论文标题
耦合梯度型准线性椭圆系统的多种溶液具有超临界生长
Multiple solutions for coupled gradient-type quasilinear elliptic systems with supercritical growth
论文作者
论文摘要
在本文中，我们考虑以下耦合梯度型椭圆形系统\ begin {equation*} \ left \ {\ oken {array} {ll} {ll} - {\ rm div}（a（x，x，x，x，x，u，u，\ nabla u）） $ω$，} \\ [10pt] - {\ rm div}（b（x，v，\ nabla v））） + b_t（x，v，v，\ nabla v）= g_v \ left（x，x，u，u，v \ oright） \ end {array} \ right。 \ end {equation*}其中$ω$是$ \ mathbb {r}^n $，$ n \ ge 2 $中的一个打开界域。我们假设某些$ \ Mathcal {C}^{1} $ - Carathéodory函数$ a，b：ω\ times \ times \ Mathbb {r} \ times \ times \ Mathbb {r}^n \ rightarrow \ rightArrow \ rightArrow \ rightBb {r} $ a_t（x，t，ξ）= \ frac {\ partial a} {\ partial t}（x，x，t，t，ξ）$，$ b（x，t，t，ξ）= \nabla_ξb（x，x，t，t，t，ξ）$，$ b_t（$ b_t（x，x，t，t，ξ）= \ freac = \ freac partial b} t}（x，t，ξ）$，然后$ g_u（x，u，v）$，$ g_v（x，u，u，v）$是$ \ mathcal {c}^{1} $ - carathéodorynorlinearity a的部分衍生物$ g：ω\ times \ mathbb {r} \ times \ times \ mathbb {r} \ rightarrow \ mathbb {r} $。粗略地说，我们假设$ a（x，t，ξ）$至少成长为$（1+ | t | t |^{s_1p_1}）|ξ|^{p_1} $，$ p_1> 1 $，$ s_1 \ ge 0 $ $（1+ | t |^{s_2p_2}）|ξ|^{p_2} $，$ p_2> 1 $，$ s_2 \ ge 0 $，而$ g（x，u，u，v）$也可以具有与$ s_1 $和$ s_2 $相关的超批评性增长。由于系数取决于溶液及其梯度本身，因此需要研究合适的Banach空间中两个不同规范的相互作用。尽管存在这些困难，但还是使用了一种变异方法来表明该系统接受了非平凡的弱界面解决方案，并且在对称性的假设下，无限很多。
In this paper we consider the following coupled gradient-type quasilinear elliptic system \begin{equation*} \left\{ \begin{array}{ll} - {\rm div} ( a(x, u, \nabla u) ) + A_t (x, u, \nabla u) = G_u(x, u, v) &\hbox{ in $Ω$,}\\[10pt] - {\rm div} ( b(x, v, \nabla v) ) + B_t(x, v, \nabla v) = G_v\left(x, u, v\right) &\hbox{ in $Ω$,}\\[10pt] u = v = 0 &\hbox{ on $\partialΩ$,} \end{array} \right. \end{equation*} where $Ω$ is an open bounded domain in $\mathbb{R}^N$, $N\ge 2$. We suppose that some $\mathcal{C}^{1}$-Carathéodory functions $A, B:Ω\times\mathbb{R}\times\mathbb{R}^N\rightarrow\mathbb{R}$ exist such that $a(x,t,ξ) = \nabla_ξ A(x,t,ξ)$, $A_t(x,t,ξ) = \frac{\partial A}{\partial t} (x,t,ξ)$, $b(x,t,ξ) = \nabla_ξ B(x,t,ξ)$, $B_t(x,t,ξ) =\frac{\partial B}{\partial t}(x,t,ξ)$, and that $G_u(x, u, v)$, $G_v(x, u, v)$ are the partial derivatives of a $\mathcal{C}^{1}$-Carathéodory nonlinearity $G:Ω\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$. Roughly speaking, we assume that $A(x,t,ξ)$ grows at least as $(1+|t|^{s_1p_1})|ξ|^{p_1}$, $p_1 > 1$, $s_1 \ge 0$, while $B(x,t,ξ)$ grows as $(1+|t|^{s_2p_2})|ξ|^{p_2}$, $p_2 > 1$, $s_2 \ge 0$, and that $G(x, u, v)$ can also have a supercritical growth related to $s_1$ and $s_2$. Since the coefficients depend on the solution and its gradient themselves, the study of the interaction of two different norms in a suitable Banach space is needed. In spite of these difficulties, a variational approach is used to show that the system admits a nontrivial weak bounded solution and, under hypotheses of symmetry, infinitely many ones.
