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On partially free boundary solutions for elliptic problems with non-Lipschitz nonlinearities. (English) Zbl 1432.35096
In this article, the elliptic equation with a non-Lipschitz right-hand side of the form \(-\Delta u=\lambda |u|^{\beta-1}u-|u|^{\alpha-1}u\) is considered. The parameters are assumed to satisfy \(\lambda>0\) and \(0<\alpha<\beta<1\). The domain under consideration is a smooth star-shaped domain of \(\mathbb{R}^N\) with \(N\geq 2\). Further, the boundary condition is given to be homogeneous Dirichlet. The authors show that this problem might have a non-negative ground state solution that violates Hopf’s maximum principle only on a non-empty proper subset of the domain’s boundary.
MSC:
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J25 Boundary value problems for second-order elliptic equations
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