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The basin of attraction method for a kind of nonlinear elliptic equations
Boundary Value Problems volume 2014, Article number: 190 (2014)
Abstract
In this paper, a new sufficient condition of the existence and uniqueness of the second order elliptic boundary value problem is given by using the basin of attraction. Some known existent results on the existence of solutions of the nonlinear elliptic equation are generalized.
MSC: 35C15.
1 Introduction
We will study the existence of solution for the second order elliptic equation
where Δ denotes the n-dimensional Laplacian.
In 1976, Elcrat [1] derived an a priori estimate for the elliptic operator defined by
and applied the result in the semilinear elliptic equation
to obtain the following existence theorem.
Theorem 1.1
[1]
Suppose that f satisfies the hypotheses of[1], Lemma 9] and that
-
(1)
, ;
-
(2)
.
Then there is a unique solution of the equationin, where ν is a positive constant, , λ will be explained by the following paper.
Comparing (1.1) with (1.2), we have , , then (1.2) becomes (1.1), and Theorem 1.1 can be expressed.
Corollary 1.1
[1]
Suppose that f satisfies the hypotheses of[1], Lemma 9] and that
-
(1)
;
-
(2)
.
Then there is a unique solution of the equationin.
In this paper, we will give a new set of sufficient condition for the existence and uniqueness of the second order elliptic equation (1.1), to which Section 3 is devoted. In our main Theorem 3.1, we replace Elcrat’s condition (2) by the condition
-
(2)
for any , , where , and Theorem 1.1 is a corollary of our main theorems.
Our proofs are different from the proofs given by Elcrat [1]. In Section 2 some functional analytic preliminaries are stated and some inequalities are derived in Section 3, which are useful for the proofs of our main theorems.
2 Preliminaries
We denote by X, Y Banach spaces, D an open connected subset of X. The following theorems will be employed to prove our main theorems.
Theorem 2.1
[2]
Letbe amapping and a local homeomorphism, let. Then for any, the path-lifting problem
has a unique continuous solutiondefined on the maximal open interval, . Moreover, the setis open inand the mappingis continuous.
Definition 2.1
[2]
In the setting of Theorem 2.1, the basin of attraction of is the set
Theorem 2.2
[3]
Letbe a local homeomorphism. Then f is a global homeomorphism of X onto Y if and only ifis defined on R for all, namely, can also be extended to −∞.
3 Main theorems
Consider the second order elliptic boundary value problem
where is measurable in x for all u and has continuous partial derivatives in u for almost all x.
We assume that Ω is a bounded domain in with piecewise smooth boundary ∂G whose principal curvatures are bounded, and we assume that the function u has square summable second derivatives on the domain Ω.
We will first study the action of the differential operator L
on the class , which may be defined as the closure of functions in that vanish on ∂G, where is for the bounded measurable functions defined in Ω.
With the above assumption L may be thought of as a linear operator mapping into . Let be the Hilbert space with the norm
where denotes the sum of the squares of the second derivatives.
Our goal is then to establish an inequality of the form
for u in . In order to obtain the desired results we will make use of the lowest eigenvalue λ for the Laplacian with homogeneous boundary conditions on Ω, which is given by
where the infimum is taken over u in .
Let
Corollary 3.1
If, then the inequality
holds, where.
Proof
From [4], Chapter 2, Section 8], we have
then for any , we can get
if , and from (3.3), it follows that
So
and if , then
So
Set , we have
The corollary is proved. □
We express (3.1) in the form
and (3.1) is equivalent to the operator equation
For any , we have
Now, we will show our main theorems.
Theorem 3.1
Suppose
-
(1)
;
-
(2)
(3.4)
where, .
Then there is a unique solution of the equation
in.
Proof
The condition implies that zero is not an eigenvalue of , so for every , the operator is invertible and P is a local homeomorphism from onto , where I denotes the identical operator.
Denote . From Corollary 3.1, we have
So we obtain
Consider the path-lifting problem for the mapping P:
It is clear that
Then
Using the Gronwall inequality, we have
which, together with (3.4), implies
By Theorem 2.2, now we need only show that can be extended to −∞.
Let , , . Then for
So is Lipschitz continuous on and can be extended to −∞, that is to say, P is a diffeomorphism from to and the theorem is proved. □
Corollary 3.2
Assume that f satisfies:
-
(1)
;
-
(2)
Uniformly in x, , where ω is a continuous map satisfying
(3.5)
Then there is a unique solution of (3.1) in.
Proof
We have from Theorem 1 in [1]
for positive constants α, β.
From (3.5),
By Theorem 3.1, there is a unique solution of (3.1) in . □
Corollary 3.3
(Elcrat)
Suppose that P is given by, that f satisfies (1) , (2) , uniformly in x.
Then there is a unique solution of the equationin.
Proof
Condition (2) implies that holds. The result of Elcrat in [5] becomes a special case of Theorem 3.1. □
References
Elcrat AR: Constructive existence for semilinear elliptic equations with discontinuous coefficients. SIAM J. Math. Anal. 1974, 5: 663-672. 10.1137/0505066
Gorni G: A criterion of invertibility in the large for local diffeomorphisms between Banach spaces. Nonlinear Anal. 1993, 21: 43-47. 10.1016/0362-546X(93)90176-S
Marco GD, Gorni G, Zampieri G: Global inversion of function: an introduction. Nonlinear Differ. Equ. Appl. 1994, 1: 229-248. 10.1007/BF01197748
Ladyzhenskaya OA, Uraltseva NN: Linear and Quasilinear Elliptic Equations. Academic Press, New York; 1968.
Feng YQ, Wang ZY, Wen CJ: Global homeomorphism and applications to the existence and uniqueness of solutions of some differential equations. Adv. Differ. Equ. 2014.
Acknowledgements
The author is grateful to the referees for their comments and references, which improved the paper. The work has been supported by the Natural Science Foundation of Jiangsu (13KJD110001).
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Feng, Yq. The basin of attraction method for a kind of nonlinear elliptic equations. Bound Value Probl 2014, 190 (2014). https://doi.org/10.1186/s13661-014-0190-7
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DOI: https://doi.org/10.1186/s13661-014-0190-7