Poincaré Problem for Nonlinear Elliptic Equations of Second Order in Unbounded Domains

In [1], I. N. Vekua propose the Poincaré problem for some second order elliptic equations, but it can not be solved. In [2], the authors discussed the boundary value problem for nonlinear elliptic equations of second order in some bounded domains. In this article, the Poincaré boundary value problem for general nonlinear elliptic equations of second order in unbounded multiply connected domains have been completely investigated. We first provide the formulation of the above boundary value problem and corresponding modified well posed-ness. Next we obtain the representation theorem and a priori estimates of solutions for the modified problem. Finally by the above estimates of solutions and the Schauder fixed-point theorem, the solvability results of the above Poincaré problem for the nonlinear elliptic equations of second order can be obtained. The above problem possesses many applications in mechanics and physics and so on.

3) The Equation (1.1) satisfies the uniform ellipticity condition, namely for any number u and w, U 1 ,  the inequality      , , , 0 , , with a non-negative constant .
0 Now, we formulate the Poincaré boundary value problem as follows.

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Problem P. In the domain D, find a solution  u z of Equation (1.1), which is continuously differentiable in D , and satisfies the boundary condition where ,

 
, n 0, K 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.
satisfying the boundary condition and the relation where j 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 are unknown real constants to be determined appropriately.In addition, for the solution   w z   is assumed to satisfy the point conditions  are distinct points, and for a non-negative constant .

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 according to the method in the proof of Theorem 4.
It is easy to see that satisfies the equation and boundary conditions .
Moreover from (2.6) and (2.7), we have If the positive constant  is small enough such that Combining (2.8) and (2.18), we obtain , then the first inequality in (2.17) implies that 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)

Solvability Results of the Poincaré Boundary Value Problem
ed in Condition C, then the nonlinear mapping G:

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We first prove a lemma.Lemma 3.1.If satisfies the condition stat- possesses the property And the inequality (3.1) is obviously true.1.1) satisfy C Theorem 3.2.Let the complex Equation ( ondition C, and the positive constant  in (1.6) and (1.7) is small enough.,  , , , z  F z u w w i.e G z u w satisfy the conditions, ition C and for any funct .Cond ions V z L  , th  is a sufficiently small positive constant, th the above solution of Problem Q is unique.ation for t is as en Proof. 1) In this case, the algebraic equ follows , , , , , stated in (2.11) and (2.12).

, , F z u w u w w Q z u w w w A z u w w
A z u w u A z u w In accordance with the method in the proof of Theore  m 1.2.5, [5], we can prove that the boundary value problem (3.7), (3.8) and (1.6) has a unique solution  provided that the positive number L  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 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 s By Lemma 3.1, we can see that 1, 2,3 as .
In accordance with the method in proof of Theorem 2.2, we can obtain the estimate in which 11 11 0 0 , , M M q p  (3.11) and the above estimate On the basis of the Schauder fixed-po ists a function , and from Th , , for the Equation (1.6) and the relation (1.8) with tion 0 the condi then the above solvability result still similar method.
2) Secondly, we discuss the case: hold by using the above   min , 1.
   In this case, (3.5) has the solution 10 t M  provided that M 9 in (3.3) is small enough.Now we consider a closed and convex subset B  in the Banach space Applying a method similar as before, we can verify that there exists a solution ) with the condition vability resul ndition, we can derive the above sol t by the similar method.

 
, , G z u w satisfies the condition (3.4), we can verify the uniqu ess of solutions in this t en heorem.In fact, if tisfies the equation and boundary conditions   . Similarly to Theorem 2.2, we can derive the following estimates of the solution , , , , , , , , 12,13 can be derived.Provided that the positive constant [2] G. C re ERENCES

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Without loss of generality, we assume that D is a circular domain in whose elements are of the form 5] G. C. Wen, D. C. Chen and Z. L. Xu, "Nonlinear Complex Analysis and Its Applications," Mathematics Monograph Series 12, Science Press, Beijing, 2008. [