Strongly indefinite functionals and multiple solutions of elliptic systems
Strongly indefinite functionals and multiple solutions of elliptic systems
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DOI：
10.1007/978-3-319-02856-9_35
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发表时间：
2003-03
影响因子：
1.3
通讯作者：
D. G. Figueiredo;Yanheng Ding
中科院分区：
文献类型：
--
作者：
D. G. Figueiredo;Yanheng Ding
研究了椭圆型方程组$$\Left\{{\Begin{数组}{*{20}L}{-\Updeltau=H_{u}(x，u，v)}\hFill&{{\Text{in}}，\Upomega，}\hFill\-\Updeltav=H_{v}(x，u，v)}\hFill&{{\Text{in}}，\quad u(X)=v(X)=0\quad{\Text{on}}\，\Partial\Upomega，}\hFill\end{ARRAY}}\右。$$其中$$\Upomega\subset{\mathbb{R}}^{N}，\，N\ge 3，$$是光滑有界域，$$H\in{\mathcal{C}}^{1}(\overline{\upomega}\Times{\mathbb{R}}^{2}，{\mathbb{R}})。$$我们假设非线性项$$H(x，\，u，\，v)\sim\Left|u\Right|^{p}+\Left|v\Right|^{q}+R(x，\，u，\，v)\，{\Text{with}}\，\mathop{\Lim}\Limits_{{\Left|{(u，v)}\Right|\to\Infty}\frac{R(x，\，u，\，V)}{{\Left|u\Right|^{p}+\Left|v\Right|^{q}=0，$$其中$$p\in(1，\，2^{*})，\，2^{*}：=2N/(N-2)，\，{\Text{and}}\，q\in(1，\，\inty)。$$因此包括了一些超临界系统。得到了非平凡解。当h(x，u，v)在(u，v)中为偶数时，我们证明了系统存在与一系列正能量相关的解序列。负能量)走向无穷大(分别为零)IFP&>；第2条(回复第2条)。所有结果都用变分方法进行了证明。证明了几个新的强不定泛函临界点定理。
We study existence and multiplicity of solutions of the elliptic system $$ \left\{{\begin{array}{*{20}l} {- \Updelta u = H_{u} (x,u,v)} \hfill & {{\text{in}}\,\Upomega,} \hfill \\ {- \Updelta v = H_{v} (x,u,v)} \hfill & {{\text{in}}\,\Upomega,\quad u(x) = v(x) = 0\quad {\text{on}}\,\partial \Upomega,} \hfill \\ \end{array}} \right. $$ where $$ \Upomega \subset {\mathbb{R}}^{N},\,N \ge 3, $$ is a smooth bounded domain and $$ H \in {\mathcal{C}}^{1} (\overline{\Upomega} \times {\mathbb{R}}^{2},{\mathbb{R}}). $$ We assume that the nonlinear term $$ H(x,\,u,\,v)\sim \left| u \right|^{p} + \left| v \right|^{q} + R(x,\,u,\,v)\,{\text{with}}\,\mathop {\lim}\limits_{{\left| {(u,v)} \right| \to \infty}} \frac{R(x,\,u,\,v)}{{\left| u \right|^{p} + \left| v \right|^{q}}} = 0, $$ where $$ p \in (1,\,2^{*}),\,2^{*} : = 2N/(N - 2),\,{\text{and}}\,q \in (1,\,\infty). $$ So some supercritical systems are included. Nontrivial solutions are obtained. WhenH(x, u, v) is even in (u,v), we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) ifp> 2 (resp.p< 2). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.