Júnior Multiplicity of positive solutions for a degenerate nonlocal problem with p-Laplacian


 We consider a nonlinear boundary value problem with degenerate nonlocal term depending on the L
 q
 -norm of the solution and the p-Laplace operator. We prove the multiplicity of positive solutions for the problem, where the number of solutions doubles the number of “positive bumps” of the degenerate term. The solutions are also ordered according to their L
 q
 -norms.


Introduction
We study the following class of degenerate nonlocal boundary value problems with p-Laplacian where Ω is a bounded smooth domain in R N , N ≥ , q , < p < +∞; a ∈ C([ , ∞)) and f ∈ C([ , t * ]), with t * large enough, are functions satisfying some conditions which we will introduce later. The above problem generalizes the one considered in Gasínski-Santos Júnior [9] to the p-Laplacian case as well as with degenerate term a possibly sign changing.
Motivations for the nonlocal problems like (P) come from the biological models of the population di usion where the velocity of the dispersion, i.e., v = −a   Ω u q dx   ∇u depends on the whole population (compare with Chipot-Rodrigues [6]). In this case u(x) denotes the population density at x and Ω is a pervious container of bacterias.
Another situation where a nonlocal model of this form is used (although in a much simpler setting, with p = q = and N = ) is a model for transversal vibrations of elastic strings where the displacements are not necessarily small (see Carrier [2]).
The main feature considered here is the degeneracy of the reaction term, namely, the function a appearing in front of the operator may have change sign several times. More precisely, we suppose that a has a multiplicity of "positive bumps" and possibly "other bumps" between them which can be negative (and as we will see later, they will not "generate" any solutions), namely: (H ) a : [ , +∞) → R is a continuous function and there exist positive numbers =: In the past the nonlocal problems with degenerate nonlocal term were considered by Ambrosetti-Arcoya [1] (degeneracy appeared at zero and at in nity) and Santos Júnior-Siciliano [15] (where the problem was variational and in obtaining multiplicity of solutions the so called area condition was exploited). Moreover, we have two papers of Gasínski-Santos Júnior [9,10], where a similar problem was considered with the Laplacian as a main operator on the left hand side. The authors proved existence, multiplicity as well as nonexistence results for the degenerate nonlocal term depending on the L q -norm of the solution. For recent developments in problems with nonstandard growth and nonuniform ellipticity with an extensive focus on regularity theory, we mention [12] and the therein references. Our paper here is the continuation of these works in the direction of a more general operator, namely the p-Laplacian. As far as we know, this is the rst time where a p−Laplacian problem ( < p < +∞) is investigated under the degenerate condition described in (H ). Although, on the right hand side, nonlinearity has a particular growth near zero, as in the case p = , see [9]. More precisely, we assume: (H ) There exists t * > such that f (t) > in ( , t * ), f (t * ) = , f ∈ C([ , t * ]) and the map ( , t * ) t → f (t)/t p− is strictly decreasing.
To obtain our goal we need to put in good order di erent properties of the minus p−Laplacian, as the (S+) property and monotonicity, see [13], that are combined in a suitable way with the Diaz-Saa's formula [7], truncation techniques, maximum principle and regularity theory, see for instance [3][4][5] and the therein references. Owing to these re ned tools we are able to realize, also for < p < +∞, the general strategy developed in [9]. In particular, we investigate the uniqueness, regularity and positivity of the solution uα of the auxiliary Let us introduce some notation. Along the paper, λ is the rst eigenvalue of the minus p-Laplacian operator with zero Dirichlet boundary condition, φ is the positive eigenfunction associated to λ normalized in W ,p (Ω)-norm and e is the positive eigenfunction associated to λ normalized in L ∞ (Ω)-norm.
The relation between the domains of a and f are stated in the following assumption: * Ω e q dx.
Finally, to prove the multiplicity result for problem (P), namely the existence of K positive solutions corresponding to the "positive bumps" of the function a, we will also need the following two assumptions: [tf (t)].

Remark 1.1. (a) All the hypotheses on a(t), namely (H ), (H ) and (H ) deal only with "even" intervals
The function a(t) can be arbitrary in "odd" intervals (can be negative, zero or positive without satisfying bound conditions like (H ) or (H )) and these intervals can be even degenerated to a point. In other words a(t) can be any continuous function de ned on a positive interval and we can split its domain into "even" intervals where the above assumptions are satis ed and "odd" intervals (possibly degenerated to a single point), where the above assumptions need not be satis ed. The number of solutions obtained in the paper will double the number of "even" intervals.  ). If γ < +∞, (H ) basically means that the peaks of a(t) in even intervals are controlled from above by variation of f at 0. On the other hand, hypothesis (H ) means that the peaks of t p a(t) in "even"intervals are controlled from below by the maximum value of (d) Finally, we wish to explicitly observe that < p < +∞ and q ≥ are not required to satisfy any particular further condition. As well as, we emphasize that the nonlinear term f has a behaviour that is prescribed only on [ , t * ], so that, nor asymptotic conditions at in nity, neither critical/subcritical growth are involved.

Multiplicity of solutions
Hereinafter, · denotes the W ,p (Ω)-norm and |·|r the L r (Ω)-norm with r ≥ . In this Section, we will produce regular positive solutions of problem (P). In particular, they will belong to int(C+), where For the sake of completeness, recall that where ν = ν(x) denotes the outer unit normal for all x ∈ ∂Ω.
The main result of the present note can be stated as follows.
In the proof of Theorem 2.1 we will consider a suitable auxiliary problem involving the following truncation function , here is the announced auxiliary problem that will play a crucial role −a (α) ∆p u = f * (u) in Ω, u = on ∂Ω.
(P k,α ) We start solving problem (P k,α ) by the variational approach. Observe that, after removing the nonlocal term of (P) we obtained a variational problem, so that we can look for solutions of (P k,α ) searching functions u ∈ W ,p (Ω) such that −a(α) . . , K} and α ∈ (t k− , t k ). Let us de ne the energy functional I k,α : W ,p (Ω) −→ R corresponding to problem (P k,α ), namely where F * (t) = t f * (s) ds. Since f * is bounded and continuous, it is clear that I k,α is coercive and weakly lower semicontinuous. Therefore I k,α has a minimizer uα which is a solution of (P k,α ). Since φ ∈ int(C+) (recall that φ is the positive eigenfunction of the minus p-Laplacian associated to rst eigenvalue λ normalized in W ,p (Ω)-norm), observing that by assumption (H ), we can pass to the limit in the previous and obtain namely, for t > su ciently small we obtain and so uα is nontrivial. First, let us verify that ≤ uα ≤ t * . Indeed, if in (2.2) we take v = −(uα) − , we get so, again recalling that a(α) > , (uα − t * ) + = and uα t * . Hence, taking in mind the de nition of f * , uα solves (P k,α,f ). Since uα is bounded, by standard regularity arguments due to Lieberman [11, Theorem 1], we conclude that uα ∈ C ,β (Ω) (for some β ∈ ( , )). Recalling that f ∈ C([ , t * ]) and observing that f (uα) ≥ , one has ∆p uα ∈ L ∞ and ∆p uα ≤ . Hence, by the Maximum Principle (see Gasiński-Papageorgiou [8, Theorem 6.2.8], Vázquez [16] or Pucci-Serrin [14]) we get that uα ∈ int(C+). Finally, suppose that vα is a second solution to (P k,α ), with uα ≠ vα. Arguing as for uα, one can point out that < vα ≤ t * in Ω, as well as that vα solves (P k,α,f ). Hence, in view of assumption (H ), from [7] one can derive ≤ a(α) and we get a contradiction.
In order to better describe the strategy that we will follow, observe that when (H ), (H ) and (H ) hold, the previous proposition allow us to de ne suitable maps S k : (t k− , t k ) → C (Ω) for every k = , . . . , K by putting S k (α) = uα for every α ∈ (t k− , t k ), where uα, that is a minimizer of I k,α , is the unique solution of (P k,α ) such that < uα ≤ t * . At this point, for every k = , . . . , K, we introduce a real function P k : (t k− , t k ) → R de ned by for all α ∈ (t k− , t k ), where q ≥ . The role of the map P k is revealed by the following claim if α ∈ Fix(P k ), then S k (α) is a solution of problem (P), where Fix(P k ) = {α ∈ (t k− , t k ) : P k (α) = α}. Indeed, let α ∈ (t k− , t k ) be such that P k (α) = α, then, recalling that S k (α) = uα is a solution of (P k,α,f ) one can conclude that −a   Ω u q α dx   ∆p uα = −a(α)∆p uα = f (uα) in Ω and the claim (C) holds. Next lemma will be useful in the proof of Proposition 2.4 which provides the continuity of the map P k . First observe that, since the map ( , t * ) t → ψ(t) = f (t)/t p− is strictly decreasing (see hypothesis (H )), there exists the inverse ψ − : ( , γ) → ( , t * ), where γ is as in (H ).
Therefore, {un} is bounded in W ,p (Ω) and, passing to a subsequence if necessary, we may assume that un u * in W ,p (Ω), un → u * in L (Ω) and un(x) → u * (x) a.e. in Ω (2.10) for some u * ∈ W ,p (Ω). Moreover, for every n ∈ N one has a(αn) Testing the previous with v = un − u * and passing to the limsup, in view of the continuity of α and f * as well as (2.10) and the Lebesgue's dominated convergence theorem, we get The (S+) property of −∆p u assures that un → u * in W ,p (Ω), so that, passing to the limit in (2.11), we can conclude that u * is a nonnegative weak solution of (P k,α ) with α = α * . We need to show that u * ≠ . By Lemma 2.3, there exists ε > , small enough, such that So, passing to the limit and using (2.10), we obtain I k,α * (u * ) − p εψ − (λ a(α * ) + ε) p Ω e p dx < .
Therefore u * ≠ . Arguing as in the proof of Proposition 2.2 we can show that u * ∈ int(C+). Since such a solution is unique, we conclude that u * = uα * = S k (α * ). Finally, again from Proposition (2.2) one has that < un ≤ t * for every n ∈ N and we can use (2.10) and the Lebesgue's dominated convergence theorem in order to conclude that P k (αn) → P k (α * ). This proves the continuity of P k .

Remark 2.5.
A deeper analysis allow us to assure a strongest convergence of the sequence {un} considered in the previous proof. Indeed, since every un solves (P k,α ), with α = αn, one has −∆p un = f * (un) a(αn) =: gn .
Recalling that f * is bounded and a(αn) is away from zero, we have for some C > . By the regularity theory (see Lieberman [11]), we have that for some C > and β ∈ ( , ). The compactness of the embedding C ,β (Ω) ⊆ C (Ω), passing to a subsequence if necessary, permits to obtain that un → u * in C (Ω), (the limit function u * is the same as in (2.10), by the uniqueness of the limit function).
Thus vα is a nontrivial function such that −a(α) Hence, by hypotheses (H ) and (H ), we have which proves (2.17).
Next, we show that for some α ∈ (t k− , t k ) we have P k (α ) < α . where uα = S k (α) is the (unique) positive solution of (P k,α ). Multiplying by uα, integrating by parts, using Cauchy-Schwarz inequality and Hölder inequality, we have Using hypothesis (H ), for some α ∈ (t k− , t k ) one has From the continuity of P k (see Proposition 2.4), since (2.17) implies the existence of α and α in (t k− , t k ) with P(α ) > t k− and P(α ) > t k , (2.18) and the intermediate value theorem for continuous real functions, we conclude that P k has at least two xed points in the interval (t k− , t k ).
Proof of Theorem 2.1 Fix k ∈ { , . . . , K}, recall claim (C) and observe that the two xed points of P k obtained in Proposition 2.7 produce two positive solutions u k, and u k, of problem (P), with u k, , u k, ∈ int(C+) and satisfying t k− < Ω u q k, dx < Ω u q k, dx < t k , for any k ∈ { , . . . , K}. This nishes the proof.
That is (H ) holds too.