THE EXISTENCE OF POSITIVE SOLUTIONS FOR AN ELLIPTIC BOUNDARY VALUE PROBLEM

By using the mountain pass lemma, we study the existence of positive solutions for the equation −Δu(x)=λ(u|u|


Introduction
Consider the boundary value problem where Ω is a bounded region with smooth boundary in R N , λ > 0 is a real parameter, and ∆ is the Laplacian operator.
The study of (1.1) is motivated by the fact that the equation has wide applications to physical models (see [4] and the references therein), our analysis is based on a method used by Rabinowitz [6] and also by Alama and Del Pino [2].
It is well known that steady-state solutions of correspond to critical points of the functional J λ : W 1,2 0 (Ω) → R, 2006 Existence of positive solutions standard norm 2 dx, ( (see, e.g., [1] or [5]).Henceforth, we will assume that, unless otherwise stated, integrals are over Ω.When J is bounded below on X, J has a minimizer on X which is a critical point of J.In many problems such as (1.1), J is not bounded below on X, however, in such cases, we may be able to use mountain pass lemma.The mountain pass lemma was introduced by Ambrosetti and Rabinowitz in 1973.
Lemma 1.1 (mountain pass lemma [3]).Let E be a Banach space over R. Let B r = {u ∈ E : u < r} and S r = ∂B r ; B 1 and S 1 will be denoted by B and S, respectively.Let (I1) there exist ρ > 0 and α > 0, such that I > 0 in B ρ − {0} and on S ρ ; (I2) there exists e ∈ E, e = 0 with I(e) = 0; (I3) if {u m } ⊂ E with the properties that I(u m ) is bounded above, and I (u m ) → 0 as m → ∞, then {u m } possesses a convergent subsequence; and if ) is a critical value of I with 0 < α ≤ b < +∞.

Main results
The corresponding Euler functional for (1.1) is given by First, we claim that J λ (u) has neither a global minimum nor a global maximum.In fact, we can choose a sequence {u n } satisfying |u n | 3 dx = 1 and |∇u n | 2 dx → +∞, so that J λ (u n ) → +∞ as n → +∞, that is, J λ (u) is not bounded from above.On the other hand, for fixed u = 0, we have as t → +∞.Hence, J λ (u) is not bounded from below, so we have provedthe following.
We now show that the mountain pass lemma can be applied in this case.
Proof.For the existence of solution to (1.1), it is sufficient to check that conditions (I1), (I2), and (I3) of mountain pass lemma are satisfied.(I1) By the Sobolev imbedding inequality ( (I2) For given fixed u > 0 in X, we consider the map t → J λ (tu).Since λ < λ 1 , using the Poincaré inequality, we have which is negative if t is large enough and is positive for sufficiently small t.Thus by continuity, there exists t 0 > 0 such that J λ (t 0 u) = 0, and so J λ (u) satisfies (I2).(I3) We take a sequence {u n } ⊂ X satisfying J λ (u n ) < M (M is a positive constant) and J λ (u n ) → 0 in X (as n → ∞).Thus, there exists N such that 2008 Existence of positive solutions for n > N. Also we have and since λ < λ 1 , it follows that {u n } is bounded in X.Hence, there exists a subsequence, again denoted by {u n }, satisfying u n u 0 weakly in X and strongly in L 2 (Ω) and L 3 (Ω).
Since J λ (u n ) → 0, we have as n → ∞.Since u n → u 0 in L 2 (Ω), (u n − u 0 ) 2 dx → 0 as n → ∞, also by Holder's inequality, we have (2.10) as u n → u 0 in L 3 (Ω), and so (u and so |∇u n | 2 dx → |∇u 0 | 2 dx.This, together with the weak convergence of {u n }, implies that {u n } is convergent strongly in X.Hence, J λ (u) satisfies the condition (I3).Therefore, by the mountain pass lemma, J λ (u) has a nontrivial critical point denoted by u * .Since J λ (u * ) = J λ (|u * |), without loss of generality, we may assume that u * is a nonnegative weak solution of (1.1) and it follows from standard regularity arguments that u * ∈ C 2 (Ω) is a classical solution of (1.1), that is, we have 12) also it is easy to deduce from the maximum principle that u * > 0 on Ω.
We now show that the problem (1.1) has no positive solution in the case where λ ≥ λ 1 .
Proof.Let λ ≥ λ 1 be arbitrary and fixed.On the contrary, we assume that u is a positive solution of (1.1), that is, Let φ > 0 be the principal eigenfunction corresponding to principal eigenvalue λ 1 , that is, and this is a contradiction as u and φ are positive and λ ≥ λ 1 , and so the problem (1.1) has no positive solution when λ ≥ λ 1 .
We now will show that when λ is sufficiently small, then the positive solution u of (1.1) corresponding to λ is very large.
Proof.Since u is the positive solution of (1.1) corresponding to λ, we have and so It is easy to see that u is a positive solution of (1.1) if and only if u is a positive solution of so we have obtained the following theorem.

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: