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Positive solutions of semilinear elliptic systems. (English) Zbl 0818.35027
The authors investigate the existence of positive solutions of the Dirichlet problem $$- \Delta u = f(v),\ -\Delta v = g(u) \text{ in } \Omega,\quad u = 0,\ v = 0 \text{ on } \partial \Omega$$ in convex bounded domains $\Omega \subset \bbfR\sp N$, where $f,g : \bbfR\sp + \to \bbfR$ are nondecreasing continuously differentiable functions. The existence of positive solutions is established by a degree argument together with a priori estimates. In the special case $f(v) = v$ the boundary value problem $\Delta\sp 2u = g(u)$ in $\Omega$, $u = 0$ and $\Delta u = 0$ on $\partial \Omega$ is considered. Finally the eigenvalue problem $\Delta\sp 2u = \lambda u + h(u)$ in $\Omega$ $(h(0) = 0)$, $u = 0$ and $\Delta u = 0$ on $\partial \Omega$ is studied looking for a branch of positive solutions bifurcating from the line of trivial solutions $(\lambda, 0)$, $\lambda \in \bbfR$.

35J65Nonlinear boundary value problems for linear elliptic equations
35B32Bifurcation (PDE)
35J55Systems of elliptic equations, boundary value problems (MSC2000)
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