On positive solutions for a class of quasilinear elliptic systems
In this paper the authors investigate the existence and properties of positive solutions for the following system of Dirichlet problems involving a class of perturbed phi-Laplace operators: \[ -(l_1(t)\varphi_1(z^{\prime}(t))^{\prime}= H_z(t, z(t), u(t)) \;\text{in} \;\;(0, 1), \] \[ -(l_2(t)\varphi_2(u^{\prime}(t))^{\prime}= H_u(t, z(t), u(t)) \;\text{in} \;\;(0, 1), \] with the boundary conditions \(z(0) = z(1) = 0\) and \( u(0) = u(1) = 0\). For this problem the proposed approach is based on variational methods, thus this problem is treated as the Euler-Lagrange equation for the functional \(J\), which is associated with the Fenchel conjugate. In the paper the authors consider the functional \(J\) on a certain subset \({W}\) of its natural domain only. In the paper it is proven that under some hypotheses, there exists \((\overline{z}, \overline{u})\in {\overline{ W}}\) being a positive solution of the problem such that \( \inf_{(z,u) \in W} J(z, u) \geq J(\overline{z}, \overline{u})\). The following is also shown. When we have an explicit definition of \({W}\) then the assumptions in the theorem above allow us to derive the solvability of the problem in \({W}\) and the fact that the solution is a minimizer of \(J\), i.e. the relation \( \inf_{(z,u) \in {W}} J(z, u) =J(\overline{z}, \overline{u})\) holds. It is shown that the results of the paper cover both sublinear and superlinear cases of nonlinearities.
scientific article; zbMATH DE number 5990903