• 대한수학회지
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• 제59권6호
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• pp.1139-1151
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• 2022

This paper studies the eigenvalues of the G(·)-Laplacian Dirichlet problem $$\{-div\;\(\frac{g(x,\;{\mid}{\nabla}u{\mid})}{{\mid}{\nabla}u{\mid}}{\nabla}u\)={\lambda}\;\(\frac{g(x,{\mid}u{\mid})}{{\mid}u{\mid}}u\)\;in\;{\Omega}, \\u\;=\;0\;on\;{\partial}{\Omega},$$ where Ω is a bounded domain in ℝ^N and g is the density of a generalized Φ-function G(·). Using the Lusternik-Schnirelmann principle, we show the existence of a nondecreasing sequence of nonnegative eigenvalues.

• 대한수학회보
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• 제35권2호
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• pp.269-278
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• 1998

Let ($G,\;{\upsilon}$) be a bounded smooth domain and reflection vector field on $\partial$G, which points uniformly into G. Under the condition that locally for some coordinate system, ${\mid}{\upsilon^i}{\mid}\;i\;=\;1,{\cdot},{\cdot}$,d - 1, where is constant depending on the Lipschitz constant of G, we have tightness for reflected diffusion with jump on G with reflection $\upsilon$ depending only on c. From this, we obtain some properties of L-harmonic function where L is a sum of Laplacian and integro one.