Multi-bump solutions for Schrödinger equation involving critical growth and potential wells
Yuxia Guo Zhongwei Tang
Discrete & Continuous Dynamical Systems - A 2015, 35(8): 3393-3415 doi: 10.3934/dcds.2015.35.3393
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.
keywords: critical exponent multi-bump solutions Schrödinger equation potential wells.
Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator
Yuxia Guo Jianjun Nie
Discrete & Continuous Dynamical Systems - A 2016, 36(12): 6873-6898 doi: 10.3934/dcds.2016099
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 $(P)$.
keywords: infinitely many solutions elliptic equation reduction. Fractional Laplacian critical exponent
Nonexistence of positive solutions for polyharmonic systems in $\mathbb{R}^N_+$
Yuxia Guo Bo Li
Communications on Pure & Applied Analysis 2016, 15(3): 701-713 doi: 10.3934/cpaa.2016.15.701
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.
keywords: moving plane polyharmonic systems. Nonexistence
Classification for positive solutions of degenerate elliptic system
Yuxia Guo Jianjun Nie
Discrete & Continuous Dynamical Systems - A 2019, 39(3): 1457-1475 doi: 10.3934/dcds.2018130

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.

keywords: Liouville theorem moving plane degenerate elliptic system
Multiple solutions for a critical quasilinear equation with Hardy potential
Fengshuang Gao Yuxia Guo
Discrete & Continuous Dynamical Systems - S 2018, 0(0): 1977-2003 doi: 10.3934/dcdss.2019128
In this paper, we investigate the following quasilinear equation involving a Hardy potential:
$\begin{array}{l}\left\{ {\begin{array}{*{20}{c}}{ - \sum\limits_{i,j = 1}^N {{D_j}} ({a_{ij}}(u){D_i}u) + \frac{1}{2}\sum\limits_{i,j = 1}^N {{{a'}_{ij}}} (u){D_i}u{D_j}u - \frac{\mu }{{|x{|^2}}}u = au + |u{|^{{2^ * } - 2}}u}&{{\rm{in}}\;\Omega ,}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{u = 0}&{{\rm{on}}\;\partial \Omega ,}\end{array}} \right.\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\rm{P}} \right)\end{array}$
$ 2^ * = \frac{2N}{N-2} $
is the Sobolev critical exponent for the embedding of
$ H_0^1(\Omega) $
$ L^p(\Omega) $
$ a>0 $
is a constant and
$ \Omega\subset \mathbb{R}^N $
is an open bounded domain which contains the origin. We will prove that under some suitable assumptions on
$ a_{ij} $
, when
$ N\geq 7 $
$ \mu\in[0,\mu^*) $
for some constant
$ \mu^* $
, problem (P) admits an unbounded sequence of solutions. To achieve this goal, we perform the subcritical approximation and the regularization perturbation.
keywords: Quasilinear equations critical exponents Hardy potential multiple bound state solutions
Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system
Jiaquan Liu Yuxia Guo Pingan Zeng
Discrete & Continuous Dynamical Systems - A 2006, 16(1): 107-119 doi: 10.3934/dcds.2006.16.107
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.
keywords: Morse index finite element approximation superlinear strongly indefinite system.

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