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### Open Access Journals

DCDS

In this paper, we consider the following Schrödinger equation with critical growth
$$-\Delta u+(\lambda a(x)-\delta)u=|u|^{2^*-2}u \quad \hbox{ in } \mathbb{R}^N, $$ where $N\geq 5$, $2^*$ is the critical Sobolev exponent, $\delta>0$ is a constant, $a(x)\geq 0$ and its zero set is not empty. We will show that if the zero set of $a(x)$ has several isolated connected components
$\Omega_1,\cdots,\Omega_k$ such that the interior of $\Omega_i (i=1, 2, ...,
k)$ is not empty and $\partial\Omega_i (i=1, 2, ..., k)$ is smooth, then for any non-empty
subset $J\subset \{1,2,\cdots,k\}$ and $\lambda$ sufficiently large, the equation admits a solution which is trapped in a
neighborhood of $\bigcup_{j\in J}\Omega_j$. Our strategy to obtain the main results is as follows: By using local mountain pass method combining with penalization of the nonlinearities, we first prove the existence of single-bump solutions which are trapped in the neighborhood of only one isolated component of zero set. Then we construct the multi-bump solution by summing these one-bump solutions as the first approximation solution. The real solution will be obtained by delicate estimates of the error term, this last step is done by using Contraction Image Principle.

DCDS

We consider the following problem:
\begin{equation*}
\left\{\begin{array}{ll}
(-\Delta)^s u=K(y)u^{p-1} \hbox { in } \ \mathbb{R}^N, \\
u>0, \ y \in \mathbb{R}^N,
\end{array}\right. (P)
\end{equation*}
where $s\in(0,\frac{1}{2})$ for $N=3$, $s\in(0,1)$ for $N\geq4$ and $p=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent. Under the condition that the function $K(y)$ has a local maximum point, we prove the existence of infinitely many non-radial solutions for the problem $(P)$.

CPAA

In this paper, we study the
monotonicity and nonexistence of positive solutions for polyharmonic
systems $\left\{\begin{array}{rlll}
(-\Delta)^m u&=&f(u, v)\\
(-\Delta)^m v&=&g(u, v)
\end{array}\right.\;\hbox{in}\;\mathbb{R}^N_+.$
By using the Alexandrov-Serrin method of moving plane combined with integral
inequalities and Sobolev's inequality in a narrow domain, we prove the monotonicity of positive solutions for
semilinear
polyharmonic systems in $\mathbb{R_+^N}.$ As a result, the nonexistence for positive weak solutions to the system are
obtained.

DCDS

In this paper, by using the Alexandrov-Serrin method of moving plane combined with integral inequalities, we obtained the complete classification of positive solution for a class of degenerate elliptic system.

DCDS-S

In this paper, we investigate the following quasilinear equation involving a Hardy potential:

$\begin{array}{l}\left\{ {\begin{array}{*{20}{c}}{ - \sum\limits_{i,j = 1}^N {{D_j}} ({a_{ij}}(u){D_i}u) + \frac{1}{2}\sum\limits_{i,j = 1}^N {{{a'}_{ij}}} (u){D_i}u{D_j}u - \frac{\mu }{{|x{|^2}}}u = au + |u{|^{{2^ * } - 2}}u}&{{\rm{in}}\;\Omega ,}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{u = 0}&{{\rm{on}}\;\partial \Omega ,}\end{array}} \right.\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\rm{P}} \right)\end{array}$ |

where

is the Sobolev critical exponent for the embedding of

into

,

is a constant and

is an open bounded domain which contains the origin. We will prove that under some suitable assumptions on

, when

and

for some constant

, problem (P) admits an unbounded sequence of solutions. To achieve this goal, we perform the subcritical approximation and the regularization perturbation.

$ 2^ * = \frac{2N}{N-2} $ |

$ H_0^1(\Omega) $ |

$ L^p(\Omega) $ |

$ a>0 $ |

$ \Omega\subset \mathbb{R}^N $ |

$ a_{ij} $ |

$ N\geq 7 $ |

$ \mu\in[0,\mu^*) $ |

$ \mu^* $ |

DCDS

We consider the second order strongly indefinite differential
system with superlinearities. By using the approximation method of
finite element, we show that bounds on solutions of the
restriction functional onto finite dimensional subspace are
equivalent to bounds on their relative Morse indices. The obtained
results can be used to establish a Morse theory for strongly
indefinite functionals with superlinearities.

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