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.
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：
[Li, Gongbao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
摘要：
We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth, -(epsilon(2)a + epsilon b integral(R3) vertical bar del u vertical bar(2))del u + V(z)u = f(u) + u(5) in R-3, u is an element of H-1(R-3), u > 0 in R-3, where e is a small positive parameter and a, b > 0 are constants, f is an element of C-1(R+, R) is subcritical, V : R-3 -> R is a locally Holder continuous function. We first prove that for epsilon(0) > 0 sufficiently small, the above problem has a weak solution u(epsilon) with exponential decay at infinity. Moreover, u(epsilon) concentrates around a local minimum point of V in Lambda as epsilon -> 0. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topological construction of the set where the potential V(z) attains its minimum.
作者机构：
[Peng, Shuangjie; Wang, Zhi-qiang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Nankai Univ, Chern Inst Math, Tianjin 300071, Peoples R China.;[Wang, Zhi-qiang] Utah State Univ, Dept Math & Stat, Logan, UT 84322 USA.
通讯机构：
[Wang, Zhi-qiang] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
摘要：
We consider the following nonlinear Schrödinger system in
$${\mathbb{R}^3}$$
$$\left\{\begin{array}{ll}-\Delta u + P(|x|)u = \mu u^{2}u + \beta v^2u,\quad x \in \mathbb{R}^3,\\-\Delta v + Q(|x|)v = \nu v^{2}v + \beta u^2v,\quad x \in \mathbb{R}^3,\end{array}\right.$$
where P(r) and Q(r) are positive radial potentials,
$${\mu > 0, \nu > 0}$$
and
$${\beta \in \mathbb{R}}$$
is a coupling constant. This type of system arises, in particular, in models in Bose–Einstein condensates theory. We examine the effect of nonlinear coupling on the solution structure. In the repulsive case, we construct an unbounded sequence of non-radial positive vector solutions of segregated type, and in the attractive case we construct an unbounded sequence of non-radial positive vector solutions of synchronized type. Depending upon the system being repulsive or attractive, our results exhibit distinct characteristic features of vector solutions.
