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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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##### References:
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