Multiple normalized solutions for the planar Schrödinger–Poisson system with critical exponential growth

Mathematische Zeitschrift (2024) 306:50 https://doi.org/10.1007/s00209-024-03432-9 Mathematische Zeitschrift Multiple normalized solutions for the planar Schrödinger–Poisson system with critical exponential growth Sitong Chen1·Vicen¸tiu D. R˘adulescu2,3,4,5,6 ·Xianhua Tang1 Received: 14 March 2023 / Accepted: 16 December 2023 / Published online: 16 February 2024 © The Author(s) 2024 Abstract The paper deals with the existence of normalized solutions for the following Schrödinger– Poisson system with L2-constraint: −u+λu+μlog |·|∗u2u=eu2−1−u2u,x∈R2, R2u2dx=c, where μ>0, λ∈Rwill arise as a Lagrange multiplier and the nonlinearity enjoys critical exponential growth of Trudinger-Moser type. By specifying explicit conditions on the energy level c, we detect a geometry of local minimum and a minimax structure for the corresponding energy functional, and prove the existence of two solutions, one being a local minimizer and one of mountain-pass type. In particular, to catch a second solution of mountain-pass type, some sharp estimates of energy levels are proposed, suggesting a new threshold of compactness in the L2-constraint. Our study extends and complements the results of Cingolani–Jeanjean (SIAM J Math Anal 51(4): 3533-3568, 2019) dealing with the power nonlinearity a|u|p−2uin the case of a>0andp>4, which seems to be the first contribution in the context of normalized solutions. Our model presents some new difficulties due to the intricate interplay between a logarithmic convolution potential and a nonlinear BVicen¸tiu D. R˘adulescu [email protected].ro Sitong Chen [email protected] Xianhua Tang [email protected] 1School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha 410083, Hunan, People’s Republic of China 2Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland 3Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 3058/10, Brno 61600, Czech Republic 4Department of Mathematics, University of Craiova, 200585 Craiova, Romania 5Simion Stoilow Institute of Mathematics of the Romanian Academy, 010702 Bucharest, Romania 6School of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China 123 50 Page 2 of 32 S. Chen et al. term of critical exponential type and requires a novel analysis and the implementation of new ideas, especially in the compactness argument. We believe that our approach will open the door to the study of other L2-constrained problems with critical exponential growth, and the new underlying ideas are of future development and applicability. Keywords Planar Schrödinger–Poisson system ·Logarithmic convolution potential · Normalized solution ·Critical exponential growth ·Trudinger–Moser inequality Mathematics Subject Classification 35J20 ·35J62 ·35Q55 1 Introduction In this paper, we study the following planar Schrödinger–Poisson equation with L2-constraint −u+λu+μlog |·|∗u2u=eu2−1−u2u,x∈R2, R2u2dx=c, (1.1) where μ>0, c>0 is a given constant, λ∈Rappears as a Lagrange parameter and is part of the unknowns. Particularly, the nonlinearity has critical exponential growth in the sense of Trudinger–Moser, which is a novelty for L2-constrained problems. Here, we recall that the nonlinear term fis said to have critical exponential growth if fsatisfies (f1) f∈C(R,R)and there exists α0>0 such that lim |t|→∞ |f(t)| eαt2=0,for all α>α 0, +∞,for all α<α 0, which is the maximal growth allowing to treat the problem variationally in H1(R2),see Adimurthi and Yadava [2] and also de Figueiredo, Miyagaki and Ruf [22]. Solutions having aprioriprescribed L2-norm are referred to as normalized solutions in the literatures. Physicists are often interested in normalized solutions because the L2-norm of such solutions is a preserved quantity of the evolution and their variational characterization can help to analyze the orbital stability or instability, see, for example, [5,39,40]. Besides that its solutions have aprioriprescribed mass, another interesting feature of (1.1)isthat a logarithmic convolution potential appears, which is unbounded and changes sign. As one will see, for prescribed c>0, a solution of problem (1.1) can be obtained as a critical point of the functional :X→Rdefined by (u)=1 2R2|∇u|2dx+μ 4R2R2 log |x−y|u2(x)u2(y)dxdy −1 2R2eu2−1−u2−u4 2dx(1.2) on the constraint Sc=u∈X:u2 2=c,(1.3) where X:= u∈H1(R2):R2 log(1+|x|)u2dx<∞.(1.4) 123 Multiple normalized solutions... Page 3 of 32 50 This problem arises when one looks for solutions of the Schrödinger-Poisson system of the type −u+λu+μφu=f(u), x∈RN, φ =u2,x∈RN(1.5) with N≥2, λ∈R\{0}and f∈C(R,R), which has a strong physical meaning because it originates in quantum mechanics models (see e.g. [8,13,33]) and in semiconductor theory [7,34,35]. The second equation (1.5) determines only up to harmonic functions, and it is natural to choose φas the negative Newton potential of u2, i.e., the convolution of u2with the fundamental solution Nof the Laplacian, which is given by N(x)=1 2πlog |x|,N=2, 1 N(2−N)ωN|x|2−N,N≥3, and ωNis the volume of the unit N-ball. With this formal inversion, system (1.5)isconverted into an equivalent nonlocal equation −u+λu+μ(N∗u2)u=f(u), x∈RN.(1.6) In the last decades, this equation has been extensively investigated by using variational methods. The majority of the literature focuses on the study of (1.6) with N=3, it seems that it is impossible to summarize it for the case that λ>0 is a fixed and assigned a parameter since the related literature is too large, we just refer to [5,6,28,38] for the case that λappears as a Lagrange parameter. In contrast with the higher-dimensional case N=3, much less is known for (1.6) with N=2. In this case, the applicability of variational methods is not straightforward because the corresponding energy functional is not well-defined on H1(R2)under the effect of the logarithmic convolution potential. This direction of research was likely brought to the attention of the community of nonlinear PDEs by the paper [21] published in 2016. In that paper, Cingolani and Weth, inspired by Stubbe [41], developed a variational framework to deal with (1.6) with N=2, within the smaller Hilbert space Xdefined by (1.4),andprovedtheexistence of ground state solutions when f(u)=|u|q−2ufor q>4. The key tool to prove the compactness is a new smart strong compactness condition (modulo translation) for Cerami sequences in the periodic setting. This tool was subsequently used by Du-Weth [23]forthe case that f(u)=|u|q−2ufor 2 <q≤4, and by Chen-Shi-Tang [14] for the more general case that f(u)∼|u|q−2ufor q>2. In recent papers [17]and[18], we introduced another axially symmetric variational framework within a natural constraint Eas := X∩u∈H1(R2):u(x):= u(x1,x2)=u(|x1|,|x2|), ∀x∈R2,(1.7) and proved respectively the existence of axially symmetric solutions for (1.6) with N=2 when f(u)∼|u|q−2ufor q>2andwhen fhas critical exponential growth satisfying (f1). Compared with the above case where λ>0 is fixed, the search of normalized solutions for (1.6) with N=2 is more challenging due to the extra need to respect the L2-constraint, which is our focus of the present paper. It seems that the first contribution to this topic was made recently by Cingolani-Jeanjean [20], in which the existence of normalized solutions for the following equation with the power nonlinearity −u+λu+μlog |·|∗u2u=a|u|p−2u,x∈R2, R2u2dx=c,(1.8) 123 50 Page 4 of 32 S. Chen et al. was established, and a complete analysis of the various cases on parameters μ, a∈Rand p>2 that may happen for (1.8) was provided. In the study of (1.8), an important role is played by the so-called L2-critical exponent 4. If p>4or2<p<4, one speaks of an L2supercritical case or an L2-subcritical case. Precisely, it was proved that (1.8) has a ground state provided that μ>0 and one of the following three conditions: i) a≤0andp>2; ii) a>0andp<4; iii) a>0, p=4andc<2/(aC4), under which the associated energy functional is bounded from below on the constraint Scfor any c>0 and a global minimum on Sccan be achieved, where the constant C4>0 comes from the Gagliardo-Nirenberg inequality (see (2.9) later). In all the other cases, although it is not possible to find a global minimizer, the interplay between a logarithmic convolution potential and a power function adds some richness to the geometric picture of the associated energy functional. In particular, when μ, a>0andp>4, it was proved that there exists an explicit value c0=c0(μ, a,p) such that for c∈(0,c0),(1.8) has two normalized solutions, one being a local minimizer and one of mountain-pass type. This is reminiscent of the recent work by Soave [39], where a similar structure has been observed for the following Schrödinger equation with combined nonlinearities of power type: −u+λu=γ|u|q−2u+|u|p−2u,x∈RN, RNu2dx=c,(1.9) with N≥1, γ>0and2<q<2+4/N<p<2∗:= 2N/(N−2), N≥3, +∞,N=1,2,see also subsequent papers [26,27,30,42] for extensions from p<2∗to Sobolev critical exponent p=2∗. However, the appearance of the nonlocal convolution term log |·|∗u2uin (1.8) exhibits some serious mathematical differences to a local nonlinear term of the form |u|q−2u. To address this trouble, Cingolani and Jeanjean used the combination of the fibration method of Pohozaev (relying on the decomposition of L2-Pohozaev manifold used in [39]) and the strong compactness condition developed by Cingolani–Weth [21], where some new estimates of energy on the dilated function su(s·)for u∈L2and s>0 belonging to Scwere given. It is worth mentioning that the argument strongly depends on the order pof power function and is not adequate for the following problem −u+λu+μlog |·|∗u2u=f(u), x∈R2, R2u2dx=c,(1.10) with the more general nonlinear term f, even the sum of power functions with super-cubic growth. In the recent preprint paper, Alves–Böer–Miyagaki [3] considered (1.10) with critical exponential growth satisfying (f1) with α0=4π. In particular, if falso satisfies (f2) f(0)=0 and there exists τ>3suchthatlim t→0|f(t)| |t|τ=0; (f3) there exists θ>6 such that f(t)t≥θF(t)>0,∀t= 0, where F(t):= t 0f(s)ds; (f4) there exist q>4andν>ν 0such that F(t)≥ν|t|q,∀t∈R, it was proved that for any c∈(0,1), there are implicit parameters μ0,ν 0>0 such that problem (1.10) has a solution for μ∈(0,μ 0)and ν>ν 0. Note that this statement is of perturbative nature in two respects: i) μ0is sufficiently small such that (1.10) can be viewed as a perturbative form of the planar Schrödinger equation; ii) ν0is sufficiently large such that the obtained mountain-pass level is small enough from which the compactness can be obtained in the same way as that of (1.8). Clearly, the perturbative argument excludes many concrete models, and circumvents the added difficulties arising from the logarithmic nature of convolution kernel and the critical 123 Multiple normalized solutions... Page 5 of 32 50 exponential growth of nonlinearity compared to (1.8)and(1.10) with μ=0. To our knowledge, it still remains open exactly how the interplay between the logarithmic convolution term log |·|∗u2uand the nonlinear term fsatisfying (f1) effects the geometry structure of the corresponding functional, which is unbounded from below on Scfor all c>0 since lim|u|→∞ |f(u)| |u|3=+∞if (f1) holds. Motivated by the study of (1.8)intheL2-critical case that a|u|p−2uwith a>0and p>4, considered in [20], a natural question arises: (Q) Is it possible to obtain an analogous structure of local minima for planar Schrödinger– Poisson problems with critical exponential growth? In the present paper, we will give an affirmative answer to above question. More precisely, after the search for a structure of local minima, differently from the perturbative argument of [3], by specifying explicit conditions on c, we study the existence of multiple normalized solutions for (1.10) with critical exponential growth, and achieve a significant extension of nonlinearity from the power type to the critical exponential type. To better illustrate our approach, we provide a concrete nonlinear model f(u)=eu2−1−u2u, which clearly satisfies condition (f1). This model is somehow inspired by Cassani–Tavares–Zhang [12]for the study of positive solutions to the Bose–Einstein type systems in R2. Compared to (1.8) with a>0andp>4 considered in [20], additional difficulties arise in the study of (1.1) since the combination of the logarithmic nature of convolution kernel and the critical exponential growth of nonlinearity mixes things up. Indeed, first, a nonlinear term of exponential type behaves like infinite series of powers nonlinear interactions, the interplay between it and the nonlocal term log |·|∗u2uis more intricate, which strongly effects the geometry structure of on Sc.Evenifsucha geometry may somehow be expected for sufficiently small values of c>0 along the research lines of [20] considering (1.8) with a|u|p−2u(a>0andp>4), the arguments of [20]are insufficient to find an explicit existence range of cfor (1.1). It requires us to develop more robust arguments in the search for a geometry of local minima for on Sc. Note that such a structure suggests the possibility to search for another solution lying at a mountain pass level, as well as a solution characterizing as a local minima. If such a structure exists, then the next most complicated part lies in the compactness analysis for minimizing sequences and (PS) sequences, since it is not clear whether lim n→∞R2|un|seαu2 n−1dx=R2|¯u|seα¯u2−1dx(1.11) for s≥2ifunuin X, despite the compactness of embedding X→Lq(R2)for all q≥2. This fact prevents us from using the compactness argument of [20]. These difficulties enforce the implementation of new ideas since the approach due to Cingolani–Jeanjean [20], treating the power case f(u)=a|u|p−2u, is not available for (1.1). In particular, instead of working directly in space X, we shall take advantage of the axially symmetric variational framework within Eas defined by (1.7), endowed with the norm given by uEas := (∇u2 2+u2 ∗)1/2,where u2 ∗=R2 log(2+|x|)u2(x)dx,(1.12) and work with the constraint ˆ Sc:= Eas ∩Sc=u∈Eas :u2 2=c.(1.13) 123 50 Page 6 of 32 S. Chen et al. As in our paper [18], if uis a critical point of restricted to ˆ Sc,thenuis a critical point of on Sc. As one will observe, besides helping to overcome the lack of compactness caused by the critical exponential growth, this type of axially symmetric setting is of extremely benefit to the proof of the L2-convergence of minimizing sequences and (PS) sequences, which is a well-identified obstacle dealing with the L2-constrained problems due to the lack of compactness for the embedding H1 rad(R2)→L2(R2). Our main results read as follows. Theorem 1.1 For any μ>0, there exists c1=c1(μ) > 0such that, for any c ∈(0,c1), (1.1)has a couple solution (uc,λ c)∈Sc×Rsuch that uc∈ˆ Sc,uc≥0,(uc)=m(c):= inf (u):u∈ˆ Sc,∇u2 2<π/3.(1.14) Theorem 1.2 For any μ>0, there exists c0=c0(μ) > 0such that, for any c ∈(0,c0), (1.1)has a second couple solution (ˆuc,ˆ λc)∈Sc×Rsuch that 0<(ˆuc)<m(c)+2π. (1.15) Remark 1.3 The condition c∈(0,c1)in Theorem 1.1 enters in the study of a geometry of local minima of , while the condition c∈(0,c0)in Theorem 1.2, which appears to be more delicate, is used in order to further ensure that a minimax structure of the mountain-pass type exists and the obtained energy level is less than m(c)+2πthat is a threshold of compactness, which is an essential and striking ingredient in our compactness argument. Define the L2-Pohozaev functional P:X→Rby P(u)=R2|∇u|2dx−μc2 4−R2u2−1eu2+1−u4 2dx.(1.16) As one will see in Lemma 3.4, any solution to (1.1) satisfies the L2-Pohozaev identity P(u)=0. Let us now sketch our research strategies and point out key elements for the proofs of Theorems 1.1 and 1.2. First, we search for a geometry of local minima for on ˆ Sc=Sc∩Eas under explicit conditions on c. For this, we introduce a crucial set Aπ/3={u∈Eas :∇u2 2<π/3}such that for any u∈ˆ Sc∩∂Aπ/3,P(u)>0 and there exists tu∈(0,1)such that P(tuutu)=0, with this important property and subtle estimates of energy, for any μ>0, we succeed in finding an explicit value c1=c1(μ) > 0 such that for any c∈(0,c1),has a geometry of local minima m(c):= inf ˆ Sc∩Aπ/3 < inf ˆ Sc∩∂Aπ/3 . (1.17) In our argument, the upper bound π/3 is not essential but brings a convenience in obtaining the explicit existence range c∈(0,c1)and proving the compactness in the following discussion. In this regard, our strategy is totally different from that of [20], since the boundary of the corresponding auxiliary set used in [20] depends on the mass u2 2=cas well as the order pof power function in (1.8), instead of being given in advance like us. Second, we prove that the local minima m(c)defined by (1.17) is attained, that is, letting {un}⊂ ˆ Sc∩Aπ/3be such that (un)→m(c),weverifythatun→uin Eas, proving Theorem 1.1. The crucial ingredient of the proof is to obtain the boundedness of {unX}, or more precisely, prove that un2 ∗=R2log(2+|x|)u2 n(x)dx≤Cfor some C>0 due to the fact that {un}⊂ ˆ Sc∩Aπ/3and the definition of ·Xgiven by (1.12). Note that at 123 Multiple normalized solutions... Page 7 of 32 50 this stage, the sign of m(c)can not be judged under the unpleasant effect of a nonlinear term of exponential type. This fact results in the failure of the method used in [20] relying on the strong compactness condition. Indeed, following the lines of [20], it is essential to verify that un→u∈L2(R2)\{0}pointwise a.e. on R2such that the strong compactness condition works which leads to the boundedness of {un∗}up to translations. But it seems impossible to make it in our case since the vanishing of {un}can not be ruled out when the situation of m(c)=0 may occur. Somewhat surprisingly, our axially symmetric variational framework allows us to avoid the obstacle since the boundedness of {un2 ∗}follows directly from the specific inequality related with the coupling term of equation restricted on Eas R2R2 log (2+|x−y|)u2(x)v2(y)dxdy ≥1 4R2 u2(x)dxR2 log(2+|x|)v2(x)dx.(1.18) This also explains why we work with Eas ∩Scat the beginning. The remaining proof of convergence is standard, since (1.11) follows directly from Trudinger-Moser inequality (see Lemma 2.3) due to the fact that ∇un2 2≤π/3<2πfor all n∈N. Last but not least, we further specify an explicit range on cto guarantee the existence of another solution of mountain-pass type, proving Theorem 1.2, which is the heart of the paper. Several crucial steps are summarized as follows. Step 1. Construct a (PS) sequence {un}⊂ ˆ Scof ˆ Scpossessing additional property P(un)→0at a mountain pass level M(c).The condition that P(un)→0helps to deduce the boundedness of {∇un2}. This step is reminiscent of the one developed by Jeanjean [25] but here the fact that has a structure of local minimum instead of a direct mountain-pass geometry and the appearance of a logarithmic convolution potential make the proof more delicate. To detect a minimax structure of ˆ Sc, we use several critical point theorems on a manifold, developed recently by us in [16] considering problem (1.9). Our approach is applicable to more general nonlinearities and totally different from that of [20] dealing with f(u)= a|u|p−2uin the case of a>0andp>4. Noting that the situation m(c)=0 can not be ruled out in advance, it is from the specific inequality (1.18) that we deduce the boundedness of {un2 ∗}, and thus {unX}is bounded and there exists ¯u∈ˆ Scsuch that, up to a subsequence, un¯uin Eas and un→¯uin Ls(R2)for s≥2. However, it is insufficient to show that ¯uis asolutionto(1.1) since it is unclear whether R2R2 log |x−y|u2 n(x)[un(y)−¯u(y)]v(y)dxdy=0,∀v∈C∞ 0(R2). (1.19) This requires to further prove the strong convergence. Inspired by the Brezis-Nirenberg problem, the crucial point in proving the compactness is to obtain a good energy estimate of the obtained (PS) sequence, which is what to do next. Step 2. Establish a precise upper estimate of the energy level M(c),given by M(c)<m(c)+2π, (1.20) such that the compactness of the obtained (PS) sequences still holds. In the unconstrained case (1.5), this kind of sharp upper estimate is known, see our papers [15,17], and the usual way to derive such strict inequality is through the use of testing functions, that is a sequence of Moser-type functions introduced by de Figueiredo, Miyagaki and Ruf [22], related with the Trudinger–Moser inequality. But, there seems no an analogue in our case due to the need to respect L2-constraint and the logarithmic nature of 123 50 Page 8 of 32 S. Chen et al. convolution kernel. This step gives firstly a counterpart in that direction, whose proof is rather complicated, and requires a lot of subtle energy estimates as well as a better understanding of structure for on ˆ Sc, see Remark 1.4 for further description. Step 3. Prove the limit (1.11)and then un→¯uin Eas,up to a subsequence. To ensure that the Trudinger–Moser inequality ii) of Lemma 2.3 works in the proof of (1.11), one needs to control appropriately the value of ∇un2 2from above, which is why one requires a sharp upper estimate of M(c)before. Unfortunately, it seems impossible to obtain ∇un2 2<4πfor large n.Instead,weprove∇(un−¯u)2 2<4πfor large nin a tactfully round-about way, of these arguments, two main difficulties are to prove P(¯u)≥0 and (¯u)≥m(c), see the proof of (4.98), and then show indirectly (1.11) with the Young’s inequality. Remark 1.4 i) To obtain a constrained (PS) sequence with additional property, the approach in [20], treating (1.8) with f(u)=a|u|p−2uin the case of a>0andp>4, not only relied on the decomposition of the L2-Pohozaev manifold into three disjoint subsets, but used the Ghoussoub minimax principle [24], where the former just works for an easy calculating form of nonlinearity, and the later requires technical topological, very complicated, arguments based on σ-homotopy stable family of compact subsets. This approach was also applied to problem (1.9) with mixed nonlinearities, see [27,29,30,39, 40,42], nevertheless, it is not available in our case due to the complex behaviors of the terms log |·|∗u2uand eu2−1−u2u. In contrast, our method does not require the decomposition of the L2-Pohozaev manifold, and our tool to detect the minimax structures just depends on the general deformation lemma on a manifold, and is technically simpler than topological arguments involved in the Ghoussoub minimax principle [24]. ii) Note that (1.20) gives a new threshold of compactness for planar Schrödinger–Poisson systems with critical exponential growth in the L2-constraint. To obtain the strict inequality (1.20), roughly speaking, we use a nice superposition of a minima obtained in Theorem 1.1 and a modified sequence of Moser-type functions with finer supports where the supports would be disjoint, see Lemma 4.4 for more details. The idea behind the proof is that the interaction decreases the involved energy value. Even if this idea is somehow inspired by [42] concerning the Sobolev critical situation in the higher dimensions, the mathematical strategies and proof techniques are different, for example, the variational characterizations of a minima are various in the use of testing functions; our tool of energy estimate is the neatly combination of the Gagliardo-Nirenberg inequality and the Trudinger-Moser inequality instead of the Sobolev inequality; extra efforts are always required to overcome the unpleasant effect due to the logarithmic nature of convolution kernel. iii) We believe that our approach may be adapted to attack more L2-constrained problems with critical exponential growth, and the new underlying ideas and the strategy of energy estimates are of future development and applicability. The paper is organized as follows. Section 2is devoted to some preliminaries. In particular, we present several critical point theorems on a manifold, we have developed recently in [16], which play a crucial role in the proofs of theorems. In Sect.3, we consider the existence of a local minima for on Eas ∩Sc, and give the proof of Theorem 1.1. In Sect.4, we study the existence of a critical point of mountain-pass type for on Eas ∩Sc, and finish the proof of Theorem 1.2. Throughout the paper, we make use of the following notations: 123 Multiple normalized solutions... Page 9 of 32 50 •H1(R2)denotes the usual Sobolev space equipped with the inner product and norm (u,v)=R2 (∇u·∇v+uv)dx,u=(u,u)1/2,∀u,v ∈H1(R2); •H1 rad(R2)denotes the space of spherically symmetric functions belonging to H1(R2): H1 rad(R2):= {u∈H1(R2)u(x)=u(|x|)a.e. in R2}; •Ls(R2)(1≤s<∞)denotes the Lebesgue space with the norm us=R2|u|sdx1/s; •For any u∈H1(R2)\{0},ut(x):= u(tx)for t>0; •For any x∈R2and r>0, Br(x):= {y∈R2:|y−x|<r}and Br=Br(0); •C1,C2,··· denote positive constants possibly different in different places, which are dependent on c>0. 2 Preliminary results As in [17], we define the following symmetric bilinear forms (u,v)→ A1(u,v):= R2R2 log (2+|x−y|)u(x)v(y)dxdy,(2.1) (u,v)→ A2(u,v):= R2R2 log 1+2 |x−y|u(x)v(y)dxdy,(2.2) (u,v)→ A0(u,v):= A1(u,v)−A2(u,v)=R2R2 log |x−y|u(x)v(y)dxdy,(2.3) where the definition is restricted, in each case, to measurable functions u,v :R2→R such that the corresponding double integral is well defined in Lebesgue sense. Noting that 0≤log(1+r)≤rfor r≥0, it follows from the Hardy–Littlewood–Sobolev inequality(see [31]or[32, p. 98]) that |A2(u,v)|≤2R2R2 1 |x−y||u(x)v(y)|dxdy≤C0u4/3v4/3(2.4) with a constant C0>0. Using (2.1), (2.2)and(2.3), we define the following energy functionals: I1:H1(R2)→[0,∞], I1(u):= A1(u2,u2)=R2R2 log (2+|x−y|)u2(x)u2(y)dxdy, I2:L8/3(R2)→[0,∞), I2(u):= A2(u2,u2)=R2R2 log 1+2 |x−y|u2(x)u2(y)dxdy, I0:H1(R2)→R∪{∞}, I0(u):= A0(u2,u2)=R2R2 log |x−y|u2(x)u2(y)dxdy. Here I2only takes finite values on L8/3(R2). Indeed, (2.4) implies |I2(u)|≤C0u4 8/3,∀u∈L8/3(R2). (2.5) 123 50 Page 16 of 32 S. Chen et al. It follows from (3.20)that∇vn2 2=o(1)and I1(vn)=o(1),andso(vn)=o(1).It follows from (3.19)thatA1(¯u2,v2 n)=o(1), and so by Lemma 2.4,vn→0inEas, i.e. un→¯uin Eas.Sinceun≥0, it follows that ¯u≥0. Step 4. Obviously u∈¯ Aπ/3and (¯u)=m(c). Next, we show that ∇¯u2 2<π 3.Let us assume by contradiction that ∇¯u2 2=π 3. Then we see directly from Corollary 3.2 that necessarily P(¯u)>0. But then we consider t0with t0<1 close to 1. Recording (3.9), it follows that t0¯ut0∈Aπ/3and (t0¯ut0)<(¯u)=m(c), providing a contradiction. Hence, Corollary 2.11 implies that |ˆ Sc (¯u)=0, and so there exists a Lagrange multiplier λc∈R such that (¯u)+λc¯u,φ=0foranyφ∈Eas. By Lemma 2.6,wehave(¯u)+λc¯u,φ=0 for any φ∈X,thatis −¯u+μlog |x|∗¯u2¯u−e¯u2−1−¯u2¯u=−λc¯u,x∈R2.(3.21) This completes the proof.  4 Proof of Theorem 1.2 In this section, we consider the existence of a critical point of mountain-pass type for on ˆ Sc=Eas ∩Sc, and give the proof of Theorem 1.2. Lemma 4.1 Let μ>0and c ∈(0,c2). For any u ∈ˆ Sc, the following exist: (i) A unique s+ u>0such that s+ uis a strict local minimum point for gu. (ii) A unique s− u>0such that s− uis a strict local maximum point for gu. Proof For any u∈ˆ Sc,letτ:= 1/∇u2and ˆu:= τuτ.Then∇ˆu2 2=1andtˆut=(tτ)utτ for t>0. Therefore, we only prove this lemma for u∈ˆ Scwith ∇u2 2=1. Fix u∈ˆ Scwith ∇u2 2=1, we have g u(t)=1 tP(tut), ∀t>0.(4.1) Let t∗>0 such that t−4 ∗R21−t2 ∗u2+t4 ∗u4et2 ∗u2−1−t4 ∗u4 2dx=1.(4.2) It follows that t2>t−2R21−t2u2+t4u4et2u2−1−t4u4 2dx,0<t<t∗(4.3) and t2<t−2R21−t2u2+t4u4et2u2−1−t4u4 2dx,t∗<t<+∞.(4.4) By (3.9)and(4.3), one has g u(t)=1 tt2−μc2 4−t−2R2t2u2−1et2u2+1−t4u4 2dx ≥1 tt2−μc2 4−t−2 2R21−t2u2+t4u4et2u2−1−t4u4 2dx 123 Multiple normalized solutions... Page 17 of 32 50 >1 2tt2−μc2 2,0<t<t∗.(4.5) If t∗<π, then from (2.14)and(4.2), we have 1=t−4 ∗R21−t2 ∗u2+t4 ∗u4et2 ∗u2−1−t4 ∗u4 2dx =∞  k=3 (k−1)2 k!u2k 2kt2(k−2) ∗ ≤2c π ∞  k=3 (k−1)24k−1(k−2)+1 (k−2)k!t2 ∗√c πk−2 +1 2π ∞  k=3 (k−1)2t2 ∗ πk−2 =2c π ∞  k=3 (k−1)24k−1(k−2)+1 (k−2)k!t2 ∗√c πk−2 +t2 ∗4π2−3πt2 ∗+t4 ∗ 2ππ−t2 ∗3.(4.6) Combining (3.3) with (4.6), we deduce t∗≥η(c). It follows from (3.3)thatη(c)is decreasing on c>0. Hence, by (3.2), we have μc2 2<μc2 2 2=η2(c2)<η 2(c)≤t2 ∗,∀c∈(0,c2). (4.7) Hence, (4.7) shows that there exists δ>0 such that t2−μc2 2>0foranyt∈(t∗−δ,t∗). Hence, by (4.5), we infer that g u(t)>0foranyt∈(t∗−δ, t∗), and thus gu(t)is increasing in (t∗−δ,t∗). Taking into account that the function gu(t)→+∞as t→0+and gu(t)→−∞as t→+∞, we conclude that there exists at least a critical point s+ u<t∗which is a local minimum point of guand a critical point s− u>t∗which is a local maximum point of gu. Since s− u>t∗, from (4.4)wederivethat (s− u)2<(s− u)−2R21−(s− u)2u2+(s− u)4u4e(s− u)2u2−1−(s− u)4u4 2dx =∞  k=3 (k−1)2 k!u2k 2k(s− u)2(k−1).(4.8) Moreover, from (3.9), (4.8) and the fact that g u(s− u)=0, we derive that g u(s− u)=1 (s− u)2(s− u)2−∞  k=3 (k−1)(2k−3) k!u2k 2k(s− u)2(k−1)+μc2 4 =2 (s− u)2(s− u)2−∞  k=3 (k−1)2 k!u2k 2k(s− u)2(k−1)<0.(4.9) Therefore s− uis a strict maximum point for gu. We have to show that s− uis unique. By contradiction we assume that there exists ˆs− u>0, another critical point of guwhich is a local maximum point. First, we observe that if 0 <ˆs− u<t∗, then from g u(ˆs− u)=0and(4.3) we obtain g u(ˆs− u)=1 (ˆs− u)2(ˆs− u)2−∞  k=3 (k−1)(2k−3) k!u2k 2k(ˆs− u)2(k−1)+μc2 4 123 50 Page 18 of 32 S. Chen et al. =2 (ˆs− u)2(ˆs− u)2−∞  k=3 (k−1)2 k!u2k 2k(ˆs− u)2(k−1)>0,(4.10) which is a contradiction. This implies that ˆs− u>t∗, and thus arguing as before we have g u(ˆs− u)<0. We derive the existence of another critical point: θu∈(ˆs− u,s− u)or θu∈(s− u,ˆs− u), which is a local minimum for gu. Taking into account (4.4), we again deduce g u(θu)<0, which is a contradiction. Therefore the point s− uis unique. Now a direct adaptation of the argument used for s− uleads us to conclude that s+ uis the unique local minimum point for gu. Lemma 4.2 Let μ>0. For any c ∈(0,c1), there exists κc>0such that M(c):= inf γ∈c max t∈[0,1](γ (t)) ≥κc>sup γ∈c max {(γ (0)), (γ (1))},(4.11) where c=γ∈C([0,1],ˆ Sc):γ(0)=uc,(γ(1)) < m(c)−1,(4.12) and ucis determined by Theorem 1.1. Proof Set κc:= infu∈∂( ˆ Sc∩Aπ/3)(u). By Theorem 1.1 and Corollary 3.2,κc>m(c)= (uc).Letγ∈cbe arbitrary. Since γ(0)=uc,and(γ (1)) < m(c)−1, necessarily in view of Theorem 1.1,γ(1)/∈ˆ Sc∩Aπ/3. By continuity of γ(t)on [0,1], there exists a t0∈(0,1)such that γ(t0)∈∂( ˆ Sc∩Aπ/3),andsomax t∈[0,1](γ (t)) ≥κc.Thus,(4.11) holds.  To apply Lemma 2.12,weletE=Eas and H=L2(R2). Define the norms of Eand H by uE:= ∇u2+u2 ∗1/2,u2 H:= 1 √cR2 u2dx1/2 ,∀u∈E.(4.13) By Lemma 2.6, after identifying Hwith its dual, we have E→H→E∗with continuous injections. Set M:= u∈E:u2 2=R2 u2dx=c.(4.14) Obviously, Lemma 2.3 shows that ∈C1(E,R),and (u), u=R2|∇u|2dx+μI0(u)−R2eu2−1−u2u2dx.(4.15) Set F(u):= 1 2eu2−1−u2−u4 2and f(u):= eu2−1−u2u. Inspired by [25], let us define a continuous map β:Eas ×R→Eas by β(v,t)(x):= etv(etx)for v∈Eas,t∈R,x∈R2,(4.16) and consider the following auxiliary functional: ˜ (v, t):= (β(v, t)) =e2t 2∇v2 2+μ 4I0(v) −μc2t 4−1 e2tR2 F(etv)dx.(4.17) We see that ˜ is of class C1,andforany(w, s)∈Eas ×R, ˜ (v, t), (w, s)=˜ (v, t), (w, 0)+˜ (v, t), (0,s) 123 Multiple normalized solutions... Page 19 of 32 50 =e2tR2∇v·∇wdx+μR2R2 log |x−y|v2(x)v(y)w(y)dxdy −1 e2tR2 f(etv)etwdx+e2ts∇v2 2−μc2s 4 +s e2tR22F(etv) −f(etv)etvdx =(β(v, t)), β(w, t) +sP(β(v, t)). (4.18) Let u(x):= β(v,t)(x)=etv(etx), φ(x):= β(w,t)(x)=etw(etx). (4.19) Then (u,φ)H=1 cR2 u(x)φ(x)dx=1 cR2 v(x)w(x)dx=(v, w)H.(4.20) This shows that φ∈Tu(ˆ Sc)⇔(w, s)∈˜ T(v,t)(ˆ Sc×R), ∀t,s∈R.(4.21) It is easy to verify that log 2+e−|t|r≥e−|t|log(2+r), ∀r>0,t∈R.(4.22) It follows from (1.12), (4.18), (4.19), (4.21)and(4.22)that |P(u)|=˜ (v, t), (0,1)≤  ˜ |ˆ Sc×R(v, t)  (4.23) and   |ˆ Sc(u)  =sup φ∈Tu(ˆ Sc) 1 φE(u), φ  =sup φ∈Tu(ˆ Sc) 1 ∇φ2 2+φ2 ∗(β(v, t)), β(w, t)  =sup φ∈Tu(ˆ Sc) 1 ∇φ2 2+φ2 ∗˜ (v, t), (w, 0) ≤sup (w,0)∈˜ T(v,t)(ˆ Sc×R) e|t| (w, 0)E×R˜ (v, t), (w, 0) ≤e|t|  ˜ |ˆ Sc×R(v, t)  .(4.24) Lemma 4.3 Let μ>0. Then for any c ∈(0,c1), there exists a sequence {un}⊂ ˆ Scsuch that (un)→M(c)>m(c), |ˆ Sc(un)→0and P(un)→0.(4.25) Proof Set ˜ c:= ˜γ∈C([0,1],ˆ Sc×R):˜γ(0)=(uc,0), ˜ ( ˜γ(1)) < m(c)−1(4.26) and ˜ M(c):= inf ˜γ∈˜ c max t∈[0,1]˜ ( ˜γ(t)). (4.27) 123 50 Page 20 of 32 S. Chen et al. For any ˜γ∈˜ c, it is easy to see that γ=β◦˜γ∈cdefined by (4.12). Let κ c:= supγ∈cmax {(γ (0)), (γ (1))}. Then it follows from (4.11)that max t∈[0,1]˜ ( ˜γ(t)) =max t∈[0,1](γ (t)) ≥κc>κ  c≥max {(γ (0)), (γ (1))} =max ˜ ( ˜γ(0)), ˜ ( ˜γ(1)). It follows that ˜ M(c)≥M(c),and ˜ M(c)=inf ˜γ∈˜ c max t∈[0,1]˜ ( ˜γ(t)) ≥κc>κ  c≥sup ˜γ∈˜ c max ˜ ( ˜γ(0)), ( ˜γ(1)).(4.28) This shows that (2.29) holds. On the other hand, for any γ∈c,let ˜γ(t):= (γ (t), 0). It is easy to verify that ˜γ∈˜ c and (γ (t)) =˜ ( ˜γ(t)), and so, we trivially have ˜ M(c)≤M(c). Thus ˜ M(c)=M(c). For any n∈N,(4.11) implies that there exists γn∈csuch that max t∈[0,1](γn(t)) ≤M(c)+1 n.(4.29) Set ˜γn(t):= (γn(t), 0). Then apply Lemma 2.12 to ˜ , there exists a sequence {(vn,tn)}⊂ ˆ Sc×Rsatisfying (i) M(c)−2 n≤˜ (vn,tn)≤M(c)+2 n; (ii) mint∈[0,1](vn,tn)−(γn(t), 0)E×R≤2 √n; (iii)   ˜ |ˆ Sc×R(vn,tn)  ≤8 √n. Let un=β(vn,tn). It follows from (4.23), (4.24) and (i)-(iii) that (4.25) holds.  Now we define the following Moser type functions wn(x)supported in B1(0) wn(x)=1 √2π⎧ ⎪ ⎨ ⎪ ⎩ √log n,0≤|x|≤1/n; log(1/|x|) √log n,1/n≤|x|≤1; 0,|x|≥1. (4.30) Computing directly, we get that ∇wn2 2=R2|∇wn|2dx=1,(4.31) wn2 2=R2|wn|2dx=log n1/n 0 rdr+1 1/n log2(1/r) log nrdr =1 4logn−1 4n2log n−1 2n2,(4.32) wn8/3 8/3=R2|wn|8/3dx=O1 log4/3n,n→∞,(4.33) wn2 ∗=R2 log(2+|x|)|wn|2dx=O1 log n,n→∞ (4.34) and |I0(wn)|=R2R2 log |x−y|w2 n(x)w2 n(y)dxdy≤O1 log2n,n→∞.(4.35) 123 Multiple normalized solutions... Page 21 of 32 50 Lemma 4.4 Let μ>0. Then for any c ∈(0,c1), there holds M(c)<m(c)+2π. (4.36) Proof Let ucbe determined by Theorem 1.1. By Theorem 1.1 and Lemma 3.4,wehave uc2 2=c,(uc)=m(c), uc(x)≥0,∀x∈R2(4.37) and −λcc=μc2 4+μI0(uc)−R2eu2 c−1−u2 c−u4 c 2dx.(4.38) Since uc∈Eas, it follows from (2.4), (2.7), (4.30), (4.32), (4.33)and(4.34)that R2R2 log |x−y|uc(x)wn(x)uc(y)wn(y)dxdy=O1 log n,n→∞,(4.39) R2R2 log |x−y|u2 c(x)w2 n(y)dxdy=O1 log n,n→∞,(4.40) R2R2 log |x−y|uc(x)wn(x)w2 n(y)dxdy=O1 log3/2n,n→∞,(4.41) R2 ucwndx=O1 √log n,n→∞ (4.42) and R2eu2 c−1−u2 cucwndx=O1 √log n,n→∞.(4.43) By (1.1), (4.32)and(4.37), one has R2∇uc·∇wndx=R2−μR2 log |x−y|u2 c(y)dy+eu2 c−1−u2 c−λcucwndx (4.44) and uc+twn2 2=c+t2wn2 2+2tR2 ucwndx =c+2tR2 ucwndx+t2O1 log n,n→∞.(4.45) Let τ:= uc+twn2/√c.Then τ2=1+2t cR2 ucwndx+t2O1 log n,n→∞ (4.46) and for any p≥1, τ−2p=1−2pt cR2 ucwndx+t2O1 log n,n→∞.(4.47) Now, we define Wn,t(x):= uc(τ x)+twn(τ x). (4.48) Then one has ∇Wn,t2 2=∇(uc+twn)2 2,Wn,t2 2=τ−2uc+twn2 2=c,(4.49) 123 50 Page 22 of 32 S. Chen et al. I0(Wn,t)=R2R2 log |x−y|[uc(τ x)+twn(τ x)]2[uc(τ y)+twn(τ y)]2dxdy =1 τ4R2R2 log |x−y|[uc(x)+twn(x)]2[uc(y)+twn(y)]2dxdy−c2log τ (4.50) and R2%eW2 n,t−1−W2 n,t−W4 n,t 2&dx =1 τ2R2e(uc+twn)2−1−(uc+twn)2−(uc+twn)4 2dx.(4.51) From (4.42)and(4.46), one has τ2=1+2t cR2 ucwndx+t2O1 log n≤1+t+t2,for large n∈N.(4.52) Now, we define n(t)by n(t)=t2 2−1 2τ2R2et2w2 n−1−t2w2 n−1 2t4w4 ndx,∀t>0.(4.53) We claim that sup t>0n(t)+t2O1 log n+t4O1 log2n ≤2π−π 2lognlog log n 32π,for large n∈N.(4.54) There are three cases to distinguish. In the sequel, we agree that all inequalities hold for large n∈Nwithout mentioning. Case i) t∈'0,√2π(.Thenby(4.35)and(4.53), we have n(t)=t2 2−1 2τ2R2et2w2 n−1−t2w2 n−1 2t4w4 ndx≤t2 2≤3π 2.(4.55) It follows that sup 0<t≤√2πn(t)+t2O1 log n+t4O1 log2n ≤2π−π 2lognlog log n 32π,for large n∈N.(4.56) Case ii) t∈'√2π,√6π. Then it follows from (4.30), (4.35)and(4.52)that 1 τ2R2et2w2 n−1−t2w2 n−1 2t4w4 ndx ≥1 2τ2B1/n et2w2 ndx≥1 16n2e(2π)−1t2log n.(4.57) 123 Multiple normalized solutions... Page 23 of 32 50 Using (4.53)and(4.57), we are led to n(t)=t2 2−1 2τ2R2et2w2 n−1−t2w2 n−1 2t4w4 ndx ≤t2 2−1 32n2e(2π)−1t2log n:= ϕn(t). (4.58) Choosing tn>0 be such that ϕ n(tn)=0, then we have 1=log n 32πn2e(2π)−1t2 nlog n.(4.59) It follows that t2 n=4π1+log(32π)−log(log n) 2logn(4.60) and ϕn(t)≤ϕn(tn)=t2 n 2−π log n,∀t≥0.(4.61) Using (4.60)and(4.61), we are led to ϕn(t)≤t2 n 2−π log n=2π−π log nlog elog n 32π, which, together with (4.58), yields n(t)≤2π−π log nlog elog n 32π. It follows that sup √2π<t≤√6πn(t)+t2O1 log n+t4O1 log2n ≤2π−π 2lognlog log n 32π,for large n∈N.(4.62) Case iii) t∈√6π,+∞. Then it follows from (4.30),(4.35)and(4.52)that n(t)+t2O1 log n+t4O1 log2n ≤t2 2−1 2τ2R2et2ω2 n−1−t2ω2 n−1 2t4ω4 ndx +t2O1 log n+t4O1 log2n ≤t2 2−π 4n2τ2e(2π)−1t2log n+t2O1 log n+t4O1 log2n ≤t2 2−π 4n2(1+t+t2)e(2π)−1t2log n+t2O1 log n+t4O1 log2n := t2 2−π 4n2(1+t+t2)e(2π)−1t2log n+ant2+bnt4(4.63) 123 50 Page 24 of 32 S. Chen et al. ≤3π−π 2n2(1+√6π+6π)e3logn+6πan+36π2bn≤3 2π, (4.64) where we have used the fact that the function φn(t):= t2 2−π 4n2(1+t+t2)e(2π)−1t2log n+ant2+bnt4 is decreasing on t∈√6π,+∞for large n. In fact, φ n(t)=(1+2an)t+4bnt3−1+t+t2tlog n−(1+2t)π 4n21+t+t22e(2π)−1t2log n. Assume that sn>0 such that φ n(sn)=0forlargen.Then 4(1+2an)sn+4bns3 n1+sn+s2 n2=1+sn+s2 nsnlog n−(1+2sn)π n2e(2π)−1s2 nlog n, which yields s2 n=4π⎧ ⎨ ⎩ 1+ log '4(1+2an)sn+4bns3 n1+sn+s2 n2( 2logn −log 1+sn+s2 nsnlog n−(1+2sn)π 2logn).(4.65) This implies that limn→∞ s2 n=4π.Soφn(t)is decreasing on t∈√6π,+∞for large n. From (4.64), one has sup √6π≤t<+∞n(t)+t2O1 log n+t4O1 log2n ≤2π−π 2lognlog log n 32π,for large n∈N.(4.66) Cases i)–iii) show that (4.54) holds. It is easy to verify the following inequality: (1+t)q≥1+qtq−1+tq,∀t≥0,q≥2.(4.67) By (4.35), (4.39)-(4.41), we have I0(uc+twn)=R2R2 log |x−y|[uc(x)+twn(x)]2[uc(y)+twn(y)]2dxdy =I0(uc)+t4I0(wn)+4tR2R2 log |x−y|u2 c(x)uc(y)wn(y)dxdy +4t2R2R2 log |x−y|uc(x)wn(x)uc(y)wn(y)dxdy +2t2R2R2 log |x−y|u2 c(x)w2 n(y)dxdy +4t3R2R2 log |x−y|uc(x)wn(x)w2 n(y)dxdy =I0(uc)+4tR2R2 log |x−y|u2 c(x)uc(y)wn(y)dxdy 123 Multiple normalized solutions... Page 25 of 32 50 +t2O1 log n+t3O1 log3/2n+t4O1 log2n.(4.68) From (1.2), (4.31), (4.37)–(4.44), (4.46)–(4.53), (4.54),(4.67)and(4.68), we have (Wn,t) =1 2∇Wn,t2 2+μ 4I0(Wn,t)−1 2R2%eW2 n,t−1−W2 n,t−W4 n,t 2&dx =1 2∇(uc+twn)2 2+μ 4τ4I0(uc+twn)−μc2 4log τ −1 2τ2R2e(uc+twn)2−1−(uc+twn)2−(uc+twn)4 2dx ≤1 2∇uc2 2+μτ−4 4I0(uc)−μc2 4log τ−1 2τ2R2eu2 c−1−u2 c−u4 c 2dx +t2 2∇wn2 2−1 2τ2R2et2w2 n−1−t2w2 n−t4w4 n 2dx+tR2∇uc·∇wndx +μτ−4tR2R2 log |x−y|u2 c(x)uc(y)wn(y)dxdy−τ−2tR2eu2 c−1−u2 cucwndx +t2O1 log n+t3O1 log3/2n+t4O1 log2n =(uc)+n(t)−μ1−τ−4 4I0(uc)−μc2 4log τ +1−τ−2 2R2eu2 c−1−u2 c−u4 c 2dx −μ1−τ−4tR2R2 log |x−y|u2 c(x)uc(y)wn(y)dxdy +1−τ−2tR2eu2 c−1−u2 cucwndx−λctR2 ucwndx +t2O1 log n+t3O1 log3/2n+t4O1 log2n ≤m(c)+n(t)−λctR2 ucwndx−μc2 4t cR2 ucwndx+t2O1 log n −μI0(uc)t cR2 ucwndx+t2O1 log n +t cR2 ucwndx+t2O1 log nR2eu2 c−1−u2 c−u4 c 2dx −μ4t cR2 ucwndx+t2O1 log ntR2R2 log |x−y|u2 c(x)uc(y)wn(y)dxdy +2t cR2 ucwndx+t2O1 log ntR2eu2 c−1−u2 cucwndx +t2O1 log n+t3O1 log3/2n+t4O1 log2n 123 50 Page 32 of 32 S. 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