### Sharp Sobolev inequality involving a critical nonlinearity on a boundary

#### Abstract

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.

$$

-\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.

#### Keywords

Neumann problem; critical Sobolev exponent; topological linking

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