作者机构：
[Guo, Yuxia] Tsinghua Univ, Dept Math Sci, Beijing 100084, 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.;[Yan, Shusen] Univ New England, Dept Math, Armidale, NSW 2351, Australia.
通讯机构：
[Guo, Yuxia] T;Tsinghua Univ, Dept Math Sci, Beijing 100084, Peoples R China.
关键词：
35J91 (primary);35B33 (secondary)
摘要：
We consider poly-harmonic equations with critical exponents. Under some conditions on the coefficient K(y) in the equations near its critical points, we prove the existence and local uniqueness of solutions with infinitely many bubbles. The local uniqueness result implies that some bubbling solutions preserve the symmetry of the scalar curvature K(y). Moreover, we also show that the conditions imposed are optimal to obtain such results.
作者机构：
[Peng, Shuangjie; Shuai, Wei] Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.;[Shuai, Wei] Chinese Univ Hong Kong, Inst Math Sci, Shatin, Hong Kong, Peoples R China.;[Wang, Qingfang] Wuhan Polytech Univ, Sch Math & Comp Sci, Wuhan 430023, Peoples R China.
通讯机构：
[Shuai, Wei] C;Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.
摘要：
This paper deals with the following system linearly coupled by nonlinear elliptic equations {-Delta u + lambda(1)u = vertical bar u vertical bar(2)*(-2)u + beta v, x is an element of Omega, -Delta u + lambda(2)u = vertical bar v vertical bar(2)*(-2)v + beta u, x is an element of Omega u = v = 0 on partial derivative Omega. Here Omega is a smooth bounded domain in R-N(N >= 3), lambda(1), lambda(2) > -lambda(1)(Omega) are constants, lambda(1)(Omega) is the first eigenvalue of (-Delta, H-0(1) (Omega)), 2* = 2N/N-2 is the Sobolev critical exponent and beta is an element of R is a coupling parameter. By variational method, we prove that this system has a positive ground state solution for some beta > 0. Via a perturbation argument, we show that this system also admits a positive higher energy solution when vertical bar beta vertical bar is small. Moreover, the asymptotic behaviors of the positive ground state and higher energy solutions as beta -> 0 are analyzed. (C) 2017 Elsevier Inc. All rights reserved.
作者机构：
[Peng, Shuangjie; Lu, Dengfeng] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Peng, Shuangjie; Lu, Dengfeng] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Lu, Dengfeng] Hubei Engn Univ, Sch Math & Stat, Xiaogan 432000, Peoples R China.
通讯机构：
[Lu, Dengfeng] C;[Lu, Dengfeng] H;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.;Hubei Engn Univ, Sch Math & Stat, Xiaogan 432000, Peoples R China.
关键词：
Asymptotic behavior;Berestycki-Lions type conditions;Fractional Laplacian system;Pohozaev manifold;Vector ground state solution
摘要：
In this paper, a class of systems of two coupled nonlinear fractional Laplacian equations are investigated. Under very weak assumptions on the nonlinear terms f and g, we establish some results about the existence of positive vector solutions and vector ground state solutions for the fractional Laplacian systems by using variational methods. In addition, we also study the asymptotic behavior of these solutions as the coupling parameter β tends to zero.
作者机构：
[Liu, Zhongyuan] Henan Univ, Sch Math & Stat, Kaifeng 475004, Henan, Peoples R China.;[Peng, Shuangjie] Cent China Normal Univ, Hubei Key Lab Math Phys, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Peng, Shuangjie] C;[Liu, Zhongyuan] H;Henan Univ, Sch Math & Stat, Kaifeng 475004, Henan, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Phys, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
摘要：
This paper is concerned with the Henon equation {-Delta u = vertical bar y vertical bar(alpha) u(p+epsilon), u > 0, in B-1(0), u = 0 on partial derivative B-1(0), where B-1(0) is the unit ball in R-N (N >= 4), p = (N + 2)/(N - 2) is the critical Sobolev exponent, alpha > 0 and epsilon > 0. We show that if epsilon is small enough, this problem has a positive peak solution which presents a new phenomenon: the number of its peaks varies with the parameter epsilon at the order epsilon(-1/(N - 1)) when epsilon -> 0(+). Moreover, all peaks of the solutions approach the boundary of B-1(0) as epsilon goes to 0(+).
作者机构：
[Liu, Zhongyuan] Henan Univ, Sch Math & Stat, Kaifeng 475004, Henan, Peoples R China.;[Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Liu, Zhongyuan] H;Henan Univ, Sch Math & Stat, Kaifeng 475004, Henan, Peoples R China.
摘要：
In this paper we study the following Henon-like equation { -Delta u = broken vertical bar broken vertical bar y broken vertical bar - 2 broken vertical bar(alpha) u(p), u > 0 in Omega u = 0, on partial derivative Omega, where alpha > 0, p =N + 2/N- 2, Omega = {y is an element of R-N We show that for a > 0 the above problem has infinitely many positive solutions concentrating simultaneously near the interior boundary {x ERN Ix I = 1) and the outward boundary {x is an element of R-N : broken vertical bar X broken vertical bar = 3), whose energy can be made arbitrarily large. (C) 2015 Elsevier Inc. All rights reserved.
作者机构：
[Peng, Shuangjie; Long, Wei; Yang, Jing] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Long, Wei] Jiangxi Normal Univ, Coll Math & Informat Sci, Nanchang 330022, Jiangxi, Peoples R China.
通讯机构：
[Long, Wei] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
会议地点：
Xiamen Univ, Xiamen, PEOPLES R CHINA
会议主办单位：
Xiamen Univ
关键词：
Fractional Laplacian;Nonlinear scalar field equation;Reduction method
摘要：
We consider the following nonlinear fractional scalar field equation $$ (-\Delta)^s u + u = K(|x|)u^p,\ \ u > 0 \ \ \hbox{in}\ \ \mathbb{R}^N, $$ where $K(|x|)$ is a positive radial function, $N\ge 2$, $0 < s < 1$, and $1 < p < \frac{N+2s}{N-2s}$. Under various asymptotic assumptions on $K(x)$ at infinity, we show that this problem has infinitely many non-radial positive solutions and sign-changing solutions, whose energy can be made arbitrarily large.
作者机构：
[Peng, Shuangjie; Long, Wei] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Long, Wei] Jiangxi Normal Univ, Coll Math & Informat Sci, Nanchang 330022, Jiangxi, Peoples R China.;[Peng, Shuangjie] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Long, Wei] C;[Long, Wei] J;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Jiangxi Normal Univ, Coll Math & Informat Sci, Nanchang 330022, Jiangxi, Peoples R China.
摘要：
This paper is concerned with the existence of multiple non-radial positive solutions for {-Delta u + (1 + beta V(y))u = vertical bar u vertical bar(p-2)u y is an element of R-N u(y) -> 0 as vertical bar y vertical bar -> + infinity where 2 < p < 2*, 2* = 2N/N-2 for N > 2 and 2* = +infinity for N = 2, beta can be regarded as a parameter and V(vertical bar y vertical bar) > 0 decays exponentially to zero at infinity. We prove that, for any positive integer k > 1, there exists a suitable range of beta such that the above problem has a non-radial positive solution with exactly k maximum points which tending to infinity as beta -> +infinity(or 0(+)).
作者机构：
[Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Peng, Yan-fang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Wang, Zhi-Qiang] Tianjin Univ, Ctr Appl Math, Tianjin 300072, Peoples R China.;[Wang, Zhi-Qiang] Utah State Univ, Dept Math & Stat, Logan, UT 84322 USA.
通讯机构：
[Wang, Zhi-Qiang] T;[Wang, Zhi-Qiang] U;Tianjin Univ, Ctr Appl Math, Tianjin 300072, Peoples R China.;Utah State Univ, Dept Math & Stat, Logan, UT 84322 USA.
关键词：
35J47;35J50;35J57
摘要：
In this paper, we study the following Dirichlet problem with Sobolev critical exponent
$$\begin{aligned} \left\{ \begin{array}{ll}\displaystyle -\Delta u=|u|^{2^*-2}u+\displaystyle \frac{\alpha }{2^*}|u|^{\alpha -2}|v|^{\beta }u,&{}\quad x\in \Omega , \\ -\Delta v=|v|^{2^*-2}v+\displaystyle \frac{\beta }{2^*}|u|^{\alpha }|v|^{\beta -2}v,&{}\quad x\in \Omega , \end{array} \right. \end{aligned}$$
where
$$\alpha , \beta >1,$$
$$\alpha +\beta =2^*:=\frac{2N}{N-2}(N\ge 3)$$
and
$$\Omega ={\mathbb {R}}^N$$
or
$$\Omega $$
is a smooth bounded domain in
$${\mathbb {R}}^N$$
. When
$$\Omega ={\mathbb {R}}^N$$
, we obtain a uniqueness result on the least energy solutions and show that a manifold of the synchronized type of positive solutions is non-degenerate for the above system for some ranges of the parameters
$$\alpha , \beta , N$$
. Our analysis also yields non-uniqueness of positive vector solutions for other parameters. Moreover, we establish a global compactness result and we extend a classical result of Coron on the existence of positive solutions of scalar equations with critical exponent on domains with nontrivial topology to the above elliptic system.
作者机构：
[Peng, Shuangjie; He, Qihan] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Peng, Shuangjie; He, Qihan] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[He, Qihan] 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：
<p>In this paper, we establish a relationship between the elliptic system <disp-formula content-type="math/mathml">
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row minus normal upper Delta u plus lamda u equals mu 1 StartAbsoluteValue u EndAbsoluteValue Superscript 2 p Baseline u plus beta 1 StartAbsoluteValue v EndAbsoluteValue Superscript q 1 Baseline StartAbsoluteValue u EndAbsoluteValue Superscript p 1 minus 1 Baseline u comma x element-of normal upper Omega comma 2nd Row minus normal upper Delta v plus lamda v equals mu 2 StartAbsoluteValue v EndAbsoluteValue Superscript 2 p Baseline v plus beta 2 StartAbsoluteValue u EndAbsoluteValue Superscript q 2 Baseline StartAbsoluteValue v EndAbsoluteValue Superscript p 2 minus 1 Baseline v comma x element-of normal upper Omega comma 3rd Row u equals v equals 0 on partial-differential normal upper Omega comma EndLayout">
<mml:semantics>
<mml:mrow>
<mml:mo>{</mml:mo>
<mml:mtable columnalign="left left" rowspacing="4pt" columnspacing="1em">
<mml:mtr>
<mml:mtd>
<mml:mo>−<!-- − --></mml:mo>
<mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>+</mml:mo>
<mml:mi>λ<!-- λ --></mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mi>μ<!-- μ --></mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">|</mml:mo>
</mml:mrow>
<mml:mi>u</mml:mi>
<mml:msup>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">|</mml:mo>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mn>2</mml:mn>
<mml:mi>p</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mi>u</mml:mi>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mi>β<!-- β --></mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">|</mml:mo>
</mml:mrow>
<mml:mi>v</mml:mi>
<mml:msup>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">|</mml:mo>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>q</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mrow>
</mml:msup>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">|</mml:mo>
</mml:mrow>
<mml:mi>u</mml:mi>
<mml:msup>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">|</mml:mo>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>−<!-- − --></mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mi>u</mml:mi>
<mml:mo>,</mml:mo>
<mml:mtext></mml:mtext>
<mml:mtext></mml:mtext>
<mml:mi>x</mml:mi>
<mml:mo>∈<!-- ∈ --></mml:mo>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mo>,</mml:mo>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mo>−<!-- − --></mml:mo>
<mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi>
<mml:mi>v</mml:mi>
<mml:mo>+</mml:mo>
<mml:mi>λ<!-- λ --></mml:mi>
<mml:mi>v</mml:mi>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mi>μ<!-- μ --></mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">|</mml:mo>
</mml:mrow>
<mml:mi>v</mml:mi>
<mml:msup>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">|</mml:mo>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mn>2</mml:mn>
<mml:mi>p</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mi>v</mml:mi>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mi>β<!-- β --></mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">|</mml:mo>
</mml:mrow>
<mml:mi>u</mml:mi>
<mml:msup>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">|</mml:mo>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>q</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:msup>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">|</mml:mo>
</mml:mrow>
<mml:mi>v</mml:mi>
<mml:msup>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">|</mml:mo>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo>−<!-- − --></mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mi>v</mml:mi>
<mml:mo>,</mml:mo>
<mml:mtext></mml:mtext>
<mml:mtext></mml:mtext>
<mml:mi>x</mml:mi>
<mml:mo>∈<!-- ∈ --></mml:mo>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mo>,</mml:mo>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mi>u</mml:mi>
<mml:mo>=</mml:mo>
<mml:mi>v</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>0</mml:mn>
<mml:mtext></mml:mtext>
<mml:mtext></mml:mtext>
<mml:mstyle displaystyle="false" scriptlevel="0">
<mml:mtext>on</mml:mtext>
</mml:mstyle>
<mml:mtext></mml:mtext>
<mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mo>,</mml:mo>
</mml:mtd>
</mml:mtr>
</mml:mtable>
<mml:mo fence="true" stretchy="true" symmetric="true" />
</mml:mrow>
<mml:annotation encoding="application/x-tex">\left \{ \begin {array}{ll} -\Delta u +\lambda u=\mu _1 |u|^{2p}u+\beta _1 |v|^{q_1} |u|^{p_1-1}u,~~x\in \Omega ,\\ -\Delta v +\lambda v=\mu _2 |v|^{2p}v+\beta _2 |u|^{q_2} |v|^{p_2-1}v,~~x\in \Omega ,\\ u=v=0~~\hbox {on}~ \partial \Omega ,\\ \end {array} \right .</mml:annotation>
</mml:semantics>
</mml:math>
\]
</disp-formula> and its corresponding single elliptic problem, where <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda element-of double-struck upper R">
<mml:semantics>
<mml:mrow>
<mml:mi>λ<!-- λ --></mml:mi>
<mml:mo>∈<!-- ∈ --></mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="double-struck">R</mml:mi>
</mml:mrow>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\lambda \in \mathbb {R}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta Subscript i Baseline greater-than 0 comma mu Subscript i Baseline greater-than 0 comma p Subscript i Baseline comma q Subscript i Baseline greater-than-or-equal-to 0 comma 1 greater-than p Subscript i Baseline plus q Subscript i Baseline equals 2 p plus 1">
<mml:semantics>
<mml:mrow>
<mml:msub>
<mml:mi>β<!-- β --></mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo>></mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>μ<!-- μ --></mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo>></mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>q</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo>≥<!-- ≥ --></mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>></mml:mo>
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mi>q</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:mn>2</mml:mn>
<mml:mi>p</mml:mi>
<mml:mo>+</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\beta _i>0, \mu _i>0, p_i,q_i\ge 0, 1>p_i+q_i =2p+1</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> for <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i equals 1 comma 2">
<mml:semantics>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>,</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
<mml:annotation encoding="application/x-tex">i=1,2</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, and <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega subset-of double-struck upper R Superscript upper N Baseline left-parenthesis upper N greater-than-or-equal-to 1 right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mo>⊂<!-- ⊂ --></mml:mo>
<mml:msup>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="double-struck">R</mml:mi>
</mml:mrow>
<mml:mi>N</mml:mi>
</mml:msup>
<mml:mspace width="thinmathspace" />
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>N</mml:mi>
<mml:mo>≥<!-- ≥ --></mml:mo>
<mml:mn>1</mml:mn>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\Omega \subset \mathbb {R}^N\,(N\ge 1)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> can be a bounded or unbounded domain. By using this fact, we can obtain many results on the existence, non-existence and uniqueness of classical vector solutions to this system via the related single elliptic problem.</p>
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.
作者机构：
[Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Pi, Huirong] E China Normal Univ, Ctr Partial Differential Equat, Shanghai 200241, Peoples R China.
通讯机构：
[Peng, Shuangjie] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Coupled nonlinear Schrödinger equations;asymptotic behavior;spike vector solutions;Lyapunov-Schmidt reduction;critical point
摘要：
We consider spike vector solutions for the nonlinear Schrödinger system \begin{equation*} \left\{ \begin{array}{ll} -\varepsilon^{2}\Delta u+P(x)u=\mu u^{3}+\beta v^2u \ \hbox{in}\ \mathbb{R}^3,\\ -\varepsilon^{2}\Delta v+Q(x)v=\nu v^{3} +\beta u^2v \ \ \hbox{in}\ \mathbb{R}^3,\\ u, v >0 \,\ \hbox{in}\ \mathbb{R}^3, \end{array} \right. \end{equation*} where $\varepsilon > 0$ is a small parameter, $P(x)$ and $Q(x)$ are positive potentials, $\mu>0, \nu>0$ are positive constants and $\beta\neq 0$ is a coupling constant. We investigate the effect of potentials and the nonlinear coupling on the solution structure. For any positive integer $k\ge 2$, we construct $k$ interacting spikes concentrating near the local maximum point $x_{0}$ of $P(x)$ and $Q(x)$ when $P(x_{0})=Q(x_{0})$ in the attractive case. In contrast, for any two positive integers $k\ge 2$ and $m\ge 2$, we construct $k$ interacting spikes for $u$ near the local maximum point $x_{0}$ of $P(x)$ and $m$ interacting spikes for $v$ near the local maximum point $\bar{x}_{0}$ of $Q(x)$ respectively when $x_{0}\neq \bar{x}_{0}$, moreover, spikes of $u$ and $v$ repel each other. Meanwhile, we prove the attractive phenomenon for $\beta < 0$ and the repulsive phenomenon for $\beta > 0$.
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.
作者机构：
[Li, Yi] Wright State Univ, Dept Math & Stat, Dayton, OH 45435 USA.;[Peng, Shuangjie] Cent China Normal Univ, Sch Math, Wuhan 430079, Peoples R China.;[Peng, Shuangjie] Cent China Normal Univ, Hubei Key Lab Math Phys, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Yi] W;Wright State Univ, Dept Math & Stat, Dayton, OH 45435 USA.
作者机构：
[Peng, Shuangjie; Deng, Yinbin; Shuai, Wei] Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.
通讯机构：
[Peng, Shuangjie] C;Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.
关键词：
Kirchhoff-type equations;Nodal solutions;Nonlocal term
摘要：
In this paper, we study the existence and asymptotic behavior of nodal solutions to the following Kirchhoff problem -(a+b integral(3)(R)vertical bar del u vertical bar(2)dx)Delta u + v(vertical bar x vertical bar)u = f(vertical bar x vertical bar, u), in R-3 , u is an element of H-1(R-3), where V(x) is a smooth function, a, b are positive constants. Because the so-called nonlocal term (integral(3)(R)vertical bar del u vertical bar(2)dx)Delta u)Au is involved in the equation, the variational functional of the equation has totally different properties from the case of b = 0. Under suitable construction conditions, we prove that, for any positive integer k, the problem has a sign-changing solution u(k)(b), which changes signs exactly k times. Moreover, the energy of u(k)(b) is strictly increasing in k, and for any sequence {b(n)}-> 0(+) (n -> + infinity), there is a subsequence {b(ns)}, such that u(k)(bs) converges in H-1 (R-3) to wk as s -> infinity, where wk also changes signs exactly k times and solves the following equation -a Delta u +V (|x|), u) = f(|x|, u), in R-3, u is an element of H-1 (R-3). (C) 2015 Elsevier Inc. All rights reserved.
作者机构：
[Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Pi, Huirong] E China Normal Univ, Ctr Partial Differential Equat, Shanghai 200241, Peoples R China.
通讯机构：
[Peng, Shuangjie] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
摘要：
<jats:p>This paper is concerned with the existence and qualitative property of solutions for a Hénon-like equation</jats:p><jats:p><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0308210513000553_inline1" /></jats:p><jats:p>where <jats:italic>Ω</jats:italic> = {<jats:italic>x</jats:italic> ∈ ℝ<jats:sup><jats:italic>N</jats:italic></jats:sup> : 1 < |<jats:italic>x</jats:italic>| < 3} with <jats:italic>N</jats:italic> ≥ 4, 2* = 2<jats:italic>N</jats:italic>/(<jats:italic>N</jats:italic> − 2), <jats:italic>τ</jats:italic> > 0 and <jats:italic>ε</jats:italic> > 0 is a small parameter. For any given <jats:italic>k</jats:italic> ∈ ℤ<jats:sup>+</jats:sup>, we construct positive solutions concentrating simultaneously at 2<jats:italic>k</jats:italic> different points for <jats:italic>ε</jats:italic> sufficiently small, among which <jats:italic>k</jats:italic> points are near the interior boundary {<jats:italic>x</jats:italic> ∈ ℝ<jats:sup><jats:italic>N</jats:italic></jats:sup> : |<jats:italic>x</jats:italic>| = 1} and the other <jats:italic>k</jats:italic> points are near the outward boundary {<jats:italic>x</jats:italic> ∈ ℝ<jats:sup><jats:italic>N</jats:italic></jats:sup> : |<jats:italic>x</jats:italic>| = 3}. Moreover, the 2<jats:italic>k</jats:italic> points tend to the boundary of <jats:italic>Ω</jats:italic> as <jats:italic>ε</jats:italic> goes to 0.</jats:p>
摘要：
In this paper, we consider the planar vortex patch problem in an incompressible steady flow in a bounded domain Omega of R-2. Let k be a positive integer and let k(j) be a positive constant, j = 1,..., k. For any given non-degenerate critical point x(0) = (x0,1,..., x(0,k)) of the Kirchhoff-Routh function defined on Omega(k) corresponding to (k1,..., k(k)), we prove the existence of a planar flow, such that the vorticity w of this flow equals a large given positive constant lambda in each small neighborhood of x(0, j), j = 1,..., k, and w = 0 elsewhere. Moreover, as lambda -> +infinity, the vorticity set {y: w(y) = lambda} shrinks to boolean OR(k)(j)(=1){x(0,j)}, and the local vorticity strength near each x(0,j) approaches k(j), j = 1,...,k. (C) 2014 Elsevier Inc. All rights reserved.
摘要：
This paper is concerned with the existence of multiple non-radial sign-changing solutions for{-Δu＋（1＋βV（y））u=｜u｜p-2u,y∈RN,{U（y）→0,当｜y｜→＋∞ where 2〈pM2N/（N-2）^＋,for N 〉 2 and 2 ＊ =＋∞ for N = 2, β can be regarded as a parameter and V（ ｜ y ｜ ） 〉 0 decays exponentially to zero at infinity. We prove that there exists a suitable range of β such that the above problem has a non-radial sign-changing solutions with exactly k maximum points and k min- imum points which tend to infinity as fl --β→- ∞ （ or 0^- ） for any positive integer k〉 1.
