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Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. (English) Zbl 1285.35051
Summary: We study the Dirichlet problem with mean curvature operator in Minkowski space \[ \text{div}\left( \frac{\nabla v}{\sqrt{1 - |\nabla v|^2}}\right) + {\lambda}[{\mu}(|x|)v^q] = 0 \quad \text{in } \mathcal B(R), \quad v = 0 \quad \text{on } \partial \mathcal B(R), \] where \({\lambda} > 0\) is a parameter, \(q > 1\), \(R > 0\), \({\mu} : [0, \infty) \to \mathbb R\) is continuous, strictly positive on \((0, \infty)\) and \(\mathcal B(R) = \{x \in \mathbb R^N : |x| < R\}\). Using upper and lower solutions and Leray-Schauder degree type arguments, we prove that there exists \({\Lambda} > 0\) such that the problem has zero, at least one or at least two positive radial solutions according to \({\lambda} \in (0, {\Lambda}), {\lambda} = {\Lambda}\) or \({\lambda} > {\Lambda}\). Moreover, \({\Lambda}\) is strictly decreasing with respect to \(R\).

MSC:
35J93 Quasilinear elliptic equations with mean curvature operator
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