.3) we see that, if \(y\in\mathcal{D}_{\mu(\lambda)}\), then
Suppose that \(y=(y_{1},\ldots,y_{n})\in\mathcal {D}_{\mu(\lambda)}\) and \(|y_{i}-y_{j}|\in[\mu(\lambda),\mu^{*}(\lambda)]\) for some \(i\neq j\). By (3.1)-(3.3), one has
By (2.11) and (3.5), for \(\lambda>0\) small enough, we obtain
for \(y=(y_{1},\ldots,y_{n})\in\mathcal{D}_{\mu(\lambda)}\) with \(|y_{i}-y_{j}|\in[\mu(\lambda),\mu^{*}(\lambda)]\) for some \(i\neq j\). On the other hand, if \(y=(y_{1},\ldots,y_{n})\in\mathcal{D}_{\mu(\lambda)}\) and \(|y_{i}-y_{j}|\rightarrow\infty\) for some \(i\neq j\), then by (2.11) and Lemmas 2.1 and 2.2, we have
where \(o(1)\) means some quantities which depend only on y and converge to 0 as \(|y_{i}-y_{j}|\rightarrow\infty\) for all \(i\neq j\). Therefore, for λ small enough,
This inequality contradicts (3.6). Thus, we obtain the result. □
We choose \(y^{k}(\lambda)=(y_{1}^{k}(\lambda),\ldots,y_{n}^{k}(\lambda))\in \mathcal {D}_{\mu(\lambda)}\) such that
Then Lemma 3.1 implies that
Therefore, for any \(1\leq i\leq n\), passing to a subsequence if necessary, we may assume either \(\lim_{k\rightarrow\infty}y_{i}^{k}(\lambda)=y_{i}^{0}(\lambda)\) with \(|y_{i}^{0}(\lambda)-y_{j}^{0}(\lambda)|\geq\mu^{*}\) for \(i\neq j\) or \(\lim_{k\rightarrow\infty}y_{i}^{k}(\lambda)=\infty\). Define
In the following, we shall prove that \(\mathcal {U}(\lambda)=\emptyset\) for \(\lambda>0\) sufficiently small and thus \(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\) attains its maximum at \((y_{1}^{0}(\lambda),\ldots,y_{n}^{0}(\lambda))\in\mathcal {D}_{\mu(\lambda)}\).
Lemma 3.2
Assume \(n\geq2\). Then there exists \(\lambda(n)>0\) such that for \(\lambda\in(0,\lambda(n))\), \(\mathcal{U}(\lambda)=\emptyset\).
Proof
We adopt an argument borrowed from Lin and Liu [39, 40]. We argue by contradiction and assume that \(\mathcal{U}(\lambda)\neq\emptyset\) along a sequence \(\lambda_{m}\rightarrow0\). Without loss of generality, we may assume \(\mathcal {U}(\lambda_{m})=\{1,\ldots,j_{n}\}\) for all \(m\in\mathbb{N}\) and for some \(1\leq j_{n}< n\). The case in which \(j_{n}=n\) can be handled similarly. For convenience of notations, we shall denote \(\lambda_{m}=\lambda\), \(y_{i}^{k}=y_{i}^{k}(\lambda_{m})\), \(y^{k}=(y_{1}^{k},\ldots,y_{n}^{k})\), \(y_{*}^{k}=(y_{j_{n}+1}^{k},\ldots,y_{n}^{k})\), and \(y_{*}^{0}=(y_{j_{n}+1}^{0},\ldots,y_{n}^{0})\) for \(k=1,2,\ldots\) . Then, as \(k\rightarrow\infty\),
and
Set
and
Similar to (3.4), we have
By (2.11), we obtain
By Lemma 2.1, one sees
Therefore, since \(|y_{i}^{k}|\rightarrow\infty\), \(i=1,\ldots,j_{n}\), as \(k\rightarrow\infty\), we obtain
Furthermore, by (3.9) and the condition (\(\mathcal{R}_{1}\)), we have
From (3.8), (3.10) and (3.11), we arrive at
Using Lemma 2.4, (3.3), and (3.7), we obtain
From Lemma 2.2, (2.12), (3.7), and (3.13), one gets
In the same way, we have
We infer from (3.14) and (3.15) that
By Lemma 2.3, the sum of the terms except \(O(\lambda^{\frac{2(p-1)}{p}})\) on the right side of (3.16) is negative. Thus, one has
Letting \(k\rightarrow\infty\), by (3.12), and using (3.17), we obtain
On the other hand, by Lemma 2.6 and (3.3), there exist \(C_{9}, C_{10}> 0\) such that
which implies for λ small enough
where \(0<\delta<\frac{1}{p}\). We choose τ such that \(0<\tau\leq\frac{p-2}{10np}\). By (\(\mathcal{R}_{2}\)), there exists \(R>0\) such that
or
For \(\lambda>0\) small enough, define
The open balls \(B(\hat{y}_{s}^{\lambda},2\mu(\lambda))\) are mutually disjoint. Thus there are \(j_{n}\) integers from \(\{1,\ldots,n\}\), denoted by \(s_{1}< s_{2}<\cdots< s_{j_{n}}\), such that
Denote \(\hat{y}_{s_{i}}^{\lambda}\) by \(y_{i}^{\lambda}\) for simplicity, \(i=1,\ldots,j_{n}\). By (3.20), (3.23), and (3.24), one has
Therefore,
Denote \(y^{\lambda}=(y_{1}^{\lambda},\ldots,y_{j_{n}}^{\lambda },y_{j_{n}+1}^{0},\ldots,y_{n}^{0})\). Set \(w_{\lambda,1}=\sum_{i=1}^{j_{n}}z(x-y_{i}^{\lambda})\), \(w_{y_{*}^{0}}=\sum_{i=j_{n}+1}^{n}z(x-y_{i}^{0})\), and \(w_{\lambda}=w_{\lambda,1}+w_{y_{*}^{0}}\). Similar to (3.8), one has
As in (3.16), we have
Together with Lemma 2.1 this implies that
By Lemma 2.6, (3.20), and (3.26), one sees that
In view of (3.27), a similar argument shows that
Combining (3.29)-(3.31), we have
Together with (3.28), it follows that
We distinguish the following two cases to finish the proof of this lemma.
-
(i)
If (3.21) holds, then by (3.25), we have, for \(i=1,\ldots,j_{n}\),
$$\begin{aligned} \int_{\mathbb{R}^{N}}a(x)w_{\lambda,1}^{2} \geq&\int_{|x-y_{i}^{\lambda}|\leq1}a(x)z^{2}\bigl(x-y_{i}^{\lambda} \bigr) \geq\int_{|x-y_{i}^{\lambda}|\leq1}e^{-\tau |x|}z^{2} \bigl(x-y_{i}^{\lambda}\bigr) \\ \geq& C_{16}e^{-\tau|y_{i}^{\lambda}|}\geq C_{16}e^{-10n\tau\ln\frac{1}{\lambda}}=C_{16} \lambda^{10n\tau}. \end{aligned}$$(3.32)Hence,
$$ \Phi_{\lambda}(u_{y^{\lambda}}+v_{\mu,\lambda,y^{\lambda}})\geq j_{n}c_{0}+\Phi_{\lambda}(w_{y_{*}^{0}}+v_{\mu,\lambda ,y_{*}^{0}})+C_{16} \lambda^{10n\tau+1} -C_{15}\lambda^{\frac{2(p-1)}{p}}. $$Since \(10n\tau+1<\frac{2(p-1)}{p}\), we obtain, for λ small enough,
$$ \Phi_{\lambda}(u_{y^{\lambda}}+v_{\mu,\lambda,y^{\lambda}})\geq j_{n}c_{0}+\Phi_{\lambda}(w_{y_{*}^{0}}+v_{\mu,\lambda ,y_{*}^{0}})+C'_{16} \lambda^{10n\tau+1}, $$which contradicts (3.18).
-
(ii)
Suppose that (3.22) holds. Similar to (3.32), one has
$$\begin{aligned} \int_{\mathbb{R}^{N}}b(x) \bigl(u_{y^{\lambda }}^{p}-u_{y_{*}^{0}}^{p} \bigr) \geq&\int_{|x-y_{1}^{\lambda}|\leq 1}b(x)z^{p} \bigl(x-y_{1}^{\lambda}\bigr) \geq\int_{|x-y_{1}^{\lambda}|\leq1}e^{-\tau |x|}z^{p} \bigl(x-y_{1}^{\lambda}\bigr) \\ \geq& C_{17}e^{-\tau|y_{1}^{\lambda}|}\geq C_{17}e^{-10n\tau\ln\frac{1}{\lambda}}=C_{17} \lambda^{10n\tau}. \end{aligned}$$Repeating the arguments of (i), we get, for λ small enough,
$$ \Phi_{\lambda}(u_{y^{\lambda}}+v_{\mu,\lambda,y^{\lambda}})\geq j_{n}c_{0}+\Phi_{\lambda}(w_{y_{*}^{0}}+v_{\mu,\lambda ,y_{*}^{0}})+C'_{17} \lambda^{10n\tau+1}. $$This contradicts (3.18).
From (i) and (ii), we know that there exists \(\lambda(n)>0\) such that, if \(0 <\lambda<\lambda(n)\), then \(\mathcal {U}(\lambda)=\emptyset\) and \(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\) reaches its maximum at some point \((y_{1}^{0},\ldots,y_{n}^{0})\in\mathcal {D}_{\mu(\lambda)}\). □
Next, we shall prove Theorem 1.1.
Proof of Theorem 1.1
For \(n\geq2\), according to Lemma 3.2, if \(0<\lambda<\lambda(n)\), then \(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\) reaches its maximum at some point \(y^{0}=(y_{1}^{0},\ldots,y_{n}^{0})\in\mathcal {D}_{\mu(\lambda)}\). Then \(u_{y^{0}}+v_{h,\lambda,y^{0}}\) is an n-bump solution of (\(\mathcal{S}_{\lambda}\)). For \(n=1\), as a consequence of Lemma 2.11(iii), if \(\lambda\in(0,\lambda_{0}]\), then
Since \(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\) is defined on all \(\mathbb{R}^{N}\), \(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\) has a critical point \(y^{0}\in\mathbb{R}^{N}\) and \(u_{y^{0}}+v_{h,\lambda,y^{0}}\) is a 1-bump solution of (\(\mathcal {S}_{\lambda}\)). By an argument similar to those in [34, 35], one sees that \(u_{y^{0}}+v_{h,\lambda,y^{0}}\) is a positive solution of (\(\mathcal{S}_{\lambda}\)). Set \(\lambda(1)=\lambda_{0}\) and \(\lambda_{1}(n)=\min\{\lambda(1),\ldots,\lambda(n)\}\). If \(0<\lambda<\lambda_{1}(n)\), then (\(\mathcal{S}_{\lambda}\)) has at least n nontrivial positive solutions. □
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Acknowledgements
This work was supported by Natural Science Foundation of China (11201186, 11071038, 11171135), NSF of Jiangsu Province (BK2012282), Jiangsu University foundation grant (11JDG117), China Postdoctoral Science Foundation funded project (2012M511199, 2013T60499).
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Fang, H., Wang, J. Existence of positive solutions for a semilinear Schrödinger equation in \(\mathbb{R}^{N}\) . Bound Value Probl 2015, 9 (2015). https://doi.org/10.1186/s13661-014-0270-8
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DOI: https://doi.org/10.1186/s13661-014-0270-8
MSC
- 35J61
- 35J20
- 35Q55
- 49J40
Keywords
- multi-bump solution
- semilinear Schrödinger equation
- variational methods