Proof
(Existence) Let
is such
that
for some
such that
on
.
For example, given the condition
(
Non-coercivity assumption 1
),
can be a large enough constant so that
.

According to the proposition
(
Energy estimates for the
bilinear form B
) there exist constants
such
that

According to the proposition
(
Existence and
uniqueness for stationary problem
) there is a sequence
such
that

We now prove that
a.e.
in
.
Indeed, the
are given by the
relationships
Note that we cannot apply the proposition
(
Monotonicity of
solution of stationary problem
) because the operators of the problems are
different and because
but
.
However, we act similarly to the proof of the proposition
(
Monotonicity of
solution of stationary problem
). We set
in the first relationship and
in the
second:
add:
and move
terms
Note that
.
Hence
Therefore, we
deduce
or
where
Hence, by the choice of
in
,

Next, we prove that
a.e. for
Indeed, we use the recursion and assume that the same is proven for
.
The
and
are given by the
relationships
Hence, by the assumption of recursion, the proposition
(
Monotonicity of
solution of stationary problem
) applies.

Next, we show that
for some constant
using the recurrence argument.

We choose
so
that
We set
in the equation
so
that
Note that
hence
We use
:
and apply the
:
By the recurrence assumption we have
and by the choice of
we have
.
Hence
.
Therefore,

We collect the previous results into the
relationship

We now complete the proof of existence as follows. We fix
in
and rewrite it using
Combining it with
and boundedness of
we
deduce

Therefore, according to the proposition
(
Weak compactness of bounded
set
),
then by the proposition
(
Rellich-Kondrachov
compactness
theorem
),
and boundedness of
and
We now pass the relationship
:
to the limit
:
thus