In this note we prove the Payne-type conjecture about the behaviour of thenodal set of least energy sign-changing solutions for the equation $-Delta_p u= f(u)$ in bounded Steiner symmetric domains $Omega subset mathbb{R}^N$under the zero Dirichlet boundary conditions. The nonlinearity $f$ is assumedto be either superlinear or resonant. In the latter case, least energysign-changing solutions are second eigenfunctions of the zero Dirichlet$p$-Laplacian in $Omega$. We show that the nodal set of any least energysign-changing solution intersects the boundary of $Omega$. The proof is basedon a moving polarization argument.
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机译：在本说明书中，我们证明了关于等式$最低能量符号更改解决方案的Payne型猜想 - delta_p u = f（u）$ ineded steiner对称域$ oomega subset mathbb { r} ^ n $在零dirichlet边界条件下。非线性$ F $被认为是超线性或谐振。在后一种情况下，更低的Energysign-yours的解决方案是$ omega $的零Dirichlet $ P $ -laplacian的第二个特征功能。我们表明，任何最少的Energysign的解决方案的节点集会与$ Omega $的边界相交。证明是一种移动的极化参数。
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