Sharp Sobolev inequality involving a critical nonlinearity on a boundary
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Neumann problem, critical Sobolev exponent, topological linkingAbstrakt
We consider the solvability of the Neumann problem for the equation $$ -\Delta u+\lambda u =0, \quad \frac {\partial u}{\partial \nu}=Q(x)|u|^{q-2}u $$ on $\partial \Omega$, where $Q$ is a positive and continuous coefficient on $\partial \Omega$, $\lambda$ is a parameter and $q= {2(N-1)}/{(N-2)}$ is a critical Sobolev exponent for the trace embedding of $H^1(\Omega)$ into $L^q(\partial \Omega)$. We investigate the joint effect of the mean curvature of $\partial \Omega$ and the shape of the graph of $Q$ on the existence of solutions. As a by product we establish a sharp Sobolev inequality for the trace embedding. In Section 6 we establish the existence of solutions when a parameter $\lambda$ interferes with the spectrum of $-\Delta$ with the Neumann boundary conditions. We apply a min-max principle based on the topological linking.Pobrania
Opublikowane
2005-03-01
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1.
CHABROWSKI, Jan & YANG, Jianfu. Sharp Sobolev inequality involving a critical nonlinearity on a boundary. Topological Methods in Nonlinear Analysis [online]. 1 marzec 2005, T. 25, nr 1, s. 135–153. [udostępniono 22.7.2024].
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