论文标题
Quasilinearschrödinger-波森系统
The quasilinear Schrödinger--Poisson system
论文作者
论文摘要
本文涉及$（p，q）$ - schrödinger--poisson system \ begin {eqnarray*} \ left \ webt \ {\ begin {array} {ll} {ll} \ displaystyle-Δ_p U+| U |^{p-2} U+λϕ | U |^{s-2} u = | U | U |^{r-2} u，＆\ Mathrm {in} \ end {array} \ right。 \ end {eqnarray*} 其中$ 1 <p <3 $，$ \ max \ left \ {1，\ frac {3p} {5p-3} {5p-3} \ right \} <q <3 $，$ p <r <p^*：= \ frac {3p {3p} $ \ max \ left \ {1，\ frac {（q^* - 1）p} {q^*} \ right \} <s <s <\ frac {（q^* - 1）p^*} {q^*} {q^*} $，$Δ_i= \ hbox u = \ hbox u = \ hbox {div} div}（ （i = p，q）$和$λ> 0 $是一个参数。这种准线性系统是新的，在文献中从未考虑过。准泊泊托泊方程溶液的独特性是通过minty-rowder定理获得的。构建了准线性系统的变异框架，并通过山间定理获得系统的非平凡解决方案。
This paper deals with the $(p,q)$--Schrödinger--Poisson system \begin{eqnarray*} \left \{\begin{array}{ll} \displaystyle -Δ_p u+|u|^{p-2}u+λϕ|u|^{s-2}u=|u|^{r-2}u,&\mathrm{in} \ \mathbb{R}^3,\\ \displaystyle -Δ_q ϕ= |u|^s, &\mathrm{in}\ \mathbb{R}^3,\\ \end{array} \right. \end{eqnarray*} where $1<p<3$, $\max \left\{1,\frac{3p}{5p-3}\right\}<q<3$, $p<r<p^*:=\frac{3p}{3-p}$, $\max\left\{1,\frac{(q^*-1)p}{q^*}\right\}<s<\frac{(q^*-1)p^*}{q^*}$, $Δ_i u=\hbox{div}(|\nabla u|^{i-2}\nabla u)\ (i=p,q)$ and $λ>0$ is a parameter. This quasilinear system is new and has never been considered in the literature. The uniqueness of solutions of the quasilinear Poisson equation is obtained via the Minty--Browder theorem. The variational framework of the quasilinear system is built and the nontrivial solutions of the system are obtained via the mountain pass theorem.
