Abstract
In this paper, we are concerned with the existence and properties of ground states for the Schrödinger–Poisson system with combined power nonlinearities
having prescribed mass
in the Sobolev critical case. Here \( a>0\), and \(\gamma >0\), \(\mu >0\) are parameters, \(\lambda \in {\mathbb {R}}\) is an undetermined parameter. By using Jeanjean’ theory, Pohozaev manifold method and Brezis and Nirenberg’s technique to overcome the lack of compactness, we prove several existence results under the \(L^2\)-subcritical, \(L^2\)-critical and \(L^2\)-supercritical perturbation \(\mu |u|^{q-2}u\), under different assumptions imposed on the parameters \(\gamma ,\mu \) and the mass a, respectively. This study can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions of a Sobolev critical Schrödinger–Poisson problem perturbed with a subcritical term in the whole space \({\mathbb {R}}^3\).
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1 Introduction and Main Results
In this paper we study the nonlinear Schrödinger–Poisson system
where \(\Psi : {\mathbb {R}}\times {\mathbb {R}}^3\rightarrow {\mathbb {C}}\) is the time-dependent wave function, \(\gamma , a\in {\mathbb {R}}\) are parameters, the nonlinear term f simulates the interaction between many particles or external nonlinear perturbations. The nonlinear Schrödinger–Poisson system (1.1) attracted much attention in the last decade, starting from the fundamental contribution [13]. System (1.1) has many physical motivations, it derived from the approximation of the Hartree-Fock equation that describes a quantum mechanical of many particles, and is highly beneficial in the quantum description of the ground states of nonrelativistic atoms and molecules [34, 35, 39], and also arises in semiconductor theory [18].
When we are concerned with the standing wave solutions \(\Psi (t, x) =e^{-i\lambda t}u(x)\), \(\lambda \in {\mathbb {R}}\), then \(u: {\mathbb {R}}^3\rightarrow {\mathbb {R}}\) must verify
At this time, there are two possible choices to deal with (1.2). One can fix \(\lambda \in {\mathbb {R}}\) and to look for solutions as critical points of the associated energy functional
where \(F(u)=\int ^u_0f(s)ds\) is the primitive integral of f. Alternatively, one can search for solutions of Eq. (1.2) with prescribed \(L^2\)-norm. At this point, the parameter \(\lambda \in {\mathbb {R}}\) cannot longer be fixed but instead appears as a Lagrange multiplier. Analogous to the first case, the solutions of (1.2) with \(\Vert u\Vert _2^2 =m > 0\) can be obtained as critical points of the energy functional
under the constraint \(L^2\)-sphere \(S_m:=\{u\in H^1({\mathbb {R}}^3): \Vert u\Vert _2^2=m^2\}.\) It is easy to check that J is a well-defined and \(C^1\) functional on \(S_m.\) This approach is relevant from the physical point of view, in particular, since the \(L^2\)-norm is a preserved quantity of the evolution and since the variational characterization of such solutions is often a strong help to analyze their orbital stability, see for example, [6, 9,10,11, 29,30,31] and references therein.
As far as we know, the first work for normalized solution to Eq. (1.2) in the case \(\gamma = 1,\) and \(f(u) = |u|^{p-2}u\) is due to Sánchez and Soler [41]. They showed that there exists a normalized solution of (1.2) provided that m is sufficiently small and \(p = \frac{8}{3}\). Since then, there are some further studies for problem (1.2) in mass subcritical case. In this case, the corresponding functional is bounded from below on \(S_m,\) then a global minimizer can be obtained for some m. See for instance, [10, 11, 31, 32]. When the nonlinearity f in (1.2) is mass supercritical, the constrained functional \(J|_{S_m}\) is no longer bounded from below and coercive. In this case, using a mountain-pass argument on \(S_m,\) Bellazzini, Jeanjean and Luo [12] proved the existence and the instability of standing waves for \(m > 0\) sufficiently small. Bartsch and de Valeriola [6], Luo [38] studied the multiplicity of normalized solutions of (1.2).
At the same time, normalized solutions for Schrödinger–Poisson–Slater equation with general nonlinearity in case \(\gamma =-1,\) has also attracted much more attention. Xie, Chen and Shi [49] showed the existence and multiplicity results of solutions when f satisfies \(\lim _{t\rightarrow 0} f (t)/ t = 0\) and \(\lim _{|t|\rightarrow \infty }F(t)/|t|^{10/3} =\infty \) under some mild conditions on f. Recently, Chen Tang and Yuan [17] investigated the existence of normalized solution by some new analytical techniques in case that f satisfies \(\lim _{t\rightarrow 0} F (t)/ t^2 = 0\) and \(\lim _{|t|\rightarrow \infty }F(t)/|t|^{10/3} =\infty \).
Very recently, Wang and Qian [45] obtained the existence of normalized ground states and infinitely many radial solutions for (1.2) with Sobolev subcritical term f, by constructing a particular bounded Palais-Smale sequence when \(\gamma < 0, a > 0.\) Meanwhile, they obtained the nonexistence result in the case \(\gamma< 0, a < 0\) and the existence result when\(\gamma > 0, a < 0\) via variational methods. In [29], Jeanjean and Trung Le specialized in the existence of normalized solutions for problem (1.2) with \(L^2\)-supercritical growth:
where \( u \in H^1({\mathbb {R}}^3), \gamma \in {\mathbb {R}}, a\in {\mathbb {R}}\) and \(p\in (\frac{10}{3},6]\). The authors dealt with the following cases:
-
(a)
If \(\gamma <0\) and \(a>0\), both in the Sobolev subcritical case \(p\in (\frac{10}{3},6)\) and in the Sobolev critical case \(p=6,\) they showed that there exists a \(c_1>0\) such that, for any \(c\in (0, c_1),\) (1.3) admits two solutions \(u^+_c\) and \(u^-_c\), which can be characterized as a local minimum and a mountain pass critical point of the associated energy functional, respectively.
-
(b)
In the case \(\gamma <0\) and \(a<0,\) they proved that, for any \(p\in (\frac{10}{3}, 6]\) and any \(c>0,\) (1.3) has a solution which is a global minimizer.
-
(c)
Finally, in the case \(\gamma>0, a>0\) and \(p=6\), they showed that (1.3) does not exist positive solutions.
When \(\gamma =1\), \(p\in (\frac{10}{3},6)\), Bellazzini, Jeanjean and Luo [12] studied the existence of normalized solutions of (1.3) by a mountain-pass argument as \(c > 0\) is sufficiently small and nonexistence as \(c> 0\) is not small. In [31], Jeanjean and Luo considered the existence of minimizers with \(L^2\)-norm for (1.3) when \(p\in [3, \frac{10}{3}],\) and they showed a threshold value of \(c> 0\) separating existence and nonexistence of minimizers. For more results on normalized solutions of Schrödinger–Poisson systems, we refer to [1, 19, 27, 29, 31, 33, 37, 38, 50,51,52,53] and references therein.
After the above literature review, we find that, only the article [29] has considered the existence of normalized solutions of (1.3) in the case \(p\in (\frac{10}{3},6]\), and \(\gamma <0\); and the no-existence of normalized solution of (1.3) with \(p=6, \gamma >0\) and \(a>0.\) Therefore, a natural and important question arising is how to obtain normalized solutions to system (1.3) in the case \(\gamma >0\), and in the presence of Sobolev critical exponent and mixed nonlinearities: \(a|u|^{p-2}u+|u|^4u\)? here \(a|u|^{p-2}u\) is a subcritical perturbation term with \(p\in (2,6)\) and \(a>0\) a parameter. We notice that, this kind of critical nonlinearities has been used by Soave [42], Wei and Wu [47] to search for the normalized solutions for the Schrödinger equation
with the prescribed \(L^2\)-norm \(\int _{{\mathbb {R}}^3} |u|^2dx=c^2\). But for the Schrödinger–Poisson system in presence of the Sobolev critical term \(|u|^4u\), coupled with a subcritical perturbation term \(a|u|^{p-2}u\), the existence of normalized solutions has not been studied in the existing literature, as far as we know. For more studies of existence of normalized solutions of the Schrödinger equation, see for example [28, 30, 42, 43, 54] and references therein.
Motivated by the works mentioned above, in this paper we focuss on studying the Schrödinger–Poisson system
having prescribed \(L^2\)-norm
where \(\lambda \in {\mathbb {R}}\) is an undetermined parameter, \(a>0\) and \(\mu ,\gamma >0\) are parameters, \(\mu |u|^{q-2}u\) is a subcritical perturbation term with \(q\in (2,6)\). For this purpose, applying the reduction argument introduced in [40], system (1.4) is equivalent to the following single equation
where \(\phi _u(x)=\frac{1}{4\pi }\int _{{\mathbb {R}}^3}\frac{|u(y)|^2}{|x-y|}dy.\) We shall look for solutions to (1.4)–(1.5), as a critical points of the action functional
under the \(L^2\)-norm constrained manifold
where \(\Psi (u)=\frac{1}{2}\int _{{\mathbb {R}}^3}u^2dx\). Physically, such type of solutions are the so-called normalized solutions to (1.4)–(1.5). In order to state our main results, we introduce some of the constants from the following Gagliardo-Nirenberg-Sobolev (GNS) inequality. That is, there exists a best constant C(p) depending on p such that for any \(u\in H^1({\mathbb {R}}^3),\)
where \(\gamma _p= \frac{3(p-2)}{2p}.\) The constant C(p) can be achieved by function \(Q_p\), see [43]. For the problem obtained from (1.4)–(1.5) by removing the critical exponent term and the nonlocal term, we obtain one of its normalized solutions by rescaling \(Q_p\). From \(\gamma _pp = 2\), we get \(p = \frac{10}{3}\) which is called the \(L^2\)-critical exponent for problem (1.4)–(1.5). Before presenting the existence result, we give the definition of ground states. If \(u^*\) is a solution to (1.4)–(1.5) having minimal energy among all the solutions which belongs to S(a) :
we say that \(u^*\) is a ground state of (1.4)–(1.5).
The following are the main results of this paper. In the \(L^2\)-subcritical case: \(2<q<\frac{10}{3}\), we have the following existence result of the normalized ground state solutions.
Theorem 1.1
Let \(2<q<\frac{10}{3}\), \(\gamma >0\), and assume that \(0<a<\min \{\alpha _1,\alpha _2\}\), where
and
where S is defined in (2.1). Then there exists \(\tilde{\mu }>0\) such that \(\mu >\tilde{\mu }\), problem (1.4)–(1.5) has a couple of solutions \((u_a,\lambda _a)\in S(a)\times {\mathbb {R}}\). Moreover,
for some suitable small constant \(k>0\), where \(\mathcal {P}(a)\) is the Pohozaev manifold defined in Lemma 2.5, the set \(\mathcal {P}(a)^+\) is defined in (3.1), and
In the \(L^2\)-critical case: \(q=\frac{10}{3}\), we have the following conclusion.
Theorem 1.2
Let \(q=\frac{10}{3}\), \(\mu >0\), and assume that \(0<a<\min \{\alpha _3,\alpha _4\}\), where
and
where \({\tilde{C}}\) is defined as (2.3), and k is defined as
Then there exist \(\tilde{\gamma _1},\tilde{\gamma _2}>0\) such that \(0<\gamma <\min \{\tilde{\gamma _1},\tilde{\gamma _2}\}\), problem (1.4)–(1.5) has a couple of solutions \((u_a,\lambda _a)\in S(a)\times {\mathbb {R}}\). Moreover,
where \(\mathcal {P}(a)^-\) is defined in (3.2).
In the \(L^2\)-supercritical case: \(\frac{10}{3}<q<6\), we have the following existence result.
Theorem 1.3
Let \(\frac{10}{3}<q<6\), \(\mu >0\), and assume that \(0<a<\alpha _5\), where
where \(k^*\) is defined in (5.1). Then there exist \(\tilde{\gamma _1},\tilde{\gamma _2}>0\) such that \(0<\gamma <\min \{\tilde{\gamma _1},\tilde{\gamma _2}\}\), problem (1.4)–(1.5) has a couple of solutions \((u_a,\lambda _a)\in S(a)\times {\mathbb {R}}\). Moreover,
Remark 1.1
In [29] Jeanjean and Le only studied the no-existence of normalized solutions of problem (1.4)–(1.5) with \(\gamma >0\) and \(\mu =0\). The existence of normalized solutions for (1.4)–(1.5) in the case \(\gamma>0,\mu >0\) and \(q\in (2,6)\) has not been studied in the existing literature. Theorems 1.1–1.3 provide a complete description of the existence of normative solutions in the \(L^2\)-subcritical, \(L^2\)-critical and \(L^2\)-supercritical perturbation \(\mu |u|^{q-2}u\), respectively.
Remark 1.2
In Theorem 1.1, we assume that the parameter \(\mu >0\) is large enough, so as to ensure that the Lagrange multiplier sequence \(\lambda _n\rightarrow \lambda <0\) as \(n\rightarrow \infty \), which plays a crucial role in our proof of the \(H^1\)-convergence of (PS)-sequence \(\{u_n\}\subset S(a)\). In Theorem 1.2 and Theorem 1.3, it is necessary for the parameter \(\gamma >0\) to be appropriately small so that the Mountain Pass level is strictly less than \(\frac{1}{3} S^{\frac{3}{2}}\). This characteristic is completely different from the critical Schrödinger–Poisson system without \(L^2\)-mass constrained, see for example [25, 26, 46, 55].
In order to prove Theorems 1.1–1.3, we apply the constrained variational methods. Note that the Sobolev critical terms \(|u|^4u\) is \(L^2\)-supercritical, the functional \(I_{\mu }\) is always unbounded from below on S(a), and this causes difficulty to treat the existence of normalized solutions on the \(L^2\)- constraint. One of the main difficulties is to prove the convergence of constrained Palais-Smale sequences: Indeed, the Sobolev critical term \(|u|^4u\) and nonlocal convolution term \(\gamma \phi _u u\), make it more complex to estimate the critical value of mountain pass, and has to consider how the interaction between the nonlocal term and the mixed nonlinearities. In particularly, the energy balance between these competing terms needs to be controlled through moderate adjustments of parameter \(\gamma >0.\) Another obstacle is that sequences of approximated Lagrange multipliers have to be controlled, since \(\lambda \) is not prescribed; and moreover, weak limits of Palais-Smale sequences could leave the constraint, since the embeddings \(H^1({\mathbb {R}}^3)\hookrightarrow L^2({\mathbb {R}}^3)\) and also \(H^1_{rad}({\mathbb {R}}^3)\hookrightarrow L^2({\mathbb {R}}^3)\) are not compact.
To overcome these difficulties, we shall employ Jeanjean’s theory [28] by showing that the mountain pass geometry of \(I_{\mu }|_{S(a)}\) allows to construct a Palais-Smale sequence of functions satisfying the Pohozaev identity, to obtain the boundedness, which is the first step to show strong \(H^1\)-convergence. To restore the loss of compactness caused by the critical growth, we shall utilize the concentration-compactness principle, mountain pass theorem and energy analysis to get the existence of normalized ground states of (1.4)–(1.5), by showing that, suitably combining some of the main ideas from [15, 42], compactness can be derived in the present setting.
This paper is organized as follows: In Sect. 2 we summarize some preliminary results which will often be used in the rest the paper. In Sect. 3, we investigate the existence of normalized ground state solutions for system (1.4)–(1.5) under the \(L^2\)-subcritical perturbation case: \(q\in (2, \frac{10}{3})\) and complete the proof Theorem 1.1. In Sect. 4, we address the presence of the normalized ground state solutions for system (1.4)–(1.5) in \(L^2\)-critical perturbation case: \(q=\frac{10}{3}\) and prove Theorem 1.2, by employing manifold and mountain road theorems. In Sect. 5, we tackle the existence of the normalized ground state solutions for problem (1.4)–(1.5) under \(L^2\)-supercritical perturbation case: \(q\in (\frac{10}{3}, 6)\) and prove Theorem 1.3.
Notations. Throughout this paper, we denote \(B_r(z)\) the open ball of radius r with center at z in \({\mathbb {R}}^3\), and \(\Vert u\Vert _p\) is the usual norm of the space \(L^p({\mathbb {R}}^3)\) for \(p\ge 1\). Moreover, we denote by \(C,C_i > 0, i=1,2,\cdots ,\) different positive constants whose values may vary from line to line and are not essential to the problem.
2 Preliminary Stuff
In this section, we will give the functional space setting and introduce some notations and useful preliminary results, which are important to proving our Theorems. Let \(H^1({\mathbb {R}}^3)\) be the completion of \(C_0^\infty ({\mathbb {R}}^3)\) with respect to the norm
And the homogeneous Sobolev space \(D^{1,2}({\mathbb {R}}^3)\) is defined by
endowed with the norm
The work space \(H^1_{rad}({\mathbb {R}}^3)\) is defined by
Let \({\mathbb {H}} = H\times {\mathbb {R}}\) with usual scalar product
and the corresponding norm
We denote the best Sobolev constant S by
It is well know that S is achieved by
for any \(\varepsilon >0\) and \(C^*\) being normalized constant such that (see [15]):
In the following, we recall some useful inequalities, which play an important part in the proof of our main results.
Proposition 2.1
(Hardy–Littlewood–Sobolev inequality [34]) Let \(l,r>1\) and \(0<\mu <N\) be such that \(\frac{1}{r}+\frac{1}{l}+\frac{\mu }{N}=2, f\in L^r({\mathbb {R}}^N)\) and \(h\in L^l({\mathbb {R}}^N)\). Then there exists a constant \(C(N,\mu ,r,l)>0\) such that
From Proposition 2.1, with \(l= r = \frac{6}{5}\), we have that:
Next, we introduce the following Gagliardo-Nirenberg inequality.
Lemma 2.2
([43]) Let \(p\in (2, 6).\) Then there exists a constant \(C(p)>0\) such that
where \(\delta _{p}=\frac{3(p-2)}{2p}.\)
Lemma 2.3
(Lemma 5.1 [23]) If \(u_n\rightharpoonup u\) in \(H^1_{rad}({\mathbb {R}}^3)\), then
and
In the sequel, we define a useful fiber map (e.g. [42]) preserving the \(L^2\)-norm
By simple calculation, we can infer that
and
Next, we define a auxiliary functional \(E:{\mathbb {H}}\rightarrow {\mathbb {R}}\) by
Besides, we have the fact that
where \(\bar{q}:=\frac{10}{3}\) is the \(L^2\)-critical exponent.
The Pohozaev manifold plays an important role in the proof of our main results, so we introduce it below [22].
Proposition 2.4
Let \(u\in H^1({\mathbb {R}}^3)\cap L^\infty ({\mathbb {R}}^3)\) be a weak solution of (1.4), then u satisfies the equality
Lemma 2.5
Let \(u\in H^1({\mathbb {R}}^3)\) be a weak solution of (1.4)–(1.5), then we can construct the following Pohozaev manifold
where
Proof
Since u is the weak solution of (1.4)–(1.5), we have that
Moreover, since u is the weak solution of system (1.4)–(1.5), we have
Combining with (2.13) and the above equality, we obtain that
The proof is completed. \(\square \)
We define \(\varphi _u(\iota ):=E(u,\iota )\) for any \(u \in S(a)\) and \(\iota \in {\mathbb {R}}\), then
Moreover, by direct calculation, we have
Therefore, we have the following lemma:
Lemma 2.6
For any \(u \in S(a)\), \(\iota \in {\mathbb {R}}\) is a critical point of \(\varphi _u(\iota )\) if and only if \((\iota \star u) \in \mathcal {P}(a)\). Particularly, \( u \in \mathcal {P}(a) \) if and only if 0 is a critical point for \(\varphi _u(\iota )\).
Finally, we state the following well-known embedding result.
Lemma 2.7
([44]) Let \(N\ge 2\). The embedding \(H^1_{rad}({\mathbb {R}}^N)\hookrightarrow L^p({\mathbb {R}}^N)\) is compact for any \(2< p < 2^*\).
Remark 2.8
([11]) The map \((u,\iota )\in {\mathbb {H}}\rightarrow (\iota \star u) \in H \) is continuous.
3 \(L^2\)-Subcritical Perturbation Case
In this section, we shall address the \(L^2\)-subcritical perturbation case: \(2<q<\frac{10}{3}\) and provide the proof of Theorem 1.1. First, we think about a decomposition of \(\mathcal {P}(a)\) as in [42, 43]. By Lemma 2.6, we define the following sets:
We can easily get that
Next, we will give some lemmas, which are useful for the proof of Theorem 1.1.
Lemma 3.1
Let \(2<q<\frac{10}{3}\), \(\mu ,\gamma >0\), and \(0<a<\alpha _1\), where
Then \(\mathcal {P}(a)^0=\varnothing \) and \(\mathcal {P}(a)\) is a smooth manifold of codimension 2 in \(H({\mathbb {R}}^3)\).
Proof
Suppose by contradiction that \(\mathcal {P}(a)^0\ne \varnothing \). Taking \(u \in \mathcal {P}(a)^0\), one has
and
Since \(\frac{\gamma }{4}\int _{{\mathbb {R}}^3}\phi _uu^2dx\ge 0\), so combining the above equalities with the GNS inequality (2.4) and (2.1), we can infer to
and
By simple calculation and the fact \(q\delta _q<2\), we have
which contradicts to \(a<\alpha _1\).
Then, we verify that \(\mathcal {P}(a)\) is a smooth manifold of codimension 2 in \(H({\mathbb {R}}^3)\). Let
for \(G(u)=\Vert u\Vert ^2_2-a^2\), with \(P_\mu \) and G of class \(C^1\) in H. Hence, we need to show that the differential \((dG(u),dP_\mu (u)):H\rightarrow {\mathbb {R}}^2\) is surjective, for every \(u \in \mathcal {P}(a)\). For this purpose, we will prove that for every \(u \in \mathcal {P}(a)\), there exists \(\varphi \in T_uS\), where
which is the tangent space of \(\mathcal {S}\) at a point \(u \in \mathcal {S}\). Then, one has \(dP_\mu (u)[\varphi ]\ne 0\). Once the existence of \(\varphi \) is established, the system
that is
is solvable with respect to \(\alpha \), \(\beta \) for every \((x,y)\in {\mathbb {R}}^2\), so the surjective is proved. Next, suppose by contrary that for \(u \in \mathcal {P}(a)\) such that a tangent vector \(\varphi \) does not exist, that is, \(dP_\mu (u)[\varphi ]=0\) for every \(\varphi \in T_uS\). Then u is a constrained critical point for the functional \(P_\mu (u)\) on S(a). Thus, by the Lagrange multipliers rule, there exists \(\nu \in {\mathbb {R}}\) such that
Then we can conclude the following Pohozaev type identity:
which is contradiction to the fact that \(u \in \mathcal {P}(a)\). \(\square \)
In virtue of the GNS inequality (2.4) and (2.1), for every \(u \in H({\mathbb {R}}^3)\bigcap S(a)\), we have
where
By the fact \(q\delta _q<2\), we can derive that \(g(0^+)=0^-\) and \(g(+\infty )=-\infty \).
In the following, we show the properties of the function g and give some technical lemmas.
Lemma 3.2
Let \(2<q<\frac{10}{3}\), \(\mu ,\gamma >0\), and \(0<a<\alpha _1\). Then the function g has a local strict minimum at negative level and a global strict maximum at positive level, and there exist two positive constants \(R_1\),\(R_2\) both depending on a, with \(R_1<R_2\), such that \(g(R_1)=g(R_2)=0\) and \(g(t)>0\) for \(t \in (R_1,R_2)\).
Proof
Note that
where
It is easy to see that \(g(t)>0\) if and only if \(m(t)>0\) for all \(t>0\). So, by direct calculation, we have
Let \(m'(t)=0\), it follows that
and we know that m is strictly increasing on \((0,t_1)\) and decreasing on \((t_1,\infty )\). Moreover, the maximum value of m on \((0,+\infty )\) is
By virtue of \(a<\alpha _1\), we deduce that there exist two constants \(R_1\) and \(R_2\) such that
Based on above analysis and the fact \(g(0^+)=0^-\), we infer that g(t) has a global maximum at positive level in \((R_1,R_2)\) and a local minimum at negative level in \((0,R_1)\). It is easy to see that \(R_1<t_1<R_2.\) Besides, by a simple calculation, we have
where
It is easy to see that \(g'(t)=0\) if and only if \(h(t)=0\) for \(t>0\). So, by direct calculation, we get
From \(h'(t)=0\), there exists a unique solution \(t_2>0\) with the expression:
and we know that h is a strictly increasing on \((0,t_2)\) and decreasing on \((t_2,+\infty )\). Hence, h has at most two zeros on \((0,+\infty )\), which are necessarily the previously found local minimum and the global maximum of g. \(\square \)
Lemma 3.3
Let \(2<q<\frac{10}{3}\), \(\mu ,\gamma >0\), and \(0<a<\alpha _1\). Then for every \(u \in S(a)\), \(\varphi _u(\iota )\) has two critical points \(s_u<t_u \in {\mathbb {R}}\) and two zeros \(c_u<d_u\) with \(s_u<c_u<t_u<d_u\). Besides,
-
(i)
\(s_u\star u \in \mathcal {P}(a)^+\), \(t_u\star u \in \mathcal {P}(a)^-\), and if \(\iota \star u \in \mathcal {P}(a)\), then either \(\iota =s_u\) or \(\iota =t_u\);
-
(ii)
\(\Vert \nabla u\Vert _2\le R_1\) for every \(\iota <c_u\) and
$$\begin{aligned} I_\mu (s_u\star u)=\min \{I_\mu (\iota \star u):\iota \in {\mathbb {R}}~\text{ and }~\Vert \nabla u\Vert _2\le R_1\}<0; \end{aligned}$$(3.5) -
(iii)
we have
$$\begin{aligned} I_\mu (t_u\star u)=\max \{I_\mu (\iota \star u):\iota \in {\mathbb {R}}\}>0, \end{aligned}$$(3.6)and \(\varphi _u(\iota )\) is strictly decreasing and concave on \((t_u,+\infty )\);
-
(iv)
the maps \(u \in \mathcal {P}(a)\mapsto s_u\times {\mathbb {R}}\) and \(u \in \mathcal {P}(a)\mapsto t_u\times {\mathbb {R}}\) are of class \(C^1\).
Proof
We claim that \(\varphi _u(\iota )\) has two critical points. In view of (3.4), one has
we can obtain \(\varphi _u(\iota )>0\) on \((\xi (R_1),\xi (R_2))\) from Lemma 3.2, where
Since \(\varphi _u(\iota )\) is a \(C^2\) function, and by the fact that \(\varphi _u(-\infty )=0^-\), \(\varphi _u(+\infty )=-\infty \), it follows that \(\varphi _u(\iota )\) has at least critical points \(s_u\), \(t_u\) with \(s_u<t_u\). Moreover, we know that \(s_u\) is a local minimum point on \((-\infty ,\xi (R_1))\) at negative level and \(t_u\) is a global maximum point at positive level. Hence, we derive to
and
Arguing as in the proof of Lemma 3.2, we can deduce that \(\varphi _u(\iota )\) has no other critical points. In view of \((\varphi _u)''(s_u)\ge 0\), \((\varphi _u)''(t_u)\le 0\) and the fact that \(\mathcal {P}(a)^0=\varnothing \), we have \(s_u\star u \in \mathcal {P}(a)^+\) and \(t_u\star u \in \mathcal {P}(a)^-\).
Next, we claim that \(\varphi _u(\iota )\) has two zeros \(c_u<d_u\). Since \(\varphi _u(s_u)<0\), \(\varphi _u(t_u)>0\) and \(\varphi _u(+\infty )=-\infty \), it is easy to get that \(\varphi _u(\iota )\) has two zeros \(c_u<d_u\) with \(s_u<c_u<t_u<d_u\). Furthermore, \(\varphi _u(\iota )\) has no other zeros. Indeed, if \(\varphi _u(\iota )\) has other zeros, then it will have other critical point, which leads to a contradiction.
Recalling that
we have \((\varphi _u)''(-\infty )=0^-\). Since \((\varphi _u)''(s_u)>0\) and \((\varphi _u)''(t_u)<0\), we get \((\varphi _u)''(\iota )\) has two zeros, which means that \(\varphi _u(\iota )\) has two inflection points. Arguing as before, \((\varphi _u)''(\iota )\) has exactly two inflection points. Hence, \(\varphi _u(\iota )\) is is strictly decreasing and concave on \((t_u,+\infty )\). The items (i)–(iii) are proved.
Finally, we will prove that maps \(u \in \mathcal {P}(a)\mapsto s_u\times {\mathbb {R}}\) and \(u \in \mathcal {P}(a)\mapsto t_u\times {\mathbb {R}}\) are of class \(C^1\). Applying the implicit function theorem, let \(\Phi (\iota ,u):=(\varphi _u)'(\iota )>0\), since \(\Phi (s_u,u)=0\) and \(\partial _\iota \Phi (s_u,u)>0\), we know that \(u \in \mathcal {P}(a)\mapsto s_u\times {\mathbb {R}}\) is of class \(C^1\). Similarly, we have \(u \in \mathcal {P}(a)\mapsto t_u\times {\mathbb {R}}\) is of class \(C^1\). \(\square \)
Thus, we can easily deduce the following conclusion.
Corollary 3.4
\(\sup _{u\in \mathcal {P}(a)^+}I_\mu (u)\le 0\le \inf _{u\in \mathcal {P}(a)^-}I_\mu (u)~~~\text{ and }~~~\mathcal {P}(a)^+\subset D_{R_1},\) where
Lemma 3.5
There holds that \(-\infty<m_\mu (a)=\inf _{u\in \mathcal {P}(a)}I_\mu (u)=\inf _{u\in \mathcal {P}(a)^+}I_\mu (u)<0,\) and
for \(\rho >0\) small enough, where
Proof
For \(u \in D_{R_1}\), in view of (3.4), we have
Besides, for any \(u \in S(a)\), we get \(\Vert \nabla u\Vert _2<R_1\) and \(I_\mu (s_u\star u)<0\). Hence, we can infer to
On one hand, since \(\mathcal {P}(a)^+\subset D_{R_1}\), we get that \(m_\mu (a)\le \inf _{\mathcal {P}(a)^+}I_\mu \). On the other hand, if \(u \in D_{R_1}\), then \(s_u\star u\in \mathcal {P}(a)^+\subset D_{R_1}\), and
which implies that \(\inf _{u\in \mathcal {P}(a)^+}I_\mu (u)\le m_\mu (a)\). Combining with the fact \(0\le \inf _{u\in \mathcal {P}(a)^-}I_\mu (u)\), we obtain
Finally, due to the continuity of g and \(g(R_1)=0\), there exists \(\rho >0\) such that
Therefore, by (3.4), we have
for any \(u \in {\overline{D}_{R_1}\setminus D_{R_1-\rho }}\). The proof is completed. \(\square \)
Proof of Theorem 1.1
First, we take a minimizing sequence \(\{v_n\}\subset H\cap S(a)\) for \(I_\mu |_{D_{R_1}}\) and assume that \(\{v_n\}\subset H_r\) are radially decreasing for every n. Otherwise, we can let \(v_n:=|v_n|^*\), which is the Schwarz rearrangement of \(|v_n|\). In view of Lemmas 3.3 and 3.5, we know that there exists a sequence \(\{s_{v_n}\}\) such that \(s_{v_n}\star v_n \in \mathcal {P}(a)^+\) and \(I_\mu (s_{v_n}\star v_n)\le I_\mu (v_n)\) for every n. Furthermore, we have \(s_{v_n}\star v_n\notin {\overline{D}_{R_1}{\setminus } D_{R_1-\rho }}\). Based on above analysis, we get a new minimizing sequence \(\{\overline{v}_n:=s_{v_n}\star v_n\}\) for \(I_\mu |_{D_{R_1}}\), satisfying
By Ekeland’s variational principle, there exists a new minimizing sequence \(\{u_n\}\), with \(\Vert u_n-\overline{v}_n\Vert \rightarrow 0\) as \(n\rightarrow \infty \), which is also a PS sequence for \(I_\mu \) on S(a). Since \(\{u_n\}\subset D_{R_1}\), we see that \(\{u_n\}\) is bounded in H. So from \(\Vert u_n-\overline{v}_n\Vert \rightarrow 0\) and the boundedness of \(\{u_n\}\), we can obtain
In fact,
and
for every \(p\in [2,6]\), where \(\theta ^1_n,~\theta ^2_n\in [0,1]\). Moreover, \(\{u_n\}\) satisfies
Then, using the Lagrange multipliers rule, there exists a sequence \(\lambda _n\in {\mathbb {R}}\) such that
Since \(\{u_n\}\subset D_{R_1}\), we have \(\{u_n\}\) is bounded in H. So there exists \(u_a \in H\), such that, for some subsequence, \(u_n\rightharpoonup u_a\) in H. In the following, we will proceed with our argument in three steps.
Step 1 We show that, up to subsequence, \(\lim _{n\rightarrow +\infty }\lambda _n=\lambda _a<0\). By (3.9) and the fact \(\{u_n\}\) is bounded in H, we get
Then, we infer to
Again by \(\{u_n\}\) is bounded in H, we see that \(\{\lambda _n\}\) is bounded. Thus, up to subsequence, there exists \(\lambda _a \in {\mathbb {R}}\) such that \(\lambda _n\rightarrow \lambda _a\in {\mathbb {R}}\). Next, we prove \(\lambda _a<0\). Before this, we show
Assume by contradiction that, \(\int _{{\mathbb {R}}^3}|u_n|^qdx\rightarrow 0\). In view of the proof of Lemma 3.2, we have \(\Vert \nabla u_n\Vert _2\le R_1<t_1\), and \(t_1=\left( \frac{3S^3(2-q\delta _q)}{6-q\delta _q}\right) ^{\frac{1}{4}}<S^{\frac{3}{4}}\). Then we deduce
From the definition of \(I_{\mu }\) and above inequality, we infer that
which is absurd.
We claim that there exists \(\tilde{\mu }>0\) independently on \(n \in {\mathbb {N}}\) such that, if \(\mu >\tilde{\mu }\), the lagrange multiplier \(\lambda _a<0\). In fact, since \(\{u_n\}\subset D_{R_1}\), by (2.3) and the GNS inequality (2.4), there exists \(T_1>0\) independently on \(n \in {\mathbb {N}}\) such that
and
where \(T_2=T_2(R_1,a)>0\). We define the constant
By the fact \(P_\mu (u_n)\rightarrow 0\), (3.11), Lemma 2.7 and \(\delta _q<1\), if \(\mu >\tilde{\mu }\), then
Thus, if \(\mu >\tilde{\mu }\), we have \(\lim _{n\rightarrow +\infty }\lambda _n=\lambda _a<0\).
Step 2 Since \(\lambda _a<0\), we define an equivalent norm of H as:
In view of the fact \(u_n\rightharpoonup u_a\) in H and (3.9), then \(u_a\) satisfies
for \(\forall v \in H\). It follows from the Pohozaev identity that \(P_\mu (u_a)=0\). Let \(v_n=u_n-u_a\rightharpoonup 0\), by Brezis–Lieb Lemma [48], we conclude
By the fact (2.5), Lemma 2.7 and \(P_\mu (v_n)=P_\mu (u_n)-P_\mu (u_a)\rightarrow 0\), we obtain
Thus, for some subsequence, we suppose that
By using (2.1), we derive to
then, one has
If \(\tau \ge S^{\frac{3}{2}}\), by (3.17), we have
In what follows, we verify that \(\tau \ge S^{\frac{3}{2}}\), which will lead to a contradiction. In fact, by the GNS inequality (2.5) and \(P_\mu (u_a)=0\), we get
where
By \(f'(t)=0\), there exists a unique \(t_3>0\) such that
with
Then, we see that f(t) is strictly decreasing on \((0,t_3)\) and increasing on \((t_3,+\infty )\). Moreover, f(t) gets the minimum on \((0,+\infty )\), that is
Define
Since \(a<\alpha _2\), we get
Combining (3.18) and (3.19), we infer that \(m_\mu (a)>0\), which is a contradiction.
Step 3 From the above analysis, we know that \(\tau =0\). In other words, we have
Then, by (3.16), we have
Combining (3.10) and (3.20), one has
Since \(u_n\rightharpoonup u_a\) in H, we have \(u_n\rightarrow u_a\) in H. Moreover, by the fact that \(I_\mu (u_a)=\inf _{u \in \mathcal {P}(a)}I_\mu (u)\), we know that \(u_a\) is a ground state.
Finally, in view of Lemma 3.5, one has
The proof is completed. \(\square \)
4 \(L^2\)-Critical Perturbation Case
In this section, we shall address the \(L^2\)-critical perturbation case: \(q=\frac{10}{3}\) and provide the proof of Theorem 1.2. To begin with, we give some useful lemmas, and show that \(E(u,\iota )\) has the mountain pass geometry on \(S_r(a)\times {\mathbb {R}}\), where \(S_r(a)=H^1_{rad}({\mathbb {R}}^3)\cap S(a)\).
Lemma 4.1
Let \(q=\frac{10}{3}\), \(\mu ,\gamma >0\) and \(u\in S(a)\), then
-
(i)
\(\Vert \nabla (\iota \star u)\Vert _2\rightarrow 0^+\) and \(I_{\mu } ((\iota \star u))\rightarrow 0^+\) if \(\iota \rightarrow -\infty \);
-
(ii)
\(\Vert \nabla (\iota \star u)\Vert _2\rightarrow +\infty \) and \(I_{\mu } ((\iota \star u))\rightarrow -\infty \) if \(\iota \rightarrow +\infty \).
Proof
By (2.10), we have
Then, it is easy to obtain
and
From (2.11), we have
From the fact \(q\delta _q=2\), it follows that
and
The proof is completed. \(\square \)
Lemma 4.2
Let \(q=\frac{10}{3}\), \(\mu ,\gamma >0\), and assume that \(0<a<\min \{\alpha _3,\alpha _4\}\), where
and
There exist \(0<k_1<k_2<k\) such that
where
Proof
Take \(k>0\), which will be determined later. Assume that \(u,v\in S_r(a)\) such that \(\Vert \nabla u\Vert ^2_2\le k\) and \(\Vert \nabla v\Vert ^2_2=2k\). By (2.1), the GNS inequality (2.4) and \(q\delta _q=2\), we derive to
and
If \(a<\alpha _3\), we can deduce that
for \(k>0\) small enough. Next, if \(a<\alpha _4\), we have
If we take
then, for \(0<k_1<k_2<k\) small enough and \(0<a<\min \{\alpha _3,\alpha _4\}\), we infer to
The proof is completed. \(\square \)
In the following, we study the characteristics of the mountain pass levels for \(E(u,\iota )\) and \(I_{\mu }(u)\). Here, we define a closed set \(I_{\mu }^d:=\{u\in S_r(a): I_{\mu }(u) \le d \}\).
Proposition 4.3
Let \(q=\frac{10}{3}\), \(\mu ,\gamma >0\), and assume that \(0<a<\min \{\alpha _3,\alpha _4\}\). Take
where
and
where
Then we have
Proof
Since \(\Gamma _a\times \{0\}\subset \widetilde{\Gamma }_a,\) it is easy to know that \(\widetilde{\sigma }_{\mu }(a)\le {\sigma }_{\mu }(a)\). Then we only need to verify \(\widetilde{\sigma }_{\mu }(a)\ge {\sigma }_{\mu }(a)\). For \(\widetilde{\zeta }(t)=(\widetilde{\zeta }_1(t),\widetilde{\zeta }_2(t))\in \widetilde{\Gamma }_a\), one has,
So, set \(\zeta (t)= (\widetilde{\zeta }_2(t)\star \widetilde{\zeta }_1(t))\), we have \(\zeta (t)\in \Gamma _a\), and so,
which implies that \(\widetilde{\sigma }_{\mu }(a)\ge {\sigma }_{\mu }(a)\). The proof is completed. \(\square \)
Next, we will verify the existence of the \((PS)_{\widetilde{\sigma }_{\mu }(a)}\) sequence for \(E(u, \iota )\) on \(S_r(a)\times {\mathbb {R}}\), which is demonstrated by a standard argument by using Ekeland’s variational principle and constructing pseudo-gradient flow (Proposition 2.2 [28]).
Proposition 4.4
Let \(\{\xi _n\}\subset \widetilde{\Gamma }_a\) be such that
then there exists a sequence \(\{(u_n, \iota _n)\}\subset S_r(a)\times {\mathbb {R}}\) satisfying
-
(i)
\(E(u_n,\iota _n)\in [\widetilde{\sigma }_{\mu }(a)-\frac{1}{n},\widetilde{\sigma }_{\mu }(a)+\frac{1}{n}];\)
-
(ii)
\(\min _{t\in [0,1]}\Vert (u_n, \iota _n)-\xi _n(t)\Vert _{{\mathbb {H}}}\le \frac{1}{\sqrt{n}}\);
-
(iii)
\(\Vert E'|_{S_r(a)\times {\mathbb {R}}}(u_n,\iota _n)\Vert \le \frac{2}{\sqrt{n}},\) i.e.,
$$\begin{aligned} |\langle E'(u_n,\iota _n),z\rangle _{{\mathbb {H}}^{-1}\times {\mathbb {H}}}|\le \frac{2}{\sqrt{n}}\Vert z\Vert _{{\mathbb {H}}}, \end{aligned}$$for all
$$\begin{aligned} z\in \widetilde{T}_{(u_n,\iota _n)}:=\{(z_1, z_2)\in {\mathbb {H}}: \langle u_n, z_1\rangle _{L^2} = 0\}. \end{aligned}$$
With the help of Proposition 4.4, we can obtain a \((PS)_{\sigma _{\mu }(a)}\) sequence for \(I_{\mu }(u)\) on \(S_r(a)\) in the following.
Proposition 4.5
Let \(q=\frac{10}{3}\), \(\mu ,\gamma >0\), and assume that \(0<a<\min \{\alpha _3,\alpha _4\}\). There exists a sequence \(\{w_n\}\subset S_r(a)\) such that
-
(i)
\(I_{\mu }(w_n)\rightarrow \sigma _{\mu }(a)\) as \(n\rightarrow \infty ;\)
-
(ii)
\(P_{\mu }(w_n)\rightarrow 0\) as \(n\rightarrow \infty ;\)
-
(iii)
\(I'_{\mu }|_{S_r(a)}(w_n)\rightarrow 0\) as \(n\rightarrow \infty \), i.e.,
$$\begin{aligned}|\langle I_{\mu }'(w_n),z\rangle _{H^{-1}\times H}|\rightarrow 0, \end{aligned}$$uniformly for all \(h\in T_{w_n}\) and \(\Vert h\Vert \le 1\), where \(T_{w_n}:=\{h\in H:~\langle w_n,h\rangle _{L^2}=0\}\).
Proof
By Proposition 4.3, we have \(\widetilde{\sigma }_{\mu }(a)={\sigma }_{\mu }(a)\). Now, we take \(\{\xi _n = ((\xi _n)_1, 0)\}\in \widetilde{\Gamma }_a\) such that
From Proposition 4.4, we know that there exists a sequence \(\{(u_n,\iota _n)\}\subset S_r(a)\times {\mathbb {R}}\) such that as \(n\rightarrow \infty \), we have
Let \( w_n =\iota _n\star u_n\), then \(I_{\mu }(w_n) = E(u_n, \iota _n)\), so item (i) follows.
Next, we show item (ii). Since
it follows that item (ii) holds.
To prove item (iii), we set \(h_n\in T_{w_n}\), then
Let \(\widetilde{h}_n(x) = e^{-\frac{3}{2}\iota _n}h_n(e^{-\iota _n}x)\), we obtain
Moreover, we get
Thus, we obtain \((\widetilde{h}_n, 0)\in \widetilde{T}_{(u_n,\iota _n)}\). On the other hand,
where the last inequality can be established by (4.3). So the item (iii) is proved. \(\square \)
Now, we construct the relationship between \(\sigma _\mu (a)\) and \(m_{\mu ,r}(a)\), where
and
Lemma 4.6
Let \(q=\frac{10}{3}\), \(\mu ,\gamma >0\), and assume that \(0<a<\min \{\alpha _3,\alpha _4\}\). Then we have
where
Proof
In the following, we split the proof into four steps.
Step 1 We verify that for each \(u \in S_r(a)\), there exists a unique \(t_u \in {\mathbb {R}}\) such that \(t_u \star u \in \mathcal {P}_r(a)\), with \(t_u\) is the strict maximum point for the function \(\varphi _u(\iota )\) on \((0,+\infty )\) at positive level. Moreover, \(\mathcal {P}_r(a)=\mathcal {P}_r(a)^-\).
In fact, by Lemma 4.1, we have
Since \(\varphi _u(\iota )\) is a \(C^2\) function, we can deduce that \(\varphi _u(\iota )\) has at least one critical point \(t_u\), with \(t_u\) is a global maximum point at positive level. In view of Lemma 2.6, we have \(t_u \star u \in \mathcal {P}_r(a)\). Next, we prove that \(\varphi _u(\iota )\) has no other critical points. Indeed, recall \((\varphi _u)'(\iota )\) and \((\varphi _u)''(\iota )\) as follow:
and
Assume by contradiction, there exists other critical point \(e_u \in {\mathbb {R}}\) with \(t_u<e_u\) and \(e_u\) is also a global maximum point of \(\varphi _u(\iota )\). Then, we see that there exists a critical point \(f_u\), such that \(t_u<f_u<e_u\) and \(f_u\) is a minimum point of \(\varphi _u(\iota )\). Consequently, we have
and
which is a contradiction.
Step 2 We show that \(I_\mu (u)\le 0\) implies \(P_\mu (u)<0\). In fact, since \(\varphi _u(0)=I_\mu (0\star u)=I_\mu (u)\le 0\), by the properties of the function \(\varphi _u(\iota )\) presented in Step 1 and by (4.5), we infer that \(t_u<0\). Besides, since
we obtain that \(P_\mu (u)<0\).
Step 3 We claim that \(m_{\mu ,r}(a)=\sigma _\mu (a)\). Indeed, let \(u \in S_r(a)\), we take \(\iota ^-\ll 0\) and \(\iota ^+\gg 0\) such that \(\iota ^-\star u\in A_{k_1}\) and \(I_\mu (\iota ^+\star u)<0\), respectively. Then we can define a path
Hence, we get
and so, we have \(m_{\mu ,r}(a)\ge \sigma _\mu (a)\). Moreover, for any \(\widetilde{\zeta }(t)=(\widetilde{\zeta }_1(t),\widetilde{\zeta }_2(t))\in \widetilde{\Gamma }_a\), one has,
Now, we define the function
Since \((\widetilde{\zeta }_2(0)\star \widetilde{\zeta }_1(0))=\widetilde{\zeta }_1(0) \in A_{k_1}\) and \((\widetilde{\zeta }_2(1)\star \widetilde{\zeta }_1(1))=\widetilde{\zeta }_1(1) \in I_{\mu }^0\), in view of Lemma 4.2 and Step 2, we have
and
Since \(\widetilde{P}_\mu (t)\) is continuous and by Remark 2.8, we infer that there exists \(t^*\in (0,1)\) so as to \(\widetilde{P}_\mu (t^*)=0\), which implies that \((\widetilde{\zeta }_2(t^*)\star \widetilde{\zeta }_1(t^*))\in \mathcal {P}_r(a)\). Consequently, one has
Hence, we have \(\sigma _\mu (a)\ge m_{\mu ,r}(a)\). In conclusion, one has \(\sigma _\mu (a)= m_{\mu ,r}(a)\).
Step 4 We claim that \(m_{\mu ,r}(a)>0\). Let \(u\in \mathcal {P}_r(a)\), we have \(P_\mu (u)=0\). By the GNS inequality (2.4) and (2.1), we infer to
And by \(q\delta _q=2\), one has
By \(a<\alpha _3\), there exists \(\rho >0\) such that
So, for any \(u\in \mathcal {P}_r(a)\), it follows that
Consequently, we obtain \(\sigma _\mu (a)>0\), which completes the proof. \(\square \)
Next, we give an upper bounded estimate for the mountain pass level \(\sigma _\mu (a)\) in the following Lemma, which plays an important role in the proof of Theorem 1.2.
Lemma 4.7
Let \(q=\frac{10}{3}\), \(\mu >0\), and assume that \(0<a<\min \{\alpha _3,\alpha _4\}\). Then there exists \(\widetilde{\gamma _1}>0\), such that \( \sigma _{\mu }(a)<\frac{1}{3}S^{\frac{3}{2}}\) for \(\gamma \in (0,\widetilde{\gamma _1})\) small enough.
Proof
Recall (2.1) and (2.2), we have the best constant S is attained by
for any \(\varepsilon >0\) and \(C^*\) being normalized constant such that
We take
where \(\varphi (x)\in C_0^{\infty }(B_2(0))\) is a radial cutoff function such that \(0\le \varphi (x)\le 1\) and \(\varphi (x)\equiv 1\) on \(B_1(0).\) Let
As showed in [15], we have
and
From Lemma 7.1 [30], we have the following estimations:
and when \(q=2\), one has
Recall the function
Similar to the first step in proving Lemma 4.6, we can conclude that \(\varphi _{v_{\varepsilon }}\) can obtain its global positive maximum at some \(\iota _{\varepsilon }\). And so, by (2.14), we have
By (4.12) and \(q\delta _q=2\), we deduce
In view of (4.7) and (4.8), we have that there exist positive constants \(C_1, C_2\) and \(C_3\) depending on s and q, such that
and
Hence, by (4.13)–(4.16) and \(q\delta _q=2\), we obtain
In the following, we shall make an upper estimation of \(\max _{\iota \in {\mathbb {R}}}\varphi _{v_\varepsilon }(\iota )\). Firstly, we define the function \(\varphi ^0_{v_{\varepsilon }}(\iota )\) as follow:
By simple calculation, we derive that the function \(\varphi ^0_{v_{\varepsilon }}(\iota )\) has a unique critical point \(\iota _\varepsilon ^0\), which is a strict maximum point given by
Applying the fact that
for any fixed \(a,b>0\). In view of (4.7) and (4.8), we infer that
Secondly, we make an estimation for \(\varphi _{v_\varepsilon }(\iota )\). By (2.3), (4.12) and Hölder inequality, we derive to
By virtue of (4.7)–(4.8), (4.10) and (4.21), we can see that \(\iota _{\varepsilon }\) can not go to \(+\infty \), namely, there exists some \(\iota ^*\in {\mathbb {R}}\) such that
Based on above analysis, by (2.3)–(2.4), (4.7)–(4.8), (4.17), (4.20), (4.22) and the fact \(q\delta _q=2\), we derive to
if we choose \(\gamma =\varepsilon ^\alpha \) for some constant \(\alpha \ge 1\), and use the fact \(0<\frac{6-q}{4}<1\).
Finally, by Lemma 4.1, we take \(\iota _1<0\) and \(\iota _2> 0\) such that \(\iota _1\star {v}_{\varepsilon } \in A_k\) and \(I_{\mu }(\iota _2\star {v}_{\varepsilon }) < 0\), respectively. Define a path
Consequently, by (4.23), we have that there exists some \(\widetilde{\gamma _1}>0\), such that
for \( \gamma \in (0,\widetilde{\gamma _1})\) small enough. \(\square \)
Now, based on the above preparation, we are ready to accomplish the proof of Theorem 1.2.
Proof of Theorem 1.2
Take a PS sequence \(\{u_n\}\) as in Proposition 4.5, we have
Using the Lagrange multipliers rule, we have that there exists a sequence \(\{\lambda _n\}\in {\mathbb {R}}\) such that
Again by Proposition 4.5, we get
which means \(I_{\mu }(u_n)\) is bounded. So we deduce to
From \(P_\mu (u_n)\rightarrow 0\) and \(q\delta _q=2\), we infer that
that is,
Combining (4.25)–(4.26), one has
Thus, we know that \(\int _{{\mathbb {R}}^3}\phi _{u_n}u^2_n dx\) and \(\int _{{\mathbb {R}}^3}|u_n|^6dx\) are bounded. Then, by (4.25) and the GNS inequality (2.4), we infer to
Since \(a<\alpha _3\), it is easy to get \(\Vert \nabla u_n\Vert _2\le R^*\) for some \(R^*>0\) independently on \(n \in {\mathbb {N}}\). Consequently, we obtain that \(\{u_n\}\) is bounded in \(H^1({\mathbb {R}}^3)\), and so, up to subsequence, there exists \(u_a\) such that
From (4.24), we have
that is
From the boundedness of \(\{u_n\}\) in \(H^1({\mathbb {R}}^3)\), we have that \(\{\lambda _n\}\) is bounded. Now, we verify
Assume by contradiction that, \(\int _{{\mathbb {R}}^3}|u_n|^qdx\rightarrow 0\). By (2.3) and the interpolation inequality, we obtain \(\int _{{\mathbb {R}}^3}\phi _{u_n}u_n^2dx\rightarrow 0\). Combining with
we get
So, \(\lambda _n\rightarrow 0\) as \(n\rightarrow \infty \). Then (4.28) becomes
Set
then,
Since \(I_{\mu }(u_n)\rightarrow \sigma _\mu (a)~~\text{ as }~~n\rightarrow \infty \), and \(\sigma _\mu (a)<\frac{1}{3}S^{\frac{3}{2}}\) from Lemma 4.7 and (4.29), we obtain
On the other hand, by virtue of the Sobolev inequality (2.1), we have
which leads to a contradiction. Hence, \(\int _{{\mathbb {R}}^3}|u_n|^qdx\rightarrow \int _{{\mathbb {R}}^3}|u_a|^qdx\ne 0\). Then, by the boundedness of \(\{\lambda _n\}\), up to subsequence, there exists \(\lambda _a\) such that \(\lambda _n\rightarrow \lambda _a\). Consequently, by \(\Vert \nabla u_n\Vert _2\le R^*\), (2.3), the GNS inequality (2.4) and \(q\delta _q=2\), we have
and
where \( T_3>0\) and \(T_4= T_4(R^*,a)\). We define the positive constant
So, we get
In view of (4.24) and \(\{u_n\}\) is bounded in \(H({\mathbb {R}}^3)\), we have
Combining with \(P_\mu (u_n)\rightarrow 0\) and Lemma 2.7, if \(\gamma \in (0,\widetilde{\gamma _2})\), one has
which proves that \(\lim _{n\rightarrow \infty }\lambda _n=\lambda _a<0\). Let \(v_n=u_n-u_a\rightharpoonup 0\), by (3.17), (2.5), Lemma 2.7 and \(P_\mu (v_n)=P_\mu (u_n)-P_\mu (u_a)\rightarrow 0\), we infer to
Up to subsequence, we assume that
So, by (2.1), we have
that is,
If \(\tau \ge S^{\frac{3}{2}}\), in view of (3.17), we derive as
Besides,
which is contradicted to Lemma 4.7. Thus, we have \(\tau =0\). By a similar argument as in the end of the proof of Theorem 1.1, we infer to
Next, we claim that \(m_\mu (a)=m_{\mu ,r}(a)\). Since \(\mathcal {P}_r(a)\subset \mathcal {P}(a)\), it is easy to see that \(m_\mu (a)\le m_{\mu ,r}(a)\). Then, we only need to verify \(m_\mu (a)\ge m_{\mu ,r}(a)\). Suppose by contradiction, there exists \(w \in \mathcal {P}(a)\backslash S_r(a)\) such that
Then, let \(v:=|w|^*\), by virtue of the schwarz rearrangement, it follows that
If \(P_\mu (v)=0\), we know \( v \in \mathcal {P}(a)\), \(v:=|w|^*\in \mathcal {P}_r(a)\) and
which is a contradiction. If \(P_\mu (v)<0\), we see that \((\varphi _v)'(0)=P_\mu (v)<0\), by the claim of Step 1 of the proof of Lemma 4.6, we have that \(t_v<0\). Since \(t_v\star v \in \mathcal {P}_r(a)\), by (4.29), we deduce to
which leads to a contradiction. Again by the the claim of step 1 of the proof of Lemma 4.6, we have \(\mathcal {P}(a)=\mathcal {P}(a)^-\). Consequently, we get
and \(u_a\) is a ground state. \(\square \)
5 \(L^2\)-Supcritical Perturbation Case
In this section, we consider the \(L^2\)-supercritical case: \(\frac{10}{3}<q<6\) and prove Theorem 1.3. For the sake of convenience, we still utilize the notations and definitions in Section 4.
In Lemma 4.1, the conclusion remains valid when \(\frac{10}{3}<q<6\). In the following, we show that \(E(u,\iota )\) has the mountain pass geometry on \(S_r(a)\times {\mathbb {R}}\).
Lemma 5.1
Let \(\frac{10}{3}<q<6\), \(\mu ,\gamma >0\), and assume that \(0<a<\alpha _5\), where
There exist \(0<k^*_1<k^*_2<k^*\) such that
where
Proof
Take \(k^*>0\), which will be determined later. Suppose that \(u,v\in S_r(a)\) such that \(\Vert \nabla u\Vert ^2_2\le k^*\) and \(\Vert \nabla v\Vert ^2_2=2k^*\). The proof here is similar to Lemma 4.2, which we briefly outline.
and
Moreover,
for \(a<\alpha _5\), and we take
then, for \(0<k^*_1<k^*_2<k^*\) small enough and \(0<a<\alpha _5\), we have
The proof is completed. \(\square \)
Lemma 5.2
Let \(\frac{10}{3}<q<6\), \(\mu ,\gamma >0\), and assume that \(0<a<\alpha _5\). Then we have
-
(i)
There exists a sequence \(\{w_n\}\in S_r(a)\) such that
$$\begin{aligned} I_{\mu }(w_n)\rightarrow \sigma _{\mu }(a)~~~ \text{ as }~~~n\rightarrow \infty , \end{aligned}$$(5.2)$$\begin{aligned} P_{\mu }(w_n)\rightarrow 0~~~ \text{ as }~~~n\rightarrow \infty , \end{aligned}$$(5.3)$$\begin{aligned} I'_{\mu }|_{S_r(a)}(w_n)\rightarrow 0~~~ \text{ as }~~~n\rightarrow \infty . \end{aligned}$$(5.4) -
(ii)
\(\sigma _\mu (a)=m_{\mu ,r}(a)>0\), where \(\sigma _\mu (a)\) and \(m_{\mu ,r}(a)\) is defined in Section 4.
The proof of this lemma is similar to that of Propositions 4.3–4.5 and Lemmas 4.2–4.6 utilizing \(q\delta _q>2\), and thus it is omitted here.
Now, we make an upper bounded estimation for the mountain pass level \(\sigma _\mu (a)\) in the following.
Lemma 5.3
Let \(\frac{10}{3}<q<6\), \(\mu ,\gamma >0\), and assume that \(0<a<\alpha _5\). Then we have \( \sigma _{\mu }(a)<\frac{1}{3}S^{\frac{3}{2}}\) for \(\gamma \in (0,\widetilde{\gamma _1})\) small enough, where \(\widetilde{\gamma _1}\) is defined in Lemma 4.7.
Proof
As in the proof of Lemma 4.7, we conclude that \(\varphi _{v_\varepsilon }(\iota )\) achieves its global positive maximum at some \(\iota _\varepsilon \), and the critical point \(\iota _\varepsilon \) is unique. In view of \((\varphi _{v_{\varepsilon }})'(\iota _{\varepsilon })=P_{\mu }(\iota _{\varepsilon }\star v_{\varepsilon })=0\), one has
Then, we consider the following possible cases.
Case 1 If \(\Vert \nabla v_{\varepsilon }\Vert ^2_2> \frac{\gamma }{4}e^{-\iota _{\varepsilon }}\int _{{\mathbb {R}}^3}\phi _{v_{\varepsilon }}v_{\varepsilon }^2dx\), we have
that is,
By \((\varphi _{v_{\varepsilon }})'(\iota _{\varepsilon })=0\), we infer that
By virtue of (4.14)–(4.16) and (5.7), we get
Case 2 If \(\Vert \nabla v_{\varepsilon }\Vert ^2_2\le \frac{\gamma }{4}e^{-\iota _{\varepsilon }}\int _{{\mathbb {R}}^3}\phi _{v_{\varepsilon }}v_{\varepsilon }^2dx\), we have
that is,
Again by \((\varphi _{v_{\varepsilon }})'(\iota _{\varepsilon })=0\), (2.3) and Hölder inequality, we infer to
By (4.7)–(4.8), we deduce that there exists constant \(C_4>0\) such that
So, in view of (4.14)–(4.16) and (5.11), we get
Based on above analysis, we will make an upper estimation for \(\varphi _{v_\varepsilon }(\iota )\). Firstly, as in Lemma 4.7, we can define the function \(\varphi _{v_\varepsilon }^0(\iota )\) and make an estimation for \(\varphi _{v_\varepsilon }^0(\iota )\), that is
where \(\iota ^0_{\varepsilon }\) is a unique strict maximum point of \(\varphi _{v_\varepsilon }^0(\iota )\). Secondly, we make an estimation for \(\varphi _{v_\varepsilon }(\iota )\). By virtue of (4.22), we know that there exists some \(\iota ^*\in {\mathbb {R}}\) such that
So, by (2.3)–(2.4), (5.11)–(5.13), (4.9)–(4.10), (4.16) and above inequality, we obtain
if we choose \(\gamma =\varepsilon ^\alpha \) for some constant \(\alpha \ge 1\), and using the fact \(0<\frac{6-q}{4}<1\).
Since \( {v}_{\varepsilon } \in S_r(a) \), from Lemma 5.2 we take \(\iota _3<0\) and \(\iota _4> 0\) such that \(\iota _3\star {v}_{\varepsilon } \in A_k\) and \(I_{\mu }(\iota _4\star {v}_{\varepsilon }) < 0\), respectively. We define a path
Consequently, by (5.14), we obtain that there exists some \(\widetilde{\gamma }_1>0\), such that
for \( \gamma \in (0,\widetilde{\gamma }_1)\) small enough, which completes the proof. \(\square \)
Now, we are ready to prove Theorem 1.3.
Proof of Theorem 1.3
By virtue of (5.4), we have that there exists a sequence \(\{\lambda _n\}\in {\mathbb {R}}\) such that
Then we claim that \(\{u_n\}\) is bounded in H. Indeed, by (5.2)and (5.3), we have
that is,
By (5.17) and the bounded of \(I_{\mu }(u_n)\), we infer to
it follows that
which implies that
are all bounded. Thus, we deduce that \(\int _{{\mathbb {R}}^3}|\nabla u_n|^2dx\) is also bounded. For convenience, we still take \(\Vert \nabla u_n\Vert _2\le R^*\). We can proceed exactly as in the proof of Theorem 1.2 utilizing Lemmas 5.2–5.3, so complete the proof of Theorem 1.3. \(\square \)
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References
Alves, C.O., Böer, E., Miyagaki, O.H.: Existence of normalized solutions for the planar Schrödinger-Poisson system with exponential critical nonlinearity. Differ. Integr. Equ. 36, 947–970 (2021)
Ambrosetti, A.: On Schrödinger-Poisson systems. Milan J. Math. 76, 257–274 (2008)
Ambrosetti, A., Ruiz, D.: Multiple bound states for the Schrödinger-Poisson equation. Commun. Contemp. Math. 10, 1–14 (2008)
Azzollini, A., Pomponio, A.: Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J. Math. Anal. Appl. 345, 90–108 (2008)
Azzollini, A., d’Avenia, P., Pomponio, A.: On the Schrödinger-Maxwell equations under the effect of a general nonlinear term. Ann. Inst. H. Poincaréé Anal. Non Linéaire 27, 779–791 (2010)
Bartsch, T., de Valeriola, S.: Normalized solutions of nonlinear Schrödinger equations. Arch. Math. (Basel) 100, 75–83 (2013)
Bartsch, T., Soave, N.: A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems. J. Funct. Anal. 272, 4998–5037 (2017)
Bartsch, T., Soave, N.: Multiple normalized solutions for a competing system of Schröddinger equations. Calc. Var. Partial Differ. Equ. 58, 22 (2019)
Bartsch, T., Jeanjean, L., Soave, N.: Normalized solutions for a system of coupled cubic Schrödinger equations on \({\mathbb{R} }^3\). J. Math. Pure Appl. 106, 583–614 (2016)
Bellazzini, J., Siciliano, G.: Scaling properties of functionals and existence of constrained minimizers. J. Funct. Anal. 261, 2486–2507 (2011)
Bellazzini, J., Siciliano, G.: Stable standing waves for a class of nonlinear Schrödinger-Poisson equations. Z. Angew. Math. Phys. 62, 267–280 (2011)
Bellazzini, J.J., Jeanjean, L., Luo, T.: Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations. Proc. Lond. Math. Soc. 107, 303–339 (2013)
Benci, V., Fortunato, D.: An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonl. Anal. 11, 283–293 (1998)
Benci, V., Fortunato, D.: Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations. Rev. Math. Phys. 14, 409–420 (2002)
Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)
Cerami, G., Vaira, G.: Positive solutions for some non-autonomous Schrödinger-Poisson systems. J. Differ. Equ. 248, 521–543 (2010)
Chen, S., Tang, X., Yuan, S.: Normalized solutions for Schrödinger-Poisson equations with general nonlinearities. J. Math. Anal. Appl. 481, 123447 (2020)
Cho, Y., Hwang, G., Kwon, S., Lee, S.: On finite time blow-up for the mass-critical Hartree equations. Proc. R. Soc. Edinb. Sect. A 145, 467–479 (2015)
Cingolani, S., Jeanjean, L.: Stationary waves with prescribed \(L^2\)-norm for the planar Schrödinger-Poisson system. SIAM J. Math. Anal. 51, 3533–3568 (2019)
Coclite, G.M.: A multiplicity result for the nonlinear Schrödinger-Maxwell equations. Commun. Appl. Anal. 7, 417–423 (2003)
D’Aprile, T., Mugnai, D.: Solitary waves for nonlinear Klei-Gordon-Maxwell and Schrödinger-Poisson equations. Proc. R. Soc. Edinb. Sect. A 134, 1–14 (2004)
D’Aprile, T., Mugnai, D.: Non-existence results for the coupled Klein-Gordon-Maxwell equations. Adv. Nonlinear Stud. 4, 307–322 (2004)
d’Avenia, P., Siciliano, G.: Nonlinear Schrödinger equation in the Bopp-Podolsky electrodynamics: solutions in the electrostatic case. J. Differ. Equ. 267, 1025–1065 (2019)
He, X.: Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations. Z. Angew. Math. Phys. 62, 869–889 (2011)
He, X., Zou, W.: Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth. J. Math. Phys. 53, 023702 (2012)
He, X., Zou, W.: Normalized solutions for Schrödinger-Poisson systems with Sobolev critical nonlinearities. arXiv:2103.01437
Huang, Y., Liu, Z., Wu, Y.: Existence of prescribed \(L^2\)-norm solutions for a class of Schrödinger-Poisson equation. Abstr. Appl. Anal. 11, 398164 (2013)
Jeanjean, L.: Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. TMA 28, 1633–1659 (1997)
Jeanjean, L., Le, T.T.: Multiple normalized solutions for a Sobolev critical Schrödinger-Poisson-Slater equation. J. Differ. Equ. 303, 277–325 (2021)
Jeanjean, L., Le, T.T.: Multiple normalized solutions for a Sobolev critical Schrödinger equation. Math. Ann. 384, 101–134 (2022)
Jeanjean, L., Luo, T.: Sharp nonexistence results of prescribed \(L^2\)-norm solutions for some class of Schrödinger-Poisson and quasi-linear equations. Z. Angew. Math. Phys. 64, 937–954 (2013)
Kikuchi, H.: Existence and stability of standing waves for Schrödinger-Poisson-Slater equation. Adv. Nonlinear Stud. 7, 403–437 (2007)
Li, Y., Zhang, B.: Critical Schrödinger-Popp-Podolsky system with prescribed mass. J. Geom. Anal. 33, 220 (2023)
Lieb, E., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence (2001)
Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case, I, II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 109–145; 223–283 (1984)
Lions, P.-L.: Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys. 109, 33–97 (1984)
Liu, Z., Zhang, Z., Huang, S.: Existence and nonexistence of positive solutions for a static Schrödinger Poisson Slater equation. J. Differ. Equ. 266, 5912–5941 (2019)
Luo, T.: Multiplicity of normalized solutions for a class of nonlinear Schrödinger-Poisson-Slater equations. J. Math. Anal. Appl. 416, 195–204 (2014)
Markowich, P., Ringhofer, C., Schmeiser, C.: Semiconductor Equations. Springer, Vienna (1990)
Ruiz, D.: The Schrödinger-Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237, 655–674 (2006)
Sachez, O., Soler, J.: Long-time dynamics of the Schrödinger-Poisson-Slater system. J. Stat. Phys. 114, 179–204 (2004)
Soave, N.: Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case. J. Funct. Anal. 279, 108610 (2020)
Soave, N.: Normalized ground states for the NLS equation with combined nonlinearities. J. Differ. Equ. 269, 6941–6987 (2020)
Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149–162 (1977)
Wang, Q., Qian, A.: Normalized solutions to the Schrödinger-Poisson-Slater equation with general nonlinearity: mass supercritical case. Anal. Math. Phys. 13, 35 (2023). https://doi.org/10.1007/s13324-023-00788-9
Wang, J., Tian, L., Xu, J., Zhang, F.: Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in \({\mathbb{R} }^3\). Calc. Var. Partial Differ. Equ. 48, 275–276 (2013)
Wei, J., Wu, Y.: Normalized solutions for Schrödinger equations for critical Sobolev exponent and mixed nonlinearities. J. Funct. Anal. 283, 109574 (2022)
Willem, M.: Minimax Theorems. Birkhauser, Boston (1996)
Xie, W., Chen, H., Shi, H.: Existence and multiplicity of normalized solutions for a class of Schrödinger-Poisson equations with general nonlinearities. Math. Methods Appl. Sci. 43, 3602–3616 (2020)
Ye, H.: The existence and the concentration behavior of normalized solutions for the \(L^2\)-critical Schrödinger-Poisson system. Comput. Math. Appl. 74, 266–280 (2017)
Ye, H., Luo, T.: On the mass concentration of \(L^2\)-constrained minimizers for a class of Schrödinger-Poisson equations. Z Angew. Math. Phys. 69, 66 (2018)
Ye, H., Luo, T.: On the mass concentration of \(L^2\)-constrained minimizers for a class of Schrödinger-Poisson equations. Z. Angew. Math. Phys. 66(66), 13 (2018)
Ye, H., Zhang, L.: Normalized solutions for Schrödinger-Poisson-Slater equations with unbounded potentials. J. Math. Anal. Appl. 452, 47–61 (2017)
Zhang, P., Han, Z.: Normalized solutions to a kind of fractional Schrödinger equation with a critical nonlinearity. Z. Angew. Math. Phys. 73, 149 (2022)
Zhao, L., Zhao, F.: Positive solutions for Schrödinger-Poisson equations with a critical exponent. Nonlinear Anal. TMA 70, 2150–2164 (2009)
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We would like to thank the anonymous reviewers for careful reading the manuscript and their valuable comments. This work is supported by NSFC (12171497, 11771468, 11971027).
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Gao, Q., He, X. Normalized Solutions for Schrödinger–Poisson Systems Involving Critical Sobolev Exponents. J Geom Anal 34, 296 (2024). https://doi.org/10.1007/s12220-024-01744-0
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DOI: https://doi.org/10.1007/s12220-024-01744-0