R E S E A R C H
Open Access
The basin of attraction method for a kind of
nonlinear elliptic equations
Yan-qing Feng
**Correspondence: [email protected]
Department of Mathematics, Changzhou Institute of Technology, Changzhou, Jiangsu 213002, P.R. China
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
Keywords: the basin of attraction; the second order elliptic equation; inequality; homeomorphism
1 Introduction
We will study the existence of solution for the second order elliptic equation
u=f(x,u), (.)
wheredenotes then-dimensional Laplacian.
In , Elcrat [] derived ana prioriestimate for the elliptic operatorLudefined by
Lu=aijuxixj+aiuxi–au
and applied the result in the semilinear elliptic equation
Lu=aijuxixj+aiuxi=f(x,u) (.)
to obtain the following existence theorem.
Theorem .[] Suppose that f satisfies the hypotheses of[, Lemma ]and that
() inffu≥√λS–λν+ε,ε> ;
() fu= O(u).
Then there is a unique solution of the equation Pu=Lu–f(x,u) = in W, (),whereν
is a positive constant,S=sup|ai– (aij)xj|,λwill be explained by the following paper. Comparing (.) with (.), we haveaij= ,ai= , then (.) becomes (.), and Theo-rem . can be expressed.
Corollary .[] Suppose that f satisfies the hypotheses of[, Lemma ]and that
() inffu≥–λ; () fu= O(u).
Then there is a unique solution of the equation Pu=u–f(x,u) = in W ,().
In this paper, we will give a new set of sufficient condition for the existence and unique-ness of the second order elliptic equation (.), to which Section is devoted. In our main Theorem ., we replace Elcrat’s condition () by the condition
() for anyu∈W, (),+∞+N(s)+dsQ(s)N(s)=∞, where
N(s) =supu
≤s[u–f
u(x,u)]–,Q(s) =sup×Rfu(x,u)and Theorem . is a corollary of our main theorems.
Our proofs are different from the proofs given by Elcrat []. In Section some functional analytic preliminaries are stated and some inequalities are derived in Section , which are useful for the proofs of our main theorems.
2 Preliminaries
We denote byX,YBanach spaces,Dan open connected subset ofX. The following the-orems will be employed to prove our main thethe-orems.
Theorem .[] Let f :D⊂X→Y be a C mapping and a local homeomorphism,let
x∈D.Then for any x∈D,the path-lifting problem
f(γx(t)) =f(x) +e–t(f(x) –f(x)), t∈R,
γx() =x, γx(t)∈D
(.)
has a unique continuous solution t→γx(t) defined on the maximal open interval Ix= (tx–,tx+), –∞ ≤tx–,tx+≤+∞.Moreover,the set{(x,t)∈D×R:t∈Ix}is open in D×R
and the mapping(x,t)→γx(t)is continuous.
Definition .[] In the setting of Theorem ., the basin of attraction ofxis the set
A=x∈D:tx+= +∞, lim
t→+∞γx(t) =x
.
Theorem .[] Let f :X→Y be a local homeomorphism.Then f is a global homeomor-phism of X onto Y if and only ifγx(t)is defined on R for all x∈A,namely,γx(t)can also be extended to–∞.
3 Main theorems
Consider the second order elliptic boundary value problem
u=f(x,u),
u(x) = , x∈∂, (.)
wheref(x,u) is measurable inxfor alluand has continuous partial derivatives inufor almost allx.
We will first study the action of the differential operatorL
Lu=u–a(x)u (.)
on the classW
,(), which may be defined as the closure of functions inC() that vanish¯
on∂G, wherea(x) is for the bounded measurable functions defined in.
With the above assumptionLmay be thought of as a linear operator mappingW ,()
intoL(). LetW, () be the Hilbert space with the norm
u
=
u+|∇u|+ Du dx,
where|Du|denotes the sum of the squares of the second derivatives. Our goal is then to establish an inequality of the form
u≤C
u+u
foruinW
,(). In order to obtain the desired results we will make use of the lowest
eigenvalueλfor the Laplacian with homogeneous boundary conditions on, which is given by
λ=inf
(∇u)
u
, (.)
where the infimum is taken overuinW ().
Let
m=inf
x∈a(x).
Corollary . Ifinfx∈a(x) > –λ,then the inequality u≤C
u+u
holds,where C= (ε+λ+ε+ )/.
Proof From [, Chapter , Section ], we have
Du dx≤
(u)dx,
then for anyε> , we can get
(∇u)dx+
audx≤–
uLu dx≤ε
udx+ ε
(Lu)dx,
(∇u)dx+
audx≤
ε +
ε
udx+ ε
if –λ<m< , and from (.), it follows that
(∇u)dx–λ
udx≤
ε +
ε
udx+ ε
(u)dx,
(∇u)dx≤
ε +
ε+λ
udx+ ε
(u)dx.
So
u
=
u+|∇u|+ Du dx
≤
ε
+ +λ+ ε
u+
ε+
(u)
and ifm≥, then
(∇u)dx≤
ε +
ε
udx+ ε
(u)dx.
So
u
=
u+|∇u|+ Du dx
≤ ε + + ε
u+
ε+
(u).
SetC= (ε+λ+ε+ )/, we have
u≤C
u+u
.
The corollary is proved.
We express (.) in the form
P(u) =u–f(x,u)
and (.) is equivalent to the operator equation
P(u) = , u∈W, ().
For anyu,φ∈W
,(), we have
P(u)(φ) =φ–fu(x,u)φ.
Now, we will show our main theorems.
Theorem . Suppose
()
for anyu∈W, (),
+∞
ds
+N(s) +Q(s)N(s)=∞holds, (.)
whereQ(u) =sup×Rfu(x,u),N(u) =sup[u–fu(x,u)]–.
Then there is a unique solution of the equation
P(u) =
in W ,().
Proof The conditioninf×Rfu(x,u) > –λimplies that zero is not an eigenvalue ofφ–
fu(x,u)φ= , so for everyu∈W
,(), the operatorP(u) =u–fuIis invertible andPis a local homeomorphism fromW
,() ontoL(), whereIdenotes the identical operator.
Denote [+fu(x,u)I]–ϕ=ψ. From Corollary ., we have
ψ ≤Cψ+ψ
=C–fu(x,u)I–ϕ+–fu(x,u)I–ϕ
≤CNuϕ+ϕ+fu(x,u)–fu(x,u)I–ϕ
≤CNuϕ+ϕ+supfu(x,u)Nuϕ
≤CNuϕ+ϕ+QuNuϕ.
So we obtain
P(u)–=–fu(x,u)I–≤C +Nu+QuNu.
Consider the path-lifting problem for the mappingP:
P(γu(t)) –P(u) =e–t(P(u) –P(u)),
γu() =u. It is clear that
γu(t) = –Pγu(t)–e–tP(u) –P(u)
.
Then
γu(t) –γu()=
t
γu(t)dt≤
t
γu(t)dt
=
t
Pγu(t)
–
e–tP(u) –P(u)dt
≤C
t
+Nγu(t)+Qγu(t)Nγu(t) e–tP(u) –P(u)dt, γu(t)≤γu()+C
t
+N
Using the Gronwall inequality, we have
γu(t)
γu()
ds
+N(s) +Q(s)N(s)≤C
t
e–tP(u) –P(u)dt≤CP(u) –P(u),
which, together with (.), implies∃M>
γu(t)≤M, t> .
By Theorem ., now we need only show thatγu(t) can be extended to –∞. Letg(–t) =γu(t),t∈(b, ],b< . Then fort,t∈(b, ]
γu(t) –γu(t)=g(–t) –g(–t)= –t
–t
g(s)ds
=
–t
–t
Pγu(t)–esP(u) –P(u)ds
≤C
–t
–t
+N(M) +Q(M)N(M) esP(u) –P(u)ds
≤C +N(M) +Q(M)N(M)ebP(u) –P(u)|t–t|.
Soγu(t) is Lipschitz continuous on (–b, ] andγu(t) can be extended to –∞, that is to say,
Pis a diffeomorphism fromW, () toL() and the theorem is proved.
Corollary . Assume that f satisfies: () inf×Rfu> –λ;
() Uniformly inx,fu ≤ω(u),whereωis a continuous map satisfying ∞
a
dt
ω(t)=∞. (.)
Then there is a unique solution of(.)in W ,().
Proof We have from Theorem in []
P(u)–≤αsup
fux,u(x) +β
for positive constantsα,β. From (.),
∞
c
dt
αsup|fu|+β =∞.
By Theorem ., there is a unique solution of (.) inW
,().
Corollary . (Elcrat) Suppose that P is given by Pu=u–f(x,u), that f satisfies
()inf×Rfu> –λ, ()fu= O(u),uniformly in x.
Proof Condition () implies thata∞ωdt(t) =∞holds. The result of Elcrat in [] becomes a
special case of Theorem ..
Competing interests
The author declares to have no competing interests.
Author’s contributions
The author read and approved the final manuscript.
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).
Received: 15 March 2014 Accepted: 24 July 2014
References
1. Elcrat, AR: Constructive existence for semilinear elliptic equations with discontinuous coefficients. SIAM J. Math. Anal.
5, 663-672 (1974)
2. Gorni, G: A criterion of invertibility in the large for local diffeomorphisms between Banach spaces. Nonlinear Anal.21, 43-47 (1993)
3. Marco, GD, Gorni, G, Zampieri, G: Global inversion of function: an introduction. Nonlinear Differ. Equ. Appl.1, 229-248 (1994)
4. Ladyzhenskaya, OA, Uraltseva, NN: Linear and Quasilinear Elliptic Equations, pp. 134-145. Academic Press, New York (1968)
5. 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). doi:10.1186/1687-1847-2014-52
doi:10.1186/s13661-014-0190-7
