arXiv subjects

Yuanze Wu

Publications and source records attributed to Yuanze Wu.

23 records · Page 2Linked to original sources

arXiv2015

On a Kirchhoff type problems with potential well and indefinite potential

In this paper, we study the following Kirchhoff type problem:% $$ \left\{\aligned&-\bigg(α\int_{\bbr^3}|\nabla u|^2dx+1\bigg)Δu+(λa(x)+a_0)u=|u|^{p-2}u&\text{ in }\bbr^3,\\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}_{α,λ})}% $$ where $4<p<6$, $α$ and $λ$ are two positive parameters, $a_0\in\bbr$ is a (possibly negative) constant and $a(x)\geq0$ is the potential well. By the variational method, we investigate the existence of nontrivial solutions to $(\mathcal{P}_{α,λ})$. To our best knowledge, it is the first time that the nontrivial solution of the Kirchhoff type problem is found in the indefinite case. We also obtain the concentration behaviors of the solutions as $λ\to+\infty$.

math.AP↗

arXiv2015

On a biharmonic equations with steep potential well and indefinite potential

In this paper, we study the following biharmonic equations:% $$ \left\{\aligned&Δ^2u-a_0Δu+(λb(x)+b_0)u=f(u)&\text{ in }\bbr^N,\\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}_λ)}% $$ where $N\geq3$, $a_0,b_0\in\bbr$ are two constants, $λ>0$ is a parameter, $b(x)\geq0$ is a potential well and $f(t)\in C(\bbr)$ is subcritical and superlinear or asymptotically linear at infinity. By the Gagliardo-Nirenberg inequality, we make some observations on the operator $Δ^2-a_0Δ+λb(x)+b_0$ in $\h$. Based on these observations, we give a new variational setting to $(\mathcal{P}_λ)$ for $a_0<0$. With this new variational setting in hands, we establish some new existence results of the nontrivial solutions to $(\mathcal{P}_λ)$ for all $a_0, b_0\in\bbr$ with $λ$ sufficiently large by the variational method. The concentration behavior of the nontrivial solutions as $λ\to+\infty$ is also obtained. It is worth to point out that it seems to be the first time that the nontrivial solution of $(\mathcal{P}_λ)$ is obtained in the case of $a_0<0$.

math.AP↗

arXiv2015

On a doubly critical Schrödinger system in $\bbr^4$ with steep potential wells

Study the following two-component elliptic system% \begin{equation*} \left\{\aligned&Δu-(λa(x)+a_0)u+u^3+βv^2u=0\quad&\text{in }\bbr^4,\\% &Δv-(λb(x)+b_0)v+v^3+βu^2v=0\quad&\text{in }\bbr^4,\\% &(u,v)\in\h\times\h,\endaligned\right.% \end{equation*} where $a_0,b_0\in\bbr$ are constants; $λ>0$ and $β\in\bbr$ are parameters and $a(x), b(x)\geq0$ are potential wells which are not necessarily to be radial symmetric. By using the variational method, we investigate the existence of ground state solutions and general ground state solutions (i.e., possibly semi-trivial) to this system. Indeed, to the best of our knowledge, even the existence of semi-trivial solutions is also unknown in the literature. We observe some concentration behaviors of ground state solutions and general ground state solutions. The phenomenon of phase separations is also excepted. It seems that this is the first result definitely describing the phenomenon of phase separation for critical system in the whole space $\bbr^4$. Note that both the cubic nonlinearities and the coupled terms of the system are all of critical growth with respect to the Sobolev critical exponent.

math.AP↗

arXiv2014

On a two-component Bose-Einstein condensate with steep potential wells

In this paper, we study the following two-component systems of nonlinear Schrödinger equations \begin{equation*} \left\{\aligned&Δu-(λa(x)+a_0(x))u+μ_1u^3+βv^2u=0\quad&\text{in }\bbr^3,\\ &Δv-(λb(x)+b_0(x))v+μ_2v^3+βu^2v=0\quad&\text{in }\bbr^3,\\ &u,v\in\h,\quad u,v>0\quad\text{in }\bbr^3,\endaligned\right. \end{equation*} where $λ,μ_1,μ_2>0$ and $β<0$ are parameters; $a(x), b(x)\geq0$ are steep potentials and $a_0(x),b_0(x)$ are sign-changing weight functions; $a(x)$, $b(x)$, $a_0(x)$ and $b_0(x)$ are not necessarily to be radial symmetric. By the variational method, we obtain a ground state solution and multi-bump solutions for such systems with $λ$ sufficiently large. The concentration behaviors of solutions as both $λ\to+\infty$ and $β\to-\infty$ are also considered. In particular, the phenomenon of phase separations is observed in the whole space $\bbr^3$. In the Hartree-Fock theory, this provides a theoretical enlightenment of phase separation in $\bbr^3$ for the 2-mixtures of Bose-Einstein condensates.

math.AP↗

arXiv2014

On the Schrödinger-Poisson system with steep potential well and indefinite potential

In this paper, we study the following Schrödinger-Poisson system: $$ \left\{\aligned&-Δu+V_λ(x)u+K(x)ϕu=f(x,u)&\quad\text{in }\bbr^3,\\ &-Δϕ=K(x)u^2&\quad\text{in }\bbr^3,\\ &(u,ϕ)\in\h\times\D,\endaligned\right.\eqno{(\mathcal{SP}_λ)} $$ where $V_λ(x)=λa(x)+b(x)$ with a positive parameter $λ$, $K(x)\geq0$ and $f(x,t)$ is continuous including the power-type nonlinearity $|u|^{p-2}u$. By applying the method of penalized functions, the existence of one nontrivial solution for such system in the less-studied case $3<p\leq4$ is obtained for $λ$ sufficiently large. The concentration behavior of this nontrivial solution for $λ\to+\infty$ are also observed. It is worth to point out that some new conditions on the potentials are introduced to obtain this nontrivial solution and the Schrödinger operator $-Δ+V_λ(x)$ may be strong indefinite in this paper.

math.AP↗