Multiple solutions for a fractional nonlinear Schrödinger equation with local potential
Wulong Liu Guowei Dai
Communications on Pure & Applied Analysis 2017, 16(6): 2105-2123 doi: 10.3934/cpaa.2017104
Using penalization techniques and the Ljusternik-Schnirelmann theory, we establish the multiplicity and concentration of solutions for the following fractional Schrödinger equation
$\left\{ \begin{align} &{{\varepsilon }^{2\alpha }}{{\left( -\Delta \right)}^{a}}u+V\left( x \right)u=f\left( u \right),\ \ x\in {{\mathbb{R}}^{N}}, \\ &u\in {{H}^{a}}\left( {{\mathbb{R}}^{N}} \right),u>0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}, \\ \end{align} \right.$
is a small parameter,
satisfies the local condition, and
is superlinear and subcritical nonlinearity. We show that this equation has at least
single spike solutions.
keywords: Penalization techniques Ljusternik-Schnirelmann theory fractional Schrödinger equation multiple solutions single spike solutions
Stability analysis of a model on varying domain with the Robin boundary condition
Xiaofei Cao Guowei Dai
Discrete & Continuous Dynamical Systems - S 2017, 10(5): 935-942 doi: 10.3934/dcdss.2017048

In this paper we develop a non-autonomous reaction-diffusion model with the Robin boundary conditions to describe insect dispersal on an isotropically varying domain. We investigate the stability of the reaction-diffusion model. The stability results of the model describe either insect survival or vanishing.

keywords: Stability bifurcation varying domain insect dispersal model Robin boundary condition
Unilateral global bifurcation for $p$-Laplacian with non-$p-$1-linearization nonlinearity
Guowei Dai Ruyun Ma
Discrete & Continuous Dynamical Systems - A 2015, 35(1): 99-116 doi: 10.3934/dcds.2015.35.99
In this paper, we establish a unilateral global bifurcation result from interval for a class of $p$-Laplacian problems. By applying above result, we study the spectrum of a class of half-quasilinear problems. Moreover, we also investigate the existence of nodal solutions for a class of half-quasilinear eigenvalue problems.
keywords: Unilateral bifurcation nodal solutions half-quasilinear problems $p$-Laplacian.
Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian
Guowei Dai Ruyun Ma Haiyan Wang
Communications on Pure & Applied Analysis 2013, 12(6): 2839-2872 doi: 10.3934/cpaa.2013.12.2839
In this paper, we establish a unilateral global bifurcation result for a class of quasilinear periodic boundary problems with a sign-changing weight. By the Ljusternik-Schnirelmann theory, we first study the spectrum of the periodic $p$-Laplacian with the sign-changing weight. In particular, we show that there exist two simple, isolated, principal eigenvalues $\lambda_0^+$ and $\lambda_0^-$. Furthermore, under some natural hypotheses on perturbation function, we show that $(\lambda_0^\nu,0)$ is a bifurcation point of the above problems and there are two distinct unbounded sub-continua $C_\nu^{+}$ and $C_\nu^{-}$, consisting of the continuum $C_\nu$ emanating from $(\lambda_0^\nu, 0)$, where $\nu\in\{+,-\}$. As an application of the above result, we study the existence of one-sign solutions for a class of quasilinear periodic boundary problems with the sign-changing weight. Moreover, the uniqueness of one-sign solutions and the dependence of solutions on the parameter $\lambda$ are also studied.
keywords: One-sign solutions. Eigenvalues Unilateral global bifurcation Periodic $p$-Laplacian
Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems
Guowei Dai Rushun Tian Zhitao Zhang
Discrete & Continuous Dynamical Systems - S 2018, 0(0): 1905-1927 doi: 10.3934/dcdss.2019125
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
$ \lambda_1 = \lambda_2, \mu_1 = \mu_2 $
, we determine the number of
$ \gamma $
-bifurcations for each
$ \beta\in(-1, +\infty) $
, and study the behavior of global
$ \gamma $
-bifurcation branches in
$ [-1, 0]\times H_r^1\left( \mathbb{R} ^N\right)\times H_r^1\left( \mathbb{R} ^N\right) $
. Moreover, several results for
$ \gamma = 0 $
, such as priori bounds, are of independent interests, which are improvements of corresponding theorems in [6] and [35].
keywords: Bifurcation Schrödinger systems positive solutions

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