期刊：
Calculus of Variations and Partial Differential Equations,2024年63(7):1-39 ISSN：0944-2669
通讯作者：
Pan, KF
作者机构：
[Luo, Peng; Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Peoples R China.;[Pan, Kefan] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Pan, KF ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
摘要：
We revisit the following nonlinear Schrödinger equation
$$\begin{aligned} -\varepsilon ^2\Delta u+ V(x)u=u^{p},\quad u>0,\;\; u\in H^1({\mathbb {R}}^N), \end{aligned}$$
where
$$\varepsilon >0$$
is a small parameter,
$$N\ge 2$$
and
$$1<p<2^*-1$$
. It is known that the Morse index gives a strong qualitative information on the solutions, such as non-degeneracy, uniqueness, symmetries, singularities as well as classifying solutions. Here we compute the Morse index of positive k-peak solutions to above problem when the critical points of V(x) are non-isolated and degenerate. We also give a specific formula for the Morse index of k-peak solutions when the critical point set of V(x) is a low-dimensional ellipsoid. Our main difficulty comes from the non-uniform degeneracy of potential V(x). Our results generalize Grossi and Servadei’s work (Ann Math Pura Appl 186: 433–453, (2007)) to very degenerate (non-admissible) potentials and show that the structure of potentials highly affects the properties of concentrated solutions.
摘要：
This paper deals with the following Cauchy problem for critical heat equation with drift term {u(t)= triangle u+del lna(x)<middle dot>del u+|u|4/n-2u,for (x,t)is an element of R(n)x(0,+infinity), u(<middle dot>,0) =u(0), for x is an element of R-n, where a(x) is a positive smooth bounded function in R-n,n >= 6. Assume that the eigenvalues of matrix A(q), denoted as sigma(i), i= 1,<middle dot><middle dot><middle dot>,n, satisfy sigma i<(1 +sigma)3n/n+ 2 (n)& sum;(i)=1 sigma(i)n. We construct sign-changing bubble tower solutions which blow up at the critical point q of a (x) in the forward time infinity of the form u(x,t)approximate to(j=1)& sum;(2)(-1)j-1 alpha(n)(mu(j)mu(2)j+|x-q|2)(n-2/2)as t ->+infinity, here 0< mu(j)(t)-> 0 exponentially as t ->+infinity.
期刊：
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS,2024年 ISSN：0219-1997
通讯作者：
Deng, YB
作者机构：
[Deng, Yinbin; Peng, Shuangjie; Deng, YB] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Deng, Yinbin; Deng, YB] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Peng, Shuangjie] Cent China Normal Univ, Key Lab Nonlinear Anal & Applicat, Wuhan 430079, Peoples R China.;[Yang, Xian] Guangxi Univ, Coll Math & Informat Sci, Nanning 550001, Peoples R China.
通讯机构：
[Deng, YB ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
摘要：
This paper deals with the following fractional Choquard equation epsilon(2s)(-Delta)(s)u+Vu=epsilon(-alpha)(I-alpha(& lowast;)|u|(p))|u|(p-2)u in R-N, where epsilon>0 is a small parameter, (-Delta)(s) is the fractional Laplacian, N>2s, s is an element of(0,1), alpha is an element of((N-4s)+,N), p is an element of[2,N+alpha/N-2s), I-alpha is a Riesz potential, V is an element of C(R-N,[0,+infinity)) is an electric potential. Under some assumptions on the decay rate of V and the corresponding range of p, we prove that the problem has a family of solutions {u(epsilon)} concentrating at a local minimum of V as epsilon -> 0. Since the potential V decays at infinity, we need to employ a type of penalized argument and implement delicate analysis on the both nonlocal terms to establish regularity, positivity and asymptotic behaviour of u(epsilon), which is totally different from the local case. As a contrast, we also develop some nonexistence results, which imply that the assumptions on V and p for the existence of u epsilon are almost optimal. To prove our main results, a general strong maximum principle and comparison function for the weak solutions of fractional Laplacian equations are established. The main methods in this paper are variational methods, penalized technique and some comparison principle developed in this paper.
期刊：
Calculus of Variations and Partial Differential Equations,2024年63(5):1-30 ISSN：0944-2669
通讯作者：
Peng, SJ
作者机构：
[Deng, Yinbin; Peng, Shuangjie; Yang, Xian] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Deng, Yinbin; Peng, Shuangjie] Cent China Normal Univ, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Peoples R China.;[Yang, Xian] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Peng, SJ ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Peoples R China.
关键词：
35J15;35A15;35J10
摘要：
We revisit the following fractional Schrödinger equation
0.1
$$\begin{aligned} \varepsilon ^{2s}(-\Delta )^su +Vu=u^{p-1},\,\,\,u>0,\ \ \ \textrm{in}\ {\mathbb {R}}^N, \end{aligned}$$
where
$$\varepsilon >0$$
is a small parameter,
$$(-\Delta )^s$$
denotes the fractional Laplacian,
$$s\in (0,1)$$
,
$$p\in (2, 2_s^*)$$
,
$$2_s^*=\frac{2N}{N-2s}$$
,
$$N>2s$$
,
$$V\in C\big ({\mathbb {R}}^N, [0, +\infty )\big )$$
is a general potential. Under various assumptions on V(x) at infinity, including V(x) decaying with various rate at infinity, we introduce a unified penalization argument and give a complete result on the existence and nonexistence of positive solutions. More precisely, we combine a comparison principle with iteration process to detect an explicit threshold value
$$p_*$$
, such that the above problem admits positive concentration solutions if
$$p\in (p_*, \,2_s^*)$$
, while it has no positive weak solutions for
$$p\in (2,\,p_*)$$
if
$$p_*>2$$
, where the threshold
$$p_*\in [2, 2^*_s)$$
can be characterized explicitly by
$$\begin{aligned} p_*=\left\{ \begin{array}{ll} 2+\frac{2s}{N-2s} &{}\quad \text{ if } \lim \limits _{|x| \rightarrow \infty } (1+|x|^{2s})V(x)=0,\\ 2+\frac{\omega }{N+2s-\omega } &{}\quad \text{ if } 0\!<\!\inf (1\!+\!|x|^\omega )V(x)\!\le \! \sup (1\!+\!|x|^\omega )V(x)\!<\! \infty \text{ for } \text{ some } \omega \!\in \! [0, 2s],\\ 2&{}\quad \text{ if } \inf V(x)\log (e+|x|^2)>0. \end{array}\right. \end{aligned}$$
Moreover, corresponding to the various decay assumptions of V(x), we obtain the decay properties of the solutions at infinity. Our results reveal some new phenomena on the existence and decays of the solutions to this type of problems.
期刊：
Calculus of Variations and Partial Differential Equations,2023年62(3):1-35 ISSN：0944-2669
通讯作者：
Peng Luo
作者机构：
[Luo, Peng; Zhou, Yang; Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Luo, Peng; Peng, Shuangjie] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Peng Luo] S;School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, China
关键词：
35A01;35B25;35J20;35J60
摘要：
We revisit the well known prescribed scalar curvature problem
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\big (1+\varepsilon K(x)\big )u^{2^*-1}, u(x)>0,~~ &{}{x\in \mathbb {R}^N},\\ u\in \mathcal {D}^{1,2}(\mathbb {R}^N),\\ \end{array}\right. } \end{aligned}$$
where
$$2^*=\frac{2N}{N-2}$$
,
$$N\ge 5$$
,
$$\varepsilon >0$$
and
$$K(x)\in C^1(\mathbb {R}^N)\cap L^{\infty }(\mathbb {R}^N)$$
. It is known that there are a number of results related to the existence of solutions concentrating at the isolated critical points of K(x). However, if K(x) has non-isolated critical points with different degenerate rates along different directions, whether there exist solutions concentrating at these points is still an open problem. We give a certain positive answer to this problem via applying a blow-up argument based on local Pohozaev identities and modified finite dimensional reduction method when the dimension of critical point set of K(x) ranges from 1 to
$$N-1$$
, which generalizes some results in Cao et al. (Calc Var Partial Differ Equ 15:403–419, 2002) and Li (J Differ Equ 120:319–410, 1995; Commun Pure Appl Math 49:541–597, 1996).
期刊：
SIAM JOURNAL ON MATHEMATICAL ANALYSIS,2023年55(2):773-804 ISSN：0036-1410
通讯作者：
Guo, YJ
作者机构：
[Peng, Shuangjie; Guo, Yujin; Luo, Yong] Cent China Normal Univ, Sch Math & Stat, POB 71010, Wuhan 430079, Peoples R China.;[Peng, Shuangjie; Guo, Yujin; Luo, Yong] Cent China Normal Univ, Hubei Key Lab Math Sci, POB 71010, Wuhan 430079, Peoples R China.
通讯机构：
[Guo, YJ ] C;Cent China Normal Univ, Sch Math & Stat, POB 71010, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, POB 71010, Wuhan 430079, Peoples R China.
关键词：
Bose–Einstein condensates;rotational velocity;ground states;free of vortices;35J20;35J60;35Q40;46N50
摘要：
We study ground states of two-dimensional Bose–Einstein condensates with repulsive ( \(a\gt 0\) ) or attractive ( \(a\lt 0\) ) interactions in a trap \(V (x)\) rotating at velocity \(\Omega\) . It is known that there exist critical parameters \(a^{\ast }\gt 0\) and \(\Omega^{\ast }:=\Omega^{\ast }(V(x))\gt 0\) such that if \(\Omega \gt \Omega^{\ast }\) , then there is no ground state for any \(a\in \mathbb{R}\) ; if \(0\le \Omega \lt \Omega^{\ast }\) , then ground states exist if and only if \(a\in (-a^{\ast },+\infty )\) . As a completion of the existing results, in this paper, we focus on the critical case where \(0\lt \Omega =\Omega^{\ast }\lt +\infty\) to classify the existence and nonexistence of ground states for any \(a\in \mathbb{R}\) . Moreover, for a suitable class of radially symmetric traps \(V(x)\) , employing the inductive symmetry method, we prove that up to a constant phase, ground states must be real valued, unique, and free of vortices as \(\Omega \searrow 0\) , no matter whether the interactions of the condensates are repulsive or not.
期刊：
Journal of Differential Equations,2023年355:16-61 ISSN：0022-0396
通讯作者：
Qing Guo<&wdkj&>Shuangjie Peng
作者机构：
[Guo, Qing] Minzu Univ China, Coll Sci, Beijing 100081, Peoples R China.;[Liu, Junyuan] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Liu, Junyuan] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Wuhan, Peoples R China.
通讯机构：
[Qing Guo] C;[Shuangjie Peng] S;College of Science, Minzu University of China, Beijing 100081, China<&wdkj&>School of Mathematics and Statistics, Central China Normal University, Wuhan, PR China
作者：
Luo, Peng;Pan, Kefan;Peng, Shuangjie;Zhou, Yang
期刊：
Journal of Functional Analysis,2023年284(12):109921 ISSN：0022-1236
通讯作者：
Peng Luo
作者机构：
[Luo, Peng; Zhou, Yang; Peng, Shuangjie; Pan, Kefan] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Luo, Peng; Peng, Shuangjie] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Peng Luo] S;School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China
关键词：
Nonlinear Schr?dinger equation;Non-isolated critical points;Existence and uniqueness;The number of peak solutions
摘要：
We revisit the following nonlinear Schrodinger equation-epsilon 2 Delta u + V(x)u = up-1, u > 0, u is an element of H1(RN), where epsilon > 0 is a small parameter, N >= 2 and 2 < p < 2*.We obtain a more accurate location for the concentrated points, the existence and the local uniqueness for positive k-peak solutions when V(x) possesses non-isolated critical points by using the modified finite dimensional reduction method based on local Pohozaev identities. Moreover, for several special potentials, with its critical point set being a low-dimensional ellipsoid, or a part of hyperboloid of one sheet or two sheets, we obtain the number and symmetry of k-peak solutions by using local uniqueness of concentrated solutions. Here the main difficulty comes from the different degenerate rate along different directions at the critical points of V(x).(c) 2023 Elsevier Inc. All rights reserved.
摘要：
We consider the following fractional Schrödinger equation:
(0.1)
$${\left( { - \Delta } \right)^s}u + V\left( y \right)u = {u^p},\,\,\,\,\,\,u > 0\,\,\,\,{\rm{in}}\,\,{\mathbb{R}^N},$$
where s ∈ (0, 1),
$$1 < p < {{N + 2s} \over {N - 2s}}$$
, and V(y) is a positive potential function and satisfies some expansion condition at infinity. Under the Lyapunov-Schmidt reduction framework, we construct two kinds of multi-spike solutions for (0.1). The first k-spike solution uk is concentrated at the vertices of the regular k-polygon in the (y1, y2)-plane with k and the radius large enough. Then we show that uk is non-degenerate in our special symmetric workspace, and glue it with an n-spike solution, whose centers lie in another circle in the (y3, y4)-plane, to construct infinitely many multi-spike solutions of new type. The nonlocal property of (−Δ)s is in sharp contrast to the classical Schrödinger equations. A striking difference is that although the nonlinear exponent in (0.1) is Sobolev-subcritical, the algebraic (not exponential) decay at infinity of the ground states makes the estimates more subtle and difficult to control. Moreover, due to the non-locality of the fractional operator, we cannot establish the local Pohozaev identities for the solution u directly, but we address its corresponding harmonic extension at the same time. Finally, to construct new solutions we need pointwise estimates of new approximation solutions. To this end, we introduce a special weighted norm, and give the proof in quite a different way.
期刊：
Calculus of Variations and Partial Differential Equations,2023年62(3):1-26 ISSN：0944-2669
通讯作者：
Lei Liu
作者机构：
[Bi, Yuchen] Univ Chinese Acad Sci, Inst Math, Acad Math & Syst Sci, Beijing 100190, Peoples R China.;[Li, Jiayu] Univ Sci & Technol China, Sch Math Sci, Hefei 230026, Peoples R China.;[Peng, Shuangjie; Liu, Lei] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Peng, Shuangjie; Liu, Lei] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Lei Liu] S;School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, People’s Republic of China
关键词：
35J60;35J57;35B44
摘要：
In this paper, we study the blow-up analysis for a sequence of solutions to the Liouville type equation with exponential Neumann boundary condition. For interior case, i.e. the blow-up point is an interior point, Li (Commun Math Phys 200(2):421–444, 1999) gave a uniform asymptotic estimate. Later, Zhang (Commun Math Phys 268(1):105–133, 2006) and Gluck (Nonlinear Anal 75(15):5787–5796, 2012) improved Li’s estimate in the sense of
$$C^0$$
-convergence by using the method of moving planes or classification of solutions of the linearized version of Liouville equation. If the sequence blows up at a boundary point, Bao–Wang–Zhou (J Math Anal Appl 418:142–162, 2014) proved a similar asymptotic estimate of Li (1999). In this paper, we will prove a
$$C^0$$
-convergence result in this boundary blow-up process. Our method is different from Gluck (2013), Zhang (2006).
期刊：
Journal of Differential Equations,2023年371:299-352 ISSN：0022-0396
通讯作者：
Yang, X
作者机构：
[Deng, Yinbin; Peng, Shuangjie; Yang, X; Yang, Xian] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Deng, Yinbin; Peng, Shuangjie] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Yang, X ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
摘要：
In this paper, we study the following Choquard equations with fractional Laplacian ⎧ ⎨ ⎩ (-⠂)su + u = (I & alpha; * |u|p)|u|p-2u in RN, lim |x|& RARR;& INFIN; u(x)=0, u & ISIN;Hs(RN), where (-⠂)s is the fractional Laplacian, I & alpha; is the Riesz potential, s & ISIN; (0, 1), 2s < N & ISIN; N, & alpha; & ISIN; (0, N) and p & ISIN; (N+& alpha; N , N+& alpha; N-2s ). Via studying limiting profiles of ground states of the above problem, we establish the uniqueness and non-degeneracy of positive ground states as & alpha; is close to 0 and & alpha; is close to N respectively. As a by-product, some uniform regularity and decay estimates for the solutions to the fractional Choquard equation, which are also of interest and importance independently, are given by taking full advantage of the Bessel kernel and employing an iterative process. & COPY; 2023 Elsevier Inc. All rights reserved.
期刊：
Journal of Differential Equations,2022年326:254-279 ISSN：0022-0396
通讯作者：
Shusen Yan
作者机构：
[Guo, Yuxia] Tsinghua Univ, Dept Math, Beijing 100084, Peoples R China.;[Musso, Monica] Univ Bath, Dept Math Sci, Bath BA2 7AY, Somerset, England.;[Peng, Shuangjie; Yan, Shusen] Cent China Normal Univ, Sch Math & Stat, Wuhan, Peoples R China.
通讯机构：
[Shusen Yan] S;School of Mathematics and Statistics, Central China Normal University, Wuhan, PR China
摘要：
We consider the following nonlinear problem & nbsp;-delta u + V (|y|)u = u(p), u > 0 in R-N, u is an element of H-1(R-N), (0.1)& nbsp;where V (r) is a positive function, 1 < p < N+2/N-2. We show that the multi-bump solutions constructed in [27] are non-degenerate in a suitable symmetric space. We also use this non-degenerate result to construct new solutions for (0.1). (C)& nbsp;2022 Elsevier Inc. All rights reserved.
作者机构：
[Qing Guo] College of Science, Minzu University of China;[Yuxia Guo] Department of Mathematical Sciences, Tsinghua University;[Shuangjie Peng] School of Mathematics and Statistics, Central China Normal University
摘要：
We consider the following fractional Schr?dinger equation:（-?）su+V （y）u=up, u> 0 in RN,（0.1）where s∈（0, 1）, 1 k is concentrated at the vertices of the regular k-polygon in the （y1, y2）-plane with k and the radius large enough. Then we show that ukis non-degenerate in our special symmetric workspace, and glue it with an n-spike solution, whose centers lie in another circle in the （y3, y4）-plane, to construct infinitely many multi-spike solutions of new type. The nonlocal property of （-?）sis in sharp contrast to the classical Schr?dinger equations. A striking difference is that although the nonlinear exponent in （0.1） is Sobolev-subcritical, the algebraic （not exponential） decay at infinity of the ground states makes the estimates more subtle and difficult to control. Moreover, due to the non-locality of the fractional operator, we cannot establish the local Pohozaev identities for the solution u directly, but we address its corresponding harmonic extension at the same time. Finally, to construct new solutions we need pointwise estimates of new approximation solutions. To this end, we introduce a special weighted norm, and give the proof in quite a different way.更多还原
期刊：
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2022年42(10):4669-4706 ISSN：1078-0947
通讯作者：
Shuangjie Peng
作者机构：
[Li, Qi] Wuhan Univ Sci & Technol, Coll Sci, Wuhan 430065, Peoples R China.;[Li, Qi] Wuhan Univ Sci & Technol, Hubei Prov Key Lab Syst Sci Met Proc, Wuhan 430065, Peoples R China.;[Peng, Shuangjie; Pan, Kefan] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Peng, Shuangjie; Pan, Kefan] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Shuangjie Peng] S;School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China
关键词：
$ k $-peak solutions;nonlinear fractional equation;Lyapunov-Schmidt reduction scheme
摘要：
<jats:p xml:lang="fr"><p style='text-indent:20px;'>This paper deals with the following nonlinear fractional equation with an external source term</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \label{eqS0.1} (-\Delta)^{s}u +u = K(x)u^{p}+f(x), \; u&gt;0, \; x\in{\Bbb R}^N, \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ N&gt;2s $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ 0&lt;s&lt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 1&lt;p&lt;2_{\ast}(s)-1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ 2_{\ast}(s) = \frac{2N}{N-2s} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ K(x) $\end{document}</tex-math></inline-formula> is a continuous function and <inline-formula><tex-math id="M6">\begin{document}$ f\in L^{2}({\Bbb R}^{N})\cap L^{\infty}({\Bbb R}^{N}) $\end{document}</tex-math></inline-formula>. Using a Lyapunov-Schmidt reduction scheme, we prove that the equation admits <inline-formula><tex-math id="M7">\begin{document}$ k $\end{document}</tex-math></inline-formula>-peak solutions for any integer <inline-formula><tex-math id="M8">\begin{document}$ k&gt;0 $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M9">\begin{document}$ f $\end{document}</tex-math></inline-formula> is small and <inline-formula><tex-math id="M10">\begin{document}$ K(x) $\end{document}</tex-math></inline-formula> satisfies some additional assumptions at infinity. The main difficulty is to improve the estimate of the remainder obtained in the reduction process.</p></jats:p>
期刊：
Electronic Research Archive,2022年30(2):585-614 ISSN：2688-1594
作者机构：
[An, Xiaoming] Sch Math & Stat, Guiyang 550025, Peoples R China.;[An, Xiaoming] Guizhou Univ Finance & Econ, Guiyang 550025, Peoples R China.;[Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Peng, Shuangjie] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
摘要：
<jats:p xml:lang="fr"><abstract><p>We study the following fractional Schrödinger equation</p>
<p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \label{eq0.1} \varepsilon^{2s}(-\Delta)^s u + V(x)u = f(u), \,\,x\in\mathbb{R}^N, \end{equation*} $\end{document} </tex-math> </disp-formula></p>
<p>where $ s\in(0,1) $. Under some conditions on $ f(u) $, we show that the problem has a family of solutions concentrating at any finite given local minima of $ V $ provided that $ V\in C( \mathbb{R}^N,[0,+\infty)) $. All decay rates of $ V $ are admissible. Especially, $ V $ can be compactly supported. Different from the local case $ s = 1 $ or the case of single-peak solutions, the nonlocal effect of the operator $ (-\Delta)^s $ makes the peaks of the candidate solutions affect mutually, which causes more difficulties in finding solutions with multiple bumps. The methods in this paper are penalized technique and variational method.</p></abstract></jats:p>
期刊：
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS,2022年152(4):879-911 ISSN：0308-2105
作者机构：
College of Science, Wuhan University of Science and Technology, Wuhan 430065, People's Republic of China;School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, People's Republic of China (qili@mails.ccnu.edu.cn);[Peng, Shuangjie] School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, People's Republic of China (sjpeng@mail.ccnu.edu.cn);[Li, Qi] College of Science, Wuhan University of Science and Technology, Wuhan 430065, People's Republic of China<&wdkj&>School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, People's Republic of China (qili@mails.ccnu.edu.cn)
摘要：
<jats:p>This paper deals with the following fractional elliptic equation with critical exponent
<jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" position="float" xlink:href="S0308210521000408_eqnU1.png" /><jats:tex-math>\[ \begin{cases} \displaystyle (-\Delta )^{s}u=u_{+}^{2_{s}^{*}-1}+\lambda u-\bar{\nu}\varphi_{1}, & \text{in}\ \Omega,\\ \displaystyle u=0, & \text{in}\ {{\mathfrak R}}^{N}\backslash \Omega, \end{cases}\]</jats:tex-math></jats:alternatives></jats:disp-formula>
where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210521000408_inline1.png" /><jats:tex-math>$\lambda$</jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210521000408_inline2.png" /><jats:tex-math>$\bar {\nu }\in {{\mathfrak R}}$</jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210521000408_inline3.png" /><jats:tex-math>$s\in (0,1)$</jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210521000408_inline4.png" /><jats:tex-math>$2^{*}_{s}=({2N}/{N-2s})\,(N>2s)$</jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210521000408_inline5.png" /><jats:tex-math>$(-\Delta )^{s}$</jats:tex-math></jats:alternatives></jats:inline-formula> is the fractional Laplace operator, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210521000408_inline6.png" /><jats:tex-math>$\Omega \subset {{\mathfrak R}}^{N}$</jats:tex-math></jats:alternatives></jats:inline-formula> is a bounded domain with smooth boundary and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210521000408_inline7.png" /><jats:tex-math>$\varphi _{1}$</jats:tex-math></jats:alternatives></jats:inline-formula> is the first positive eigenfunction of the fractional Laplace under the condition <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210521000408_inline8.png" /><jats:tex-math>$u=0$</jats:tex-math></jats:alternatives></jats:inline-formula> in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210521000408_inline9.png" /><jats:tex-math>${{\mathfrak R}}^{N}\setminus \Omega$</jats:tex-math></jats:alternatives></jats:inline-formula>. Under suitable conditions on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210521000408_inline10.png" /><jats:tex-math>$\lambda$</jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210521000408_inline11.png" /><jats:tex-math>$\bar {\nu }$</jats:tex-math></jats:alternatives></jats:inline-formula> and using a Lyapunov-Schmidt reduction method, we prove the fractional version of the Lazer-McKenna conjecture which says that the equation above has infinitely many solutions as <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210521000408_inline12.png" /><jats:tex-math>$|\bar \nu | \to \infty$</jats:tex-math></jats:alternatives></jats:inline-formula> .</jats:p>
作者机构：
[Li, Qi] Wuhan Univ Sci & Technol, Coll Sci, Wuhan 430065, Peoples R China.;[Shuai, W; Shuai, Wei; Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Shuai, W; Shuai, Wei; Peng, Shuangjie] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Li, Qi] Wuhan Univ Sci & Technol, Hubei Prov Key Lab Syst Sci Met Proc, Wuhan 430065, Peoples R China.
通讯机构：
[Shuai, W ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
作者：
Deng, Yinbin;Peng, Shuangjie;Zhang, Xinyue;Zhou, Yang
期刊：
Journal of Differential Equations,2022年341:150-188 ISSN：0022-0396
通讯作者：
Xinyue Zhang
作者机构：
[Deng, Yinbin; Zhou, Yang; Peng, Shuangjie; Zhang, Xinyue] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Deng, Yinbin; Zhou, Yang; Peng, Shuangjie; Zhang, Xinyue] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Xinyue Zhang] S;School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, PR China
摘要：
In this paper, we first deduce the following Sobolev inequality with logarithmic term: sup{integral(B) vertical bar u vertical bar(2)*vertical bar ln (tau + vertical bar u vertical bar)vertical bar(vertical bar x vertical bar beta) dx : u is an element of H-0,rad(1)(B), parallel to del u parallel to(L2(B)) = 1} < infinity, (0.1) B (0.1) where beta > 0, tau >= 0 are constants, B is the unit ball in R-N, N >= 3, and 2* = 2N/ (N - 2) is the critical Sobolev exponent. Then we show that the supremum in (0.1) is attained when 0 < beta < min{N/2, N - 2} and 1 <= tau < infinity. The inequality (0.1) can be used to prove the existence of positive solution for the following supercritical problem: {-Delta u = u(2)*(-1)(ln(tau + u))(vertical bar x vertical bar beta) + g(vertical bar x vertical bar, u), u > 0 in B, (0.2) u = 0 on partial derivative B, where g(r, u) is an element of C([0, 1) x R) is a subcritical perturbation. As a consequence, we can deduce the existence of positive solution for the supercritical problem with non-power nonlinearity: {- Lambda u= u(2)*(-1)(ln(tau + u))(vertical bar x vertical bar beta), u > 0 in B, (0.3) u=0 on partial derivative B. This is somewhat surprising, because the problem (0.3) has no nontrivial solution by Pohozaev's identity if the variable exponent vertical bar x vertical bar(beta) is replaced by any non-negative constant. (C) 2022 Elsevier Inc. All rights reserved.
期刊：
Calculus of Variations and Partial Differential Equations,2021年60(6):1-27 ISSN：0944-2669
通讯作者：
Shuangjie Peng
作者机构：
[Peng, Shuangjie; Luo, Yong; Guo, Yujin] Cent China Normal Univ, Sch Math & Stat, POB 71010, Wuhan 430079, Peoples R China.;[Peng, Shuangjie; Luo, Yong; Guo, Yujin] Cent China Normal Univ, Hubei Key Lab Math Sci, POB 71010, Wuhan 430079, Peoples R China.
通讯机构：
[Shuangjie Peng] S;School of Mathematics and Statistics, and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, P. R. China
摘要：
We study ground states of two-dimensional Bose-Einstein condensates with attractive interactions in a trap V(x) rotating at the velocity
$$\Omega $$
. It is known that there exists a critical rotational velocity
$$0<\Omega ^*:=\Omega ^*(V)\le \infty $$
and a critical number
$$0<a^*<\infty $$
such that for any rotational velocity
$$0\le \Omega <\Omega ^*$$
, ground states exist if and only if the coupling constant a satisfies
$$a<a^*$$
. For a general class of traps V(x), which may not be symmetric, we prove in this paper that up to a constant phase, there exists a unique ground state as
$$a\nearrow a^*$$
, where
$$\Omega \in (0,\Omega ^*)$$
is fixed.
