Multiplicity Results and Global Bifurcations for a Degenerate Elliptic Equation

We consider the question of determining the exact number of solutions of the problem E λ below We distinguish two cases: whether 1 < p 2 or 2 < p < ∞ . In the first case, we shall show that the problem E λ behaves like in the semi-linear case which corresponds to p = 2.. In the second case and under some conditions which will be specified later; we shall show that the spectrum for problem E λ consists of a collection of intervals In, n = 1, 2, 3, whose ends points are members of the sequences (λn)n, (μn)n where the first one is the sequence of eigenvalues for the pseudo-Laplacian operator and the other one is a sequence of a kind of eigenvalues for E λ. And each time λ = μn, there exists secondary bifurcating continuum εn of singular solutions which are diffeomorphic to [0, π]n, n = 1, 2..


Introduction
We investigate the non trivial solutions of this problem and more particularly we consider the question of determining the exact number of solutions of this problem.We distinguish two cases: whether 0 p 2 or 2 < p < ∞.In the first case, we shall show that the problem E p λ behaves like in the semi-linear case which corresponds to p = 2. Indeed, we shall show that bifurcation can appear in the point (0, λ) only it λ belongs to the set {λ k } +∞ k=1 , where is the eigenvalue of the problem ℘ (λ) For more details about the problem ℘ (λ) the reader is referred to see [1].
We shall always refer to the first case as the semi-linear case and to the second case as the superlinear case.It is interesting to see what becomes of the notion of point spectrum of E p λ when one goes from the semi-linear to the superlinear case.In the superlinear case and under some conditions which will be precise later : we shall show that the spectrum for problem E p λ consists of a collection of intervals I n , n = 1, 2, 3 . . . . . ., whose ends points are members of the sequences {λ n }, {μ n } where the first one is the sequence of eigenvalues for ℘ (λ) and the other one is a sequence of a kind of eigenvalues for E p λ .And each time λ = μ n , there exists secondary bifurcating continuum ε n of singular solutions which are diffeomorphic to [0, π] n , n = 1, 2 . . . .The approach used in this paper is based only on the elementary tools of the theory or ordinary differential equations and essentially the use of a similar type of the time map for problem E p λ .Even there has been a significant amount of work that has been done on degenerate elliptic equations, and in particular, bifurcation problems for p-Laplacian equations, not only for one-dimensional problems but in higher dimensions as well, in the last 25 years.Authors who have contributed to this field include, for instance, Laurent Veron, M.Guedda [16], Pavel Drabek [3], [8], [9], [10], [11], [12], Bryan Rynne [4], [5], [13], [14], [15], and many others.However, until recently, the basic spectral properties obtained in the case p > 2 and the exact numbers of solutions of problem E p λ and especially the existence of collection of intervals I n , n = 1, 2, 3 . . .had not been obtained.
Before passing to the study of the problem (E 2 λ ) in the next section, we now state explicitly the main's theorems.
As has been said above, we shall show the striking difference between the nonlinear (p = 2) and semi-linear p = 2 case.
We consider a nonlinearity f which satisfies the following hypotheses: Hypotheses governing the function f .
Throughout the remaining part of this work we shall assume that the function f appearing in (E p λ ) satisfies the following hypotheses: Where (a − , a + ) is the maximal interval centered on the origin such that: Furthermore, we suppose in case a − and a + are finite that In case both a − and a + are finite, the condition H6 ) implies H4 ); 2. When p = 2, one can replace the condition H5 ) by a sufficient condition, namely:

Examples of function f
Here is some examples of functions which satisfy the above conditions:

Summary of the main results
We shall distinguish two cases: p ≤ 2 and p > 2.
First case A: has precisely 2n+1 solutions where λ n is the eigenvalue of (℘(λ)) and α is given by H 3 ) Thus under the first case A; the bifurcation picture is as portrayed in figure 1.By a regular solution of problem E p λ , we mean every solution u which has a finite number of zeros in (0, π), and whose first derivative vanishes only once between two consecutive zeros.

Definition 2
By a singular solution of problem E p λ , we mean every solution u which has a finite number of zero in (0, π), and such that between two consecutive zeros, the first derivative vanishes at least in a subinterval.

Remark
We will show later that there are only those two kinds of solutions.Next, we will treat three sub cases which go with the following additional hypotheses: i a ± are both infinite; ii a ± are both finite, and the two following integrals are infinite , is finite and the two following integrals are finite Under the second case B and the statements i) or ii) we obtain: λ has precisely 2n + 1 regular solutions and no singular solutions: Thus under the second case B and the statements i) or ii) above, the bifurcation picture is as portrayed in figure 2. We notice that under the first case A and under the second case B and the statements i) or ii), the problem E p λ behaves like in the semi-linear case (p ≤ 2).
Under the statement iii), we consider the following real number whose significance only appears later: Under The second case B, and iii), we shall distinguish two cases depending on whether: r(p, F ) < 1 or r(p, F ) ≥ 1 First case: r(p, F ) < 1 We obtain Then problem E p λ has precisely three regular solutions and no singular solutions Then problem E p λ has precisely three regular solutions and each time λ = μ n there exists a secondary bifurcating continuum ε n of non trivial singular solutions which are diffeomorphic to ]0, π[ n , n = 1, 2, . . .where In this case, we have:   We conclude this work by exhibiting an example which justifies our statements about the real number r(p, F ).
If we take f (u) = −|u| 2(p−1) u + |u| p−2 u, p > 2, then we shall verify easily that f satisfies all the hypotheses H 1 ), H 2 ), H 3 ), H 4 ), H 5 , H 6 , and iii) and we find that: Where Γ(x) = +∞ 0 t x−1 e x−1 e −t dt The graph of r(p, F ) is as portrayed in figure 5 Figure 5: We notice that the real number r(p, F ) can take every possible positif value.In fact, p * corresponds to the separation value of the two cases considered above; that is

Remark
When p * > 2, there exists for some value of the parameter λ(for example between μ 1 and λ 2 α in theorem 3) what the physicists call a Dead core where the solution is necessarily constant, since between μ 1 and λ 2 α the only non trivial solution is the singular one [2], [7].

I. Study of the problem
One can show that such solution is also a solution in the sense of the definition (1) above.
ii) If a ± are both infinite, then we have for all solutions of the problem E p λ .
Proof i) Let x 0 be a point such that u(x 0 ) = maxu(x).In a such point, we have Indeed, suppose the contrary, that is is strictly increasing for every x ∈ ϑ, which in turn yields that du dx is strictly increasing in ϑ.And this contradicts the fact that u possess a maximum in x 0 .This yields f (u(x 0 )) ≥ 0, so that u(x 0 ) ≤ a + In the same way, one can show that in a point y 0 such that u(y 0 ) = min x u(x) and one has a − ≤ u(y 0 ).This together with the results above prove i).
ii) It is an immediate consequence of the definition of F .It will be convenient below to write the problem E p λ as a first order system.For that purpose, let us perform the change of variables: Let us now consider now the following system E p λ (δ): For every δ, the system E p λ (δ) has almost a solution (u, w) defined entirely on R.

Proof
It split in several steps.

First step.
The system E p λ (δ) can also be put in an integral equation form: (u, w) is a solution of E p λ (δ) means that in R, (u, w) satisfy (9) and (10) which, in turn, are equivalent to one of the following integral equations By taking into account the hypothesis H 3 , we show easily that for x sufficiently small T 1 • T 2 is a contraction on the Banach space C([0, x]).And hence by fixed point theorem the equation (11) has one and only one local solution.
Consequently, for such an x, Applying the general theory of first order system with continuous right hand, we can extend u over a maximal interval [0, T ] [5].

Second step Lemma 2
Let (u, w) be a solution of the system E p λ (δ), then we have u ≡ 0, w ≡ 0, or the zeros of u and the zeros of w are isolated.

Proof
Indeed, let (u(t), w(t)) be a non trivial solution of E p λ (δ), t 0 be a zero of u: u(t 0 ) = 0.If we have u (t 0 ) = 0, then u would be a solution of the Cauchy problem: From above, we deduce that the only solution of this system is the trivial one.This contradiction shows that u (t 0 ) = 0 and hence the zeros of u(t) are isolated.On the other hand, let x0 be such that w(x 0 ) = 0, from the first equation of E p λ (γ), we get u (x 0 ) = 0.If u(x 0 ) = 0, then one deduce as above that u(t) ≡ 0. Consequently u (x 0 ) = 0 and w (x 0 ) = −f (u(x 0 )), so two situations can occur.i) a − < u(x 0 ) < a + , then w (x 0 ) = 0 and one deduce easily that the zeros of w are isolated.
ii) If u(x 0 ) is equal to at least one of the numbers a ± then w(x 0 ) = 0, consequently then zeros of w may not be isolated.

Now let us consider now the following function
), then we have.

Proposition
V * is a first integral of the system E p λ (δ):

Proof
Indeed, by multiplying the first equation of the system E p λ (δ) by dw dt and the second equation by du dt , we obtain: From which one deduces by using the identity , and hence

Remark
The existence of the first integral V * implies automatically then that solutions (u, w) of E p λ (δ) are defined entirely on R.

II. The structure of the set of solutions of E p λ (δ).
Now let us study the system E p λ (δ) in more details.The main tool is the phase plane.We interpret E p λ (δ) as a dynamical system defined on the phase plane (u, w).The point of the trajectory has Cartesian coordinates (u, v) and we call such point a representative point.We note that u and w are function of t, that is the curve in (u, w) plane may be regarded as given in parametric form with t as parameter.And the totality of all the solutions of E p λ (δ) may be described in the phase portrait of the system.

Remark
By analogy, the identity (3) expresses the law of conservation of energy; 1 p * |w| p * represents the changes in kinetic energy, F (u) represents the potential energy of the system E p λ (δ).Now let us proceed to description of the energy levels of E p λ (δ) that is: which is a continuously differentiable curve in the neighborhood of each one of its points, except the equilibrium position.In fact, we shall use the implicit function theorem: We have If one of these derivatives is different from zero in neighborhood of the considered point, then ξ p δ is represented by the graph of a differential curve of the form: Let a ± be the singular points and set F (a ± ) = E ± .
We propose to study the general aspect of the trajectory near the singular points.
One sees that for every a : 1 p * |w| p * = E − F (u) so, every time the potential curve is slightly larger than the line E, 1  p * |w| p * is negative, which is none sense, it follows that the representative point in the phase plane, never reaches those points where the curve F (u) dominates the line E. In the common points of F and the line E, the velocity of the system is equal zero.

Inflexion Points
Let a be a point such that Figure 7: Then du dt (t 0 ) = 0, so the representation point never reaches the points which are larger than a.Then it must leaves the point of abscissa a and moves towards the left.So there exists t 1 > t 0 such that. And On the other hand, as dw dt (t 0 ) = −f (a) < 0, it follows that the velocity at the point of abscissa b is given by The representative point reaches the point of abscissa b at a time t 1 given by We must check the convergence of this integral.By hypothesis, F is at least twice continuously differentiable.So it may be expanded in Taylor series about the point a And ϕ(u) = 0 ∀u ∈ [b, a], because on the contrary for such value of u, E = F (a) which is impossible by (18).
So there exists a real number k such that Let's put But by virtue of (24), one has The representative point reaches then the point of abscissa b on a finite time.And going so on, one can show that, the trajectory corresponding to E, is closed in the phase plane.

Stationary points
Let a + be the abscissa of a point where the curve of F and the line E are tangent: Figure 9: Let t 0 be such that: We suppose that the line E + dominates the trajectory on some interval on the left of a + : ∃ b < a + such that We also suppose that One can write By virtue of (28) one has The function ϕ is continuous on [b, a + ], so it is bounded: For b ≤ α < a + let I α be defined as follows In term of (30) one has du Two cases should be considered: We obtain immediately from above that lim That is the representation point reaches the point of abscissa a + in an infinite time.

Second case B: p > 2
The convergence or the divergence of the integral above depends on the aspect of the function F in the neighborhood of the point a + .That is two situations can occur: Finally, there are two cases which we shall suppose depending on whether a ± are either finite or infinite.
i-a ± are both finite.In this case, the portrait phase looks like

III. Bifurcations equations associated with E p λ (δ)
Let us recall the main hypotheses made on the nonlinearity f .For this, let F : R → R be the function defined by We suppose that there exists unique numbers a − and a + such that : Further, we suppose that outside the interval (a − , a + ), one has: We introduce two quantities E ± , through: We note that F is strictly increasing on [0, a + ) and strictly decreasing on (a − , 0].So on [0, a + ), the function F has an inverse And on (a + , 0], the function F has an inverse We define two functions l ± (E) in the following way: More precisely, the significance of l ± (E) is as follows.Let us consider a trajectory on the phase plane corresponding to a given constant E: a + are both finite Consider l + (E) for given E, 0 < E < E + .We make the change of variables This leads us to the relation: Similarly, if one put: We can prove:

Proposition 3
The functions l ± (E) are continuous on (0, E ± ).Furthermore Where α is given by H3 and B(x, y) is the beta function.

Proof
From H3 it follows that: Given this δ, we can find η with 0 < η < E + such that if, 0 ≤ E ≤ η, then m + (E) ≤ δ.This follows from continuity of m + (E).Now, we have by integrating (40): Hence elevating it to the power 1 p : Therefore as one obtains: From (40) one has: From this and the definition of l + (E), there follows: By taking into account that , we obtain: In an entirely similar manner, one obtains Let us now investigate the behavior of l ± (E) when E tends to E ± .For this, we shall distinguish several cases: In this case, we make no more hypotheses on f outside H 1 )...H 6 ).
In this case, we shall distinguish the following sub-cases: i. a ± are both infinite; ii. a ± are both finite, and the two following integrals are infinite: iii. a ± are both infinite and the two following integrals are finite: Then we have: 1.Under the statements A or B, i), ii), one has: 2. Under the statements B, iii), one has We render that proof only for l + (E), the proof for l − (E) is similar.For this, firstly let's suppose that a + < +∞, then under the condition A, an entirely similar argument like the one made in §IIshows that lim While that under the statements B, i), ii), the result is an immediate consequence of the hypotheses (45) et (46).Now suppose that a + = +∞, then by virtue of hypotheses made on f , f is strictly positive for u ∈ (0, +∞) and Hence by H3 : Therefore, Choose any ε sufficiently small with 0 < ε < ξ, then there exists an u 0 ∈ (0, +∞) such that: This together with (53) imply that there exists a ū ∈ [u 0 , +∞) such that and referring to (34), we see that and after some computation, one obtains 2. The result is immediate by virtue of ( 50) and ( 51) The function l + (E) are differentiable in their domain (0, E ± ) and are stricly increasing in (0, E ± ).

Proof
We render it only for l + (E).One has. dl where Let's define a function : R → R by setting Which is the case by virtue of H5 .
And one finds readily that the corresponding phase portrait looks like: Figure 16: It follows from the above study, that under the case p > 2 and a ± are both finite, two possibilities can occur relatively to the heteroclinic orbits.That is the curves which pass through the single points (a − , 0) and (a + , 0) ; either one can traverse the heteroclinic orbit in an infinite time or in a finite time, the later case leads us to consider two kinds of solutions.

Definition 1
By a regular solution of problem E p λ , we mean every solution u which has a finite number of zeros in (0, π) and whose first derivative vanishes only once between two consecutive zeros.

Definition 2
By a singular solution of problem E p λ , we mean every solution u which has a finite number of zeros in (0, π), and such that between two consecutive zeros, the first derivative vanishes at least in a subinterval.
From this, the claims about the number and disposition of solutions follow.More precisely.If p−1 p | du dx | p + F (u) = E, then the equation E p λ (δ) admits a regular solution u satisfying u(λ 1 p π) = 0 If and only if there exists an integer number n and a constant E such that one of the following bifurcation equations is satisfied Furthermore, one must of course include the singular solutions which appear in the case where E = F (a + ) (resp E = F (a − )).We shall return to them later in more details Before discussing in more details the bifurcation equations, we shall add other hypotheses on f in thecase B, iii), namely: And let us consider the following real number whose significance only appears later: where Let us now study the bifurcation equations.For this, we have: Let λn = λn α ; n = 1, 2, .. where λ n is the eigenvalue of the problem ℘(λ) and α is given by H3 ), then we have: is the unique solution of the equation (60).

II)
Under the statement B, iii), we shall distinguish two sub cases: depending on whether r(p, F ) < 1 or r(p, F ) ≥ 1.

And lim
E→min(E − ,E + ) By using the expression of μ n and λn+1 , one obtains (after computations) Then for fixed p, let us check for the value of n for which the above inequality is satisfied.
After some computations, one obtain n ≤ 1 This leads us to distinguish two cases: First case: r(p, F ) < 1 Under this case, one sees that the inequality (68) is never satisfied.That is we have always.
That is the set of interval I n are overlapping; hence we deduce tha final desired results of first case I.

Second case:r(p, F ) ≥ 1
In this case, one has always From which, we deduce the desired results of II(2 case).Let us confine attention for the moment to the singular solutions.Let us recall that they correspond to the case where E = Ē, hence they appear every time λ = μ n , where μ n are defined previously by (63).Referring to the below figure which shows a heteroclinic trajectory associated to Ē in the phase-plane (u, w):  Thus, we notice that all a continuum of solutions ε 1 take place; it is diffeomorphic to [0, π] because one can index it by the parameter α, which is the time that the representative point (u(t), v(t)) may put in the equilibrium point (a, 0): Let us consider the solutions which correspond to a one tour Thus, we see this time, that all a continuum of solution ε 2 take place, which is diffeomorphic to [0, π] 2 , for one can index ε 2 by parameters α and β.
So by continuing on, one sees that every time λ = μ n , a continuum ε n of solutions which is diffeomorphic to [0, π] n appears.
By collecting all the informations above, we can state the following theorem: α , then problem (E p λ ) has only the trivial solution     As the heteroclinic trajectory is closed, one deduce that the solution u n is periodic, the period T n is given by From the fact that μ n = 2 p (n p (p−1)) μ p p l p (F ), on deduce that In this case we note that the period of the motion is independent of the amplitude.Furthermore the zeros of the functions μ n are given by the formula: x j n = j π n j = 0, 1, 2, ..., n (72) Let us consider the Dirichlet's problem associated with nonlinear degenerate elliptic equation.E p λ They considered the following problem:(E 2 λ ) u = λf (u) x ∈]0, π[ u (0) = u (π) = 0 λ > 0Where f is a nonlinearity which satisfies the following hypotheses:H1 ) : f : R → R is of classe C 2 H2 ) : f (0) = 0 H3 ) : f (0) > 0 H4 ) : lim |u|→∞ sup f (u) u −1 < 0 H5 ) : sgn f = −sgn uThey obtain the following resultsTheoremi If λ ≤ 1 f (0) , then problem (E 2 λ) has only the trivial solution;

Remark 1 One can also define the solutions
of E p λ in the weak sense: u is a solution if and only if: u ∈ W 1,p 0 (0, π) and satisfies the equation E p λ in the sense of distribution if and only if π 0