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CPAA

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 $0$.

CPAA

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

is bounded,

are continuous positive radial potentials, and

is a coupling constant. We proved that if

satisfy some suitable conditions, the above problem has infinitely many non-radial segregated solutions.

$A(y)=A(|y|)∈ C^1(\mathbb{R}^3,\mathbb{R})$ |

$λ_1(|y|),λ_2(|y|)$ |

$β∈ \mathbb{R}$ |

$A(y),λ_1(y),λ_2(y)$ |

DCDS

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

is a positive and continuous potential and

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

such that for

, 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 ].

$K(x)$ |

$f(u)$ |

$\epsilon_{0}>0$ |

$0<\epsilon<\epsilon_{0}$ |

DCDS

Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations

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.

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