ON A CLASS OF SEMILINEAR ELLIPTIC EQUATIONS WITH BOUNDARY CONDITIONS AND POTENTIALS WHICH CHANGE SIGN

We study the existence of nontrivial solutions for the problem ∆u = u, in a bounded smooth domain Ω ⊂ RN, with a semilinear boundary condition given by ∂u/∂ν = λu− W(x)g(u), on the boundary of the domain, where W is a potential changing sign, g has a superlinear growth condition, and the parameter λ∈ ]0,λ1]; λ1 is the first eigenvalue of the Steklov problem. The proofs are based on the variational and min-max methods.


Introduction
In this paper, we study the existence of nontrivial solutions of the following problem: where Ω is a bounded domain set of R N , N ≥ 3 with smooth boundary ∂Ω, ∆u = ∇ • (∇u) is the Laplacian and ∂/∂ν is the outer normal derivative; the parameter λ ∈ ]0,λ 1 ], where λ 1 is the first eigenvalue of the Steklov problem (see [5]), W ∈ C(Ω) different from zero such a solution is where e 1 denotes a positive eigenfunction of Laplacian related to the first eigenvalue, with p ∈ ]2,2 * [, 2 * = 2N/(N − 2) if N > 2, 2 * = +∞ if N = 2. Also, in [3], it was proved that (1.2) is a necessary and sufficient condition to obtain a positive solution; recently, Margone in [14], proved some results of existence in case that 0 < λ ≤ λ 1 , close to λ 1 ; by using mountain pass lemma (see [4]) and linking-type theorem (see [15]).Finally, in [1], Alama and Delpino proved under some restriction on the sign of W(x) the existence of nontrivial solution, by using two different approach: one involving min-max methods, the other Morse theory methods.However, nonlinear boundary conditions have only been considered in recent years, for the Laplacian with boundary conditions, see, for example [6,7,8,12,13,16], where the authors discussed mountain pass theorem on an order interval with Dirichlet boundary condition.For elliptic systems with nonlinear boundary conditions, see [9,10].
The main purpose of this work is to study one problem of Neumman boundary value, in the case λ = λ 1 because if λ < λ 1 , it is easy to prove that the functional Φ λ has a condition of mountain pass structure.We show two results of existence obtained as critical points of the functional related at (P λ ), by using mountain pass lemma introduced in [4] and linking-type theorem introduced in [15].
The rest of this paper is organized as follows: in Section 2, we cite the main results and in Section 3, we prove the main results.

Main results
In the sequel, we consider the following functional: (2.1) Then, we show the following existence results for (P λ ).
Theorem 2.1.Let g be a continuous real-valued function on R such that the following assumptions hold: sufficiently large, and for some where λ 2 is the second eigenvalue of the Steklov problem, and M. Ouanan and A. Touzani 97 ) ∂Ω W(x)e r 1 dσ < 0, where e 1 is a positive eigenfunction related to λ 1 , then (P λ ) has a positive solution u λ for any λ ∈ (0,λ 1 ]. Remarks 2.2.(i) Condition (G6) was introduced by Girardi and Matzeu (see [11]) and plays a crucial role in the proof of Palais-Smale condition.
Remark 2.4.Note that the solution found in Theorem 2.3 is surely not always positive because (W 1 ) does not hold.Moreover, condition (W 2 ), which appears in Theorem 2.3, is in some sense complementary to (W 1 ) if g is a power.

Proof of the main results
It is well known that the solutions of (P λ ) are critical points of the functional In order to prove the main results, we apply the mountain pass theorem (see [4]) and a suitable version of the linking-type theorem (see [15]) to the functional Φ λ .
Proof.Let (u n ) n ⊂ H 1 (Ω) be a Palais-Smale sequence, namely, there exist c 1 and c 2 such that sup We are going to show that (u n ) n is bounded in H 1 (Ω).By assumptions (G3) and (G6), and from (3.3) and (3.4), we get for some constant c R > 0 depending on the number R of (G3), Set X 1 = vect(e 1 ), then, there exist Using the variational characterization of λ 2 , (3.5) becomes where n is an infinitesimal sequence of positive numbers.
On the other hand, using variational characterization of λ 1 , it follows that On the other side, by (2.2) and taking into acount that n → 0, we deduce that 1,2 = λ 1 ∂Ω e 2 1 = 0, which is an absurdum as we know that e 1 is the principal eigenvector related with λ 1 .
On the other side, combining (W 0 ) and (3.9), it follows that either In the first case, we take φ regular nonnegative function with meas(suppφ then, by (G6) and (3.15), we get for some positive constant c, s+1 dσ > 0. (3.17) Finally, we have proved that (u n ) n is bounded, this implies the existence of a subsequence weakly converging in H 1 (Ω).On the other side, thanks to (G2) and the compact embedding H 1 (Ω) L r (∂Ω) for r ∈ ]2,2(N − 1)/(N − 2)[, we have the strong convergence.
Lemma 3.2.The origin is a strict locale minimizer of Φ λ .
Proof.First, remark that each u ∈ H 1 (Ω) can be written as u = te 1 + v, where t ∈ R, and Choosing e 1 such that ∂Ω e 2 1 dσ = 1/λ 1 , one gets, for all u satisfying u 1,2 ≤ 1/2 e 1 ∞ , Hence, by variational characterization of the eigenvalues of the Laplacian with boundary conditions and for a suitable function F(t,v), we obtain where by (G4), (3.21) On the other hand, using arrangement-finite theorem, there exists a function 0 so by (G2), while, if |te 1 + θv(x)| ≤ 1, using again (G2), one obtains where C 1 , C 2 are two positive constants.Hence, using Sobolev trace embedding, for < A/C 1 , we deduce For r > 2, the least expression is strictly positive as v 1,2 is close to 0.
Proof of Theorem 2.1.We will study only the case λ = λ 1 because if λ < λ 1 , it is easily proved that the functional Φ λ has a condition of mountain pass structure.Now, it suffices to prove that there exist u ∈ H 1 (Ω) such that u 1,2 > ρ, ρ large enough satisfying Φ λ (u) < 0 which completes the proof of Theorem 2.3.
Using (G4), we obtain (3.28) Then, there exists t 0 > 0 large enough, such that u = t 0 φ.Hence, using mountain pass lemma, there exists a critical point u of Φ λ1 at the level where is the class of the path joining the origin to u.The positivity of u can be checked by a standard argument based on (3.29) (which yields the nonnegativity of u) and by the strong maximum principle of Vazquez [17] (which yields the strict positivity of u).
The proof of Theorem 2.3 is based on Lemma 3.1 and the following version of the linking theorem, see [15].
As for the proof of (J2), first of all, we note that, as also observed in [15], it is enough to prove the following two properties: (a) Φ λ1 (te 1 ) ≤ 0 for all t ∈ R; (b) there exist v ∈ X 2 \{0} and ρ 0 > ρ such that Φ λ1 (u) ≤ 0 for all u ∈ X 1 ⊕ [v] and |u| ≥ ρ 0 .For (a), we have which is not positive by (W 2 ), and (a) follows.

M. Ouanan and A. Touzani 103
On the other side, let v be a sufficiently regular function in X 2 \{0} such that supp v ⊂ Ω\D and meas(supp therefore, by (W 3 ), one gets We observe now that the map is an isomorphism and that as it easily can be deduced from the fact that −te 1 (x) = δv(x) in Ω\D if δ 2 + t 2 = 0 (indeed e 1 (x) > 0 everywhere on Ω, while v has a compact support in Ω\D) therefore, as all the norms are equivalents in a finite dimensional space, we get, for some positive constant c, hence, Φ λ satisfies the assumptions of Proposition 3.3, which completes the proof of Theorem 2.3.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.