Abstract
In this paper, we consider the following non-autonomous Schrödinger–Bopp–Podolsky system
By using some original analytic techniques and new estimates of the ground state energy, we prove that this system admits a ground state solution under mild assumptions on V and f. In the final part of this paper, we give a min-max characterization of the ground state energy.
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1 Introduction
Consider the following Schrödinger–Bopp–Podolsky system
where \(u,\phi :{\mathbb {R}}^3 \rightarrow {\mathbb {R}}\), \(\omega , a>0\), \(q\ne 0\).
This nonlinear system appears when we couple a Schrödinger field \(\psi =\psi (t,x)\) with its electromagnetic field in the Bopp–Podolsky electromagnetic theory, and, in particular, in the electrostatic case for standing waves \(\psi (t,x)=e^{i\omega t} u(x)\).
System (1.1) has a strong physical meaning especially in the Bopp–Podolsky theory, developed independently by Bopp [3] and Podolsky [24]. The Bopp–Podolsky theory is a second order gauge theory for the electromagnetic field. As the Mie theory [22] and its generalizations given by Born and Infeld [4,5,6,7], it was introduced to solve the “infinity problem”, which appears in the classical Maxwell theory. In fact, by the well-known Gauss law (or Poisson equation), the electrostatic potential \(\phi \) for a given charge distribution whose density is \(\rho \) satisfies the equation
If \(\rho =4\pi \delta _{x_0}\), with \(x_0\in {\mathbb {R}}^3\), the fundamental solution of (1.2) is \({\mathcal {G}}(x-x_0)\), where
and the electrostatic energy is
Thus, Eq. (1.2) is replaced by
in the Born-Infeld theory and by
in the Bopp–Podolsky theory. In both cases, if \(\rho =4\pi \delta _{x_0}\), we are able to write explicitly the solutions of the respective equations and to see that their energy is finite. In particular, when we consider the differential operator \(-\Delta + a^2\Delta ^2\), we have that \({\mathcal {K}}(x-x_0)\), with
is the fundamental solution of the equation
Then \({{\mathcal {K}}}\) has no singularity in \(x_0\) since it satisfies
and its energy is
Moreover, the Bopp–Podolsky theory may be interpreted as an effective theory for short distances (see [20]), while for large distances it is experimentally indistinguishable from the Maxwell theory. Thus, the Bopp–Podolsky parameter \(a>0\), which has dimension of the inverse of mass, can be interpreted as a cut-off distance or can be linked to an effective radius for the electron. For more physical details we refer the reader to the recent papers [1, 2, 9, 10, 16, 17] and to references therein.
The differential operator \(-\Delta +\Delta ^2\) appears in various different interesting mathematical and physical situations; see [19] and the references therein.
Before stating our results, few preliminaries are in order. We introduce here the space \({\mathcal {D}}\) as the completion of \({\mathcal {C}}^{\infty }_{c}({\mathbb {R}}^{3})\) with respect to the norm \(\sqrt{\Vert \nabla \phi \Vert _{2}^2+a^2\Vert \Delta \phi \Vert _2^2}\); see Sect. 2 for more properties on this space.
For fixed \(a>0\) and \(q\ne 0\), we say that a pair \(( u, \phi )\in H^{1}({\mathbb {R}}^{3})\times {\mathcal {D}}\) is a solution of problem (1.1) if
We say that a solution \(( u,\phi )\) is nontrivial whenever \( u \not \equiv 0\); a solution is called a ground state solution if its energy is minimal among all nontrivial solutions. As described in Sect. 2, to solve problem (1.1) is equivalent to solving
whose solutions correspond to critical points of the energy functional defined in \(H^1({\mathbb {R}}^3)\) by
where \(F(u)=\int _0^uf(t)\mathrm {d}t\). A solution is called a ground state solution if its energy is minimal among all nontrivial solutions.
In this paper, we also consider the following “limit” system with a general nonlinearity f
To the best of our knowledge, there is no result on the existence of ground state solutions for systems (1.1) and (1.5). Inspired by [11, 12, 14, 25], we will seek a ground state solution of Nehari–Poho\(\breve{\mathrm{z}}\)aev type for systems (1.1) and (1.5).
To state our results, we introduce the following assumptions:
-
(V1)
\(V\in {\mathcal {C}}({\mathbb {R}}^3, [0, \infty ))\) and \(V_{\infty }:=\lim _{|y|\rightarrow \infty }V(y)= \sup _{x\in {\mathbb {R}}^3}V(x)>0\);
-
(V2)
\(V\in {\mathcal {C}}^1({\mathbb {R}}^3, {\mathbb {R}})\), \(\nabla V(x)\cdot x\in L^{\infty }({\mathbb {R}}^3)\), \(2V(x)+\nabla V(x)\cdot x \ge 0\) and \(\liminf _{|x|\rightarrow \infty }[2V(x)+\nabla V(x)\cdot x]>0\);
-
(F1)
\(f\in {\mathcal {C}}({\mathbb {R}}, {\mathbb {R}})\), and there exist constants \({\mathcal {C}}>0\) and \(p\in (2,6)\) such that
$$\begin{aligned} |f(t)|\le {\mathcal {C}}\left( 1+|t|^{p-1}\right) , \ \ \ \ \forall \ t\in {\mathbb {R}}; \end{aligned}$$ -
(F2)
\(f(t)=o(t)\) as \(t\rightarrow 0\);
-
(F3)
\(F(t)\ge 0\) for all \(t\in {\mathbb {R}}\) and \(\lim _{|t|\rightarrow \infty }\frac{F(t)}{|t|^3}=\infty \);
-
(F4)
the function \(\frac{2f(t)t-3F(t)}{t^3}\) is nondecreasing on \((-\infty ,0)\) and \((0,+\infty )\).
Our first result is as follows.
Theorem 1.1
Assume that (V1), (V2) and (F1)–(F4) hold. Then problem (1.1) admits a ground state solution.
Remark 1.2
There are many functions which satisfy (V1) and (V2). An example is given by \(V(x) = 1- \frac{\sin ^2 |x|}{1+ |x|}\).
For the constant potential case, we replace the monotonicity condition (F4) with the super-quadratic condition which is easier to verify:
-
(F5)
\(f(t)t\ge 3F(t)\) for all \(t\in {\mathbb {R}}\), and there exist \(\kappa >3/2\) and \(r_0, {\mathcal {C}}_0>0\) such that
$$\begin{aligned} \left| \frac{f(t)}{t}\right| ^{\kappa }\le {\mathcal {C}}_0[f(t)t-3F(t)], \ \ \ \ \forall \ |t|\ge r_0. \end{aligned}$$
Our second result is as follows.
Theorem 1.3
Assume that (F1)–(F3) and (F5) hold. Then problem (1.5) admits a ground state solution.
Finally, we give the min-max property of the ground state energy of \({\mathcal {I}}\). To this end, we introduce the following monotonicity condition.
-
(V3)
\(V\in {\mathcal {C}}^1({\mathbb {R}}^3)\), and the function \(t\mapsto t^2[V(tx)-\nabla V(tx)\cdot (tx)]\) is increasing on \((0, +\infty )\) for every \(x\in {\mathbb {R}}^3\).
We define the Nehari–Poho\(\breve{\mathrm{z}}\)aev manifold as follows:
where \({\mathcal {P}}(u)\) is the Poho\(\breve{\mathrm{z}}\)aev functional of (1.3) defined by
If \(u\in H^1({\mathbb {R}}^3)\) is a critical point of \({\mathcal {I}}\), then u satisfies \({\mathcal {P}}(u)=0\); see [18, A.14] for more details. Then every nontrivial solution of (1.1) is contained in \({\mathcal {M}}\). In this direction, we have the following theorem.
Theorem 1.4
Assume that (V1), (V3), (F1)–(F4) hold. Then problem (1.1) admits a ground state solution \({\bar{u}}\in H^1({\mathbb {R}}^3)\) such that
where \(u_t(x):=u(tx)\).
Remark 1.5
We observe that the function \(V(x)=1-\frac{1}{\left( 1+|x|\right) ^\alpha }\) with \(\alpha >0\) satisfies hypotheses (V1) and (V3).
For the limiting problem related to (1.3), that is, (1.3) with \(V(x)\equiv V_{\infty }\), we further weaken (F4) to the following condition:
- (F4\('\)):
-
there exists a constant \(\theta \in [0,1)\) such that the function \(\frac{4f(t)t-6F(t)-\theta V_{\infty } t}{2t^3}\) is nondecreasing on \((-\infty ,0)\) and \((0,+\infty )\).
To state the following result, we define the energy functional in \(H^1({\mathbb {R}}^3)\) by
and the Nehari–Poho\(\breve{\mathrm{z}}\)aev manifold by
where \({\mathcal {P}}^{\infty }(u)\) is the Poho\(\breve{\mathrm{z}}\)aev functional defined by
We have the following corollary.
Corollary 1.6
Assume that (F1)–(F3) and (F4\('\)) hold. Then problem (1.5) admits a ground state solution \({\bar{u}}\in H^1({\mathbb {R}}^3)\) such that
Remark 1.7
Our more general conditions (F1)–(F4) or (F4\('\)) on the function f(u) allow many other examples different to the pure power nonlinearity considered in [18]. For example, the function \(f(u) = 3|u|u\ln (1+u^2) + \frac{2|u|^3u}{1+u^2}\) satisfies (F1)–(F4). The function \(f(u) = a|u|^{3/2}u+b|u|^{1/2}u\) with a, \(b > 0\) satisfies (F1)–(F3) and (F4\('\)) with \(\theta = \frac{2}{3}\) when \(15\sqrt{10}a \ge 14 b^{3/2} > 0\) but it does not fulfill (F4).
To prove Theorem 1.4, that is, to obtain a ground solution for Eq. (1.1) with (V1) and (V3), we first choose a minimizing sequence \(\{u_n\}\) of \({\mathcal {I}}\) on \({\mathcal {M}}\), which satisfies
Next, we show that the sequence \(\{u_n\}\) is bounded in \(H^1({\mathbb {R}}^3)\).
Due to lack of global compactness and adequate information on \({\mathcal {I}}'(u_n)\) and in order to avoid relying the radial compactness, we establish a crucial inequality related to \({\mathcal {I}}(u)\), \({\mathcal {I}}(u_t)\) and \({\mathcal {J}}(u)\) (Lemma 3.4), which plays a crucial role in our arguments, see Lemmas 3.8, 3.9, 3.13, 3.14 and 4.5 . With the help of this inequality, we then can recover the compactness for the minimizing sequence \(\{u_n\}\) and show that \(\{u_n\}\) converges weakly to some \({\bar{u}}\in H^1({\mathbb {R}}^3){\setminus }\{0\}\) and \({\mathcal {I}}({\bar{u}})=\inf _{{\mathcal {M}}}{\mathcal {I}}\) by using Lions’ concentration-compactness, the “least energy squeeze approach” and some subtle analysis. Finally, we take advantage of a quantitative deformation lemma and the intermediate value theorem to show that \({\bar{u}}\) is a critical point of \({\mathcal {I}}\), as the Lagrange multiplier theorem does not work, because \({\mathcal {M}}\) is not a \({\mathcal {C}}^1\)-manifold, .
To prove Theorem 1.1, we use the monotonicity technique explored by Jeanjean [21] to parameterize the nonlinearity f. In such a way, we build a parametrization of the energy functional associated to (1.1) and give some energy relations of problems (1.1) and (1.5) which play a key role in getting the critical point of (1.1), see Lemma 4.5. Moreover, in order to show that a critical point associated to the parametrization functional is indeed a solution to the original problem, we also need give a delicate estimation for the parametrization problem. Finally, we study the constant potential case by using weaker conditions.
Throughout the paper we make use of the following notations:
-
Under (V1), \(H^1({\mathbb {R}}^3)\) denotes the Sobolev space equipped with the inner product and norm
$$\begin{aligned} (u, v)=\int _{{\mathbb {R}}^3}[\nabla u \nabla v+V(x)uv]\mathrm {d}x, \ \ \Vert u\Vert =(u, u)^{1/2}, \ \ \forall \ u,v\in H^1({\mathbb {R}}^3); \end{aligned}$$ -
\(L^s({\mathbb {R}}^3) (1\le s< \infty )\) denotes the Lebesgue space with the norm \(\Vert u\Vert _s =\left( \int _{{\mathbb {R}}^3}|u|^s\mathrm {d}x\right) ^{1/s}\);
-
For any \(x\in {\mathbb {R}}^3\) and \(r>0\), \(B_r(x):=\{y\in {\mathbb {R}}^3: |y-x|<r \}\);
-
\(S=\inf _{u\in D^{1,2}({\mathbb {R}}^3){\setminus }\{0\}}\Vert \nabla u\Vert _2^2/\Vert u\Vert _6^2\);
-
\(C_1, C_2,\cdots \) denote positive constants possibly different in different places.
2 Variational setting
We start with some preliminary basic results. Let us consider the nonlinear Schrödinger Lagrangian density
where \(\psi :{\mathbb {R}}\times {\mathbb {R}}^3\rightarrow {\mathbb {C}}\), \(\hbar ,m>0\), and let \((\phi ,\mathbf{A})\) be the gauge potential of the electromagnetic field \((\mathbf{E},\mathbf{H})\), namely \(\phi :{\mathbb {R}}^3\rightarrow {\mathbb {R}}\) and \(\mathbf{A}:{\mathbb {R}}^3\rightarrow {\mathbb {R}}^3\) satisfy
The coupling of the field \(\psi \) with the electromagnetic field \((\mathbf{E},\mathbf{H})\) through the minimal coupling rule, namely the study of the interaction between \(\psi \) and its own electromagnetic field, can be obtained by replacing in \({\mathcal {L}}_{\mathrm{Sc}}\) the derivatives \(\partial _t\) and \(\nabla \) respectively with the covariant ones
q being a coupling constant. This leads to consider
Now, to get the total Lagrangian density, we have to add to \({\mathcal {L}}_{\mathrm{CSc}}\) the Lagrangian density of the electromagnetic field.
The Bopp–Podolsky Lagrangian density (see [24, Formula (3.9)]) is
Thus, the total action is
where \({\mathcal {L}}:={\mathcal {L}}_{\mathrm{CSc}} + {\mathcal {L}}_{\mathrm{BP}}\) is the total Lagrangian density.
Let \({\mathcal {D}}\) be the completion of \({\mathcal {C}}_c^\infty ({\mathbb {R}}^3)\) with respect to the norm \(\Vert \cdot \Vert _{{\mathcal {D}}}\) induced by the scalar product
Then \({\mathcal {D}}\) is a Hilbert space continuously embedded into \(D^{1,2}({\mathbb {R}}^3)\) and consequently in \(L^6({\mathbb {R}}^3)\).
We notice the following auxiliary properties; see Lemmas 3.1 and 3.2 in [18].
Lemma 2.1
The space \({\mathcal {D}}\) is continuously embedded in \(L^\infty ({\mathbb {R}}^3)\).
The next property gives a useful characterization of the space \({\mathcal {D}}\).
Lemma 2.2
The space \({\mathcal {C}}^{\infty }_{c}({\mathbb {R}}^{3})\) is dense in
normed by \(\sqrt{\langle \phi ,\phi \rangle _{{\mathcal {D}}}}\) and, therefore, \({\mathcal {D}}={\mathcal {A}}\).
For every fixed \(u\in H^1({\mathbb {R}}^3)\), the Riesz representation theorem implies that there is a unique solution \(\phi _u\in {\mathcal {D}}\) of the second equation in (1.1). To write explicitly such a solution (see also [24, Formula (2.6)]), we consider
We have the following fundamental properties.
Lemma 2.3
[18, Lemma 3.3] For all \(y\in {\mathbb {R}}^3\), \({\mathcal {K}}(\cdot -y)\) solves in the sense of distributions
Moreover,
-
(i)
if \(g\in L^1_{\mathrm{loc}}({\mathbb {R}}^3)\) and, for a.e. \(x\in {\mathbb {R}}^3\), the map \(y\in {\mathbb {R}}^3\mapsto g(y)/|x-y|\) is summable, then \({\mathcal {K}}*g \in L^1_{\mathrm{loc}}({\mathbb {R}}^3)\);
-
(ii)
if \(f\in L^s ({\mathbb {R}}^3)\) with \(1\le s< 3/2\), then \({\mathcal {K}}*g\in L^q({\mathbb {R}}^3)\) for \(q\in (3s/(3-2s),+\infty ]\).
In both cases, \({\mathcal {K}}*g\) solves
in the sense of distributions, and we have the following distributional derivatives:
Fix \(u\in H^1({\mathbb {R}}^3)\), the unique solution in \({\mathcal {D}}\) of the second equation in (1.1) is
Actually the following useful properties hold.
Lemma 2.4
[18, Lemma 3.4] For every \(u\in H^{1}({\mathbb {R}}^{3})\) we have:
-
(1)
for every \(y\in {\mathbb {R}}^3\), \(\phi _{u( \cdot +y)} = \phi _{u}( \cdot +y)\);
-
(2)
\(\phi _{u}\ge 0\);
-
(3)
for every \(s\in (3,+\infty ]\), \(\phi _{u}\in L^{s}({\mathbb {R}}^{3})\cap {\mathcal {C}}_{0}({\mathbb {R}}^{3})\);
-
(4)
for every \(s\in (3/2,+\infty ]\), \(\nabla \phi _{u} = \nabla {\mathcal {K}} * u^{2}\in L^{s}({\mathbb {R}}^{3})\cap {\mathcal {C}}_{0}({\mathbb {R}}^{3})\);
-
(5)
\(\phi _u\in {\mathcal {D}}\);
-
(6)
\(\Vert \phi _{u}\Vert _{6}\le C \Vert u\Vert ^{2}\);
-
(7)
\(\phi _{u}\) is the unique minimizer of the functional
$$\begin{aligned} E(\phi ) = \frac{1}{2} \Vert \nabla \phi \Vert _{2}^{2} +\frac{a^{2}}{2} \Vert \Delta \phi \Vert _{2}^{2}-\int _{{\mathbb {R}}^3}\phi u^{2}\mathrm {d}x, \quad \phi \in {\mathcal {D}}. \end{aligned}$$
Moreover, if \(v_{n}\rightharpoonup v\) in \(H^{1}({\mathbb {R}}^{3})\), then \(\phi _{v_{n}} \rightharpoonup \phi _{v}\) in \({\mathcal {D}}\).
Under hypotheses (V1), (F1) and (F2), the energy functional defined in \(H^{1}({\mathbb {R}}^{3})\times {\mathcal {D}} \) by
is continuously differentiable and its critical points correspond to the weak solutions of problem (1.1). Indeed, if \( (u, \phi )\in H^{1}({\mathbb {R}}^{3})\times {\mathcal {D}}\) is a critical point of \({\mathcal {S}}\), then
and
In order to avoid the difficulty generated by the strongly indefiniteness of the functional \({\mathcal {S}}\), we apply a reduction procedure. Noting that \(\partial _{\phi } {\mathcal {S}}\) is a \({\mathcal {C}}^{1}\) functional, if \(G_{\Phi }\) is the graph of the map \(\Phi : u\in H^{1}({\mathbb {R}}^{3})\mapsto \phi _{u}\in {\mathcal {D}}\), an application of the implicit function theorem gives
Jointly with (2.3) and (2.4), the functional \({\mathcal {I}}(u):={\mathcal {S}}( u,\phi _u)\) has the reduced form
which is of class \({\mathcal {C}}^{1}\) on \(H^{1}({\mathbb {R}}^{3})\) and, for all \(u,v\in H^{1}({\mathbb {R}}^{3})\)
Moreover, the following statements are equivalent:
-
(i)
the pair \((u,\phi )\in H^{1}({\mathbb {R}}^{3})\times {\mathcal {D}}\) is a critical point of \({\mathcal {S}}\), that is, \((u,\phi )\) is a solution of problem (1.1);
-
(ii)
u is a critical point of \({\mathcal {I}}\) and \(\phi =\phi _{u}\).
Hence, if \(u\in H^1({\mathbb {R}}^3)\) is a critical point of \({\mathcal {I}}\), then the pair \((u, \phi _{u})\) is a solution of (1.1). For the sake of simplicity, in many cases we just say \(u\in H^1({\mathbb {R}}^3)\), instead of \((u, \phi _{u})\in H^1({\mathbb {R}}^3)\times {\mathcal {D}}\), is a solution of (1.1).
3 Proof of Theorem 1.3
In this section, we give the proof of Theorem 1.3.
By a simple calculation, we have the following two lemmas.
Lemma 3.1
Let \(b>0\). Then
and
Lemma 3.2
-
(i)
Assume that (V1) and (V3) hold. Then
$$\begin{aligned} 3\left[ V(x)-tV(t^{-1}x)\right] -(1-t^3)[V(x)-\nabla V(x)\cdot x] > 0, \ \ \ \ \forall \ t\in [0, 1)\cup (1, +\infty ). \end{aligned}$$(3.3) -
(ii)
Assume that (F1) and (F4) hold. Then
$$\begin{aligned} \frac{2(1-t^3)}{3}f(\tau )\tau +(t^3-2)F(\tau )+\frac{1}{t^3}F(t^2\tau )\ge 0, \ \ \ \ \forall \ t> 0, \ \tau \in {\mathbb {R}}. \end{aligned}$$(3.4) -
(iii)
Assume that (F1) and (F4\('\)) hold. Then
$$\begin{aligned}&\frac{2(1-t^3)}{3}f(\tau )\tau +(t^3-2)F(\tau )+\frac{1}{t^3}F(t^2\tau )\nonumber \\&\quad +\frac{\theta _0}{6}(1-t)^2(2+t)V_{\infty }\tau ^2\ge 0, \ \ \ \ \forall \ t> 0, \ \tau \in {\mathbb {R}}. \end{aligned}$$(3.5)
Note that if \(t\rightarrow 0\) in (3.4) and (3.5), then
and
Lemma 3.3
Assume that (V1) and (V3) hold. Then
Proof
Arguing by contradiction, we assume that there exist a sequence \(\{x_n\}\subset {\mathbb {R}}^3\) and \(\delta >0\) such that
Now, we distinguish two cases: i) \(\nabla V(x_n)\cdot x_n\ge \delta \) for all \(n\in {\mathbb {N}}\) and ii) \(\nabla V(x_n)\cdot x_n\le -\delta \) for all \(n\in {\mathbb {N}}\).
Case i) \(\nabla V(x_n)\cdot x_n\ge \delta \) for all \(n\in {\mathbb {N}}\). In this case, by (3.3), one has
Since
there exists \(t_1>1\) such that
Then it follows from (V1), (3.9) and (3.11) that
which is an obvious contradiction.
Case ii) \(\nabla V(x_n)\cdot x_n\le -\delta \) for all \(n\in {\mathbb {N}}\). In this case, (3.3) gives
From (3.10), there exists \(0<t_2<1\) such that
Then it follows from (V1), (3.13) and (3.14) that
which is again an obvious contradiction. This completes the proof. \(\square \)
Since \({\mathcal {J}}(u)=2{\mathcal {I}}'(u)[u]-{\mathcal {P}}(u)\) for \(u\in H^1({\mathbb {R}}^3)\), we have
Define the function
Lemma 3.4
Assume that (V1), (V3), (F1) and (F4) hold. Then
where \(u_t(x)=u(tx)\).
Proof
For \(u\in H^1({\mathbb {R}}^3)\) and \(t> 0\), one has
Thus, (2.5), (3.1), (3.3), (3.4), (3.16), (3.17) and (3.19) imply that for all \(u\in H^1({\mathbb {R}}^3)\) and all \(t> 0\)
This shows (3.18). \(\square \)
Remark that (3.18) with \(t\rightarrow 0\) gives
For the limiting problem, corresponding to (2.5) and (3.16), we define the following functionals in \(H^1({\mathbb {R}}^3)\):
and
From Lemma 3.4, we deduce the following two properties.
Corollary 3.5
Assume that (V1), (V3), (F1) and (F4) hold. Then for \(u\in {\mathcal {M}}\)
Corollary 3.6
Assume that (F1) and (F4) hold. Then
By using (3.5) instead of (3.4), as in the proof of Lemma 3.4, we have the following lemma.
Lemma 3.7
Assume that (F1) and (F4\('\)) hold. Then
Lemma 3.8
Assume that (V1), (V3) and (F1)–(F4) hold. Then for any \(u\in H^1({\mathbb {R}}^3){\setminus }\{0\}\), there exists a unique \(t_u>0\) such that \(t_u^2u_{t_u}\in {\mathcal {M}}\).
Proof
Let \(u\in H^1({\mathbb {R}}^3){\setminus }\{0\}\) be fixed and define the function \(\zeta (t):={\mathcal {I}}(t^2u_t)\) on \((0, \infty )\). Using (3.16) and (1.6), it is easily checked that
By (V1) and (F1)–(F3), we have \(\lim _{t\rightarrow 0^+}\zeta (t)=0\), \(\zeta (t)>0\) for \(t>0\) small and \(\zeta (t)<0\) for t large. Therefore, \(\max _{t\in (0, \infty )}\zeta (t)\) is achieved at \(t_0=t_u>0\), so that \(\zeta '(t_0)=0\) and \(t_0^2u_{t_0}\in {\mathcal {M}}\).
Next, we claim that \(t_u\) is unique for any \(u\in H^1({\mathbb {R}}^3){\setminus }\{0\}\). In fact, for any given \(u\in H^1({\mathbb {R}}^3){\setminus }\{0\}\), let \(t_1, t_2>0\) be such that \(\zeta '(t_1)= \zeta '(t_2)=0\). Then \({\mathcal {J}}(t_1^2u_{t_1})={\mathcal {J}}(t_2^2u_{t_2})=0\). Jointly with (3.18), we have
and
Then (3.1), (3.25) and (3.25) give \(t_1=t_2\). Therefore, \(t_u> 0\) is unique for any \(u\in H^1({\mathbb {R}}^3){\setminus }\{0\}\). \(\square \)
Combining Corollary 3.5 with Lemma 3.8, w obtain the following min-max property.
Lemma 3.9
Assume that (V1), (V3) and (F1)–(F4) hold. Then
Lemma 3.10
Assume that (V1), (V3) and (F1)–(F4) hold. Then
-
(i)
there exists \(\rho >0\) such that \(\Vert u\Vert \ge \rho , \ \forall \ u\in {\mathcal {M}}\);
-
(ii)
\(m=\inf _{{\mathcal {M}}} {\mathcal {I}}>0\).
Proof
-
(i).
In view of [13, Lemma 2.5], if V satisfies (V1) and (V3), then there exist \(\varrho _1, \varrho _2>0\) such that
$$\begin{aligned}&2V(x)+\nabla V(x)\cdot x \ge \varrho _1, \ \ \ \ \forall \ x\in {\mathbb {R}}^3, \end{aligned}$$(3.27)$$\begin{aligned}&V(x)-\nabla V(x)\cdot x \ge \varrho _2, \ \ \ \ \forall \ x\in {\mathbb {R}}^3. \end{aligned}$$(3.28)Since \({\mathcal {J}}(u)=0\) for \(u\in {\mathcal {M}}\), by (3.2), (3.16), (3.28) and the Sobolev embedding theorem, we have
$$\begin{aligned} \frac{\min \{3,\varrho _2\}}{2}\Vert u\Vert ^2\le & {} \frac{3}{2}\Vert \nabla u\Vert _2^2+\frac{1}{2}\int _{{{\mathbb {R}}}^3}[V(x)-\nabla V(x)\cdot x]u^2\mathrm {d}x\nonumber \\&\ \ +\frac{3q^2}{4}\int _{{{\mathbb {R}}}^3}\int _{{{\mathbb {R}}}^3} \frac{1-e^{-\frac{|x-y|}{a}} -\frac{|x-y|}{3a}e^{-\frac{|x-y|}{a}}}{|x-y|}u^2(x)u^2(y)\mathrm {d}x\mathrm {d}y\nonumber \\\le & {} \int _{{{\mathbb {R}}}^3}[2f(u)u-3F(u)]\mathrm {d}x\nonumber \\\le & {} \frac{\min \{3,\varrho _2\}}{4}\Vert u\Vert ^{2}+C_1\Vert u\Vert ^{p}, \ \ \ \ \forall \ u\in {\mathcal {M}}, \end{aligned}$$which implies
$$\begin{aligned} \Vert u\Vert \ge \rho :=\left( \frac{\min \{3,\varrho _2\}}{4C_1}\right) ^{1/(p-2)}, \ \ \ \ \forall \ u\in {\mathcal {M}}. \end{aligned}$$(3.29) -
(ii).
Let \(\{u_n\}\subset {\mathcal {M}}\) be such that \({\mathcal {I}}(u_n)\rightarrow m\). There are two possible cases: 1) \(\inf _{n\in {\mathbb {N}}}\Vert u_n\Vert _2>0\) and 2) \(\inf _{n\in {\mathbb {N}}}\Vert u_n\Vert _2=0\).
Case 1) \(\inf _{n\in {\mathbb {N}}}\Vert u_n\Vert _2:=\rho _1>0\). In this case, (3.20) and (3.27) yield
$$\begin{aligned} m+o(1)={\mathcal {I}}(u_n)={\mathcal {I}}(u_n)-\frac{1}{3}{\mathcal {J}}(u_n)\ge \frac{\varrho _1}{6}\rho _1^2>0. \end{aligned}$$(3.30)Case 2) \(\inf _{n\in {\mathbb {N}}}\Vert u_n\Vert _2=0\). By (3.29), passing to a subsequence, we have
$$\begin{aligned} \Vert u_n\Vert _2\rightarrow 0, \ \ \ \ \Vert \nabla u_n\Vert _2\ge \frac{1}{2}\rho . \end{aligned}$$(3.31)Let \(t_n=\Vert \nabla u_n\Vert _2^{-2/3}\). Then (3.31) implies that \(\{t_n\}\) is bounded. Using (F1), (F2) and the Sobolev inequality, there exists \(C_2>0\) such that
$$\begin{aligned} \left| \int _{{\mathbb {R}}^3}F(u)\mathrm {d}x\right| \le C_2\Vert u\Vert _2^2+\frac{S^3}{4}\Vert u\Vert _6^{6} \le C_2\Vert u\Vert _2^2+\frac{1}{4}\Vert \nabla u\Vert _2^{6}, \ \ \ \ \forall \ u\in H^1({\mathbb {R}}^3). \end{aligned}$$(3.32)Since \({\mathcal {J}}(u_n)=0\) for all \(n\in {\mathbb {N}}\), then (3.18), (3.19), (3.31) and (3.32) give
$$\begin{aligned} m+o(1)= & {} {\mathcal {I}}(u_n)\ge {\mathcal {I}}(t_n^2(u_n)_{t_n})\\= & {} \frac{t_n^3}{2}\Vert \nabla u_n\Vert _2^2+\frac{t_n}{2}\int _{{{\mathbb {R}}}^3}V(t_n^{-1}x)u_n^2\mathrm {d}x\\&+\frac{q^2t_n^3}{4}\int _{{{\mathbb {R}}}^3}\int _{{{\mathbb {R}}}^3}\frac{1-e^{-\frac{|x-y|}{at_n}}}{|x-y|} u_n^2(x)u_n^2(y)\mathrm {d}x\mathrm {d}y\\&\ \ -\frac{1}{t_n^3}\int _{{{\mathbb {R}}}^3}F(t_n^2u_n)\mathrm {d}x\\\ge & {} \frac{t_n^3}{2}\Vert \nabla u_n\Vert _2^2-C_2t_n\Vert u_n\Vert _2^2-\frac{t_n^9}{4}\Vert \nabla u_n\Vert _2^6 \nonumber \\= & {} \frac{1}{4}t_n^3\Vert \nabla u_n\Vert _2^2\left[ 2-\left( t_n^3\Vert \nabla u_n\Vert _2^2\right) ^{2}\right] +o(1)=\frac{1}{4}+o(1). \end{aligned}$$Cases 1) and 2) show that \(m=\inf _{{\mathcal {M}}}{\mathcal {I}}>0\). This completes the proof. \(\square \)
Lemma 3.11
Assume that (V1), (V3) and (F1)–(F4) hold. Then \(m^{\infty }:=\inf _{{\mathcal {M}}^{\infty }} {\mathcal {I}}^{\infty }\ge m\).
Proof
Arguing by contradiction, suppose that \(m>m^{\infty }\). Let \(\varepsilon :=m-m^{\infty }\). Then there exists \(u_{\varepsilon }^{\infty }\) such that
In view of Lemma 3.8, there exists \(t_\varepsilon >0\) such that \(t_\varepsilon ^2(u_{\varepsilon }^{\infty })_{t_\varepsilon }\in {\mathcal {M}}\). Thus, it follows from (V1), (2.5), (3.3), (3.21), (3.24) and (3.33) that
This contradiction shows that \(m^{\infty }\ge m\). \(\square \)
By combining [18, Lemma B.2] and [23, 26], we obtain the following Brezis-Lieb type lemma, see [8].
Lemma 3.12
Assume that (V1), (V2), (F1) and (F2) hold. If \(u_n\rightharpoonup {\bar{u}}\) in \(H^1({\mathbb {R}}^3)\), then up to a subsequence
Lemma 3.13
Assume that (V1), (V3) and (F1)–(F4) hold. Then m is achieved.
Proof
Let \(\{u_n\}\subset {\mathcal {M}}\) be such that \({\mathcal {I}}(u_n)\rightarrow m\). Since \({\mathcal {J}}(u_n)=0\), then (3.20) and (3.27) yield
This shows that \(\{\Vert u_n\Vert _2\}\) is bounded. Now we assert that \(\{\Vert \nabla u_n\Vert _2\}\) is also bounded. Arguing by contradiction, suppose that \(\Vert \nabla u_n\Vert _2 \rightarrow \infty \). From (F1), (F2) and the Sobolev inequality, there exists \(C_2>0\) such that
Let \(t_n=\left( 8m/\Vert \nabla u_n\Vert _2^2\right) ^{1/3}\). Since \({\mathcal {J}}(u_n)=0\), it follows from (3.18), (3.19) and (3.38) that
This contradiction shows that \(\{\Vert \nabla u_n\Vert _2\}\) is also bounded and the assertion holds. Hence \(\{u_n\}\) is bounded in \(H^1({\mathbb {R}}^3)\). Thus, there exists \({\bar{u}}\in H^1({\mathbb {R}}^3)\) such that, passing to a subsequence, \(u_n\rightharpoonup {\bar{u}}\) in \(H^1({\mathbb {R}}^3)\), \(u_n\rightarrow {\bar{u}}\) in \(L_{\mathrm {loc}}^s({\mathbb {R}}^3)\) for all \(1\le s<6\) and \(u_n\rightarrow {\bar{u}}\) a.e. in \({\mathbb {R}}^3\). There are two possible cases: i) \({\bar{u}}=0\) and ii) \({\bar{u}}\ne 0\).
Case i) \({\bar{u}}=0\), i.e. \(u_n\rightharpoonup 0\) in \(H^1({\mathbb {R}}^3)\), \(u_n\rightarrow 0\) in \(L_{\mathrm {loc}}^s({\mathbb {R}}^3)\) for all \(1\le s<6\) and \(u_n\rightarrow 0\) a.e. in \({\mathbb {R}}^3\). Using (V1) and (3.8), it is easily checked that
From (2.5), (3.16), (3.21), (3.22) and (3.40), we derive
From [26, Lemma 1.21], we deduce that there exist \(\delta >0\) and a sequence \(\{y_n\}\subset {\mathbb {R}}^3\) such that \(\int _{B_1(y_n)}|u_n|^2\mathrm {d}x> \delta \). Let \({\hat{u}}_n(x)=u_n(x+y_n)\). Then we have \(\Vert {\hat{u}}_n\Vert =\Vert u_n\Vert \) and
Therefore, there exists \({\hat{u}}\in H^1({\mathbb {R}}^3){\setminus }\{0\}\) such that, passing to a subsequence,
Let \(w_n={\hat{u}}_n-{\hat{u}}\). Then (3.43) and Lemma 3.12 yield
We define the functional \(\Psi ^{\infty }: H^1({\mathbb {R}}^3)\rightarrow {\mathbb {R}}\) by
By (3.21), (3.22), (3.42), (3.44) and (3.45), we have
If there exists a subsequence \(\{w_{n_i}\}\) of \(\{w_n\}\) such that \(w_{n_i}=0\), then
Thus, we assume that \(w_n\ne 0\) for all \(n\in {\mathbb {N}}\). We claim that \({\mathcal {J}}^{\infty }({\hat{u}})\le 0\). Otherwise, if \({\mathcal {J}}^{\infty }({\hat{u}})>0\), then (3.46) implies \({\mathcal {J}}^{\infty }(w_n) < 0\) for large n. In view of Lemma 3.8, there exists \(t_n>0\) such that \(t_n^2(w_n)_{t_n}\in {\mathcal {M}}^{\infty }\) for large n. From (3.21), (3.22), (3.23), (3.46) and Lemma 3.11, we obtain
which contradicts the fact that \(\Psi ^{\infty }({\hat{u}})>0\). Hence, \({\mathcal {J}}^{\infty }({\hat{u}})\le 0\) and the claim holds. In view of Lemma 3.8, there exists \(t_{\infty }>0\) such that \(t_{\infty }^2{\hat{u}}_{t_{\infty }}\in {\mathcal {M}}^{\infty }\). Now (3.23), (3.41), (3.42), (3.45), Fatou’s lemma and Lemma 3.11 yield
which implies again the validity of (3.47) also in this case. In view of Lemma 3.8, there exists \({\hat{t}}>0\) such that \({\hat{t}}^2{\hat{u}}_{{\hat{t}}}\in {\mathcal {M}}\). Moreover, it follows from (V1), (2.5), (3.21), (3.47) and Corollary 3.5 that
This shows that m is achieved at \({\hat{t}}^2{\hat{u}}_{{\hat{t}}}\in {\mathcal {M}}\).
Case ii) \({\bar{u}}\ne 0\). We define the functional \(\Psi : H^1({\mathbb {R}}^3)\rightarrow {\mathbb {R}}\) by
In this case, similarly to the proof of (3.47), by using \({\mathcal {I}}, {\mathcal {J}}\) and \(\Psi \) instead of \({\mathcal {I}}^{\infty }, {\mathcal {J}}^{\infty }\) and \(\Psi ^{\infty }\), we deduce that \({\mathcal {I}}({\bar{u}})=m\) and \({\mathcal {J}}({\bar{u}})=0\). \(\square \)
Lemma 3.14
Assume that (V1), (V3) and (F1)–(F4) hold. If \({\bar{u}}\in {\mathcal {M}}\) and \({\mathcal {I}}({\bar{u}})=m\), then \({\bar{u}}\) is a critical point of \({\mathcal {I}}\).
Proof
Assume that \({\mathcal {I}}'({\bar{u}})\ne 0\). Then there exist \(\delta >0\) and \(\rho >0\) such that
It is easy to check that
Then there exists \(\delta _1>0\) such that
Using (V1), (V3) and (F1)–(F3), it is easy to prove that there exist \(T_1\in (0,1)\) and \(T_2\in (1,\infty )\) such that
In view of Lemma 3.4, we have
The rest of the proof is similar to that of [11, Lemma 2.14]. For the sake of completeness, we give the details. Let
and \(\varepsilon :=\min \{\beta _0/24, 1, \rho \delta /8\}\). From [26, Lemma 2.3], there exists a deformation \(\eta \in {\mathcal {C}}([0, 1]\times H^1({\mathbb {R}}^3), H^1({\mathbb {R}}^3))\) such that
-
(i)
\(\eta (1, u)=u\) if \({\mathcal {I}}(u)<m-2\varepsilon \) or \({\mathcal {I}}(u)>m+2\varepsilon \);
-
(ii)
\(\eta \left( 1, {\mathcal {I}}^{m+\varepsilon }\cap B({\bar{u}}, \delta )\right) \subset {\mathcal {I}}^{m-\varepsilon }\);
-
(iii)
\({\mathcal {I}}(\eta (1, u))\le {\mathcal {I}}(u), \ \forall \ u\in H^1({\mathbb {R}}^3)\);
-
(iv)
\(\eta (1, u)\) is a homeomorphism of \(H^1({\mathbb {R}}^3)\).
Note that Corollary 3.5 implies that \({\mathcal {I}}\left( t^2{\bar{u}}_t\right) \le {\mathcal {I}}({\bar{u}})=m\) for all \(t> 0\). Then (3.49) and ii) give
On the other hand, (3.51) and iii) yield
where
Combining (3.52) with (3.53), we have
Define the function \(\Psi _0(t):={\mathcal {J}}\left( \eta \left( 1, t^2{\bar{u}}_t\right) \right) \) for all \(t> 0\). It follows from (3.51) and i) that \(\eta (1, t^2{\bar{u}}_t)=t^2{\bar{u}}_t\) for \(t=T_1\) and \(t=T_2\), which, together with (3.50), implies
Since \(\Psi _0(t)\) is continuous on \((0, \infty )\), then we have that \(\eta \left( 1, t^2{\bar{u}}_t\right) \cap {\mathcal {M}}\ne \emptyset \) for some \(t_0\in [T_1, T_2]\), contradicting the definition of m. \(\square \)
Proof of Theorem 1.4
In view of Lemmas 3.13 and 3.14, there exists \({\bar{u}}\in {\mathcal {M}}\) such that
This shows that \({\bar{u}}\) is a ground state solution of (1.1) such that \({\mathcal {I}}({\bar{u}})=m=\inf _{{\mathcal {M}}}{\mathcal {I}}\). \(\square \)
Remark 3.15
As in the proof of Theorem 1.4, by replacing Lemma 3.4 with Lemma 3.7, we then obtain Corollary 1.6.
4 Proof of Theorem 1.1
In this section, we give the proof of Theorem 1.1. Without loss of generality, we consider that \(V(x)\not \equiv V_{\infty }\).
Proposition 4.1
[21] Let X be a Banach space and let \(J\subset {\mathbb {R}}^+\) be an interval, and
be a family of \({\mathcal {C}}^1\)-functionals on X such that
-
(i)
either \(A(u)\rightarrow +\infty \) or \(B(u)\rightarrow +\infty \), as \(\Vert u\Vert \rightarrow \infty \);
-
(ii)
B maps every bounded set of X into a set of \({\mathbb {R}}\) bounded below;
-
(iii)
there are two points \(v_1, v_2\) in X such that
$$\begin{aligned} {\tilde{c}}_{\lambda }:=\inf _{\gamma \in {\tilde{\Gamma }}}\max _{t\in [0, 1]}\Phi _{\lambda }(\gamma (t))>\max \{\Phi _{\lambda }(v_1), \Phi _{\lambda }(v_2)\}, \end{aligned}$$(4.1)
where
Then, for almost every \(\lambda \in J\), there exists a sequence \(\{u_n(\lambda )\}\) such that
-
(i)
\(\{u_n(\lambda )\}\) is bounded in X;
-
(ii)
\(\Phi _{\lambda }(u_n(\lambda ))\rightarrow {\tilde{c}}_{\lambda }\);
-
(iii)
\(\Phi _{\lambda }'(u_n(\lambda ))\rightarrow 0\) in \(X^*\), where \(X^*\) is the dual of X.
For \(\lambda \in [1/2, 1]\) we introduce two families of \({\mathcal {C}}^1\)-functionals on \(H^1({\mathbb {R}}^3)\) defined by
In view of [18, A.14], we obtain the following useful identity.
Lemma 4.2
Assume that (V1), (V2) and (F1)–(F3) hold. Let u be a critical point of \({\mathcal {I}}_{\lambda }\) in \(H^1({\mathbb {R}}^3)\), then the following Poho\(\breve{\mathrm{z}}\)aev-type identity holds
Let us set \({\mathcal {J}}_{\lambda }(u):=2 {\mathcal {I}}_{\lambda }'(u)[u]-{\mathcal {P}}_{\lambda }(u)\) for all \(\lambda \in [1/2, 1]\). Then
Similarly, for all \(\lambda \in [1/2,1]\), if u is a critical point of \({\mathcal {I}}^{\infty }_{\lambda }\), then u satisfies the following Poho\(\breve{\mathrm{z}}\)aev-type identity:
We also let
Define
By Lemma 3.7, we have the following lemma.
Lemma 4.3
Assume that (F1), (F3) and (F4) hold. Then
In view of Corollary 1.6, \({\mathcal {I}}_1^{\infty }={\mathcal {I}}^{\infty }\) has a minimizer \(u_1^{\infty }\ne 0\) on \({\mathcal {M}}_1^{\infty }={\mathcal {M}}^{\infty }\), i.e.
Noting that (1.5) is autonomous, \(V\in {\mathcal {C}}({\mathbb {R}}^3, {\mathbb {R}})\) and \(V(x)\le V_{\infty }\) but \(V(x)\not \equiv V_{\infty }\), we can find \({\bar{x}}\in {\mathbb {R}}^3\) and \({\bar{r}}>0\) such that
after suitable translations to \(u_1^{\infty }\).
By (V1), we have \(V_{\max }:=\max _{x\in {\mathbb {R}}^3}V(x)\in (0,\infty )\). Let
Then it follows from (3.19) and (4.10) that there exists \(T>0\) such that
Lemma 4.4
Assume that (V1), (V2) and (F1)–(F3) hold. Then
-
(i)
there exists \(T>0\), independent of \(\lambda \), such that \({\mathcal {I}}_{\lambda }(T^2(u_1^{\infty })_{T})<0\) for all \(\lambda \in [1/2, 1]\);
-
(ii)
there exists a positive constant \(\kappa _0 \), independent of \(\lambda \), such that for all \(\lambda \in [1/2, 1]\),
$$\begin{aligned} c_{\lambda }:=\inf _{\gamma \in \Gamma }\max _{t\in [0, 1]}{\mathcal {I}}_{\lambda }(\gamma (t))\ge \kappa _0 >\max \{{\mathcal {I}}_{\lambda }(0), {\mathcal {I}}_{\lambda }(T^2(u_1^{\infty })_{T})\}, \end{aligned}$$where
$$\begin{aligned} \Gamma =\left\{ \gamma \in {\mathcal {C}}([0, 1], H^1({\mathbb {R}}^3)): \gamma (0)=0, \gamma (1)=T^2(u_1^{\infty })_{T}\right\} ; \end{aligned}$$ -
(iii)
\(c_{\lambda }\) is bounded for \(\lambda \in [1/2, 1]\) and \(\limsup _{\lambda \rightarrow \lambda _0}c_{\lambda }\le c_{\lambda _0}\) for all \(\lambda _0\in (1/2, 1]\);
-
(iv)
if f further satisfies (F4), then \(m_{\lambda }^{\infty }\) are non-increasing on \(\lambda \in [1/2, 1]\).
The proof of Lemma 4.4 is standard, so we omit it. Moreover, similarly to proof of [15, Lemma 4.5], we have the following lemma.
Lemma 4.5
Assume that (V1), (V2) and (F1)–(F4) hold. Then there exists \({\bar{\lambda }}\in [1/2, 1)\) such that \(c_{\lambda }<m_{\lambda }^{\infty }\) for all \(\lambda \in ({\bar{\lambda }}, 1]\).
Lemma 4.6
Assume that (V1), (V2) and (F1)–(F4) hold. Then for almost every \(\lambda \in ({\bar{\lambda }},1]\), there exists \(u_{\lambda }\in H^1({\mathbb {R}}^3){\setminus }\{0\}\) such that
Proof
By Proposition 4.1, for almost every \(\lambda \in [1/2,1]\), there exists a bounded sequence \(\{u_n(\lambda )\} \subset H^1({\mathbb {R}}^3)\), which we denote it by \(\{u_n\}\) for simplicity, such that
Similarly to the proof of [18, Lemma 4.5], using Lemma 3.12, we then deduce that there exist \(u_{\lambda }\in H^1({\mathbb {R}}^3)\), an integer \(l\in {\mathbb {N}}\cup \{0\}\), a sequence \(\{y_n^k\}\subset {\mathbb {R}}^3\) and \(w^k\in H^1({\mathbb {R}}^3)\) for \(1\le k\le l\) such that \(u_n\rightharpoonup u_{\lambda }\) in \(H^1({\mathbb {R}}^3)\), \({\mathcal {I}}_{\lambda }'(u_{\lambda })=0\), \(({\mathcal {I}}_{\lambda }^{\infty })'(w^k)=0\) and \({\mathcal {I}}_{\lambda }^{\infty }(w^k)\ge m_{\lambda }^{\infty }\) for \(1\le k\le l\),
Since \({\mathcal {I}}_{\lambda }'(u_{\lambda })=0\), then \({\mathcal {J}}_{\lambda }(u_{\lambda })=0\). It follows from (V2), (3.6), (4.2) and (4.5) that
If \(l\ne 0\), then
which contradicts Lemma 4.5. Thus, \(l = 0\), and (4.16) implies that \(u_n\rightarrow u_{\lambda }\) in \(H^1({\mathbb {R}}^3)\) and \({\mathcal {I}}_{\lambda }(u_{\lambda })=c_{\lambda }\) for almost every \(\lambda \in ({\bar{\lambda }},1]\). \(\square \)
Lemma 4.7
Assume that (V1), (V2) and (F1)–(F4) hold. Then there exists \({\bar{u}}\in H^1({\mathbb {R}}^3){\setminus }\{0\}\) such that
Proof
In view of Lemma 4.4 (ii) and (iii) and Lemma 4.6, there exist two sequences \(\{\lambda _n\}\subset ({\bar{\lambda }}, 1]\) and \(\{u_{\lambda _n}\}\subset H^1({\mathbb {R}}^3)\), which we denoted it by \(\{u_n\}\) for brevity, such that
Now we assert that \(\{u_n\}\) is bounded in \(H^1({\mathbb {R}}^3)\).
By (4.2), (4.5), (4.19) and Lemma 4.4 (iii), one has
By (V2), there exist constants \(\varrho _0,R_0>0\) such that
Then it follows from (3.6), (4.20) and (4.21) that
which implies that \(\{\Vert u_n\Vert _2\}\) is bounded.
Next, we prove that \(\{\Vert \nabla u_n\Vert _2\}\) is also bounded. Arguing by contradiction, suppose that \(\Vert \nabla u_n\Vert _2 \rightarrow \infty \). By (V1), (V2), (4.22) and Lemma 4.4 (iii), one has
for some constant \(M_0>0\). Let \(t_n=\min \left\{ 1, 2(M_0/\Vert \nabla u_n\Vert _2^2)^{1/3}\right\} \). Then \(t_n \rightarrow 0\). Thus, it follows from (4.2), (4.3), (4.5), (4.7), (4.9) and (4.23) that
As in the proof of (3.39), we then deduce a contradiction by using (4.24). Hence, \(\{u_n\}\) is bounded in \(H^1({\mathbb {R}}^3)\), and the assertion holds.
Similarly to the proof of Lemma 4.6, there exists \({\bar{u}}\in H^1({\mathbb {R}}^3){\setminus }\{0\}\) such that (4.18) holds. \(\square \)
Proof of Theorems 1.1
Define
Then Lemma 4.7 shows that \({\mathcal {K}}\ne \emptyset \) and \({\hat{m}}\le c_1\). For any \(u\in {\mathcal {K}}\), (3.16), (4.5) and Lemma 4.2 imply \({\mathcal {J}}(u)={\mathcal {J}}_1(u)=2{\mathcal {I}}'(u)[u]-{\mathcal {P}}(u)=0\). By (2.5), (3.16) and (4.21), one has
which implies \({\hat{m}}\ge 0\). Since \({\mathcal {I}}'(u)[u]=0\) for \(u\in {\mathcal {K}}\), we then deduce from (F1), (F2) and the Sobolev embedding theorem that there exists \(\alpha _0>0\) such that
Let \(\{u_n\}\subset {\mathcal {K}}\) be such that \({\mathcal {I}}'(u_n)=0\) and \({\mathcal {I}}(u_n) \rightarrow {\hat{m}}\). In view of Lemma 4.5, we have \({\hat{m}}\le c_1<m_1^{\infty }\). Similarly to the proof of Lemma 4.6, we deduce that there exists \({\hat{u}}\in H^1({\mathbb {R}}^3)\) such that \(u_n\rightarrow {\hat{u}}\) in \(H^1({\mathbb {R}}^3)\), \({\mathcal {I}}'({\hat{u}})=0\) and \({\mathcal {I}}({\hat{u}}) = {\hat{m}}\). Moreover, (4.25) leads to \({\hat{u}}\ne 0\). Hence, \({\hat{u}}\in H^1({\mathbb {R}}^3)\) is a ground state solution of (1.1). \(\square \)
Proof of Theorems 1.3
As in the proof of Lemma 4.6, for almost every \(\lambda \in [1/2,1]\), there exists a bounded sequence \(\{u_n(\lambda )\} \subset H^1({\mathbb {R}}^3)\), which we denote it by \(\{u_n\}\) for simplicity, and a positive constant \(\kappa _0^{\infty }\), independent of \(\lambda \), such that
Using (F1), (F2), (4.26) and [26, Lemma 1.21], we can prove that there exists a sequence \(y_n\in {\mathbb {R}}^3\) such that \(\int _{B_1(y_n)}|u_n|^2\mathrm {d}x> 0\). Let \({\bar{u}}_n(x)=u_n(x+y_n)\). Then \(\Vert {\bar{u}}_n\Vert =\Vert u_n\Vert \) and there exists \({\bar{u}}\in H^1({\mathbb {R}}^3){\setminus }\{0\}\) such that \({\bar{u}}_n\rightharpoonup {\tilde{u}}\) in \(H^1({\mathbb {R}}^3)\). Note that
By a standard argument, for almost every \(\lambda \in [1/2,1]\), there exists \(u_{\lambda }\in H^1({\mathbb {R}}^3){\setminus }\{0\}\) such that
From (4.28), there exist two sequences \(\{\lambda _n\}\subset [1/2, 1]\) and \(\{u_{\lambda _n}\}\subset H^1({\mathbb {R}}^3)\), which we denote the latter by \(\{u_n\}\), such that
Similarly to (4.20), we have
which implies
and
Next, we claim that \(\{\Vert \nabla u_n\Vert _2\}\) is also bounded. Arguing by contradiction, suppose that \(\Vert \nabla u_n\Vert _2\rightarrow \infty \). Set \(v_n=u_n/\Vert u_n\Vert \), then \(\Vert v_n\Vert =1\), and (4.31) implies \(\Vert v_n\Vert _2\rightarrow 0\). If \(\delta _0:=\limsup _{n\rightarrow \infty }\sup _{y\in {\mathbb {R}}^3}\int _{B_1(y)}|v_n|^2\mathrm {d}x=0\), then by [26, Lemma 1.21], \(v_n\rightarrow 0\) in \(L^{s}({\mathbb {R}}^3)\) for \(2<s<6\).
Since \(\Vert v_n\Vert _2\rightarrow 0\), we have
Set \(\kappa '=\kappa /(\kappa -1)\). Then (F5), (4.31) and the Hölder inequality yield
Since \(({\mathcal {I}}_{\lambda _n}^{\infty })'(u_n)[u_n]=0\) by (4.29), then (4.33) and (4.34) yield
This contradiction shows that \(\delta _0=\limsup _{n\rightarrow \infty }\sup _{y\in {\mathbb {R}}^3} \int _{B_1(y)}|v_n|^2\mathrm {d}x>0\). Going if necessary to a subsequence, we may assume that there exists a sequence \(\{y_n\}\subset {\mathbb {R}}^3\) such that \(\int _{B_{1}(y_n)}|v_n|^2\mathrm {d}x> \frac{\delta _0}{2}\) for all \(n\in {\mathbb {N}}\). Let \(w_n(x)=v_n(x+y_n)\). Then \(\Vert w_n\Vert =\Vert v_n\Vert =1\), and for all \(n\in {\mathbb {N}}\)
Then there exists \(w\in H^1({\mathbb {R}}^3){\setminus }\{0\}\) such that, passing to a subsequence, \(w_n\rightharpoonup w\) in \(H^1({\mathbb {R}}^3)\), \(w_n\rightarrow w\) in \(L^{s}_{\mathrm {loc}}({\mathbb {R}}^3)\) for all \(1 \le s<6\), \(w_n\rightarrow w\) a.e. in \({\mathbb {R}}^3\). Let us define \({\tilde{u}}_n(x)=u_n(x+y_n)\). Then \({\tilde{u}}_n/\Vert u_n\Vert =w_n\rightarrow w\) a.e. in \({\mathbb {R}}^3\) and \(w\ne 0\). For \(x\in \{y\in {\mathbb {R}}^3 : w(y)\ne 0\}\), we have \(\lim _{n\rightarrow \infty }|{\tilde{u}}_n(x)|=\infty \). By (F1) and (F2), there exists \(M_1>0\) such that
Note that (4.29) and (4.32) lead to
and
From (F3), (4.3), (4.6), (4.37), (4.38), Lemma 4.2 and Fatou’s lemma, we derive
This contradiction shows that \(\{u_n\}\) is bounded in \(H^1({\mathbb {R}}^3)\) and the claim holds.
As in the proof of Lemma 4.6, there exists \({\bar{u}}\in H^1({\mathbb {R}}^3){\setminus }\{0\}\) such that
Set
The above argument shows that \({\mathcal {K}}^{\infty }\ne \emptyset \).
For any \(u\in {\mathcal {K}}^{\infty }\), Lemma 4.2 implies \({\mathcal {J}}^{\infty }(u)=2({\mathcal {I}}^{\infty })'(u)[u]-{\mathcal {P}}^{\infty }(u)=0\). By (F5) and (3.45), we have
which implies \({\hat{m}}^{\infty }\ge 0\). Since \(({\mathcal {I}}^{\infty })'(u)[u]=0\) for \(u\in {\mathcal {K}}^{\infty }\), we easily deduce from (F1), (F2) and the Sobolev embedding theorem that there exists \(\alpha _{\infty }>0\) such that
Let \(\{u_n\}\subset {\mathcal {K}}^{\infty }\) be such that \(({\mathcal {I}}^{\infty })'(u_n)=0\) and \({\mathcal {I}}^{\infty }(u_n) \rightarrow {\hat{m}}^{\infty }\). Since \(({\mathcal {I}}^{\infty })'(u_n)[u_n]=0\), we can deduce from (4.39) and [26, Lemma 1.21] that \(\{u_n\}\) is non-vanishing, and so up to a subsequence, there exists a sequence \(\{y_n\}\subset {\mathbb {R}}^3\) such that \(\int _{B_{1}(y_n)}|u_n|^2\mathrm {d}x>0\). Let \({\hat{u}}_n(x)=v_n(x+y_n)\). Then there exists \({\hat{u}}\in H^1({\mathbb {R}}^3){\setminus }\{0\}\) such that \(u_n\rightharpoonup {\hat{u}}\) in \(H^1({\mathbb {R}}^3)\), \(({\mathcal {I}}^{\infty })'({\hat{u}})=0\) and \({\mathcal {I}}^{\infty }({\hat{u}})\ge {\hat{m}}^{\infty }\). Moreover, it follows from (F5), (3.21), (3.22) and Fatou’s lemma that
which implies \({\mathcal {I}}^{\infty }({\hat{u}})= {\hat{m}}^{\infty }\). Hence, \({\hat{u}}\in H^1({\mathbb {R}}^3)\) is a ground state solution of problem (1.5). The proof is now complete. \(\square \)
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Sitong Chen is supported by the National Natural Science Foundation of China (No. 12001542), Lin Li is supported by the National Natural Science Foundation of China (No. 11601046), Chongqing Science and Technology Commission (No. cstc2016jcyjA0310). Xianhua Tang is supported by the National Natural Science Foundation of China (No. 11971485). The research of Vicenţiu D. Rădulescu was supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI–UEFISCDI, project number PCE 137/2021, within PNCDI III.
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Chen, S., Li, L., Rădulescu, V.D. et al. Ground state solutions of the non-autonomous Schrödinger–Bopp–Podolsky system. Anal.Math.Phys. 12, 17 (2022). https://doi.org/10.1007/s13324-021-00627-9
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DOI: https://doi.org/10.1007/s13324-021-00627-9
Keywords
- Schrödinger–Bopp–Podolsky system
- Ground state solution
- Least energy squeeze method
- Nehari–Poho\(\breve{\mathrm{z}}\)aev manifold
- Concentration-compactness