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Uniqueness of solutions of nonlinear Dirichlet problems. (English) Zbl 0803.35057

Summary: We prove the uniqueness of solution for the problem \[ - \Delta u = u^ p + \lambda u \quad \text{in} \quad \Omega, \qquad u>0 \quad \text{in} \quad \Omega, \qquad u = 0 \quad \text{on} \quad \partial \Omega, \] where \(\Omega \subset \mathbb{R}^ n\) is the unit ball with \(n \geq 3\), \(1<p \leq(n+2)/(n-2)\) and \(\lambda>0\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J25 Boundary value problems for second-order elliptic equations
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