CPAA
Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents
Yanfang Peng Jing Yang
Communications on Pure & Applied Analysis 2015, 14(2): 439-455 doi: 10.3934/cpaa.2015.14.439
We study the following elliptic equation with two Sobolev-Hardy critical exponents \begin{eqnarray} & -\Delta u=\mu\frac{|u|^{2^{*}(s_1)-2}u}{|x|^{s_1}}+\frac{|u|^{2^{*}(s_2)-2}u}{|x|^{s_2}} \quad x\in \Omega, \\ & u=0, \quad x\in \partial\Omega, \end{eqnarray} where $\Omega\subset R^N (N\geq3)$ is a bounded smooth domain, $0\in\partial\Omega$, $0\leq s_2 < s_1 \leq 2$ and $2^*(s):=\frac{2(N-s)}{N-2}$. In this paper, by means of variational methods, we obtain the existence of sign-changing solutions if $H(0)<0$, where $H(0)$ denote the mean curvature of $\partial\Omega$ at $0$.
keywords: mean curvature sign-changing solutions. Critical exponents
CPAA
Segregated vector Solutions for nonlinear Schrödinger systems with electromagnetic potentials
Jing Yang
Communications on Pure & Applied Analysis 2017, 16(5): 1785-1805 doi: 10.3934/cpaa.2017087
In this paper, we study the following nonlinear Schrödinger system in
$\mathbb{R}^3$
$\left\{ \begin{array}{*{35}{l}} {{(\frac{\nabla }{i}-A(y))}^{2}}u+{{\lambda }_{1}}(|y|)u=|u{{|}^{2}}u+\beta |v{{|}^{2}}u,&x\in {{\mathbb{R}}^{3}}, \\ {{(\frac{\nabla }{i}-A(y))}^{2}}v+{{\lambda }_{2}}(|y|)v=|v{{|}^{2}}v+\beta |u{{|}^{2}}v,&x\in {{\mathbb{R}}^{3}}, \\\end{array} \right.$
where
$A(y)=A(|y|)∈ C^1(\mathbb{R}^3,\mathbb{R})$
is bounded,
$λ_1(|y|),λ_2(|y|)$
are continuous positive radial potentials, and
$β∈ \mathbb{R}$
is a coupling constant. We proved that if
$A(y),λ_1(y),λ_2(y)$
satisfy some suitable conditions, the above problem has infinitely many non-radial segregated solutions.
keywords: Segregated vector solutions electromagnetic fields reduction method
DCDS
Positive solutions for a nonlinear Schrödinger-Poisson system
Chunhua Wang Jing Yang
Discrete & Continuous Dynamical Systems - A 2018, 38(11): 5461-5504 doi: 10.3934/dcds.2018241
In this paper, we study the following nonlinear Schrödinger-Poisson system
$\left\{\begin{array}{ll} -\Delta u+u+\epsilon K(x)\Phi(x)u = f(u),& x\in \mathbb{R}^{3} , \\ -\Delta \Phi = K(x)u^{2},\,\,& x\in \mathbb{R}^{3}, \\\end{array}\right.$
where
$K(x)$
is a positive and continuous potential and
$f(u)$
is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Under some suitable conditions, which are given in section 1, we prove that there exists some
$\epsilon_{0}>0$
such that for
$0<\epsilon<\epsilon_{0}$
, the above problem has infinitely many positive solutions by applying localized energy method. Our main result can be viewed as an extension to a recent result Theorem 1.1 of Ao and Wei in [3] and a result of Li, Peng and Wang in [26].
keywords: Schrödinger-Poisson system nonsymmetric potential reduction
DCDS
Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations
Wei Long Shuangjie Peng Jing Yang
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 917-939 doi: 10.3934/dcds.2016.36.917
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.
keywords: Fractional Laplacian reduction method. nonlinear scalar field equation

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