Multiplicity and Bifurcation of Solutions for a Class of Asymptotically Linear Elliptic Problems on the Unit Ball

(u − b) 2 + ε which is not increasing. The main goal of this paper is to study the exact number and bifurcation structure of the solutions of (Pλ) on a unit ball Ω, with a general asymptotically linear function f. Some results in this paper (see Section 3) can be viewed as an extension and improvement of that in [7], but the argument approach here is very different to that in [7]. As byproducts, we also get some new results which also hold for general domain Ω (see Section 2). The paper is organized as follows. In Section 2, we study the existence and multiplicity of solutions for problem (Pλ) on a general bounded domain, with some new results complementing those existing in the literature. In Section 3, we study the exact number and global bifurcation structure of positive solutions of (Pλ) on a unit ball.


Introduction
In this paper, we are concerned with positive solutions of the following elliptic equation subject to homogeneous Dirichlet boundary condition −Δ =  () , in Ω,  = 0, on Ω, (  ) where Ω is a smooth bounded domain in   ,  is a positive parameter,  ∈  2 (Ω) ∩ (Ω), and the function  satisfies the following.
The main goal of this paper is to study the exact number and bifurcation structure of the solutions of (  ) on a unit ball Ω, with a general asymptotically linear function .Some results in this paper (see Section 3) can be viewed as an extension and improvement of that in [7], but the argument approach here is very different to that in [7].As byproducts, we also get some new results which also hold for general domain Ω (see Section 2).The paper is organized as follows.In Section 2, we study the existence and multiplicity of solutions for problem (  ) on a general bounded domain, with some new results complementing those existing in the literature.In Section 3, we study the exact number and global bifurcation structure of positive solutions of (  ) on a unit ball.
Proof.If not, assume that  is a solution of (  ) for some  >  1 /.Multiplying (  ) by  1 > 0, the normalized positive eigenfunction with respect to the first eigenvalue  1 of −Δ subject to homogenous Dirichlet boundary condition, and then integrating by parts, we get which is a contradiction.

We begin by show the following.
Lemma 3.There exists a number  1 / ≤ Λ ≤  1 /, such that (  ) has at least a solution for  < Λ and has no solution for  > Λ.
By Lemmas 1 and 2,  1 / ≤ Λ ≤  1 /.We need just to prove that if (  ) has a solution, then (  ) also has a solution for all 0 <  < .This can be done by a simple argument of subsup solution method, since it is easy to see that any solution of (  ) is a super solution of (  ) and  ≡ 0 a subsolution.It is easy to see that  * ≡ 0 is a subsolution of (  ), then a standard sub-super solution method's argument and comparison theorems give the following lemma.Lemma 4. If (  ) is solvable, then one has a minimal solution   , that is, for any solution V of (  ),   ≤ V.Moreover,   is increasing with respect to .Lemma 5.If  ∈ (0,  1 /), then the solution of (  ) is unique.
By mean value theorem, V satisfies where V lies between V 1 and V 2 .Multiplying V and integrating, we get which implies that V ≡ 0. The proof is complete.
If ( Λ ) has a solution, let  Λ denote the minimal solution of (  ).By Lemma 4, (b) For clarity, the proof will be divided into 3 steps.
Step 1.The existence and uniqueness of solutions of (  ) for  =  1 /.The existence follows directly from Lemma 4. Note that   < , and the uniqueness can be proved in a similar way as in the proof of Lemma 5.
Step 2. The existence and uniqueness of solutions of (  ) for  = Λ.
Hence Λ =  1 , that is, Λ =  1 /, a desired contradiction.Now in a similar way, the boundedness of Then subject to a subsequence, we may suppose that there exits  * , such that Then by letting  → ∞, we get and the existence is proved.Now we prove the uniqueness.Let  Λ be the minimal solution of ( Λ ) and  a different solution.Then  := − Λ > 0 satisfies where denotes the first eigenvalue of the operator −Δ − Λ  ( Λ + ) subject to the Dirichlet boundary condition, as defined in Lemma 1.Since   ( Λ ) <   ( Λ + ) in Ω, we have that  1 (−Δ−Λ  ( Λ )) >  1 (−Δ−Λ  (  +)) = 0, which implies that the operator −Δ − Λ  ( Λ ) is nondegenerate.Then by the Implicit Function Theorem, the solution of (  ) forms a cure in a neighborhood of (Λ,  Λ ), which is clearly contradicted to the definition of Λ in (7).
On the other hand, let  1 be the eigenfunction with ∫ Ω  2  1  = 1 of the first eigenvalue  1 of the following equation: Since   is the minimal solution, it follows from Lemmas 4 and 6 that  1 > 0. Then Hence   ( 1 ) > 0 when  is small enough.Now it is easy to see that   is not empty.In fact, take  * =  1 for some large , and  * =  for some small  > 0, such that respectively.Define a continuous function  on [0, 1], namely, Then (0) > 0, (1) < 0, and hence there exist  0 ∈ (0, 1) such that ( 0 ) = 0, that is,   ( 0  * + (1 −  0 ) * ) = 0, and Integrating (29) with respect to V from 0 to V, we get Therefore, on that is,   (V) is bounded from below.And then we obtain a nonminimal positive solution of (  ) by using the Nehari variational method.The proof is complete.
Remark 8.The solutions that we get from the above discussion are weak ones, but a standard elliptic regularity argument shows that they are indeed classical solutions.
Proof.(i) If  ∞ ≤ 0, then () ≥  for all  ≥ 0. We prove that (  ) has no solution and hence Λ =  1 /.Suppose the contrary that  is a solution (  ) for  =  1 /, then Let  be a positive eigenfunction of the first eigenvalue  of −Δ on Ω with Dirichlet boundary condition, that is Multiplying (32) by  > 0, and integrating by parts, we get which yields that () = , contradicting the fact that (0) > 0.
On the other hand, if  ∞ > 0, by we get that  ∞ > 0. Then the conclusion follows for Theorem 9.

Exact Number and Global Bifurcation of Solutions on a Unit Ball
From Theorem 7, the exact number of solutions (  ) is now clear in the case of Λ =  1 /; that is, the solution is unique if it exists.On the other hand, it is far from known in general exactly how may solutions of (  ) for  ∈ ( 1 /, Λ) if Λ >  1 /.Using the bifurcation approach developed in [12][13][14], and also the idea and techniques developed in [7], we solve this problem on the unit ball under some conditions.Throughout this section, we suppose that Ω is the unit ball in   centered with the origin.
The next remarkable results regarding (  ) are due to Gidas et al. [15] and Lin and Ni [16].
(2) Suppose  ∈  1 ().If  is a positive solution to (  ), and  is a solution of the linearized problem (43) (if it exists), then  is also radially symmetric and satisfies

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The next lemma also plays a key role in this section.
We will also need the following theorem of bifurcation from infinity.
To make bifurcation argument work, crucial thing is the following result.
Let  be a solution of problem (  ), then  is called a degenerate solution if the corresponding linearized equation has a nontrivial solution.Now suppose that  satisfies (F1), (F2).As in the end of Section 2, let If  ∞ > 0, then there exists a unique real number  > 0, such that Lemma 14. Suppose that  ∞ > 0. If  is a degenerate solution of (  ), then (0) > .
Proof .Suppose the contrary that (0) ≤ , then Let  be a nontrivial solution of the corresponding linearized equation (43).From (  ) and (43), we get It appears from ( 46) and (47) that  must change sign in Ω.
On the other hand, using integration by parts, we have a contradiction.
Proof.By Theorem 10, Λ >  1 /, and Theorem 7 tells us that (  ) has a unique solution (Λ,  Λ ) for  = Λ, and Implicit Function Theorem implies that (Λ,  Λ ) is a degenerate solution.By Theorem 15, non-trivial solution  of the corresponding linearized equation (43) does not change sign in Ω, and we may suppose that  is positive in Ω.Then Crandall-Rabinowitz's bifurcation theorem [20] and the discussion prior to this theorem imply that the solutions near (Λ,  Λ ) form a smooth curve which turns to the left in the phase space.We may call the part of the smooth solution curve {(, )} with (0) >  Λ (0) the upper branch, and the rest the lower branch.We denote the upper branch by   and the lower branch by   .
For the upper branch, as long as (,   ) nondegenerate, the Implicit Function Theorem ensures that we can continue to extend this solution curve in the direction of decreasing .We still denote the extension by (,   ).This process of continuation towards smaller values of  will not encounter any other degenerate solutions.This is because, if, say, (,   ) becomes degenerate at  =  0 , the discussion prior to this theorem implies that all the solutions near ( 0 ,   0 ) must lie to the left side of it, which is a contradiction.Lemma 12 tells us that  →   (0) is decreasing.So in the progress of extension of (,   ) towards smaller values of , there are only the following two possibilities.
But case (i) cannot happen, since (0,  0 ) is obviously not a solution of (  ).Hence case (ii) happens.We assert that λ =  1 /.In fact, let {  } be an arbitrary sequence such that   → λ.Denote   =‖   ‖ ∞ , V  =   /  , then   → ∞ and ΔV  +    (  V  )   = 0, in Ω, V = 0, on Ω. (61) Since (  V  )/  is bounded, by Sobolev Imbedding Theorems and standard regularity of elliptic equation, it is easy to see that {V  } has a subsequence, still denoted by {V  }, such that V  → V in  2, (Ω) ( → ∞), for some V ∈  2, (Ω), V > 0 in Ω. Letting  → ∞ in (61), we get which implies that λ =  1 /.Now we study the structure of the lower branch.As in the case of upper branch, as long as (,   ) nondegenerate, the Implicit Function Theorem ensures that we can continue to extend this solution curve in the direction of decreasing .We still denote the extension by (,   ).This process of continuation towards smaller values of  will not encounter any other degenerate solutions.Lemma 12 implies that  →   (0) is increasing.So in the progress of extension of (,   ) towards smaller values of , there are only the following two possibilities.
As before, case (i) will not happen.Then case (ii) happens.By (0) > 0, it is easy to see that  0 = 0.That is to say, the lower branch of solutions extends till the origin (0, 0) in the phase plane.
By the above argument, we obtain a smooth positive solution curve which consists of an upper branch {(,   )} and a lower branch {(,   )}.The lower branch starts from (Λ,  Λ ) and stops at (0, 0), and  →   (0) is a strictly increasing function.The upper branch {(,   )} starts from (Λ,  Λ ) and stops at ( 1 /, ∞), and  →   (0) is a strictly decreasing function with   (0) blowing up as  →  1 / + 0. By Lemma 12, all solutions of (  ) are contained in this smooth solution curve, and the complete bifurcation diagram can be described as in Figure 3.The proof is complete.

Figure 3 :
Figure 3: Precise bifurcation diagram on a unit ball.