Multiple positive solutions for some nonlinear elliptic systems
The following system of nonlinear elliptic equations: \[ k_1\Delta u+V_u(u,v)=0 \quad \text{in }\Omega,\qquad k_2\Delta v+V_v(u,v)=0 \quad\text{in }\Omega, \tag{E} \] \[ {\partial u\over\partial n}={\partial v\over\partial n}=0 \quad \text{on }\partial\Omega,\qquad u(x)>0,\;v(x)>0 \quad\text{in }\Omega \] is considered, where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N\geq 1\) with smooth boundary, \(k_1\), \(k_2\) are positive constants and the potential \(V\in C^2(\mathbb{R}^2,\mathbb{R})\). Assuming that the system (E) has a variational structure, the author finds conditions under which this system either has at least one non-constant positive solution or it has at least two non-constant positive solutions. One of the hypotheses in the main theorem says that there exist \(u_0,v_0>0\) such that \((u_0,v_0)\) is a non-degenerate saddle point of \(V\). The other hypotheses are related to the instability of \((u_0,v_0)\) or to the Morse index of \((u_0,v_0)\) of the corresponding functional. When \(N=1\) a stronger result is obtained, namely that (E) possesses at least \(2(m-1)\) non-constant positive solutions for a suitable \(m\geq 2\). The approach used in the proofs is essentially the same as the one employed by \textit{H. Hofer} [Math. Ann. 261, 493-514 (1982; Zbl 0488.47034)] in the case of the multiplicity of solutions of a scalar elliptic problem \(-\Delta u=g(u)\) in \(\Omega\) under Dirichlet boundary condition. The present author uses the mountain pass theorem, Morse indices as well as the Leray-Schauder degree theory. Moreover the relations between the system (E) and the Lotka-Volterra competition model appearing in mathematical biology are discussed.
scientific article; zbMATH DE number 1157133