Poincaré Problem for Nonlinear Elliptic Equations of Second Order in Unbounded Domains ()
1. Formulation of the Poincaré Boundary Value Problem
Let D be an 
-connected domain including the infinite point with the boundary 
in
where
. Without loss of generality, we assume that D is a circular domain in
, where the boundary consists of 
circles
, 
and
. In this article, the notations are as the same in References [1-8]. We consider the second order equation in the complex form
(1.1)
satisfying the following conditions.
Condition C. 1)
, 
are continuous in 
for almost every point 
and 
for 
2) The above functions are measurable in 
for all continuous functions 
in
, and satisfy
(1.2)
in which 
are non-negative constants.
3) The Equation (1.1) satisfies the uniform ellipticity condition, namely for any number 
and w, U1, 
the inequality
for almost every point 
holds, where 
is a non-negative constant.
4) For any function
, 
, 
satisfies the condition
in which 
satisfy the condition
(1.3)
with a non-negative constant
.
Now, we formulate the Poincaré boundary value problem as follows.
Problem P. In the domain D, find a solution 
of Equation (1.1), which is continuously differentiable in
, and satisfies the boundary condition
(1.4)
in which 
is any unit vector at every point on
, 
and 
are known functions satisfying the conditions
(1.5)
where
, 
, 
, 
are non-negative constants.
If 
and 
on
, where n is the outward normal vector on
, then Problem P is the Dirichlet boundary value problem (Problem D). If 
and 
on
, then Problem P is the Neumann boundary value problem (Problem N), and if
, and 
on
, then Problem P is the regular oblique derivative problem, i.e. the third boundary value problem (Problem III or O). Now the directional derivative may be arbitrary, hence the boundary condition is very general.
The integer
is called the index of Problem P. When the index 
Problem P may not be solvable, and when 
the solution of Problem P is not necessarily unique. Hence we consider the well-posedness of Problem P with modified boundary conditions.
Problem Q. Find a continuous solution 
of the complex equation
(1.6)
satisfying the boundary condition
(1.7)
and the relation
(1.8)
where 
are appropriate real constants such that the function determined by the integral in (1.8) is single-valued in
, and the undetermined function 
is as stated in
in which
, 
are unknown real constants to be determined appropriately. In addition, for 
the solution 
is assumed to satisfy the point conditions
(1.9)
where
are distinct points, and 
are all real constants satisfying the conditions
(1.10)
for a non-negative constant
.
2. Estimates of Solutions for the Poincaré Boundary Value Problem
First of all, we give a prior estimate of solutions of Problem Q for (1.6).
Theorem 2.1. Suppose that Condition C holds and ε = 0 in (1.6) and (1.7). Then any solution 
of Problem Q for (1.6) satisfies the estimates
(2.1)
(2.2)
in which
Proof. Noting that the solution 
of Problem Q satisfies the equation and boundary conditions
(2.3)
(2.4)
(2.5)
according to the method in the proof of Theorem 4.3, Chapter II, [2] or Theorem 2.2.1, [5], we can derive that the solution 
satisfies the estimates
(2.6)
(2.7)
where
and
From (1.8), it follows that
(2.8)
(2.9)
in which 
is a non-negative constant. Moreover, it is easy to see that
(2.10)
Combining (2.6)-(2.10), the estimates (2.1) and (2.2) are obtained.
Theorem 2.2. Let the Equation (1.6) satisfy Condition C and 
in (1.6)-(1.7) be small enough. Then any solution 
of Problem Q for (1.6) satisfies the estimates
(2.11)
(2.12)
here 
are as stated in Theorem 2.1,
Proof. It is easy to see that 
satisfies the equation and boundary conditions
(2.13)
(2.14)
(2.15)
Moreover from (2.6) and (2.7), we have
(2.16)
and from (2.8)-(2.10), it follows that
(2.17)
If the positive constant 
is small enough such that
, then the first inequality in (2.17) implies that
(2.18)
Combining (2.8) and (2.18), we obtain
(2.19)
which is the estimate (2.11). As for (2.12), it is easily derived from (2.9) and the second inequality in (2.17), i.e.
(2.20)
3. Solvability Results of the Poincaré Boundary Value Problem
We first prove a lemma.
Lemma 3.1. If 
satisfies the condition stated in Condition C, then the nonlinear mapping G:
defined by 
is continuous and bounded
(3.1)
where 
Proof. In order to prove that the mapping
:
Defined by 
is continuous, we choose any sequence of functions 
such that
as 
Similarly to Lemma 2.2.1, [5], we can prove that
possesses the property
(3.2)
And the inequality (3.1) is obviously true.
Theorem 3.2. Let the complex Equation (1.1) satisfy Condition C, and the positive constant 
in (1.6) and (1.7) is small enough.
1) When
, 
, Problem Q for (1.6) has a solution
, where
, 
, 
is a constant as stated before.
2) When 
Problem Q for (1.6) has a solution
, where 
provided that
(3.3)
is sufficiently small.
3) If 
satisfy the conditions, i.e. Condition C and for any functions 
and
, there are
(3.4)
where
is a sufficiently small positive constant, then the above solution of Problem Q is unique.
Proof. 1) In this case, the algebraic equation for t is as follows
(3.5)
where M6, M7 are constants as stated in (2.11) and (2.12). Because
, 
, the Equation (3.5) has a unique solution 
Now we introduce a bounded, closed and convex subset B* of the Banach space 
whose elements are of the form 
satisfying the condition
(3.6)
We choose a pair of functions 
and substitute it into the appropriate positions of
, 
in (1.6) and the boundary condition (1.7), and obtain
(3.7)
(3.8)
where
In accordance with the method in the proof of Theorem 1.2.5, [5], we can prove that the boundary value problem (3.7), (3.8) and (1.6) has a unique solution
. Denote by 
the mapping from 
to
. Noting that
provided that the positive number 
is sufficiently small, and noting that the coefficients of complex Equation (3.7) satisfy the same conditions as in Condition C, from Theorem 2.2, we can obtain
(3.9)
This shows that T maps B* onto a compact subset in B*. Next, we verify that T in B* is a continuous operator. In fact, we arbitrarily select a sequence 
in B*, such that
(3.10)
By Lemma 3.1, we can see that
(3.11)
Moreover, from
,
it is clear that 
is a solution of Problem Q for the following equation
(3.12)
(3.13)
(3.14)
In accordance with the method in proof of Theorem 2.2, we can obtain the estimate
(3.15)
in which 
From (3.10), (3.11) and the above estimate, we obtain
as 
On the basis of the Schauder fixed-point theorem, there exists a function 
such that
, and from Theorem 2.2, it is easy to see that
, 
, and
is a solution of Problem Q for the Equation (1.6) and the relation (1.8) with the condition
,
.
In addition, if 
in
where 
then the above solvability result still hold by using the above similar method.
2) Secondly, we discuss the case: 
In this case, (3.5) has the solution 
provided that M9 in (3.3) is small enough. Now we consider a closed and convex subset 
in the Banach space 
i.e.
(3.16)
Applying a method similar as before, we can verify that there exists a solution
of Problem Q for (1.6) with the condition 
Moreover, if 
in D, where
, 
, j = 1, 2. Under the same condition, we can derive the above solvability result by the similar method.
3) When 
satisfies the condition (3.4), we can verify the uniqueness of solutions in this theorem. In fact, if
, 
are two solutions of Problem Q for the Equation (1.6), then
satisfies the equation and boundary conditions
(3.17)
(3.18)
(3.19)
in which
. Similarly to Theorem 2.2, we can derive the following estimates of the solution 
for complex Equation (3.17):
(3.20)
(3.21)
where
are two non-negative constants, 
Moreover the estimate
(3.22)
can be derived. Provided that the positive constant 
is small enough such that
, from (3.22) it follows
, i.e. 
in D. This completes the proof of the theorem.
From the above theorem, the next result can be derived.
Theorem 3.3. Under the same conditions as in Theorem 3.2, the following statements hold.
1) When the index K > N, Problem P for (1.1) has N solvability conditions, and the solution of Problem P depends on 
arbitrary real constants.
2) When 
Problem P for (1.1) is solvable, if 
solvability conditions are satisfied, and the solution of Problem P depends on 
arbitrary real constants.
3) When K < 0, Problem P for (1.1) is solvable under 
conditions, and the solution of Problem P depends on 1 arbitrary real constant.
Moreover, we can write down the solvability conditions of Problem P for all other cases.
Proof. Let the solution 
of Problem Q for (1.6) be substituted into the boundary condition (1.7) and the relation (1.8). If the function
, i.e.
and
, 
, then we have 
in D and the function 
is just a solution of Problem P for (1.1). Hence the total number of above equalities is just the number of solvability conditions as stated in this theorem. Also note that the real constants b0 in (1.8) and 
in (1.9) are arbitrarily chosen. This shows that the general solution of Problem P for (1.1) includes the number of arbitrary real constants as stated in the theorem.