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In this paper we study the Hartree-Fock type system as follows: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+V\left( x\right) u+\rho \left( x\right) \phi _{\rho ,\left(u,v\right) }u=\left\vert u\right\vert ^{p-2}u+\beta \left\vert v\right\vert^{\frac{p}{2}}\left\vert u\right\vert ^{\frac{p}{2}-2}u & \text{ in }\mathbb{R}^{3}, \\ -\Delta v+V\left( x\right) v+\rho \left( x\right) \phi _{\rho ,\left( u,v\right) }v=\left\vert v\right\vert ^{p-2}v+\beta \left\vert u\right\vert ^{\frac{p}{2}}\left\vert v\right\vert ^{\frac{p}{2}-2}v & \text{ in }\mathbb{R}^{3}, \end{array} \right. \end{equation*} where $\phi _{\rho ,\left( u,v\right) }=\int_{\mathbb{R}^{3}}\frac{\rho \left( y\right) \left( u^{2}(y)+v^{2}\left( y\right) \right) }{|x-y|}dy,$ the potentials $V(x),\rho (x)$ are positive continuous functions in $\mathbb{R}^{3},$ the parameter $\beta \in \mathbb{R}$ and $2 more...

In this paper, we investigate vectorial standing waves with prescribed mass for the Hartree-Fock type system (HF system) with the double coupled feature. Such system is viewed as an approximation of the Coulomb system with two particles appeared in quantum mechanics. By exploring the interaction of the double coupled terms, we prove the exis?tence/nonexistence and symmetry of vectorial energy ground states for the corresponding stationary problem. Furthermore, we obtain the relation between vectorial energy ground states and vectorial action ground states in some cases. Finally, we establish conditions for global well-posedness and finite time blow-up to HF system with the initial data, and prove orbital stability/strong instability of standing waves. more...

In this paper we study the Hartree-Fock type system as follows: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+u+\lambda \phi _{u,v}u=\left\vert u\right\vert ^{p-2}u+\beta \left\vert v\right\vert ^{\frac{p}{2}}\left\vert u\right\vert ^{\frac{p}{2}% -2}u & \text{ in }\mathbb{R}^{3}, \\ -\Delta v+v+\lambda \phi _{u,v}v=\left\vert v\right\vert ^{p-2}v+\beta \left\vert u\right\vert ^{\frac{p}{2}}\left\vert v\right\vert ^{\frac{p}{2}% -2}v & \text{ in }\mathbb{R}^{3},% \end{array}% \right. \end{equation*}% where $\phi _{u,v}(x)=\int_{\mathbb{R}^{3}}\frac{u^{2}(y)+v^{2}\left( y\right) }{|x-y|}dy,$ the parameters $\lambda,\beta >0$ and $2 more...

We study the Schr\"{o}dinger-Poisson type system: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+\lambda u+\left( \mu _{11}\phi _{u}-\mu _{12}\phi _{v}\right) u=% \frac{1}{2\pi }\int_{0}^{2\pi }\left\vert u+e^{i\theta }v\right\vert ^{p-1}\left( u+e^{i\theta }v\right) d\theta & \text{ in }\mathbb{R}^{3}, \\ -\Delta v+\lambda v+\left( \mu _{22}\phi _{v}-\mu _{12}\phi _{u}\right) v=% \frac{1}{2\pi }\int_{0}^{2\pi }\left\vert v+e^{i\theta }u\right\vert ^{p-1}\left( v+e^{i\theta }u\right) d\theta & \text{ in }\mathbb{R}^{3},% \end{array}% \right. \end{equation*}% where $10$. Novel approaches are employed to prove the existence of a positive solution for $1 more...

In this paper, we study a class of Schr\"{o}dinger-Poisson (SP) systems with general nonlinearity where the nonlinearity does not require Ambrosetti-Rabinowitz and Nehari monotonic conditions. We establish new estimates and explore the associated energy functional which is coercive and bounded below on Sobolev space. Together with Ekeland variational principle, we prove the existence of ground state solutions. Furthermore, when the `charge' function is greater than a fixed positive number, the (SP) system possesses only zero solutions. In particular, when `charge' function is radially symmetric, we establish the existence of three positive solutions and the symmetry breaking of ground state solutions. more...

We investigate the NLS equation with competing Hartree-type and power-type nonlinearities \begin{equation*} \begin{array}{ll} i\partial _{t}\psi +\Delta \psi +\gamma (I_{\alpha }\ast |\psi |^{p})|\psi |^{p-2}\psi +\mu |\psi |^{q-2}\psi =0, & \text{ }\forall (t,x)\in \mathbb{R\times R}^{N},% \end{array}% \end{equation*}% where $\gamma \mu <0$. We establish conditions for the local well-posedness in the energy space. Under the double $L^{2}$-supercritical case, we prove the existence and multiplicity of standing waves with prescribed mass by developing a constraint method when $\gamma <0,\mu >0$ and $\gamma >0,\mu <0, $ respectively. Moreover, we prove weak orbital stablility and strong instability of standing waves by considering a suitable local minimization problem and by analyzing the fibering mapping, respectively. A new analysis of the fibering mapping is performed in this work. We believe that it is innovative as it was not discussed at all in any previous results. The lower bound rate of blow-up solutions for the Cauchy problem is given as well. Due to the different \textquotedblleft strength" of the two types of nonlinearities, we find some essential differences in our results between these two competing cases. We will be dealing with two major scenarios that are totally different from each other due to their diverse geometric structure. This leads to surprising findings. Additionally, the competing pure power-type nonlinearities case can be derived from our study thanks to a good choice of the kernel of the Hartree term. more...

We investigate a class of Kirchhoff type equations involving a combination of linear and superlinear terms as follows: \begin{equation*} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+1\right) \Delta u+\mu V(x)u=\lambda f(x)u+g(x)|u|^{p-2}u\quad \text{ in }\mathbb{R}^{N}, \end{equation*}% where $N\geq 3,20$ and $\mu $ sufficiently large, we obtain that at least one positive solution exists for $% 0<\lambda\leq\lambda _{1}(f_{\Omega}) $ while at least two positive solutions exist for $\lambda _{1}(f_{\Omega })< \lambda<\lambda _{1}(f_{\Omega})+\delta_{a}$ without any assumption on the integral $% \int_{\Omega }g(x)\phi _{1}^{p}dx$, where $\lambda _{1}(f_{\Omega })>0$ is the principal eigenvalue of $-\Delta $ in $H_{0}^{1}(\Omega )$ with weight function $f_{\Omega }:=f|_{\Omega }$, and $\phi _{1}>0$ is the corresponding principal eigenfunction. When $N\geq 3$ and $20$ small and $0<\lambda <\lambda _{1}(f_{\Omega })$; $% (ii)$ under the classical assumption $\int_{\Omega }g(x)\phi _{1}^{p}dx<0$, at least three positive solutions exist for $a>0$ small and $\lambda _{1}(f_{\Omega })\leq \lambda<\lambda _{1}(f_{\Omega})+\overline{\delta }% _{a} $; $(iii)$ under the assumption $\int_{\Omega }g(x)\phi _{1}^{p}dx>0$, at least two positive solutions exist for $a>a_{0}(p)$ and $\lambda^{+}_{a}< \lambda<\lambda _{1}(f_{\Omega})$ for some $a_{0}(p)>0$ and $\lambda^{+}_{a}\geq0$. more...

The existence and multiplicity of solutions for a class of non-local elliptic boundary value problems with superlinear source functions are investigated in this paper. Using variational methods, we examine the changes arise in the solution behaviours as a result of the non-local effect. Comparisons are made of the results here with those of the elliptic boundary value problem in the absence of the non-local term under the same prescribed conditions to highlight this effect of non-locality on the solution behaviours. Our results here demonstrate that the complexity of the solution structures is significantly increased in the presence of the non-local effect with the possibility ranging from no permissible positive solution to three positive solutions and, contrary to those obtained in the absence of the non-local term, the solution profiles also vary depending on the superlinearity of the source functions., Comment: 35 pages, 6 figures more...

We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:% \begin{equation*} \left\{ \begin{array}{l} -M\left( \int_{\mathbb{R}^{3}}\left\vert \nabla u\right\vert ^{2}dx\right) \Delta u+\mu V\left( x\right) u=Q(x)\left\vert u\right\vert ^{p-2}u+\lambda f\left( x\right) u\text{ in }\mathbb{R}^{N}, \\ u\in H^{1}\left( \mathbb{R}^{N}\right) ,% \end{array}% \right. \end{equation*}% where $N\geq 3,20\right) ,$ the potential $V$ is a nonnegative function in $\mathbb{R}% ^{N}$ and the weight function $Q\in L^{\infty }\left( \mathbb{R}^{N}\right) $ with changes sign in $\overline{\Omega }:=\left\{ V=0\right\} .$ We mainly prove the existence of at least two positive solutions in the cases that $% \left( i\right) $ $20$ sufficiently large, where $\lambda _{1}\left( f_{\Omega }\right) $ is the first eigenvalue of $-\Delta $ in $% H_{0}^{1}\left( \Omega \right) $ with weight function $f_{\Omega }:=f|_{% \overline{\Omega }},$ whose corresponding positive principal eigenfunction is denoted by $\phi _{1}.$ Furthermore, we also investigated the non-existence and existence of positive solutions if $a,\lambda $ belongs to different intervals., Comment: 34 pages more...

Consider a nonlinear Kirchhoff type equation as follows \begin{equation*} \left\{ \begin{array}{ll} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+b\right) \Delta u+u=f(x)\left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{N}, \\ u\in H^{1}(\mathbb{R}^{N}), & \end{array}% \right. \end{equation*}% where $N\geq 1,a,b>0,2 more...

It is well known that a single nonlinear fractional Schr\"odinger equation with a potential $V(x)$ and a small parameter $\varepsilon $ may have a positive solution that is concentrated at the nondegenerate minimum point of $V(x)$. In this paper, we can find two different positive solutions for two weakly coupled fractional Schr\"odinger systems with a small parameter $\varepsilon $ and two potentials $V_{1}(x)$ and $V_{2}(x)$ having the same minimum point are concentrated at the same point minimum point of $V_{1}(x)$ and $V_{2}\left(x\right) $. In fact that by using the energy estimates, Nehari manifold technique and the Lusternik-Schnirelmann theory of critical points, we obtain the multiplicity results for a class of fractional Laplacian system. Furthermore, the existence and nonexistence of least energy positive solutions are also explored., Comment: 33pages more...

In this paper, we study a class of generalized extensible beam equations with a superlinear nonlinearity \begin{equation*} \left\{ \begin{array}{ll} \Delta ^{2}u-M\left( \Vert \nabla u\Vert _{L^{2}}^{2}\right) \Delta u+\lambda V(x) u=f( x,u) & \text{ in }\mathbb{R}^{N}, \\ u\in H^{2}(\mathbb{R}^{N}), & \end{array}% \right. \end{equation*}% where $N\geq 3$, $M(t) =at^{\delta }+b$ with $a,\delta >0$ and $b\in \mathbb{% R}$, $\lambda >0$ is a parameter, $V\in C(\mathbb{R}^{N},\mathbb{R})$ and $% f\in C(\mathbb{R}^{N}\times \mathbb{R},\mathbb{R}).$ Unlike most other papers on this problem, we allow the constant $b$ to be nonpositive, which has the physical significance. Under some suitable assumptions on $V(x)$ and $f(x,u)$, when $a$ is small and $\lambda$ is large enough, we prove the existence of two nontrivial solutions $u_{a,\lambda }^{(1)}$ and $% u_{a,\lambda }^{(2)}$, one of which will blow up as the nonlocal term vanishes. Moreover, $u_{a,\lambda }^{(1)}\rightarrow u_{\infty}^{(1)}$ and $% u_{a,\lambda }^{(2)}\rightarrow u_{\infty}^{(2)}$ strongly in $H^{2}(\mathbb{% R}^{N})$ as $\lambda\rightarrow\infty$, where $u_{\infty}^{(1)}\neq u_{\infty}^{(2)}\in H_{0}^{2}(\Omega )$ are two nontrivial solutions of Dirichlet BVPs on the bounded domain $\Omega$. It is worth noting that the regularity of weak solutions $u_{\infty}^{(i)}(i=1,2)$ here is explored. Finally, the nonexistence of nontrivial solutions is also obtained for $a$ large enough. more...

In this paper, we study the existence of nodal solutions for the non-autonomous Schr\"{o}dinger--Poisson system: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+u+\lambda K(x) \phi u=f(x) |u|^{p-2}u & \text{ in }\mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2} & \text{ in }\mathbb{R}^{3},% \end{array}% \right. \end{equation*}% where $\lambda >0$ is a parameter and $2 more...

We are concerned with a class of Kirchhoff type equations in $\mathbb{R}^{N}$ as follows: \begin{equation*} \left\{ \begin{array}{ll} -M\left( \int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right) \Delta u+\lambda V\left( x\right) u=f(x,u) & \text{in }\mathbb{R}^{N}, \\ u\in H^{1}(\mathbb{R}^{N}), & \end{array}% \right. \end{equation*}% where $N\geq 1,$ $\lambda>0$ is a parameter, $M(t)=am(t)+b$ with $a,b>0$ and $m\in C(\mathbb{R}^{+},\mathbb{R}^{+})$, $V\in C(\mathbb{R}^{N},\mathbb{R}^{+})$ and $f\in C(\mathbb{R}^{N}\times \mathbb{R}, \mathbb{R})$ satisfying $\lim_{|u|\rightarrow \infty }f(x,u) /|u|^{k-1}=q(x)$ uniformly in $x\in \mathbb{R}^{N}$ for any $2 more...

In this paper, we study the following two-component systems of nonlinear Schr\"odinger equations \begin{equation*} \left\{\aligned&\Delta u-(\lambda a(x)+a_0(x))u+\mu_1u^3+\beta v^2u=0\quad&\text{in }\bbr^3,\\ &\Delta v-(\lambda b(x)+b_0(x))v+\mu_2v^3+\beta u^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 $\lambda,\mu_1,\mu_2>0$ and $\beta<0$ 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 $\lambda$ sufficiently large. The concentration behaviors of solutions as both $\lambda\to+\infty$ and $\beta\to-\infty$ 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., Comment: 39 pages more...

In this paper, we study the following Schr\"odinger-Poisson system: $$ \left\{\aligned&-\Delta u+V_\lambda(x)u+K(x)\phi u=f(x,u)&\quad\text{in }\bbr^3,\\ &-\Delta\phi=K(x)u^2&\quad\text{in }\bbr^3,\\ &(u,\phi)\in\h\times\D,\endaligned\right.\eqno{(\mathcal{SP}_{\lambda})} $$ where $V_\lambda(x)=\lambda a(x)+b(x)$ with a positive parameter $\lambda$, $K(x)\geq0$ 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 more...

The purpose of this paper is to study the indefinite Kirchhoff type problem: \begin{equation*} \left\{ \begin{array}{ll} M\left( \int_{\mathbb{R}^{N}}(|\nabla u|^{2}+u^{2})dx\right) \left[ -\Delta u+u\right] =f(x,u) & \text{in }\mathbb{R}^{N}, \\ 0\leq u\in H^{1}\left( \mathbb{R}^{N}\right), & \end{array} \right. \end{equation*} where $N\geq 1$, $M(t)=am\left( t\right) +b$, $m\in C(\mathbb{R}^{+})$ and $ f(x,u)=g(x,u)+h(x)u^{q-1}$. We require that $f$ is \textquotedblleft local\textquotedblright\ sublinear at the origin and \textquotedblleft local\textquotedblright\ linear at infinite. Using the mountain pass theorem and Ekeland variational principle, the existence and multiplicity of nontrivial solutions are obtained. In particular, the criterion of existence of three nontrivial solutions is established. more...

In this paper, we study the problem: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+u+\lambda K\left( x\right) \phi u=a\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{3}, \\ -\Delta \phi =K\left( x\right) u^{2} & \ \text{in }\mathbb{R}^{3}, \end{array} \right. \end{equation*} where $\lambda >0$ and $20$, we establish some existence results of positive solutions, depending on the parameter $\lambda$. More importantly, we prove the existence of ground state solutions for the case $3.18\thickapprox \frac{1+\sqrt{73}}{3} more...