DCDS
Asymptotic behavior of the positive solutions for an elliptic equation with Hardy term
Yinbin Deng Qi Gao
Discrete & Continuous Dynamical Systems - A 2009, 24(2): 367-380 doi: 10.3934/dcds.2009.24.367
We give an explicit formula for exponential decay properties of positive solutions for a class of semilinear elliptic equations with Hardy term in the whole space Rn.
keywords: Hardy term Asymptotic formula elliptic equation.
DCDS
Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents
Kun Cheng Yinbin Deng
Discrete & Continuous Dynamical Systems - A 2017, 37(1): 77-103 doi: 10.3934/dcds.2017004

This paper is concerned with constructing nodal radial solutions for generalized quasilinear Schrödinger equations in $\mathbb{R}^N$ with critical growth which arise from plasma physics, fluid mechanics, as well as the self-channeling of a high-power ultashort laser in matter. We find the critical exponents for a generalized quasilinear Schrödinger equations and obtain the existence of sign-changing solution with k nodes for any given integer $k ≥ 0$.

keywords: Quasilinear Schrödinger equations critical exponents nodal solutions mountain pass lemma constrained minimization problem
CPAA
Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity
Yinbin Deng Wei Shuai
Communications on Pure & Applied Analysis 2014, 13(6): 2273-2287 doi: 10.3934/cpaa.2014.13.2273
This paper is concerned with the existence of positive solutions for a class of quasilinear Schrödinger equations in $R^N$ with critical growth and potential vanishing at infinity. By using a change of variables, the quasilinear equations are reduced to semilinear one. Since the potential vanish at infinity, the associated functionals are still not well defined in the usual Sobolev space. So we have to work in the weighted Sobolev spaces, by Hardy-type inequality, then the functionals are well defined in the weighted Sobolev space and satisfy the geometric conditions of the Mountain Pass Theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution $v$. In the proof that $v$ is nontrivial, the main tool is classical arguments used by H. Brezis and L. Nirenberg in [13].
keywords: critical growth positive solution. Quasilinear Schrödinger equations potential vanishing weighted Sobolev space
DCDS
Positive ground state solutions for a quasilinear elliptic equation with critical exponent
Yinbin Deng Wentao Huang
Discrete & Continuous Dynamical Systems - A 2017, 37(8): 4213-4230 doi: 10.3934/dcds.2017179
In this paper, we study the following quasilinear elliptic equation with critical Sobolev exponent:
$ -\Delta u +V(x)u-[\Delta(1+u^2)^{\frac 12}]\frac {u}{2(1+u^2)^\frac 12}=|u|^{2^*-2}u+|u|^{p-2}u, \quad x\in {{\mathbb{R}}^{N}}, $
which models the self-channeling of a high-power ultra short laser in matter, where N ≥ 3; 2 < p < 2* = $\frac{{2N}}{{N -2}}$ and V (x) is a given positive potential. Combining the change of variables and an abstract result developed by Jeanjean in [14], we obtain the existence of positive ground state solutions for the given problem.
keywords: Ground state solutions quasilinear elliptic equation critical exponent
DCDS
Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$
Yinbin Deng Wei Shuai
Discrete & Continuous Dynamical Systems - A 2018, 38(6): 3139-3168 doi: 10.3934/dcds.2018137
We are interested in the existence of sign-changing multi-bump solutions for the following Kirchhoff equation
$ - (a + b\int_{{\mathbb{R}^3}} {|\nabla u{|^2}dx} )\Delta u + \lambda V(x)u = f(u),\;x \in {\mathbb{R}^3},$
where
$λ$
>0 is a parameter and the potential
$V(x)$
is a nonnegative continuous function with a potential well
$Ω: = int(V^{-1}(0))$
which possesses
$k$
disjoint bounded components
$Ω_1,Ω_2,···,Ω_k$
. Under some conditions imposed on
$f(u)$
, multiple sign-changing multi-bump solutions are obtained. Moreover, the concentration behavior of these solutions as
$λ→ +∞$
are also studied.
keywords: Kirchhoff-type equations multiple sign-changing solutions multi-bump solutions concentration behavior
DCDS
Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent
Yinbin Deng Shuangjie Peng Li Wang
Discrete & Continuous Dynamical Systems - A 2012, 32(3): 795-826 doi: 10.3934/dcds.2012.32.795
In this paper, we consider the following problem $$ \left\{ \begin{array}{ll} -\Delta u+u=u^{2^{*}-1}+\lambda(f(x,u)+h(x))\ \ \hbox{in}\ \mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),\ \ u>0 \ \hbox{in}\ \mathbb{R}^{N}, \end{array} \right. (\star) $$ where $\lambda>0$ is a parameter, $2^* =\frac {2N}{N-2}$ is the critical Sobolev exponent and $N>4$, $f(x,t)$ and $h(x)$ are some given functions. We prove that there exists $0<\lambda^{*}<+\infty$ such that $(\star)$ has exactly two positive solutions for $\lambda\in(0,\lambda^{*})$ by Barrier method and Mountain Pass Lemma and no positive solutions for $\lambda >\lambda^*$. Moreover, if $\lambda=\lambda^*$, $(\star)$ has a unique solution $(\lambda^{*}, u_{\lambda^{*}})$, which means that $(\lambda^{*}, u_{\lambda^{*}})$ is a turning point in $H^{1}(\mathbb{R}^{N})$ for problem $(\star)$.
keywords: variational methods. critical exponent Multiple solutions
DCDS
Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity
Yinbin Deng Yi Li Wei Shuai
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 683-699 doi: 10.3934/dcds.2016.36.683
In this paper, we study the existence of positive solution for the following p-Laplacain type equations with critical nonlinearity \begin{equation*} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -\Delta_p u + V (x)|u|^{p-2}u = K(x)f(u)+P(x)|u|^{p^*-2}u, \quad x\in\mathbb{R}^N,\\ u \in \mathcal{D}^{1,p}(\mathbb{R}^N), \end{array} \right. \end{equation*} where $\Delta_p u = div(|\nabla u|^{p-2} \nabla u),\ 1 < p < N,\ p^* =\frac {Np}{N-p}$, $V(x)$, $K(x)$ are positive continuous functions which vanish at infinity, $f$ is a function with a subcritical growth, and $P(x)$ is a bounded, nonnegative continuous function. By working in the weighted Sobolev spaces, and using variational method, we prove that the given problem has at least one positive solution.
keywords: p-Laplacain type equations weighted Sobolev space critical growth variational method. vanishing potential
DCDS
Infinitely many radial solutions to elliptic systems involving critical exponents
Yinbin Deng Shuangjie Peng Li Wang
Discrete & Continuous Dynamical Systems - A 2014, 34(2): 461-475 doi: 10.3934/dcds.2014.34.461
In this paper, by an approximating argument, we obtain infinitely many radial solutions for the following elliptic systems with critical Sobolev growth $$ \left\lbrace\begin{array}{l} -\Delta u=|u|^{2^*-2}u + \frac{η \alpha}{\alpha+β}|u|^{\alpha-2}u |v|^β + \frac{σ p}{p+q} |u|^{p-2}u|v|^q , \ \ x ∈ B , \\ -\Delta v = |v|^{2^*-2}v + \frac{η β}{\alpha+ β } |u|^{\alpha }|v|^{β-2}v + \frac{σ q}{p+q} |u|^{p}|v|^{q-2}v , \ \ x ∈ B , \\ u = v = 0, \ \ &x \in \partial B, \end{array}\right. $$ where $N > \frac{2(p + q + 1) }{p + q - 1}, η, σ > 0, \alpha,β > 1$ and $\alpha + β = 2^* = : \frac{2N}{N-2} ,$ $p,\,q\ge 1$, $2\le p +q<2^*$ and $B\subset \mathbb{R}^N$ is an open ball centered at the origin.
keywords: elliptic systems Radial solution (PS) condition. critical exponent
DCDS
Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$
Yinbin Deng Qi Gao Dandan Zhang
Discrete & Continuous Dynamical Systems - A 2007, 19(1): 211-233 doi: 10.3934/dcds.2007.19.211
This paper is concerned with the existence and nodal character of the nontrivial solutions for the following equations involving critical Sobolev and Hardy exponents:

$-\Delta u + u - \mu \frac{u}{|x|^2}=|u|^{2^*-2}u + f(u),$
$u \in H^1_r (\R ^N),(1)$

where $2^$*$=\frac{2N}{N-2}$ is the critical Sobolev exponent for the embedding $H^1_r (\R ^N) \rightarrow L^{2^}$*$ (\R ^N)$, $\mu \in [0, \ (\frac {N-2}{2})^2)$ and $f: \R \rightarrow\R $ is a function satisfying some conditions. The main results obtained in this paper are that there exists a nontrivial solution of equation (1) provided $N\ge 4$ and $\mu \in [0, \ (\frac {N-2}{2})^2-1] $ and there exists at least a pair of nontrivial solutions $u^+_k$, $u^-_k$ of problem (1) for each k $\in N \cup \{0\}$ such that both $u^+_k$ and $u^-_k$ possess exactly k nodes provided $N\ge 6$ and $\mu \in [0, \ (\frac {N-2}{2})^2-4]$.

keywords: critical Sobolev and Hardy exponents; node solutions.
CPAA
Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion
Yinbin Deng Yi Li Xiujuan Yan
Communications on Pure & Applied Analysis 2015, 14(6): 2487-2508 doi: 10.3934/cpaa.2015.14.2487
This paper is concerned with a type of quasilinear Schrödinger equations of the form \begin{eqnarray} -\Delta u+V(x)u-p\Delta(|u|^{2p})|u|^{2p-2}u=\lambda|u|^{q-2}u+|u|^{2p2^{*}-2}u, \end{eqnarray} where $\lambda>0, N\ge3, 4p < q < 2p2^*, 2^*=\frac{2N}{N-2}, 1< p < +\infty$. For any given $k \ge 0$, by using a change of variables and Nehari minimization, we obtain a sign-changing minimizer with $k$ nodes.
keywords: radial solutions Quasilinear Schrödinger equations nodal solutions.

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