MULTIPLE POSITIVE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEMS WITH SIGN-CHANGING NONLINEARITIES

where Ω is a smooth bounded domain in RN , N ≥ 1, λ > 0 is a parameter, and f is a C1 sign-changing sublinear function. They showed using sub-super solutions arguments and recent results from semipositone problems that there are λ and λ such that (1.1) has no positive solution for λ < λ and at least two positive solutions for λ≥ λ. More recently, in [8], the author extends these results to the quasilinear problem


Introduction
In a recent paper, [7], the authors studied the existence, multiplicity, and nonexistence of positive classical solutions of the semilinear elliptic boundary value problem where Ω is a smooth bounded domain in R N , N ≥ 1, λ > 0 is a parameter, and f is a C 1 sign-changing sublinear function.
They showed using sub-super solutions arguments and recent results from semipositone problems that there are λ and λ such that (1.1) has no positive solution for λ < λ and at least two positive solutions for λ ≥ λ.
More recently, in [8], the author extends these results to the quasilinear problem 1048 Quasilinear problems with sign-changing nonlinearities (H 3 ) there exists t 0 > 0 such that F(x,t 0 ) > 0, (H 4 ) limsup t→∞ F(x,t)/t p ≤ 0 uniformly in x, where F(x,t) = t 0 f (x,s)ds.The purpose of this article is twofold.Applying variational methods, we first extend the results in [8] to quasilinear elliptic systems of the form where (F u ,F v ) stands for the gradient of a given potential F, and second, we want to see to what extent these variational techniques can be adapted to deal with the nonlinear boundary condition case where ∂/∂ν is the outer unit normal derivative.Systems of the form (1.3) are usually called gradient systems and have been widely studied in the past.See, for example, [2] for a comprehensive analysis of such systems.This gradient structure allows us to treat (1.3) variationally.Other kinds of elliptic systems that can be treated variationally are the so-called Hamiltonian systems, see [3].
However, as far as we know, all the results for (1.3) assume, to begin with, that F u ,F v ≥ 0 for u,v ≥ 0.
For problem (1.4), in a previous paper, [4], the authors studied the problem where the nonlinearity g was assumed to be of power type, that is, essentially the case g(x,t) = |t| q−2 t was considered, so again g(x,t) ≥ 0 for t ≥ 0.
The main results of this paper can be formulated as follows.
The rest of the paper is organized as follows: in Section 2 we deal with problem (1.3) and in Section 3 with (1.4).

Gradient systems
In this section, we deal with problem (1.3).First, we prove the nonexistence result.To this end, we assume that F(x,u,v) is a Carathéodory function on x ∈ Ω, and F u , F v are also Carathéodory functions satisfying for some constant C > 0.
Julián Fernández Bonder 1049 We have the following theorem.
For the proof we need the following observation.We denote by λ r the best constant in the Sobolev embedding W 1,r 0 (Ω) L r (Ω).We have so if we denote λ p,q = min{λ p ,λ q }, we obtain and, moreover, one can easily see that λ p,q is optimal.
Proof of Theorem 2.1.
3) has a positive solution (u,v), multiplying the first equation of (1.3) by u, the second by v, and integrating by parts and adding up, we get Thus, using (2.1), we obtain and hence λ ≥ λ p,q /C by (2.3), proving Theorem 2.1.Now, we prove the multiplicity result.To this end, along with (2.1), we also have to assume that Under these assumptions, we have the following theorem.
For the proof of Theorem 2.2, we use critical point theory.Set F(x,u,v) = 0 for u,v < 0, and consider the C 1 functional 1050 Quasilinear problems with sign-changing nonlinearities Observe that if (u,v) is a critical point of Ᏺ λ , denoting by u − and v − the negative parts of u and v, respectively, hence we have that u,v ≥ 0. Furthermore, by [10], u,v ∈ C 1,α (Ω) and so, by Harnack inequality (see [11]), it follows that either u,v > 0 or u ≡ v ≡ 0. Therefore, nontrivial critical points of Ᏺ λ are positive solutions of (1.4).By (F 3 ) and (2.1), there is a constant and hence where | • | d denotes the d-dimensional Lebesgue measure in R N , so Ᏺ λ is bounded from below and coercive.Therefore, as Ᏺ λ is weakly lower semicontinuous, we obtain a global minimizer (u 1 , v 1 ).We show that, if λ is big enough, this minimizer is nontrivial.
Proof.We consider a sufficiently large compact subset Ω of Ω and take functions Then, we obtain if Ω is big enough.Hence, Ᏺ λ (u 0 ,v 0 ) < 0 for λ large enough.
We will obtain a critical point (u 2 ,v 2 ) with Ᏺ λ (u 2 ,v 2 ) > 0 via the mountain pass lemma, which would complete the proof since (2.11) By (2.1), Hölder's inequality, and Sobolev embedding, where r = N p/(N − p) if p < N and r > p if p ≥ N, and s = Nq/(N − q) if q < N and s > q if q ≥ N. So, in order to finish the proof we need to show that (2.13) as we wanted to show.Now, we are in position to finish the proof of Theorem 2.2.
Proof of Theorem 2.2.As Ᏺ λ is coercive, every Palais-Smale sequence is bounded and hence contains a convergent subsequence as usual.Now, the mountain pass lemma gives a critical point (u 2 ,v 2 ) of Ᏺ λ at the level where } is the class of paths joining the origin to (u 1 ,v 1 ) (see [9]).

The nonlinear boundary condition case
In this section, we deal with the nonlinear boundary condition case, problem (1.4).The main ideas and structures of the proofs are the same as in the previous section, so we only sketch them and stress the differences between the two cases.
We begin with the nonexistence result.To this end, we assume that g is a Carathéodory function on ∂Ω × [0,∞) satisfying for some 1 ≤ r ≤ p and some constants C,c > 0.
1052 Quasilinear problems with sign-changing nonlinearities We have the following theorem.
For the proof, we need some knowledge on the following eigenvalue problem: This problem was studied in [4,6] (see also [5]).It was proved there that problem (3.2) has a first positive eigenvalue λ 1 given by where dσ is the boundary measure.In the linear case, p = 2, problem (3.2) is known as the Steklov problem (see [1]).Now we prove the multiplicity result.The assumptions in this case are as follows: let G(x,t) = t 0 g(x,s)ds, and assume the following: Theorem 3.2.Assume (3.1) and (G 1 ), (G 2 ), and (G 3 ) hold.Then, there is a λ such that (1.4) has at least two positive solutions u 1 > u 2 for λ ≥ λ.
Observe that for problem (1.4) we can prove that the two solutions are ordered.We believe that this should hold also for (1.3), but the truncation argument used in the proof does not work because it destroys the variational structure of (1.3).
Again, set g(x,t) = 0 for t < 0, and consider the C 1 functional Julián Fernández Bonder 1053 Arguing as before, if u is a critical point of Ᏻ λ , denoting by u − the negative part of u, hence we have that u ≥ 0. Furthermore, by [10], u ∈ C 1,α (Ω) and so, by the strong maximum principle and Hopf 's lemma (see [12]), it follows that either u > 0 in Ω or u ≡ 0. Therefore, nontrivial critical points of Ᏺ λ are positive solutions of (1.4).Observe that in this case, the solution u is positive up to the boundary.By (G 3 ) and (3.1), there is a constant and hence so Ᏻ λ is bounded from below and coercive.Therefore, as Ᏻ λ is weakly lower semicontinuous, we obtain a global minimizer u 1 .Once again, if λ is big enough, this minimizer is nontrivial.Lemma 3.3.There is a λ such that inf Ᏻ λ < 0, and hence u 1 = 0, for λ ≥ λ.
Proof.Take the constant function u 0 ≡ t 0 , where t 0 is as in (G 2 ).
The main difference in the arguments arrives at this point.As we mentioned before, by a truncation argument we can prove that the two solutions are ordered.In fact, fix λ ≥ λ.Let Then consider Proof of Theorem 3.2.The same argument used for Ᏻ λ shows that Ᏻ λ is also coercive, so every Palais-Smale sequence of Ᏻ λ is bounded and hence contains a convergent subsequence as usual.Now, the mountain pass lemma gives a critical point u 2 of Ᏻ λ at the level where is the class of paths joining the origin to u 1 .

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.
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