SIGN-CHANGING AND MULTIPLE SOLUTIONS FOR THE p-LAPLACIAN

where ∆pu= div(|∇u|p−2∇u) is the p-Laplacian, 1 < p <∞. By W1,p(Ω) we denote the usual Sobolev space with dual space (W1,p(Ω))∗, and W 0 (Ω) denotes its subspace whose elements have generalized homogeneous boundary values and whose dual space is given by W−1,p(Ω). We assume the following growth and asymptotic behaviour of the nonlinear right-hand side f of (1.1): (H1) f : Ω×R→R is a Carathéodory function satisfying ∣∣ f (x, t)∣∣≤ C(|t|p−1 + 1), f (x, t)= a(t+)p−1− b(t−)p−1 + g(x, t), (1.2)


Introduction
Let Ω ⊂ R N , N ≥ 1, be a bounded domain with a smooth boundary ∂Ω.In this paper, we consider the quasilinear elliptic boundary value problem where ∆ p u = div(|∇u| p−2 ∇u) is the p-Laplacian, 1 < p < ∞.By W 1,p (Ω) we denote the usual Sobolev space with dual space (W 1,p (Ω)) * , and W 1,p 0 (Ω) denotes its subspace whose elements have generalized homogeneous boundary values and whose dual space is given by W −1,p (Ω).We assume the following growth and asymptotic behaviour of the nonlinear right-hand side f of (1.1): (H1) f : where lim t→0 g(x,t) |t| p−1 = 0 uniformly in x. (1. 3) The set Σ p of those points (a,b) ∈ R 2 for which the asymptotic problem has a nontrivial solution, is called the Fučík spectrum of the p-Laplacian on Ω, where u + = max{u,0} and u − = max{−u,0}.The Fučík spectrum was introduced in the semilinear case p = 2 by Dancer [8] and Fučík [14] who recognized its significance for the solvability of problems with jumping nonlinearities.In the semilinear ODE case p = 2, N = 1, Fučík [14] showed that Σ 2 consists of a sequence of hyperbolic-like curves passing through the points (λ l ,λ l ), where {λ l } l∈N are the eigenvalues of −d 2 /dx 2 , with one or two curves going through each point.Drábek [12] has recently shown that Σ p has this same general shape for all p > 1 in the ODE case.
In the PDE case N ≥ 2, much of the work to date on Σ p has been done for the semilinear case p = 2.It is now known that Σ 2 consists, at least locally, of curves emanating from the points (λ l ,λ l ) (see, e.g., [2,7,8,10,14,25]).Schechter [28] has shown that Σ 2 contains two continuous and strictly decreasing curves through (λ l ,λ l ), which may coincide, such that the points in the square (λ l−1 ,λ l+1 ) 2 that are either below the lower curve or above the upper curve are not in Σ 2 , while the points between them may or may not belong to Σ 2 when they do not coincide.
In the quasilinear PDE case p = 2, N ≥ 2, it is known that the first eigenvalue λ 1 of −∆ p is positive, simple, and admits a positive eigenfunction ϕ 1 (see Lindqvist [20]), so Σ p clearly contains the two lines λ 1 × R and R × λ 1 .In addition, σ(−∆ p ) has an unbounded sequence of variational eigenvalues {λ l } satisfying a standard min-max characterization, and Σ p contains the corresponding sequence of points {(λ l ,λ l )}.A first nontrivial curve Ꮿ in Σ p through (λ 2 ,λ 2 ) asymptotic to λ 1 × R and R × λ 1 at infinity was recently constructed and variationally characterized by a mountain-pass procedure by Cuesta et al. [6] (see Figure 1.1).More recently, unbounded sequences of curves (analogous to the lower and upper curves of Schechter) have been constructed and variationally characterized by min-max procedures by Micheletti and Pistoia [26] for p ≥ 2 and by the second author [27] for all p > 1.
The main goal of this paper is to identify the set of points (a,b) relative to the Fučík spectrum which ensure the existence of sign-changing solutions of (1.1).More precisely, assuming the existence of a positive supersolution u and a negative subsolution u of (1.1) and (a,b) located above the curve Ꮿ, we prove the existence of at least three nontrivial solutions within the order interval [u,u]; a positive solution, a negative solution, and a sign-changing solution.
There are many existence and multiplicity results for (1.1) in the literature (see, e.g., [5,6,9,13,23,27]).However, to the best of our knowledge, the first results on sign-changing solutions were obtained only recently by Li and Zhang [17].In their paper the authors assume that p > N and f is independent of x and locally Lipschitz in t.All these assumptions can be relaxed by our approach S. Carl and K. Perera 615 which is very different from that of Li and Zhang.Our main result, which will be proved in Section 3 (Theorem 3.1), improves upon their results.

Preliminaries
We denote the norm in W 1,p (Ω) and L p (Ω) by • and • p , respectively, and recall the notion of sub-and supersolutions.Definition 2.1.A function u ∈ W 1,p (Ω) is a supersolution of (1.1) if the following holds: Similarly, u is a subsolution of (1.1) if the reversed inequalities of Definition 2.1 hold with u replaced by u.Here L p + (Ω) stands for the positive cone of L p (Ω).Consider the boundary value problem (2.1) Besides the hypothesis (H1) we will assume the following hypotheses to hold throughout the rest of the paper.
(H2) There exist a positive supersolution u and a negative subsolution u of (1.1), and the point (a,b) ∈ R 2 is above the curve Ꮿ of the Fučík spectrum.(H3) Any solution u of (2.1) with h ∈ L ∞ (Ω) belongs to C 1 (Ω).
Remark 2.2.(i) Assuming the existence of super-and subsolutions as in hypothesis (H2) is a weaker assumption than the usual condition on the jumping nonlinearity at infinity.
Proof.Passing to a subsequence (again denoted by (u n )), we may assume that u n → u a.e. and in L p (Ω).By Egoroff 's theorem, for any µ > 0 there is a measurable subset The first integral on the right-hand side of (2.4) tends to zero by the asymptotic behavior (1.3) of g, (2.5), and Lebesgue's dominated convergence theorem (observe that the integrand is majorized by C(|u(x)| + δ) p−1 for any δ > 0 due to the uniform convergence in Ω µ ).The second integral is bounded by which proves (2.2).Observing that the elementary inequality holds, where 0 ≤ τ n (x) ≤ 1, which yields we see that (2.3) follows similarly.Proof.In the proof we focus on the existence of a positive solution only since the existence of a negative solution can be shown in a similar way.
As is well known, solutions of (1.1) are the critical points of where F(x,t) = t 0 f (x,s)ds.Let f be the following truncated nonlinearity: (2.10) and F its associated primitive given by (2.11) Consider the functional 618 Sign-changing and multiple solutions whose critical points are the solutions of the auxiliary boundary value problem Obviously, Φ : W 1,p 0 (Ω) → R is bounded from below, weakly lower semicontinuous, and coercive.Thus, there is a global minimizer, that is, a critical point u of Φ (2.14) We will show that this global minimizer is in fact a positive solution of (1.
which shows that u − = 0, and thus which implies that ∇(u − u) + = 0, and thus u ≤ u.This shows that the global minimizer u of the functional Φ satisfies u ∈ [0,u], and thus u is a solution of (1.1) due to the definition of f .Since a > λ 1 , we get by hypothesis (H1) that As u is a global minimizer of Φ, it follows Φ(u) ≤ Φ(εϕ) < 0, and thus u must be a positive solution of (1.1).Proof.We are going to prove the existence of the least positive solution only, since the proof of the existence of the greatest negative solution is analogous.In view of Lemma 2.5, there exists a positive solution u ∈ [0,u], and applying S. Carl and K. Perera 619 Lemma 2.3 there is a ε > 0 small enough such that εϕ 1 ≤ u, where ϕ 1 is the eigenfunction that belongs to the first eigenvalue λ 1 of −∆ p .Since a > λ 1 , one readily verifies that εϕ 1 is a subsolution of problem (1.1) for sufficiently small ε > 0. Thus there is an ε 0 > 0 such that ε 0 ϕ 1 and u forms an ordered pair of sub-and supersolutions.Applying [3, Corollary 5.1.2]on the existence of extremal solutions for general quasilinear elliptic problems, we obtain the existence of a least and greatest solution of (1.1) with respect to the order interval [ε 0 ϕ 1 ,u].We denote the least solution within this interval by u 0 .Now let (ε n ) ∞ n=0 be a decreasing sequence with ε n → 0 as n → ∞, and denote by u n the corresponding least solution of (1.1) with respect to the order interval [ε n ϕ 1 ,u].Then obviously (u n ) is a decreasing sequence of least positive solutions of (1.1) which converges to its nonnegative pointwise limit u * in L p (Ω).We will show that u * is in fact the least positive solution, that is, u * = u + .First we verify that u * is a solution of (1.1).Since the u n are solutions of (1.1) we get from the equation which by the growth condition (H1) and the boundedness in L p (Ω) of the sequence (u n ) implies its boundedness in W 1,p 0 (Ω), that is, u n ≤ c.Thus there exists a subsequence weakly convergent in W 1,p 0 (Ω), and due to the strong convergence of (u n ) in L p (Ω) even the entire sequence is weakly convergent in W 1,p 0 (Ω) with weak limit u * .From (1.1) with the test function u n − u * , we obtain (2.20) The weak convergence of (u n ) and (2.20) along with the S + -property of the operator −∆ p (see, e.g., [3, Chapter D]) yield its strong convergence in W 1,p 0 (Ω).This allows the passage to the limit in (1.1) with u replaced by u n , and hence u * is a solution of problem (1.1).To show that u * > 0, our argument is by contradiction.Suppose u * = 0, that is, u n → 0 in W 1,p 0 (Ω).Since u n > 0 we may consider ũn := u n / u n which satisfies

21)
By definition ũn = 1, so there is a subsequence (ũ n ) that converges weakly in W 1,p 0 (Ω) and strongly in L p (Ω) to ũ due to the compact embedding of W 1,p 0 (Ω) ⊂ L p (Ω). Taking in (2.21) as special test function ϕ = ũn − ũ, we get for the righthand side of (2.21) as n → ∞, because the terms in parentheses are L q (Ω)-bounded.Hence (2.21) implies which due to the S + -property of −∆ p implies the strong convergence of ũn → ũ in W 1,p 0 (Ω).Moreover, the third integral term on the right-hand side of (2.21) converges to zero by Lemma 2.4, so we may pass to the limit to get that is, ũ satisfies the boundary value problem

.25)
Since ũn = 1 and ũn > 0, by Lemma 2.3 we have the same properties for ũ, which, however, contradicts the fact that a nontrivial solution of (2.25) changes sign.So far we have shown that the limit u * of the least solutions u n ∈ [ε n ϕ 1 ,u] is a positive solution of (1.1).Finally, to prove that u * is the least positive solution, let w be any positive solution of (1.1).Then by Lemma 2.3 there is a ε n > 0 for n sufficiently large such that ε n ϕ 1 ≤ w which by definition of the sequence of least solutions (u n ) yields u * ≤ u n ≤ w, which proves that u * = u + is in fact the least positive one.

Main result
Theorem 3.1.Let hypotheses (H1), (H2), and (H3) be satisfied.Then the boundary value problem (1.1) has at least three nontrivial solutions: a positive solution, a negative solution, and a sign-changing solution. Proof.Let S. Carl and K. Perera 621 and consider Arguments similar to those in the proof of Lemma 2.5 show that critical points of Φ + are solutions of (1.1) in the order interval [0,u + ], so 0 and u + are the only critical points of Φ + by Lemmas 2.3 and 2.7.Now, Φ + is bounded from below and coercive, and since a > λ 1 , so Φ + has a global minimizer at a negative critical level.It follows that u + is the (strict) global minimizer of Φ + and As before, critical points of Φ are solutions of (1.1) in the order interval [u − ,u + ], so it follows from Lemmas 2.3 and 2.7 that any nontrivial critical point other than u ± is a sign-changing solution.
Proof.We only consider u + as the argument for u − is similar.Suppose that there is a sequence u j → u + in W 1,p 0 (Ω), u j = u + with Φ(u j ) ≤ Φ(u + ).By (1.2) and (1.3) we have so If u − j = 0, then u + j = u + and contradicting the fact that u + is the unique global minimizer of Φ + , so u − j = 0. We will show that u − j p > C u − j p p , j large.(3.8)Assuming this for the moment, we have the contradiction Φ + (u + j ) < Φ + (u + ).To see that (3.8) holds, we first note that the measure of the set Ω j = {x ∈ Ω : u j (x) < 0} goes to zero.To see this, given ε > 0, take a compact subset Then where c = min Ω ε u + > 0, so |Ω ε j | → 0. Since Ω j ⊂ Ω ε j ∪ (Ω \ Ω ε ) and ε > 0 is arbitrary, the claim follows.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

Lemma 2 . 5 .
Problem (1.1) has a positive solution u > 0 within the order interval [0,u] and a negative solution u < 0 within the order interval [u,0].

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation