ON AN ASYMPTOTICALLY LINEAR ELLIPTIC DIRICHLET PROBLEM

where Ω is a bounded domain in RN (N ≥ 1) with smooth boundary ∂Ω. The conditions imposed on f (x, t) are as follows: (f1) f ∈ C(Ω×R,R); f (x,0)= 0, for all x ∈Ω. (f2) lim|t|→0( f (x, t)/t)= μ, lim|t|→∞( f (x, t)/t)= uniformly in x ∈Ω. Since we assume (f2), problem (1.1) is called asymptotically linear at both zero and infinity. This kind of problems have captured great interest since the pioneer work of [1]. For more information, see [2, 3, 4, 5, 6, 7, 8, 11, 12] and the references therein. Obviously, the constant function u= 0 is a trivial solution of problem (1.1). Therefore, we are interested in finding nontrivial solutions. Let F(x,u) = ∫ u 0 f (x,s)ds. It follows from (f1) and (f2) that the functional J(u)= 1 2 ∫

Obviously, the constant function u = 0 is a trivial solution of problem (1.1).Therefore, we are interested in finding nontrivial solutions.Let F(x,u) = u 0 f (x,s)ds.It follows from (f 1 ) and (f 2 ) that the functional is of class C 1 on the Sobolev space H 1 0 := H 1 0 (Ω) with norm and the critical points of J are solutions of (1.1).Thus we will try to find critical points of J.In doing so, we have to prove that the functional J satisfies the (PS) condition.
We denote by 0 , then the problem is resonant at infinity.This case is more delicate.To ensure that J satisfies the (PS) condition usually one needs to assume additional conditions, such as the wellknown Landesman-Lazer condition, see, for example, [3,4]; the angle condition at infinity, see [2].
Recently, in the case 0 ≤ µ < λ 1 < , Zhou [12] obtained a positive solution of problem (1.1) under (f 2 ) and the following conditions: Note that our assumption (f 1 ) is weaker than (H 1 ).And condition (H 2 ) is a strong assumption.
In this paper, we prove that (f 1 ) and (f 2 ) are sufficient to obtain a positive solution and a negative solution of problem (1.1).Our main result is the following.
Note that in Theorem 1.1, even in the resonant case, we do not need to assume any additional conditions to ensure that J satisfies the (PS) condition.Thus Theorem 1.1 greatly improves previous results, such as Zhou's [12].This fact is interesting.The proof of Theorem 1.1 will be stated in Section 2.
We can also consider the asymptotically linear Dirichlet problem for the p-Laplacian It is known that − p has a smallest eigenvalue (see [5]), that is, the principle eigenvalue, λ p 1 , which is simple and has an associated eigenfunction (1.6) Assuming (f 1 ) and the following condition: (1.4) has at least two nontrivial solutions, one is positive, the other is negative.
(2) Obviously, Theorem 1.1 is a special case of Theorem 1.2.But we would rather state the proof of Theorem 1.1 separately, because the proof is very simple and clear.

Proof of Theorem 1.1
In this section, we will always assume that (f 1 ) and (f 2 ) hold and give the proof of Theorem 1.1.
Consider the following problem: where where Lemma 2.1.J + satisfies the (PS) condition.
Proof.Let {u n } ⊂ H 1 0 be a sequence such that It is easy to see that Set φ = u n , we have where • 2 is the standard norm in L 2 := L 2 (Ω).We claim that u n 2 is bounded.For otherwise, we may assume that u n 2 → +∞.Let v n = u n / u n 2 , then v n 2 = 1.Moreover, from (2.7) we have That is, v n is bounded.So, up to a subsequence, we have (2.9) From (2.6) it follows that where v + = max{0,v}.From this and the regularity theory we have The maximum principle implies that v = v + ≥ 0. But > λ 1 and hence v ≡ 0 which contradicts with v 2 = 1.
Since u n 2 is bounded, from (2.7) we get the boundness of u n .A standard argument shows that {u n } has a convergent subsequence.Therefore, J + satisfies the (PS) condition.Proof of Theorem 1.1.By Lemmas 2.1, 2.2, and the Mountain Pass Theorem [9, Theorem 2.2], the functional J + has a critical point u + with J + (u + ) ≥ β.But J + (0) = 0, that is, u + = 0. Then u + is a nontrivial solution of (2.1).From the strong maximum principle, u + > 0. Hence u + is also a positive solution of (1.1).
Similarly, we obtain a negative solution u − of (1.1).The proof is completed.

Proof of Theorem 1.2 and final remarks
In this section, we sketch the proof of Theorem 1.2 and give some remarks.First, we recall the concept Fučik spectrum and a related result.
The Fučik spectrum of p-Laplacian with Dirichlet boundary condition is defined as the set Σ p of those (a,d) ∈ R 2 such that has a nontrivial solution, where u + = max{u,0} and u − = max{−u,0}.By [5], we know that if (a,d) ∈ Σ p and (a,d We will also need the following lemma, which is due to Zhang and Li [11,Lemma 3]. satisfies the (PS) condition, where H(u) = u 0 h(t)dt.Sketch of the proof of Theorem 1.2.As in Section 2, consider the trancated problem where f + is defined as in (2.2).Due to the maximum principle (see [10]), solutions of (3.4) are positive, thus are solutions of (1.4).We have Since > λ p 1 , one deduces directly from the definition of Fučik spectrum that ( ,0) / ∈ Σ p , thus by Lemma 3.1, we deduce that the C 1 -functional satisfies the (PS) condition on the Sobolev space W 1,p 0 (Ω) with norm where F + (x,t) = t 0 f + (x,s)ds.As [7, Lemma 2.3], the functional J + admits the "Mountain Pass Geometry."Thus J + has a nonzero critical point, which is a nontrivial solution of (3.4).From the strong maximum principle (see [10]), it is also a positive solution of (1.4).
Similarly, we obtain a negative solution of (1.4).

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Remark 3.2.Problems (1.1) and (1.4) can be resonant at infinity, this is the main difficulty in verifying the (PS) condition.But after trancating, the problems are not resonant with respect to the Fučik spectrum.Thus, from the Fučik spectrum point of view, the corresponding functionals of the trancated problems satisfies the (PS) condition naturally.And our limit conditions at zero allow us to use the trancation technique and apply the Mountain Pass Theorem.These are the main ingredient of this work.
Remark 3.3.In fact, let P := {u ∈ H 1 0 : u(x) ≥ 0, a.e.}, the functional J does not satisfies the (PS) condition on the whole space H 1 0 whenever = λ i , i > 1, but from our proof J satisfies the (PS) condition on P. That is, the unbounded (PS) sequences do not belong to P. This idea may be used to weaken the compact conditions for other problems.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: (a) there exist ρ,β > 0 such that J + (u) ≥ β for all u ∈ H 1 0 with u = ρ; (b) J + (tφ 1 ) → −∞ as t → +∞.Proof.See the proof of [12, Lemma 2.5].Now, we are in a position to state the proof of Theorem 1.1.