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### Open Access Journals

CPAA

We study symmetry and asymptotic behavior of ground state solutions for the doubly coupled nonlinear Schrödinger elliptic system

$\left\{ {\begin{array}{*{20}{l}}{ - \Delta u + {\lambda _1}u + \kappa v = {\mu _1}{u^3} + \beta u{v^2}}&{\quad {\rm{ in}}\;\;\Omega ,}\\{ - \Delta v + {\lambda _2}v + \kappa u = {\mu _2}{v^3} + \beta {u^2}v}&{\quad {\rm{ in}}\;\;\Omega ,}\\{u = v = 0\;on\;\;\partial \Omega \;({\rm{or}}\;u,v \in {H^1}({\mathbb{R}^N})\;{\rm{as}}\;\Omega = {\mathbb{R}^N}),}&{}\end{array}} \right.$ |

where

is a smooth domain. First we establish the symmetry of ground state solutions, that is, when

is radially symmetric, we show that ground state solution is foliated Schwarz symmetric with respect to the same point. Moreover, ground state solutions must be radially symmetric under the condition that

is a ball or the whole space

. Next we investigate the asymptotic behavior of positive ground state solution as

, which shows that the limiting profile is exactly a minimizer for

(the minimized energy constrained on Nehari manifold corresponds to the limit systems). Finally for a system with three equations, we prove the existence of ground state solutions whose all components are nonzero.

$ N≤3, Ω\subseteq\mathbb{R}^N$ |

$ Ω$ |

$ Ω$ |

$ \mathbb{R}^N$ |

$ κ\to 0^-$ |

$ c_0$ |

PROC

In this paper, we use Nehari manifold to
extend the anti-maximum principle of Laplacian operator to an
existence theorem for p-Laplacian ($p\not=2$), then consider the
existence of nonnegative solutions to semipositone quasilinear
elliptic problems $-\Delta_p u=\lambda f(u), x\in \Om; u>0, x\in
\Om; u=0, x\in \Po$.

DCDS

In this paper, we use Lyapunov-Schmidt method and Morse theory to study semilinear elliptic boundary value problems with resonance at infinity, and
get new multiple solutions theorems.

DCDS

We are concerned with the existence of positive solutions for a
coupled Schrödinger system
\begin{equation*}
\left\{ \begin{aligned}
&-\Delta{u}_1+{\lambda}_1 {u}_1={\mu}_1 {u}_1^3+\varepsilon \beta(x) {u}_1 {u}_2^2 & ~~in &~~~~ \mathbb{R}^3,\\
&-\Delta{u}_2+{\lambda}_2 {u}_2={\mu}_2 {u}_2^3+\varepsilon \beta(x) {u}_1^2 {u}_2 & ~~in & ~~~~\mathbb{R}^3,\\
&{u}_1>0, ~~{u}_2>0& ~~in & ~~~~\mathbb{R}^3,\\
&{u}_1\in H^1(\mathbb{R}^3),~~{u}_2\in H^1(\mathbb{R}^3),
\end{aligned} \right.
\end{equation*}
where ${\lambda}_1,{\lambda}_2,{\mu}_1,{\mu}_2$ are positive
constants. We use perturbation methods to prove that if $\beta \in
L^r(\mathbb{R}^3)(r\geq 3)$ doesn't change sign, as corresponding
$\varepsilon$ is sufficiently small the system has a positive
solution of which both components are positive. Our results is also
true for domain $\mathbb{R}^{2}$ and for domain $ \mathbb {R}^{N}, N
\geq 4 $ when the similar system is subcritical.

DCDS-S

We study a class of nonlocal eigenvalue problems related to certain boundary value problems that arise in many application areas. We construct a nondecreasing and unbounded sequence of eigenvalues that yields nontrivial critical groups for the associated variational functional using a nonstandard minimax scheme that involves the $\mathbb{Z}_2$-cohomological index. As an application we prove a multiplicity result for a class of nonlocal boundary value problems using Morse theory.

DCDS-S

In this paper, we consider the following coupled elliptic system

$ \begin{equation} \left\{ \begin{array}{ll} -\Delta u+\lambda_1 u = \mu_1 u^3+\beta uv^2-\gamma v &\text{in } \mathbb{R}^N, \\ -\Delta v+\lambda_2 v = \mu_2 v^3+\beta vu^2-\gamma u &\text{in } \mathbb{R}^N, \\ u(x), v(x)\rightarrow 0 \text{ as } \vert x\vert\rightarrow+\infty. \end{array} \right.\nonumber \end{equation} $ |

Under symmetric assumptions

, we determine the number of

-bifurcations for each

, and study the behavior of global

-bifurcation branches in

. Moreover, several results for

, such as priori bounds, are of independent interests, which are improvements of corresponding theorems in [6 ] and [35 ].

$ \lambda_1 = \lambda_2, \mu_1 = \mu_2 $ |

$ \gamma $ |

$ \beta\in(-1, +\infty) $ |

$ \gamma $ |

$ [-1, 0]\times H_r^1\left( \mathbb{R} ^N\right)\times H_r^1\left( \mathbb{R} ^N\right) $ |

$ \gamma = 0 $ |

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