Three positive solutions of N-dimensional p-Laplacian with indefinite weight

This paper is concerned with the global behavior of components of positive radial solutions for the quasilinear elliptic problem with indefinite weight div(φp(∇u)) + λh(x) f (u) = 0, in B, u = 0, on ∂B, where φp(s) = |s|p−2s, B is the unit open ball of RN with N ≥ 1, 1 < p < ∞, λ > 0 is a parameter, f ∈ C([0, ∞), [0, ∞)) and h ∈ C(B̄) is a sign-changing function. We manage to determine the intervals of λ in which the above problem has one, two or three positive radial solutions by using the directions of a bifurcation.

It is known that the existence of three positive solutions for one-dimensional p-Laplacian problem with indefinite weight (ϕ p (u (x))) + λh(x) f (u(x)) = 0, x ∈ I, u(0) = 0 = u (1) (1.3) was mainly studied by three positive solutions theorem in Amann [1] based on the method of lower and upper solutions.Notice that even they established the basic three positive solutions theorem for the indefinite weight case, they could only apply it for a positive weight case.This implies that it is difficult to construct upper and lower solutions for the indefinite weight.
Variational approach [3,16] can also be applied to get three solutions, however, this method does not guarantee positivity or nontriviality of all solutions at most cases.
To overcome the difficulties mentioned above, very recently, Sim and Tanaka [17] employed a bifurcation technique to show the existence of three positive solutions for the onedimensional p-Laplacian problem (1.3) with h ∈ H(I), see [17,Theorem 1.1] for more details.
Up to our knowledge, the existence of three positive radial solutions have never been established for N-dimensional p-Laplacian problem (1.1) (or (1.2)) with indefinite weight h on the unit ball of R N .For example, Dai, Han and Ma [4]only showed the existence of one positive radial solution of (1.1) (or (1.2)) with indefinite weight.So it is the main purpose of this paper to obtain a similar result to Sim and Tanaka [17] for (1.1) (or (1.2)) with h ∈ H(B).Indeed, problem with indefinite weight arises from the selection-migration model in population genetics.In this model, h(r) changes sign corresponding to the fact that an allele A 1 holds an advantage over a rival allele A 2 at same points and is at a disadvantage at others; the parameter λ corresponds to the reciprocal of diffusion, for detail, see [8].
For other results on the study of positive solutions of N-dimensional p-Laplacian problem (1.1) or (1.2) we refer the reader to [4][5][6][7][9][10][11].It is worth noting that Dai, Han and Ma [4] studied the unilateral global bifurcation phenomena for (1.2) with indefinite weight and constructed the eigenvalue theory of the following problem with indefinite weight (1.4) Let µ 1 be the first positive eigenvalue of (1.4).Then from the variational characterization of µ 1 , it follows that where For the spectrum of the p-Laplacian operator with indefinite weight we refer the reader to [2,14].
We turn now to a more detailed statement of our assumptions and main conclusions.Throughout the paper we shall assume, without further comment, the following hypotheses concerning the function f : (H4) there exists s 0 > 0 such that min s∈[s 0 , 2s 0 ] f (s) ] h(r), ν 1 (p) and ν 2 (p) are the first two zeros of the initial value problem It is well known [15] that (1.5) has a unique solution Φ defined on [0, ∞), which is oscillatory.Let 0 be the zeros of Φ.These zeros are simple and ν n (p) → +∞ as n → +∞.
Note that (H2) implies Combining this with (H3), we can deduce that there exists f * > 0 satisfying To wit, our principal result can now be stated.Theorem 1.1.Assume (H1)-(H4) hold.Let h ∈ H(I).Then the pair ( µ 1 f 0 , 0) is a bifurcation point of problem (1.2), and there is an unbounded continuum C of the set of positive solutions of problem such that the continuum C grows to the right from the bifurcation point ( µ 1 f 0 , 0), to the left at (λ * , u λ * ) and to the right at (λ * , u λ * ).
From Theorem 1.1, we can easily derive the following corollary, which gives the ranges of parameter guaranteeing problem (1.2) has one, two or three positive solutions (see Figure 1).

Corollary 1.2. Assume (H1)-(H4) hold. Let h ∈ H(I). Then there exist λ
2) has at least three positive solutions if Remark 1.3.Note that (H2) has been used in [17] studying the one-dimensional p-Laplacian with a sign-changing weight.Indeed, under (H2) there is an unbounded continuum C which is bifurcating from µ 1 f 0 .Conditions (H1)-(H3) and h ∈ H(I) push the direction of continuum C to the right near u = 0.Moreover, it follows from (H3) and (H4) that the nonlinearity is superlinear at some point and is sublinear near ∞, which make continuum C turn to the left at some point and to the right near λ = ∞.Nevertheless, assumptions (H2) and (H4) are technical and need to be further improved.

Remark 1.4. Let us consider the nonlinear function
where a > 1, m > ln a.It is not difficult to prove that f satisfies (H1), (H2) and (H3) with We can easily verify that g is increasing on and is decreasing on and so (H4) is satisfied.Then f satisfies all of the conditions in Theorem 1.1.
The contents of this paper have been distributed as follows.In Section 2, we establish a global bifurcation phenomena from the trivial branch with the rightward direction.In Section 3, we show that the bifurcation curve grows to the left at some point under (H4) condition.In Section 4, we get the second turn of the bifurcation curve which grows to right near λ = ∞.Moreover, we give the proof of Theorem 1.1.

Global bifurcation phenomena with the rightward direction
We firstly introduce the following important result, which is proved in [4, Theorem 3.1], see also [5,Theorem 2.1] or [13] studying the semilinear problem.Lemma 2.1 (See [4]).Let h ∈ H(I).Assume g : I × R × R satisfies Carathéodory condition in the first two variable, g(r, s, 0) ≡ 0 for (r, s) ∈ I × R and uniformly for r ∈ I and λ on bounded sets.Then from each (λ ν k , 0) bifurcates an unbounded continuum C ν k of solutions to problems (r with exactly k − 1 simple zeros, where λ ν k is the eigenvalue of (1.4) and ν ∈ {−1, 1}.Now, we are ready to show the unbounded continuum C of positive solutions to problem (1.2).Thanks to (1.6), there exists δ > 0 such that f (s Let us consider the auxiliary problem as a bifurcation problem from the trivial solution u ≡ 0.
From Lemma 2.1, we can easily obtain the following result.
Proof.Put v n := u n u n .Then v n = 1, and hence v n ∞ and v n ∞ are bounded.Applying the Arzelà-Ascoli theorem, a subsequence of {v n } uniformly converges to a limit v.We again denote by {v n } the subsequence.Observe that v (0) = 0 = v(1) and v = 1.Now, from the equation of (1.2) with λ = λ n and u = u n , we obtain Dividing the both sides of (2.6) by u n p−1 , we get whence also On the other hand, it can be easily seen that Then, by virtue of (1.7), it follows from Lebesgue's dominated convergence theorem that w n (r) tends to w(r), Consequently, combining (2.8) and Lebesgue's dominated convergence theorem, we can deduce which is equivalent to (1.4) with λ = µ 1 , and hence v ≡ φ 1 .
Proof.Multiplying − div(ϕ p (∇u)) = µ 1 h(x)ϕ p (u) by φ 1+α 1 and integrating it over B, we obtain The next result establishes that the continuum C grows to the right from ( µ 1 f 0 , 0).Lemma 2.6.Let the hypotheses of Lemma 2.2 hold.Then there exists δ > 0 such that (λ, u) ∈ C and |λ − µ Proof.For contradiction we assume that there exists a sequence {(λ n , u n )} ⊂ C satisfying From Lemma 2.4, there exists a subsequence of {u n }, we again denote it by {u n }, such that u n u n converges uniformly to φ 1 on [0, 1], here φ 1 > 0 is the eigenfunction corresponding to µ 1 satisfying φ 1 = 1.Multiplying the equation of (1.1) applied to (λ n , u n ) by u n and integrating over B, we see that It follows from the definition of µ 1 that Together with (H2), Lemma 2.5 and Lebesgue's dominated convergence theorem, then gives Consequently, λ n > µ 1 f 0 , which contradicts (2.9).

Direction turn of bifurcation
In this section, in view of the condition (H4), we show that the continuum C grows to the left at some point.
Proof.It can be easily seen from Lemma 3.1 that For contradiction we assume λ ≥ µ 1 / f 0 .Then it follows from (H4) that, for all r ∈ J, λh(r) f (u(r)) Recall that Φ is a unique solution of (1.5).Then v is a solution of On the other hand, u is a solution of Applying the Sturm comparison Theorem [15, Lemma 4.1], we can deduce that u has at least one zero on J, an obvious contradiction.
Remark 3.3.It follows from Lemma 3.2 that there exists a direction turn of the bifurcation continuum C which grows to the left at some point (λ * , u λ * ) ∈ C.

Second turn and proof of main result
In this section, we prove that there is the second direction turn of bifurcation and complete the proof of Theorem 1.1.
Then there exists a constant C > 0 independent of u such that Proof.An easy integration for (1.2) now yields Together this with (1.7), we are lead to the result follows at once.
The following result provides us with a lower bound for the parameter.Proof.Let r 0 be a point of a maximum of u.According to (4.1), we obtain and hence, λ ≥ C −(p−1) , where C is a constant defined in Lemma 4.1.
Remark 4.3.Lemma 4.2 implies that the bifurcation continuum C can not intersect with the X axis.
The next result shows that there is an upper estimate of the C 1 -norm of positive solutions of (1.2).Lemma 4.4.Let the hypotheses of Lemma 4.1 hold.Suppose J ⊂ (0, ∞) is a compact interval.Then there exists M J > 0 such that all possible positive solutions u of (1.2) with λ ∈ J satisfy u ≤ M J .
Proof.Now we proceed as in [12], repeating the arguments for completeness.Put J := [a, b].For contradiction, we suppose that there exists a sequence {u n } of positive solutions of (1.2) Proof.Let r 0 be a point of a maximum of u.As in the proof of Lemma 3.1, it readily follows that u (r) > 0, r ∈ [0, 1] \ (r 1 , r 2 ), u < 0, r ∈ (r 1 , r 2 ).
Then, by the standard argument, there exists at least one positive solution of (1.2) with λ = λ * and λ = λ * , respectively.Clearly, C turns to the left at (λ * , u λ * ) and to the right at (λ * , u λ * ).