Existence and uniqueness of solution for a nonhomogeneous nonlocal problem
aa r X i v : . [ m a t h . A P ] J a n EXISTENCE AND UNIQUENESS OF SOLUTION FOR ANONHOMOGENEOUS NONLOCAL PROBLEM
CAMIL S. Z. REDWAN, JO ˜AO R. SANTOS J ´UNIOR, AND ANTONIO SU ´AREZ
Abstract.
In this paper we investigate a class of elliptic problems involving a nonlocalKirchhoff type operator with variable coefficients and data changing its sign. Underappropriated conditions on the coefficients, we have shown existence and uniqueness ofsolution. Introduction
In this paper we are concerned with uniqueness of nontrivial classic solution to the followingclass of nonlocal elliptic equations(P) (cid:26) − (cid:0) a ( x ) + b ( x ) R Ω |∇ u | dx (cid:1) ∆ u = h ( x ) in Ω, u = 0 on ∂ Ω,where Ω ⊂ IR N , N ≥
2, is a bounded domain with smooth boundary, a, b ∈ C ,γ (Ω), γ ∈ (0 , a ( x ) ≥ a > b ( x ) ≥ b > h ∈ C ,γ (Ω) is given.When functions a, b are positive constants, problem (P) is the N -dimensional stationaryversion of a hyperbolic problem proposed in [5] to model small transversal vibrations of anelastic string with fixed ends which is composed by a homogeneous material. Such equation isa more realistic model than that provided by the classic D’Alembert’s wave equation becausetakes account the changing in the length of the string during the vibrations. The hyperbolicKirchhoff problem (with a, b constants) began receiving special attention mainly after thatin [6] the author used an approach of functional analysis to attack it.At least in our knowledge, the first work in studying uniqueness questions to problem (P)with a, b constants was [2]. It is an immediate consequence of Theorem 1 in [2] that if h is aH¨older continuous nonnegative (nonzero) function then problem (P), with a, b constants, has aunique positive solution. In [2] functions h sign changing are not considered. In the case that a, b are not constant, problem (P) is yet more relevant in an applications point of a view becauseits unidimensional version models small transversal vibrations of an elastic string composedby non-homogeneous materials (see [4], section 2). In [4] (see Theorem 1) the authors provedthat for each h ∈ L ∞ (Ω) ( h
0) given, problem (P) admits at least a nontrivial solution.Moreover, same article tells us that if h has defined sign ( h ≤ h ≥
0) such a solutionis unique. Unfortunately, since their approach was based in a monotonicity argument whichdoes not work when h is a sign changing function, they were not able to say anything aboutuniqueness in this case. Indeed, at least in our knowledge, actually the uniqueness of solutionto problem (P) in the general case is an open problem. Mathematics Subject Classification.
Key words and phrases.
Kirchhoff type equation, sublinear problem, topological method.Camil S. Z. Redwan was partially supported by Fapespa, Brazil. Jo˜ao R. Santos J´unior was partiallysupported by CNPq-Proc. 302698/2015-9, Brazil. Antonio Su´arez has been partially supported by MTM2015-69875-P (MINECO/FEDER, UE) and CNPq-Proc. 400426/2013-7.
At this article we have obtained sufficient conditions on the quotient a/b to ensure uniquenessof solution when function h , given, changes its sign. The main results of this paper are as follows Theorem 1.1.
If there exists θ > such that a/b = θ in Ω , then for each h ∈ C ,γ (Ω) problem (P) has a unique solution. Theorem 1.2.
Let a, b ∈ C ,γ (Ω) , h ∈ C ,γ (Ω) is sign changing and suppose c = a/b notconstant. ( i ) If ∆ c ≥ |∇ c | /c in Ω , then, for each h ∈ C ,γ (Ω) given, problem (P) has a uniquenontrivial classic solution. ( ii ) If ∆ c < |∇ c | /c in some open Ω ⊂ Ω then, for each h ∈ C ,γ (Ω) given, problem (P) has a unique nontrivial classic solution, provided that |∇ c | ∞ c M √ λ c L ≤ / , where λ is the first eigenvalue of laplacian operator with Dirichlet boundary condition, c L = min x ∈ Ω c ( x ) , c M = max x ∈ Ω c ( x ) and |∇ c | ∞ = max x ∈ Ω |∇ c ( x ) | . First theorem above generalizes Theorem 1 in [2] because it is true to functions h signchanging or not. The second theorem above complements Theorem 1 in [4].The paper is organized as follows.In Section 2 we present some abstracts results, notations and definitions. In Section 3 weinvestigated a nonlocal eigenvalue problem which seems to be closely related with uniquenessquestions to problem (P). In Section 4 we prove Theorems 1.1 and 1.2. Moreover, an alternativeproof of the existence and uniqueness result in [4] is supplied.2. Preliminaries
In this section we state some results and fix notations used along of paper.
Definition 2.1.
We say that a function h is signed in Ω if h ≥ in Ω or h ≤ in Ω . Definition 2.2.
An application
Ψ : E → F defined in Banach spaces is locally invertible in u ∈ E if there are open sets A ∋ u in E and B ∋ Ψ( u ) in F such that Ψ : A → B is a bijection.If Ψ is locally invertible in any point u ∈ E it is said that Ψ : E → F is locally invertible. Definition 2.3.
Let
M, N be metric spaces. We say that a map
Ψ : M → N is proper if Ψ − ( K ) = { u ∈ M : Ψ( u ) ∈ K } is compact in M for all compact set K ⊂ N . Below we enunciate the classic local and global inverse function theorems, whose proofs canbe found, for instance, in [1].
Theorem 2.4 (Local Inverse Theorem) . Let
E, F be two Banach spaces. Suppose Ψ ∈ C ( E, F ) and Ψ ′ ( u ) : E → F is a isomorphism. Then Ψ is locally invertible at u and itslocal inverse, Ψ − , is also a C -function. Theorem 2.5 (Global Inverse Theorem) . Let
M, N be two metric spaces and Ψ ∈ C ( M, N ) a proper and locally invertible function on all of M . Suppose that M is arcwise connected and N is simply connected. Then Ψ is a homeomorphism from M onto N . Next, we state another classical result which will be used in our arguments and whose proofcan be found, for instance, for a more general class of problems, in [3].
XISTENCE AND UNIQUENESS OF SOLUTION FOR A NONHOMOGENEOUS NONLOCAL PROBLEM 3
Proposition 2.6.
Let m ∈ L ∞ (Ω) , m ( x ) > in a set of positive measure. Then, problem (2.1) (cid:26) − div ( A ( x ) ∇ u ) = λm ( x ) u in Ω , u = 0 on ∂ Ω ,where A ∈ L ∞ (Ω) and A ( x ) ≥ m for some positive constant m , has a smallest positiveeigenvalue λ ( m ) which is simple and corresponding eigenfunctions do not change sign in Ω . Throughout this paper X is the Banach space X = { u ∈ C ,γ (Ω) : u = 0 on ∂ Ω } with norm k u k X = k u k C (Ω) + max | β | =2 [ D β u ] γ , where γ ∈ (0 , β = ( β , . . . , β N ) ∈ IN N , | β | = β + . . . + β N , k u k C (Ω) = X ≤| β |≤ k D β u k C (Ω) and [ D β u ] γ = sup x,y ∈ Ω x = y | D β u ( x ) − D β u ( y ) || x − y | γ . Moreover Y will denote the Banach space C ,γ (Ω) with norm k f k Y = k f k C (Ω) + [ f ] γ , where k f k C (Ω) = max x ∈ Ω | f ( x ) | .Hereafter same symbol C denotes different positive constants.3. A nonlocal eigenvalue problem
In this section we are interested in studying the following nonlocal eigenvalue problem( EP ) − div (cid:18) ∇ uc + |∇ u | (cid:19) = λ (cid:26) − div (cid:20) ∇ c ( c + |∇ u | ) (cid:21)(cid:27) u in Ω, u = 0 on ∂ Ω,where Ω ⊂ IR N is bounded smooth domain, λ is a positive parameter and c ∈ C (Ω) is apositive (not constant) function. As we will see in the next section, problem ( EP ) arisesnaturally when one studies questions of uniqueness to the problem (P).Before to state the main results of this section, we observe that Lemma 3.1.
The set A := (cid:26) α > − div (cid:20) ∇ c ( c + α ) (cid:21) > in some open Ω ⊂ Ω (cid:27) is not empty if, and only if, there is an open ˆΩ ⊂ IR N such that (3.1) ∆ c < |∇ c | c in ˆΩ . Proof.
Differentiating we get(3.2) − div (cid:20) ∇ c ( c + α ) (cid:21) = − c + α ) ∆ c + 2( c + α ) |∇ c | . Now, note that(3.3) − div (cid:20) ∇ c ( c + α ) (cid:21) > if, and only if,(3.4) ∆ c < |∇ c | ( c + α ) in Ω . It is clear that the existence of a positive number α satisfying (3.4) is equivalent to inequalityin (3.1). (cid:3) Remark 1.
In previous Lemma we have shown also that A = ∅ if, and only if, (3.5) ∆ c ≥ |∇ c | c in Ω . Certainly, there are many positive functions c ∈ C (Ω) verifying (3.5) . For instance, setting c = δe + 1 , where < δ ≤ min { / (4 |∇ e | ∞ ) , / (2 | e | ∞ ) } and (3.6) (cid:26) ∆ e = 1 in Ω , e = 0 on ∂ Ω ,we conclude that c > and satisfies (3.5) . Remark 2.
An interesting question when (3.1) holds is about the topology of set A . In thisdirection, the proof of Lemma 3.1 allows us to say that A contains ever a neighborhood (0 , α ) . Now we are ready to claim the following result.
Theorem 3.2.
Suppose that (3.1) holds. For each α ∈ A , problem ( EP ) has a unique solution ( λ α , u α ) such that λ α > , u α > and |∇ u α | = α .Proof. From Lemma 3.1,
A 6 = ∅ . Since c ∈ C (Ω) and b > α ∈ A , the eigenvalue problem( P α ) − div (cid:18) ∇ uc + α (cid:19) = λ (cid:26) − div (cid:20) ∇ c ( c + α ) (cid:21)(cid:27) u in Ω, u = 0 on ∂ Ω,has a positive smallest eigenvalue λ α whose associated eigenspace V α is unidimensional and itseigenfunctions have defined sign. Choosing u ∈ V α such that u > |∇ u | = α , the resultfollows. (cid:3) Remark 3.
In particular, if (3.1) holds then (3.7) Z Ω u α (cid:26) − div (cid:20) ∇ c ( c + α ) (cid:21)(cid:27) dx = 1 λ α Z Ω |∇ u α | c + α dx, ∀ α ∈ A . Corollary 3.3.
Suppose (3.1) . For each α ∈ A , the following inequality holds λ α ≥ √ λ ( c L + α ) |∇ c | ∞ ( c M + α ) , where λ is the first eigenvalue of laplacian operator with Dirichlet boundary condition, c L = min x ∈ Ω c ( x ) , c M = max x ∈ Ω c ( x ) and |∇ c | ∞ = max x ∈ Ω |∇ c ( x ) | .Proof. From Remark 3, we get(3.8) λ α = Z Ω |∇ u α | c + α dx Z Ω u α (cid:26) − div (cid:20) ∇ c ( c + α ) (cid:21)(cid:27) dx . XISTENCE AND UNIQUENESS OF SOLUTION FOR A NONHOMOGENEOUS NONLOCAL PROBLEM 5
Observe that(3.9) Z Ω |∇ u α | c + α dx ≥ αc M + α . Moreover, by using the Divergence Theorem, Z Ω u α (cid:26) − div (cid:20) ∇ c ( c + α ) (cid:21)(cid:27) dx = 2 Z Ω u α ∇ u α ∇ c ( c + α ) dx ≤ |∇ c | ∞ Z Ω u α |∇ u α | dx ( c L + α ) . From H¨older and Poincar´e inequalities, we conclude that(3.10) Z Ω u α (cid:26) − div (cid:20) ∇ c ( c + α ) (cid:21)(cid:27) dx ≤ |∇ c | ∞ α √ λ ( c L + α ) . From (3.8), (3.9) and (3.10) we have λ α ≥ √ λ ( c L + α ) |∇ c | ∞ ( c M + α ) , for all α ∈ A . (cid:3) Uniqueness results
In order to apply Theorem 2.5 we define operator Ψ : X → Y byΨ( u ) = (cid:18) a ( x ) + b ( x ) Z Ω |∇ u | dx (cid:19) ∆ u. In the sequel, we will denote M (cid:0) x, |∇ u | (cid:1) = a ( x ) + b ( x ) R Ω |∇ u | dx for short, where |∇ u | = R Ω |∇ u | dx . The proof of main results of this paper will be divided in variouspropositions. Proposition 4.1.
Operator
Ψ : X → Y is proper.Proof. It is sufficient to prove that if { h n } ⊂ Y is a sequence converging to h ∈ Y and { u n } ⊂ X is another sequence with Ψ( u n ) = − h n then { u n } has a convergent subsequence in X . For this,note that the equality Ψ( u n ) = − h n is equivalent to(4.1) − ∆ u n = h n M (cid:0) x, |∇ u n | (cid:1) . Observe that h n /M (cid:0) ., |∇ u n | (cid:1) ∈ Y because h n ∈ Y , M (cid:0) ., |∇ u n | (cid:1) ∈ Y and M (cid:0) x, |∇ u n | (cid:1) ≥ a .Moreover,(4.2) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h n ( x ) M (cid:0) x, |∇ u n | (cid:1) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C (Ω) ≤ k h n k C (Ω) /a , ∀ n ∈ IN . From k h n k C (Ω) ≤ k h n k Y , (4.2) and from boundedness of { h n } in Y follows that (cid:8) h n /M (cid:0) x, |∇ u n | (cid:1)(cid:9) is bounded in C (Ω). Thus, the continuous embedding from C ,γ (Ω)into C (Ω) and equality in (4.1) tell us that { u n } is bounded in C ,γ (Ω) (see Theorem 0.5in [1]). Finally, by compact embedding from C ,γ (Ω) into C (Ω), we conclude that there exists u ∈ C (Ω) such that, passing to a subsequence,(4.3) u n → u in C (Ω) . C. S. Z. REDWAN, J. R. SANTOS JR., AND A. SU ´AREZ
Last convergence leads to(4.4) |∇ u n ( x ) | → |∇ u ( x ) | uniformly in x ∈ Ω . Whence(4.5) |∇ u n | → |∇ u | . In the follows, we show that(4.6) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h n M (cid:0) ., |∇ u n | (cid:1) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y ≤ C, for some positive constant C . In fact, since { h n } ⊂ Y and (cid:8) M (cid:0) ., |∇ u n | (cid:1)(cid:9) ⊂ Y , with M ( x, t ) ≥ a > t ≥
0, a straightforward calculation shows us that " h n M (cid:0) ., |∇ u n | (cid:1) γ ≤ a (cid:16) k h n k C (Ω) (cid:2) M (cid:0) ., |∇ u n | (cid:1)(cid:3) γ + (cid:13)(cid:13) M (cid:0) ., |∇ u n | (cid:1)(cid:13)(cid:13) C (Ω) [ h n ] γ (cid:17) . From k h n k C (Ω) , [ h n ] γ ≤ C ,(4.7) (cid:2) M (cid:0) ., |∇ u n | (cid:1)(cid:3) γ ≤ [ a ] γ + [ b ] γ |∇ u n | ≤ [ a ] γ + C [ b ] γ and(4.8) (cid:13)(cid:13) M (cid:0) ., |∇ u n | (cid:1)(cid:13)(cid:13) C (Ω) ≤ k a k C (Ω) + k b k C (Ω) |∇ u n | ≤ k a k C (Ω) + C k b k C (Ω) it follows that " h n M (cid:0) ., |∇ u n | (cid:1) γ ≤ Ca (cid:16) [ a ] γ + C [ b ] γ + k a k C (Ω) + C k b k C (Ω) (cid:17) = Ca k a k Y + C a k b k Y . Being (cid:8) h n /M (cid:0) x, |∇ u n | (cid:1)(cid:9) bounded in C (Ω), the last inequality proves the assertion in (4.6).By (4.1), (4.6) and Theorem 0.5 in [1], sequence { u n } is bounded in X . By compactembedding from X in C (Ω), passing to a subsequence, we get(4.9) u n → u in C (Ω) . By (4.9), passing to the limit in n → ∞ in (4.1) we have(4.10) − ∆ u = hM (cid:0) x, |∇ u | (cid:1) . Last equality and Theorem 0.5 in [1] allow us to conclude that u ∈ X .Finally, by linearity of laplacian, we have(4.11) − ∆( u n − u ) = h n M (cid:0) x, |∇ u n | (cid:1) − hM (cid:0) x, |∇ u | (cid:1) . From (4.11) and Theorem 0.5 in [1] we conclude that u n → u in X . (cid:3) Proposition 4.2.
Let a, b ∈ C ,γ (Ω) and u ∈ X . If (4.12) Z Ω b ( x ) u ∆ uM ( x, |∇ u | ) dx = 1 / holds then Ψ is locally invertible in u . XISTENCE AND UNIQUENESS OF SOLUTION FOR A NONHOMOGENEOUS NONLOCAL PROBLEM 7
Proof.
We are interested in using Theorem 2.4 to prove this Lemma. It is standard to showthat Ψ ∈ C ( X, Y ) andΨ ′ ( u ) v = 2 b ( x )∆ u Z Ω ∇ u ∇ vdx + M ( x, |∇ u | )∆ v. Remain us proving that Ψ ′ ( u ) : X → Y is an isomorphism. It is clear that if u = 0 there isnothing to prove. Now, if u = 0, observes that Ψ ′ ( u ) is an isomorphism if, and only if, for each g ∈ Y given, there is a unique v ∈ X such that Ψ ′ ( u ) v = − g , this is(4.13) − M ( x, |∇ u | )∆ v = g ( x ) + 2 b ( x )∆ u Z Ω ∇ u ∇ vdx. From Divergence Theorem, equation in (4.13) is equivalent to(4.14) − M ( x, |∇ u | )∆ v = g ( x ) − b ( x )∆ u Z Ω u ∆ vdx. Consequently, Ψ ′ ( u ) is an isomorphism if, and only if, for each g ∈ Y given, there is a unique v ∈ X such that(4.15) ∆ v = 2 b ( x )∆ u R Ω u ∆ vdxM ( x, |∇ u | ) − g ( x ) M ( x, |∇ u | ) . To study equation (4.15) we define the mapping T : Y → Y by(4.16) T ( w ) = 2 b ( x )∆ u R Ω uwdxM ( x, |∇ u | ) − g ( x ) M ( x, |∇ u | )and we note that, since for each w ∈ Y problem(LP) (cid:26) ∆ z = w ( x ) in Ω, z = 0 on ∂ Ω,has a unique solution z ∈ X , looking for solutions of (4.15) is equivalent to find fixed points of T . Denoting t = R Ω uwdx , it follows that w is a fixed point of T if, and only if,(4.17) w = T ( w ) = t b ( x )∆ uM ( x, |∇ u | ) − g ( x ) M ( x, |∇ u | ) . Therefore w is a fixed point of T if, and only if, T (cid:18) t b ( x )∆ uM ( x, |∇ u | ) − g ( x ) M ( x, |∇ u | ) (cid:19) = t b ( x )∆ uM ( x, |∇ u | ) − g ( x ) M ( x, |∇ u | ) . From (4.16), we get2 b ( x )∆ uM ( x, |∇ u | ) Z Ω u (cid:20) t b ( x )∆ uM ( x, |∇ u | ) − g ( x ) M ( x, |∇ u | ) (cid:21) dx = t b ( x )∆ uM ( x, |∇ u | ) . Since b > u u = 0), T admits a fixed point if, and only if,2 Z Ω u (cid:20) t b ( x )∆ uM ( x, |∇ u | ) − g ( x ) M ( x, |∇ u | ) (cid:21) dx = t, namely,(4.18) t (cid:20)Z Ω b ( x ) u ∆ uM ( x, |∇ u | ) dx − (cid:21) = 2 Z Ω g ( x ) uM ( x, |∇ u | ) dx. C. S. Z. REDWAN, J. R. SANTOS JR., AND A. SU ´AREZ
Equality (4.18) say us that if (4.12) occurs then T has a unique fixed point w given by w = t b ( x )∆ uM ( x, |∇ u | ) − g ( x ) M ( x, |∇ u | ) , with t = 2 Z Ω g ( x ) uM ( x, |∇ u | ) dx/ (cid:20)Z Ω b ( x ) u ∆ uM ( x, |∇ u | ) dx − (cid:21) . (cid:3) Remark 4.
Equality (4.18) shows us that Ψ ′ ( u ) : X → Y is not surjective if Z Ω b ( x ) u ∆ uM ( x, |∇ u | ) dx = 1 / . In fact, in this case, functions g ∈ Y such that Z Ω g ( x ) uM ( x, |∇ u | ) dx = 0 are not in the range of Ψ ′ ( u ) . Actually, it is possible to get the same result of (existence and) uniqueness provided in [4] forsigned functions as a consequence of Global Inverse Theorem and previous Proposition. Thisis exactly the content of next corollary.
Corollary 4.3.
For each signed function h ∈ Y given, problem (P) has a unique solution.Proof. First of all, we define the sets P = { u ∈ X : ∆ u ≥ } ⊂ X and P = { h ∈ Y : h ≥ } ⊂ Y. Consider P ∪ ( − P ) and P ∪ ( − P ) as metric spaces whose metrics are induced from X and Y , respectively.It is clear that P ∪ ( − P ) is arcwise connected (because P and − P are convex sets and P ∩ ( − P ) = { } ) closed in X . On the other hand, since P ∪ ( − P ) is the union of the closedcone of nonnegative functions of Y with the closed cone of nonpositive functions of Y , followsthat P ∪ ( − P ) is simply connected.From Ψ( P ) ⊂ P and Ψ( − P ) ⊂ ( − P ), it follows that Ψ is well defined from P ∪ ( − P ) to P ∪ ( − P ).Moreover, being Ψ proper from X to Y (see Proposition 4.1) and P ∪ ( − P ) and P ∪ ( − P )are closed metric spaces in X and Y , respectively, it follows that Ψ is proper from P ∪ ( − P )to P ∪ ( − P ).Note that if u ∈ P (resp. − P ) then, as u is (the unique) solution to problem(4.19) (cid:26) ∆ u = ∆ u in Ω, u = 0 on ∂ Ω.Follows from maximum principle that u ≤ u ≥ Z Ω b ( x ) u ∆ uM ( x, |∇ u | ) dx ≤ , ∀ u ∈ P ∪ ( − P ) . Therefore, from Proposition 4.2, Ψ : P ∪ ( − P ) → P ∪ ( − P ) is locally invertible. The resultfollows now from Global Inverse Theorem. (cid:3) XISTENCE AND UNIQUENESS OF SOLUTION FOR A NONHOMOGENEOUS NONLOCAL PROBLEM 9
Next corollary does not ensure uniqueness of solution for problem (P) when function h givenis sign changing, but it tells us that there is a unique solution with “little variation” if h ∈ Y given (signed or not) has “little variation”. Corollary 4.4.
There are positive constants ε, δ such that for each h ∈ Y with k h k Y < ε ,problem (P) has a unique solution u with k u k X < δ .Proof. It is sufficient to note that when u = 0 the integral in previous proposition is null. (cid:3) We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1.
Since X and Y are Banach spaces then X is arcwise connected and Y is simply connected.Moreover, from Proposition 4.1, operator Ψ is proper and by Divergence Theorem Z Ω b ( x ) u ∆ uM ( x, |∇ u | ) dx = 1 θ + |∇ u | Z Ω u ∆ udx = − |∇ u | θ + |∇ u | < , ∀ u ∈ X. The result follows directly from Proposition 4.2 and Global Inverse Theorem. (cid:3)
Next proposition provides us a sufficient condition on functions a and b for that (4.12) occurswhen a/b is not constant. Proposition 4.5.
Let a, b ∈ C ,γ (Ω) and c = a/b . ( i ) If ∆ c ≥ |∇ c | /c in Ω , then (4.20) Z Ω b ( x ) u ∆ uM ( x, |∇ u | ) dx ≤ , ∀ u ∈ X. ( ii ) If ∆ c < |∇ c | /c in some open Ω ⊂ Ω then (4.21) Z Ω b ( x ) u ∆ uM ( x, |∇ u | ) dx < / , ∀ u ∈ X, provided that |∇ c | ∞ c M √ λ c L ≤ / . Proof.
Putting b in evidence in the integral (4.12), we get Z Ω b ( x ) u ∆ uM ( x, |∇ u | ) dx = Z Ω u ∆ uc + |∇ u | dx, where c = c ( x ) = a ( x ) /b ( x ). From Divergence Theorem, we have Z Ω u ∆ uc + |∇ u | dx = − Z Ω ∇ (cid:18) uc + |∇ u | (cid:19) ∇ udx. Since ∇ (cid:18) uc + |∇ u | (cid:19) = 1 c + |∇ u | ∇ u − u ( c + |∇ u | ) ∇ c, we conclude that Z Ω b ( x ) u ∆ uM ( x, |∇ u | ) dx = − Z Ω |∇ u | c + |∇ u | dx + Z Ω u ∇ u ∇ c ( c + |∇ u | ) dx = − Z Ω |∇ u | c + |∇ u | dx + 12 Z Ω ∇ ( u ) ∇ c ( c + |∇ u | ) dx. Using again the Divergence Theorem(4.22) Z Ω b ( x ) u ∆ uM ( x, |∇ u | ) dx = − Z Ω |∇ u | c + |∇ u | dx + 12 Z Ω u (cid:26) − div (cid:20) ∇ c ( c + |∇ u | ) (cid:21)(cid:27) dx. (i) At this case, from Lemma 3.1 (see also Remark 1), A = ∅ and, consequently, for each u ∈ X we have Z Ω u (cid:26) − div (cid:20) ∇ c ( c + |∇ u | ) (cid:21)(cid:27) dx ≤ . Whence, by (4.22), Z Ω b ( x ) u ∆ uM ( x, |∇ u | ) dx ≤ , ∀ u ∈ X. (ii) In this case A 6 = ∅ . If u ∈ X is such that |∇ u |
6∈ A we saw already that Z Ω b ( x ) u ∆ uM ( x, |∇ u | ) dx ≤ . Now, if u ∈ X is such that |∇ u | ∈ A then, from (4.22) and Proposition 3.2 (see also Remark3), we obtain(4.23) Z Ω b ( x ) u ∆ uM ( x, |∇ u | ) dx ≤ (cid:18) λ α − (cid:19) Z Ω |∇ u | c + α dx, where α := |∇ u | . If α is such that 1 / ≤ λ α then, by (4.23), R Ω b ( x ) u ∆ u/M ( x, |∇ u | ) dx ≤ < λ α < / α = |∇ u | and from Corollary 3.3 that,(4.24) Z Ω b ( x ) u ∆ uM ( x, |∇ u | ) dx < |∇ c | ∞ ( c M + α ) √ λ ( c L + α ) − g ( α ) . We have that g (0) = |∇ c | ∞ c M / √ λ c L − g ′ ( α ) = |∇ c | ∞ ( c L − c M − α ) √ λ ( c L + α ) < , ∀ α > . Therefore g is decreasing and, from (4.24), we conclude that if |∇ c | ∞ c M √ λ c L ≤ (cid:3) Bellow we give the proof of our main uniqueness result to problem (P) which covers signchanging functions.
Proof of Theorem 1.2.
It follows directly from Proposition 4.1, Proposition 4.2, Proposition4.5 and Global Inverse Theorem. (cid:3)
Theorems 1.1 and 1.2 seem to indicate that in the case that h is sign changing the uniquenessof solution to the problem (P) is, in some way, related with the variation of a/b . In any way,remains open the question to know what happens with the number of solutions of (P) in thecase that h is sign changing, ∆ c < |∇ c | /c in some open Ω ⊂ Ω and |∇ c | ∞ c M / √ λ c L islarge. XISTENCE AND UNIQUENESS OF SOLUTION FOR A NONHOMOGENEOUS NONLOCAL PROBLEM 11
References [1] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis,
Cambridge University Press , 1993. 2, 5, 6[2] C. O. Alves, F. J. S. A. Corrˆea and T. F. Ma, Positive solutions for a quasilinear elliptic equation ofKirchhoff type,
Comput. Math. Appl. (2005) 85-93. 1, 2[3] D. G. de Figueiredo, Positive solutions of semilinear elliptic problems in differential equations, volume 957of Lecture Notes in Mathematics, Springer , Berlin-New York, 1982. 2[4] G. M. Figueiredo, C. Morales-Rodrigo, J. R. Santos J´unior and A. Su´arez, Study of a nonlinear Kirchhoffequation with non-homogeneous material,
J. Math. Anal. Appl. (2014) 597-608 1, 2, 8[5] G. Kirchhoff, Mechanik,
Teubner , Leipzig, 1883. 1[6] J. L. Lions, On some question on boundary value problem of mathematical physics, in: G.M. de LaPenha, L.A. Medeiros (Eds.),
Contemporary Developments of Continuum Mechanics and Partial DifferentialEquations , North-Holland, Amsterdam, (1978) 285-346. 1(C. S. Z. Redwan)
Faculdade de Matem´aticaInstituto de Ciˆencias Exatas e NaturaisUniversidade Federal do Par´aAvenida Augusto corrˆea 01, 66075-110, Bel´em, PA, Brazil
E-mail address : [email protected] (J. R. Santos Jr.) Faculdade de Matem´aticaInstituto de Ciˆencias Exatas e NaturaisUniversidade Federal do Par´aAvenida Augusto corrˆea 01, 66075-110, Bel´em, PA, Brazil
E-mail address : [email protected] (A. Su´arez) Departamento de Ecuaciones Diferenciales y An´alisis Num´ericoFacultad de Matem´aticasUniversidad de SevillaC/. Tarfia s/n, 41012, Sevilla, Spain.
E-mail address ::