摘要：
In this paper, we mainly investigate ground states of trapped attractive Bose-Einstein condensates (BEC) passing an obstacle in the plane, which can be described by an L-2-critical constraint minimization problem in an exterior domain Omega = R-2\omega, where the bounded convex domain omega subset of R-2 with smooth boundary denotes the region of the obstacle. It is shown that minimizers (i.e. ground states) exist, if and only if the interaction strength a satisfies a < a* = parallel to Q parallel to(2)(2), where Q > 0 is the unique positive radial solution of Delta u - u + u(3) = 0 in R-2. If the trapping potential V(x) attains its global minima only along the whole boundary partial derivative Omega, the limit behavior of minimizers is analyzed as a NE arrow a* by employing the Pohozaev identity and the delicate energy analysis, where the mass concentration occurs at the flattest critical point of V (x) on a partial derivative Omega. (C) 2020 Published by Elsevier Inc.
作者机构：
[Deng, Yinbin; Shuai, Wei] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Deng, Yinbin; Shuai, Wei] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Jin, Qingfei] Jianghan Univ, Dept Math, Wuhan 430056, Peoples R China.
通讯机构：
[Shuai, Wei] 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.
关键词：
Choquard System;Positive;Ground State Solution;Nonlocal Term;35J91;35A01;35J20
摘要：
<jats:title>Abstract</jats:title>
<jats:p>We study the existence of positive ground state solution for Choquard systems. In the autonomous case, we prove the existence of at least one positive ground state solution by the Pohozaev manifold method and symmetric-decreasing rearrangement arguments. Moreover, we show that each positive ground state solution is radial symmetric. While, in the nonautonomous case, a positive ground state solution is obtained by using a monotonicity trick and a global compactness lemma. We remark that, under our assumptions of the nonlinearity <jats:inline-formula id="j_ans-2020-2099_ineq_9999">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:msub>
<m:mi>W</m:mi>
<m:mi>u</m:mi>
</m:msub>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2099_inl_001.png" />
<jats:tex-math>{W_{u}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, the search of ground state solutions cannot be reduced to the study of critical points of a functional restricted to a Nehari manifold.</jats:p>
作者机构：
[Deng, Yinbin] Guangxi Univ, Sch Math & Informat, Nanning 530004, Peoples R China.;[Deng, Yinbin; Zhang, Shen] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Huang, Wentao] East China JiaoTong Univ, Sch Sci, Nanchang 330013, Jiangxi, Peoples R China.
通讯机构：
[Deng, Yinbin] G;[Deng, Yinbin] C;Guangxi Univ, Sch Math & Informat, Nanning 530004, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
关键词：
Critical Growth;Ground State Solutions;Quasilinear Schrödinger Equation
摘要：
<jats:title>Abstract</jats:title>
<jats:p>We study the following generalized quasilinear Schrödinger equation:</jats:p>
<jats:p>
<jats:disp-formula id="j_ans-2018-2029_eq_9999">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mrow>
<m:mrow>
<m:mrow>
<m:mrow>
<m:mo>-</m:mo>
<m:mrow>
<m:mo stretchy="false">(</m:mo>
<m:mrow>
<m:msup>
<m:mi>g</m:mi>
<m:mn>2</m:mn>
</m:msup>
<m:mo></m:mo>
<m:mrow>
<m:mo stretchy="false">(</m:mo>
<m:mi>u</m:mi>
<m:mo stretchy="false">)</m:mo>
</m:mrow>
<m:mo></m:mo>
<m:mrow>
<m:mo>∇</m:mo>
<m:mo></m:mo>
<m:mi>u</m:mi>
</m:mrow>
</m:mrow>
<m:mo stretchy="false">)</m:mo>
</m:mrow>
</m:mrow>
<m:mo>+</m:mo>
<m:mrow>
<m:mi>g</m:mi>
<m:mo></m:mo>
<m:mrow>
<m:mo stretchy="false">(</m:mo>
<m:mi>u</m:mi>
<m:mo stretchy="false">)</m:mo>
</m:mrow>
<m:mo></m:mo>
<m:msup>
<m:mi>g</m:mi>
<m:mo>′</m:mo>
</m:msup>
<m:mo></m:mo>
<m:mrow>
<m:mo stretchy="false">(</m:mo>
<m:mi>u</m:mi>
<m:mo stretchy="false">)</m:mo>
</m:mrow>
<m:mo></m:mo>
<m:msup>
<m:mrow>
<m:mo stretchy="false">|</m:mo>
<m:mrow>
<m:mo>∇</m:mo>
<m:mo></m:mo>
<m:mi>u</m:mi>
</m:mrow>
<m:mo stretchy="false">|</m:mo>
</m:mrow>
<m:mn>2</m:mn>
</m:msup>
</m:mrow>
<m:mo>+</m:mo>
<m:mrow>
<m:mi>V</m:mi>
<m:mo></m:mo>
<m:mrow>
<m:mo stretchy="false">(</m:mo>
<m:mi>x</m:mi>
<m:mo stretchy="false">)</m:mo>
</m:mrow>
<m:mo></m:mo>
<m:mi>u</m:mi>
</m:mrow>
</m:mrow>
<m:mo>=</m:mo>
<m:mrow>
<m:mi>h</m:mi>
<m:mo></m:mo>
<m:mrow>
<m:mo stretchy="false">(</m:mo>
<m:mi>u</m:mi>
<m:mo stretchy="false">)</m:mo>
</m:mrow>
</m:mrow>
</m:mrow>
<m:mo rspace="12.5pt">,</m:mo>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>∈</m:mo>
<m:msup>
<m:mi>ℝ</m:mi>
<m:mi>N</m:mi>
</m:msup>
</m:mrow>
</m:mrow>
<m:mo>,</m:mo>
</m:mrow>
</m:math>
<jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2018-2029_fig_001.png" />
<jats:tex-math>-(g^{2}(u)\nabla u)+g(u)g^{\prime}(u)|\nabla u|^{2}+V(x)u=h(u),\quad x\in% \mathbb{R}^{N},</jats:tex-math>
</jats:alternatives>
</jats:disp-formula>
</jats:p>
<jats:p>where <jats:inline-formula id="j_ans-2018-2029_ineq_9999">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mi>N</m:mi>
<m:mo>≥</m:mo>
<m:mn>3</m:mn>
</m:mrow>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2018-2029_inl_001.png" />
<jats:tex-math>{N\geq 3}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, <jats:inline-formula id="j_ans-2018-2029_ineq_9998">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mi>g</m:mi>
<m:mo>:</m:mo>
<m:mrow>
<m:mi>ℝ</m:mi>
<m:mo>→</m:mo>
<m:msup>
<m:mi>ℝ</m:mi>
<m:mo>+</m:mo>
</m:msup>
</m:mrow>
</m:mrow>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2018-2029_inl_002.png" />
<jats:tex-math>{g\colon\mathbb{R}\rightarrow\mathbb{R}^{+}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is an even differentiable function such that <jats:inline-formula id="j_ans-2018-2029_ineq_9997">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mrow>
<m:msup>
<m:mi>g</m:mi>
<m:mo>′</m:mo>
</m:msup>
<m:mo></m:mo>
<m:mrow>
<m:mo stretchy="false">(</m:mo>
<m:mi>t</m:mi>
<m:mo stretchy="false">)</m:mo>
</m:mrow>
</m:mrow>
<m:mo>≥</m:mo>
<m:mn>0</m:mn>
</m:mrow>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2018-2029_inl_003.png" />
<jats:tex-math>{g^{\prime}(t)\geq 0}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> for all <jats:inline-formula id="j_ans-2018-2029_ineq_9996">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mi>t</m:mi>
<m:mo>≥</m:mo>
<m:mn>0</m:mn>
</m:mrow>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2018-2029_inl_004.png" />
<jats:tex-math>{t\geq 0}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, <jats:inline-formula id="j_ans-2018-2029_ineq_9995">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mi>h</m:mi>
<m:mo>∈</m:mo>
<m:mrow>
<m:msup>
<m:mi>C</m:mi>
<m:mn>1</m:mn>
</m:msup>
<m:mo></m:mo>
<m:mrow>
<m:mo stretchy="false">(</m:mo>
<m:mi>ℝ</m:mi>
<m:mo>,</m:mo>
<m:mi>ℝ</m:mi>
<m:mo stretchy="false">)</m:mo>
</m:mrow>
</m:mrow>
</m:mrow>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2018-2029_inl_005.png" />
<jats:tex-math>{h\in C^{1}(\mathbb{R},\mathbb{R})}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is a nonlinear function including critical growth and lower power subcritical perturbation, and the potential <jats:inline-formula id="j_ans-2018-2029_ineq_9994">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mrow>
<m:mi>V</m:mi>
<m:mo></m:mo>
<m:mrow>
<m:mo stretchy="false">(</m:mo>
<m:mi>x</m:mi>
<m:mo stretchy="false">)</m:mo>
</m:mrow>
</m:mrow>
<m:mo>:</m:mo>
<m:mrow>
<m:msup>
<m:mi>ℝ</m:mi>
<m:mi>N</m:mi>
</m:msup>
<m:mo>→</m:mo>
<m:mi>ℝ</m:mi>
</m:mrow>
</m:mrow>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2018-2029_inl_006.png" />
<jats:tex-math>{V(x)\colon\mathbb{R}^{N}\rightarrow\mathbb{R}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is positive. Since the subcritical perturbation does not satisfy the (AR) condition, the standard variational method cannot be used directly. Combining the change of variables and the monotone method developed by Jeanjean in [L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on <jats:inline-formula id="j_ans-2018-2029_ineq_9993">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:msup>
<m:mi>𝐑</m:mi>
<m:mi>N</m:mi>
</m:msup>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2018-2029_inl_007.png" />
<jats:tex-math>{\mathbf{R}}^{N}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, Proc. Roy. Soc. Edinburgh Sect. A 129 1999, 4, 787–809], we obtain the existence of positive ground state solutions for the given problem.</jats:p>
期刊：
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S,2019年12(7):1929-1954 ISSN：1937-1632
通讯作者：
Deng, Yinbin
作者机构：
[Deng, Yinbin] Guangxi Univ, Sch Math & Informat, Nanning 530004, Peoples R China.;[Deng, Yinbin] Cent China Normal Univ, Dept Math, Wuhan 430079, Hubei, Peoples R China.;[Huang, Wentao] East China JiaoTong Univ, Sch Sci, Nanchang 330013, Jiangxi, Peoples R China.
通讯机构：
[Deng, Yinbin] G;[Deng, Yinbin] C;Guangxi Univ, Sch Math & Informat, Nanning 530004, Peoples R China.;Cent China Normal Univ, Dept Math, Wuhan 430079, Hubei, Peoples R China.
关键词：
Fractional Kirchhoff equation;Nehari-Pohozaev manifold;least energy solutions;critical growth
摘要：
<p style='text-indent:20px;'>We study the following fractional Kirchhoff type equation:</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE1">\begin{document}$ \begin{equation*} \begin{array}{ll} \left \{ \begin{array}{ll} \Big(a+b\int_{ \mathbb{R} ^3}|(-\Delta)^\frac{s}{2}u|^2dx\Big)(-\Delta )^s u+V(x)u = f(u)+|u|^{2^*_s-2}u, \ x\in \mathbb{R} ^3, \\ u\in H^s( \mathbb{R} ^3), \end{array} \right . \end{array} \end{equation*} $\end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ a, \ b>0 $\end{document}</tex-math></inline-formula> are constants, <inline-formula><tex-math id="M2">\begin{document}$ 2^*_s = \frac{6}{3-2s} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ s\in(0, 1) $\end{document}</tex-math></inline-formula> is the critical Sobolev exponent in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R} ^3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ V $\end{document}</tex-math></inline-formula> is a potential function on <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{R} ^3 $\end{document}</tex-math></inline-formula>. Under some more general assumptions on <inline-formula><tex-math id="M7">\begin{document}$ f $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ V $\end{document}</tex-math></inline-formula>, we prove that the given problem admits a least energy solution by using a constrained minimization on Nehari-Pohozaev manifold and monotone method.</p>
期刊：
Calculus of Variations and Partial Differential Equations,2019年58(1):1-26 ISSN：0944-2669
通讯作者：
Deng, Yinbin
作者机构：
[Deng, Yinbin] Cent China Normal Univ, Dept Math, Wuhan 430079, Hubei, Peoples R China.;[Guo, Yuxia] Tsinghua Univ, Dept Math Sci, Beijing, Peoples R China.;[Yan, Shusen] Univ New England, Dept Math, Armidale, NSW 2351, Australia.
通讯机构：
[Deng, Yinbin] C;Cent China Normal Univ, Dept Math, Wuhan 430079, Hubei, Peoples R China.
摘要：
We study the existence of infinitely many solutions for the following quasilinear elliptic equations with critical growth:
where
$$ b_{ij}\in C^{1}(\mathbb {R},\mathbb {R})$$
satisfies the growth condition
$$|b_{ij}(t)|\sim |t|^{2s-2}$$
at infinity,
$$s\ge 1$$
,
$$\Omega \subset \mathbb {R}^N$$
is an open bounded domain with smooth boundary, a is a constant. Here we use the notations:
$$D_i=\frac{\partial }{\partial x_i}, b'_{ij}(t)=\frac{db_{ij}(t)}{dt}.$$
We will study the effect of the terms
$$a|v|^{2s-2}v$$
and
$$b_{ij}(v)$$
on the existence of an unbounded sequence of solutions for (P). Here, we do not assume the crucial global monotone condition. We overcome the difficulties caused by the lack of such monotone condition by performing various kinds of changes of variables.
期刊：
Journal of Mathematical Physics,2018年59(1):011503 ISSN：0022-2488
通讯作者：
Deng, Yinbin
作者机构：
[Deng, Yinbin; Guo, Yujin] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Lu, Lu] Zhongnan Univ Econ & Law, Sch Stat & Math, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Deng, Yinbin] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
关键词：
ground states;Schrodinger equation
摘要：
We study ground states of mass critical Schrodinger equations with spatially inhomogeneous nonlinearities in R-2 by analyzing the associated L-2-constraint Gross-Pitaevskii energy functionals. In contrast to the homogeneous case where m(x) equivalent to 1, we prove that both the existence and nonexistence of ground states may occur at the threshold a* depending on the inhomogeneity of m(x). Under some assumptions on m(x) and the external potential V(x), the uniqueness and radial symmetry of ground states are analyzed for almost every a is an element of[0, a*). When there is no ground state at the threshold a*, the limit behavior of ground states as a NE arrow a* is also investigated if V(x) reaches its global minimum in a domain Omega with positive Lebesgue measure and m(x) attains its global maximum at finite points. We show that all the mass concentrates at a flattest global maximum of m(x) within Omega. Published by AIP Publishing.
期刊：
ADVANCES IN DIFFERENTIAL EQUATIONS,2018年23(1-2):109-134 ISSN：1079-9389
通讯作者：
Deng, Yinbin
作者机构：
[Deng, Yinbin] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Deng, Yinbin] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
关键词：
In this paper;we consider the existence of sign-changing solutions for fractional elliptic equations of the form \begin{equation*} \left\{\begin{array}{ll} (-\Delta)^s u=f(x;u) & \text{in}\ \Omega;\\ u=0 & \text{in}\ \mathbb R^N\setminus \Omega;\end{array} \right. \end{equation*} where $s\in(0;1)$ and $\Omega\subset \mathbb R^N$ is a bounded smooth domain. Since the non-local operator $(-\Delta)^s$ is involved in the equation;the variational functional of the equation has totally different properties from the local cases. By introducing some new ideas;we prove;via variational method and the method of invariant sets of descending flow;that the problem has a positive solution;a negative solution and a sign-changing solution under suitable conditions. Moreover;if $f(x;u)$ satisfies a monotonicity condition;we show that the problem has a least energy sign-changing solution with its energy is strictly larger than that of the ground state solution of Nehari type. We also obtain an unbounded sequence of sign-changing solutions if $f(x;u)$ is odd in $u$. Published: January/February 2018 First available in Project Euclid: 26 October 2017 zbMATH: 06822195 MathSciNet: MR3718170 Digital Object Identifier: 10.57262/ade/1508983363 Subjects: Primary: 35J60;35R11;47J30;58E05
摘要：
In this paper, we consider the existence of sign-changing solutions for fractional elliptic equations of the form where s ∈ (0, 1) and Ω ⊂ ℝN is a bounded smooth domain. Since the non-local operator (-Δ)s is involved in the equation, the variational functional of the equation has totally different properties from the local cases. By introducing some new ideas, we prove, via variational method and the method of invariant sets of descending flow, that the problem has a positive solution, a negative solution and a sign-changing solution under suitable conditions. Moreover, if f(x; u) satisfies a monotonicity condition, we show that the problem has a least energy sign-changing solution with its energy is strictly larger than that of the ground state solution of Nehari type. We also obtain an unbounded sequence of sign-changing solutions if f(x, u) is odd in u.
期刊：
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2018年38(6):3139-3168 ISSN：1078-0947
通讯作者：
Shuai, Wei
作者机构：
[Deng, Yinbin; Shuai, Wei] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Deng, Yinbin; Shuai, Wei] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.;[Shuai, Wei] Chinese Univ Hong Kong, Inst Math Sci, Shatin, Hong Kong, Peoples R China.
通讯机构：
[Shuai, Wei] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.;Chinese Univ Hong Kong, Inst Math Sci, Shatin, Hong Kong, Peoples R China.
摘要：
We are interested in the existence of sign-changing multi-bump solutions for the following Kirchhoff equation -(a + b integral(R3) vertical bar del u vertical bar(2) dx)Delta u + lambda V(x)u = f(u), x is an element of R-3, where lambda > 0 is a parameter and the potential V(x) is a nonnegative continuous function with a potential well Omega := int(V (-1)(0)) which possesses k disjoint bounded components Omega(1), Omega(2), . . . , Omega(k). Under some conditions imposed on f(u), multiple sign-changing multi-bump solutions are obtained. Moreover, the concentration behavior of these solutions as lambda -> +infinity are also studied.
期刊：
Journal of Differential Equations,2018年264(6):4006-4035 ISSN：0022-0396
通讯作者：
Deng, Yinbin;Shuai, Wei
作者机构：
[Deng, Yinbin; Shuai, Wei; Peng, Shuangjie; Deng, YB; Shuai, W] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Deng, Yinbin; Shuai, Wei; Peng, Shuangjie; Deng, YB; Shuai, W] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.;[Shuai, Wei] Chinese Univ Hong Kong, Inst Math Sci, Shatin, Hong Kong, Peoples R China.
通讯机构：
[Deng, YB; Shuai, W; Shuai, Wei] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.;Chinese Univ Hong Kong, Inst Math Sci, Shatin, Hong Kong, Peoples R China.
摘要：
This paper investigates the existence and asymptotic behavior of nodal solutions to the following gauged nonlinear Schrodinger equation {-Delta u+omega u+(h(2)(vertical bar x vertical bar)/vertical bar x vertical bar(2) +integral(+infinity)(vertical bar x vertical bar) h(s)/su(2) (s)ds) = lambda vertical bar u vertical bar(p-2)u, x is an element of R-2, u(x) = u(vertical bar x vertical bar) is an element of H-1 (R-2), where omega, lambda > 0, p > 6 and h(s) = 1/2 integral(s)(0) ru(2)(r)dr is the so-called Chern-Simons term. We prove that for any positive integer k, the problem has a signchanging solution u(k)(lambda) which changes sign exactly ktimes. Moreover, the energy of u(k)(lambda) is strictly increasing in k, and for any sequence {lambda(n)} -> +infinity (n ->infinity), there exists a subsequence {lambda(ns)}, such that (lambda(ns))(1/p-2)u(k)(lambda ns) converges in H-1(R-2) to w(k) as s ->infinity, where w(k) also changes sign exactly ktimes and solves the following equation -Delta u + omega u =vertical bar u vertical bar(p-2)u, u is an element of H-1 (R-2). (c) 2017 Elsevier Inc. All rights reserved.
期刊：
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2017年37(8):4213-4230 ISSN：1078-0947
通讯作者：
Deng, Yinbin
作者机构：
[Deng, Yinbin; Huang, Wentao] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Deng, Yinbin] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Ground state solutions;quasilinear elliptic equation;critical exponent
摘要：
In this paper, we study the following quasilinear elliptic equation with critical Sobolev exponent: $ -\Delta u +V(x)u-[\Delta(1+u^2)^{\frac 12}]\frac {u}{2(1+u^2)^\frac 12}=|u|^{2^*-2}u+|u|^{p-2}u, \quad x\in {{\mathbb{R}}^{N}}, $ which models the self-channeling of a high-power ultra short laser in matter, where N ≥ 3; 2 < p < 2* = $\frac{{2N}}{{N -2}}$ and V (x) is a given positive potential. Combining the change of variables and an abstract result developed by Jeanjean in [14], we obtain the existence of positive ground state solutions for the given problem.
摘要：
This paper is concerned with constructing nodal radial solutions for generalized quasilinear Schrödinger equations in RN with critical growth which arise from plasma physics, uid mechanics, as wells the self-channeling of a high-power ultashort laser in matter. We nd the critical exponents for a generalized quasilinear Schrödinger equations and obtain the existence of sign-changing solution with k nodes for any given integer k ≥ 0.
期刊：
Journal of Mathematical Physics,2016年57(3):031503 ISSN：0022-2488
通讯作者：
Deng, Yinbin;Guo, Yuxia;Liu, Jiaquan
作者机构：
[Deng, Yinbin] Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.;[Guo, Yuxia] Tsinghua Univ, Dept Math, Beijing 100084, Peoples R China.;[Liu, Jiaquan] Peking Univ, LMAM, Sch Math Sci, Beijing 100871, Peoples R China.
通讯机构：
[Deng, Yinbin] C;[Guo, Yuxia] T;[Liu, Jiaquan] P;Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.;Tsinghua Univ, Dept Math, Beijing 100084, Peoples R China.
关键词：
elliptic equations;perturbation theory
摘要：
In this paper, we consider the following quasilinear elliptic equation with Hardy potential and Dirichlet boundary condition: -Sigma(N)(i,j=1) D-j(a(i j)(x, u)D(i)u) + 1/2 Sigma(N)(i,j=1) D(s)a(i,j)(x, u)D(i)uD(j)u - lambda|x|(-2)u = f (x, u) in Omega, where Omega subset of R-N (N >= 3) is a smooth bounded domain, D-i = partial derivative/partial derivative x(i), D(s)a(i j)(x, s) = partial derivative/partial derivative s a(i j)(x, s), and 0 <= lambda < lambda* := (N-2/2)(2), and lambda|x|(-2) is called the Hardy potential. By using the perturbation method, we prove the existence of infinitely many solutions for the above problem. (C) 2016 AIP Publishing LLC.
摘要：
We consider the quasilinear Schrodinger equations of the form where ε > 0 is a small parameter, the nonlinearity g(u) ∈ C~1(R) is an odd function with subcritical growth and V (x) is a positive Holder continuous function which is bounded from below, away from zero, and inf_Λ V (x) < inf_(?Λ) V (x) for some open bounded subset Λ of R~N. We prove that there is an ε0 > 0 such that for all ε ∈ (0, ε0], the above mentioned problem possesses a sign-changing solution uε which exhibits concentration profile around the local minimum point of V (x) as ε → 0+.
期刊：
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2016年36(2):683-699 ISSN：1078-0947
通讯作者：
Deng, Yinbin
作者机构：
[Deng, Yinbin; Shuai, Wei] Huazhong Normal Univ, Dept Math, Wuhan 430079, Peoples R China.;[Li, Yi] Wright State Univ, Dept Math & Stat, Dayton, OH 45435 USA.
通讯机构：
[Deng, Yinbin] H;Huazhong Normal Univ, Dept Math, Wuhan 430079, Peoples R China.
会议名称：
2013 Workshop on Variational Problems and Evolution Equations
会议时间：
JUL 22-25, 2013
会议地点：
Xiamen Univ, Xiamen, PEOPLES R CHINA
会议主办单位：
Xiamen Univ
关键词：
p-Laplacain type equations;weighted Sobolev space;critical growth;variational method.;vanishing potential
摘要：
In this paper, we study the existence of positive
solution for the following p-Laplacain type equations with critical nonlinearity
\begin{equation*}
\left\{
\renewcommand{\arraystretch}{1.25}
\begin{array}{ll}
-\Delta_p u + V （x)|u|^{p-2}u = K(x)f(u)+P(x)|u|^{p^*-2}u, \quad
x\in\mathbb{R}^N,\\
u \in \mathcal{D}^{1,p}(\mathbb{R}^N),
\end{array}
\right.
\end{equation*}
where $\Delta_p u = div(|\nabla u|^{p-2} \nabla u),\ 1 < p < N,\ p^* =\frac
{Np}{N-p}$, $V(x)$, $K(x)$ are positive continuous functions which vanish at
infinity, $f$ is a function with a subcritical growth, and $P(x)$ is a bounded,
nonnegative continuous function.
By working in the weighted Sobolev spaces, and using variational method, we
prove that the given problem has at least one positive solution.
摘要：
This paper is concerned with the solitary wave solutions for a generalized quasilinear Schrodinger equation in RN involving critical exponents, which have appeared from plasma physics, as well as high-power ultashort laser in matter. We find the related critical exponents for a generalized quasilinear Schrodinger equation and obtain its solitary wave solutions by using a change of variables and variational argument. (C) 2015 Elsevier Inc. All rights reserved.
摘要：
This paper is concerned with the positive solutions for generalized quasilinear Schrodinger equations in R-N with critical growth which have appeared from plasma physics, as well as high-power ultrashort laser in matter. By using a change of variables and variational argument, we obtain the existence of positive solutions for the given problem. (C) 2014 Elsevier Inc. All rights reserved.
期刊：
Journal of Mathematical Physics,2015年56(6):061503 ISSN：0022-2488
通讯作者：
Deng, Yinbin
作者机构：
[Deng, Yinbin; Lu, Lu; Shuai, Wei] Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.
通讯机构：
[Deng, Yinbin] C;Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.
关键词：
Bose-Einstein condensation;HF calculations
摘要：
We consider L-2-constraint minimizers of mass critical Hartree energy functionals in R-N with N >= 3. We prove that minimizers exist if and only if the parameter a > 0 satisfies a < a* =parallel to Q parallel to(2)(2), where Q is a positive radially symmetric ground state of Delta u - u + (integral(RN)|u(y)|(2)/|x-y|(2)dy)(u) = 0 in R-N. The blow-up behavior of minimizers as a approaches a* is also analyzed, for which all the mass concentrates at a global minimum point x(0) of the external potential V(x). (C) 2015 AIP Publishing LLC.
