Solutions for resonant elliptic systems with nonodd or odd nonlinearities
Let \(\Omega\) be a bounded smooth domain in \(\mathbb{R}^N\), \(\lambda, \gamma,\delta \in\mathbb{R}\), \(F\in C^1(\overline\Omega \times\mathbb{R}^2, \mathbb{R})\), \(A^*= \left(\begin{smallmatrix} \lambda & \delta \\ \delta & \gamma\end{smallmatrix} \right)\), \(\sigma(A^*)= \{\xi,\zeta\}\), \(A^*e_1=\xi e_1\), \(A^*e_2=\zeta e_2\), \(e_1,e_2\in \mathbb{R}^2\), \(e_1\bullet e_2=0\), \(| e_1|=| e_2|=1\), \(\sigma(A^*) \cap\sigma (-\Delta)\neq \emptyset\). Theorem 1. We suppose that there exist \(C>0\), \(0<\alpha<1\) such that \(|\nabla F(.,y) |\leq C(| y|^\alpha +1)\), a.e. for \(y\in\mathbb{R}^2\) and there exist \(\theta_1,\theta_2\in C(\overline \Omega,\mathbb{R})\) such that \(\theta_1\geq 0\) a.e., \(\int_\Omega\theta_1 dx\neq 0\), \[ \theta_1 \leq\liminf_{y\to\infty} {\nabla F(.,y) \bullet y\over| y |^{1+ \alpha}} \leq\limsup_{y\to\infty} {\nabla F(.,y) \bullet y\over| y |^{1+\alpha}} \leq\theta_2 \text{ a.e.} \] Then there is a solution of \[ -\Delta u=\lambda u+\delta v+\partial_1 F(.,u,v),\quad -\Delta v=\delta u+\gamma v+ \partial_2F(.,u,v) \text{ in }\Omega,\quad u,v\in H^1_0 (\Omega).\tag{*} \] If moreover \(F(.,0)=0\) a.e. and there exist \(\delta_1\) and \(\delta_2\) such that \(\max\{ \lambda_j- \zeta,\lambda_{k-1} -\lambda_k\} \leq\delta_1\leq \delta_2<0\), where \(\xi=\lambda_k\) for \(k>1\), \(\lambda_j<\zeta \leq\lambda_{j+1}\) and \(\sigma(-\Delta)= \{\lambda_k: k\in \mathbb{Z}_+\}\), \(\lambda_0\equiv-\infty< \lambda_1< \cdots< \lambda_n <\lambda_{n+1} <\dots\), and such that \(\delta_1\leq\inf \{2F(.,y)/ | y|^2: y\neq 0\}\), \(\limsup_{y\to 0}2F(.,y)/ | y|^2\leq\delta_2\) a.e., then the system (*) has at least one nontrivial solution. Theorem 2. We assume that \(\nabla F(.,-y)=-\nabla F(.,y)\) for all \(y\in\mathbb{R}^2\), \(F(.,0)=0\) and that \(\liminf_{y\to 0}2F(.,y)/ | y|^2= +\infty\) a.e., we define \(A(u,v)=(u-\xi(-\Delta)^{-1}u\), \(v-\zeta (-\Delta)^{-1}v)\) for \(u\), \(v\in H^1_0(\Omega)\), \(I(u,v)= (1/2)\langle A(u,v),(u,v) \rangle-\int_\Omega F(.,ue_1+ ve_2)dx\). If \(\lim_{y\to\infty} \nabla F(.,y)=0\) a.e. and there exists \(R_0>0\) such that \(F(.,y) \leq 0\) a.e. for \(y\in\mathbb{R}^2\), \(| y|\geq R_0\) or if there exists \(M_0>0\) such that \(|\nabla F(.,y) |\leq M_0\) a.e. for \(y\in\mathbb{R}^2\) and there exists \(R_0>0\) such that \(F(.,y)-{1\over 2} \nabla F(.,y)\bullet y\geq h\) a.e. for \(y\in\mathbb{R}^2\), \(| y|\geq R_0\), where \(h\in C(\overline\Omega, \mathbb{R})\) and \(\int_\Omega hdx>0\), then the system (*) has infinitely many solutions \((u_n,v_n)\), \(n\in\mathbb{N}\) and \(I(u_n,v_n)\to 0^-\).
scientific article; zbMATH DE number 1215630