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On a class of singular superlinear elliptic systems in a ball. (English) Zbl 1410.35031

Author’s abstract: We establish the existence of large positive radial solutions for the elliptic system \[ \begin{cases} -\Delta u= \lambda f(v) & \text{in}\;B, \\ -\Delta v= \lambda g(u) & \text{in}\;B, \\ u=v=0 \quad & \text{on}\;\partial B, \end{cases} \] when the parameter \(\lambda>0\) is small, where \(B\) is the open unit ball \(\mathbb{R}^N\), \(N>2\), \(f,g: (0,\infty) \to \mathbb{R}\) are possibly singular at \(0\) and \(f(u) \sim u^p\), \(g(v) \sim v^q\) at \(\infty\) for some \(p,q>0\) with \(pq>1\). Our approach is based on fixed point theory in a cone.

MSC:

35J57 Boundary value problems for second-order elliptic systems
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
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