Elliptic Weighted Problem with Indefinite Asymptotically Linear Nonlinearity

where,  is a regular bounded subset N and N  2, a(x) is a nonnegative function and f(x, t) is allowed to be sign-changing. We employ variational techniques to prove the existence of a nontrivial solution for the problem (P), under some suitable assumptions, when the nonlinearity is asymptotically linear. Then, we prove by the same method the existence of positive solution when the function f is superlinear and subcritical at infinity.


Introduction with Main Results
In the present paper, we investigate the existence result for the following nonlinear elliptic equation involving sign-changing nonlinearity: where,  N , N  2, is a smooth bounded open set, a(x) is a nonnegative function and also the function f(x, t) is an indefinite nonlinearity.
This problem is nonlinear and in Physics, Dynamics and Biology nonlinear problems have many interest since they are able to explain the evolution of a system. If we change some parameters or the nonlinearity, the system undergoes transitions mainly the existence of solutions: We have the bifurcation phenomena. The problem (1.1) is the stationary position of the problem induced in 1952 by (Turing, 1952): which modules the interaction between species and chemicals in a morphogenesis phenomenon in Biology, where u is the density and f(x, t) represents the diffusion-interaction of substances. The term -div(a(x)u) indicatess the substance of diffusion through the given system.
When the function a(x) is a smooth on  and f(x, t) = g(t), with the same conditions (G1)-(G4), the problem (1.1) was studied by (Saanouni and Trabelsi, 2016a). The condition g(0) > 0 was capital in their work. On the other hand, the problem (1.1) was treated by (Zhou, 2002), when a is a constant function but the asymptotically linear nonlinearity depends on x and t. More precisely, the author consider the case when: for a.e. x. and he proved also that the bifurcation phenomena occurs.
As a recent work, we can cite that (Li and Huang, 2019) a generalized quasilinear Schrödinger equations with asymptotically linear nonlinearities. They supposed that the nonlinearities f(t) depend only on t and they proved that the problem has positive solutions. For the superlinear nonlinearities, the Schrödinger equations was investigated by (Li et al., 2020). Throughout this paper, we assume different type of conditions. The nonlinearity f(x, t) does not have to be positive and it is asymptotically linear (ℓ finite) or super linear at  . More precisely, we make the following assumptions: Also, the weight function a(x) is nonnegative and for this reason we will use weighted Sobolev spaces. We remark that a non-trivial solution for the Eq. (1.1) is nonzero a critical point of the following functional: In order to investigate the existence of nonzero critical point of J, we will apply Mountain Pass Theorem introduced by (Ambrosetti and Rabinowitz, 1973). The most difficult property that J has to satisfy the compactness condition, which is also called the Palais-Smale condition and often, one requires a technical condition introduced as this introduced in (Ambrosetti and Rabinowitz, 1973;Rabinowitz, 1986) and called Ambrosetti -Rabionowitz condition, that is: for some  > 2 and t0 > 0.
Sometimes, other type of condition were made as in the following papers (Costa and Magalhaes, 1994;Costa and Miyagaki, 1995;Jeanjean, 1999;Schechter, 1995;Stuart and Zhou, 1996;1999). When the nonlinearity is asymptotically linear, we can not suppose the condition (AR) because it gives: In this study, we will not use (AR) or any assumption when we prove the existence of critical point for the functional J.
Our results state as follows.
In the present paper C and Ci denotes positive constants, which may change from line to another.

Variational Formulation
Consider  N , N  2, a regular bounded open set and throughout this paper, we denote: Let a(x)L 1 () be a nonnegative function and followed by (Calanchi et al., 2017), set: Weighted Sobolev spaces have been developed and studied for a long time and we can refer to (Drabek et al., 1997;Kufner, 1985).
For completeness, recall that the space   Ha  be functional of class C 1 defined by: Hence, a solution of the problem (1.1) can be found as critical point of functional J. Before starting the Mountain Pass Theorem, we introduce the following definition.

Definition 2.2
Let H be a Banach space and a functional JC 1 (H, ). We say J satisfies the Palais Smale (PS) condition if any sequence {un} H such that J(un) converges in and J(un)  0 in H, the dual space of H, the sequence {un} has a convergent subsequence. (Ambrosetti and Rabinowitz, 1973 div a x in

Proof of the Theorem 1.1
We begin by proving the two geometric properties.

Lemma 3.1
Assume that (V 1) -(V 3) hold and ℓ(0, +). Then, we can find two positive numbers  > 0 and  > 0 satisfying: In the next lemma, we prove the second geometry property.
Then, there exists w  

Proof
For t > 0, consider the function: which gives 1  ℓ. This finish the proof of (i  (x, u) in L 2 () and so: As in (Meyers, 1963), we prove that the operator L = -div(a(x)  ) is an isomorphism between    By using step 1, we know that: From (3.14), the sequence {gn} is bounded on  and so it is weakly star convergent in L  (), up to subsequence to a function g.
By (V 3) and the fact that un(x) not equal to zero a.e. in , the function g(x) = ℓ, for a.e x. Now, if we consider the second term of (3.16), we have: hence w = c1 and ℓ = 1. This is impossible since we have supposed that 1 < ℓ < . We deduce {un} is bounded in Step 3 (Conclusion) By Proposition 2.1, the Eq. (1.1) has a nontrivial solution.
(iii) Suppose that ℓ = 1. Set u a positive solution for (1.1) or negative solution. If we take  = 1 in (2.3), we get: , and (V 5), the constant C in the inequality (3.14) will be equal to ℓ and since 1 > 0, we obtain f(x, u) = ℓu a.e. in . This means f(x, u) = 1u and then u is an eigenfunction associated to the simple eigenvalue 1, so u = c1 for some constant c > 0 or c < 0 according to the sign of u.

Proof of the Theorem 1.2
First, we begin by proving the geometric properties.

Proof
We have:     So, the Lemma 4.2 is proved. Similar to the proof of the Lemma 2.3 in (Zhou, 2002), we can prove the following result.  d, this is for any c > 0.
So, the Theorem 1.2 is proved. 2 The Theorem 1.2 holds also when the function f(x, t) is superlinear and subcritical at - and the proof is the same.

Conclusion
A weighted elliptic problem with indefinite asymptotically linear nonlinearity is investigated in the present paper. Under suitable conditions, we prove the existence or the nonexistence of nontrivial solutions. Then, we consider the same problem when the nonlinearity is super-linear and in the same time subcritical and we prove an existence result. We use variational method in the proof of the existence of such solutions without using the Ambrosetti-Rabionowitz condition (AR) or any other condition of the same type.