Sharp Sobolev inequality involving a critical nonlinearity on a boundary

Jan Chabrowski, Jianfu Yang


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.


Neumann problem; critical Sobolev exponent; topological linking

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