Nonresonance Conditions on the Potential for a Nonlinear Nonautonomous Neumann Problem

The aim of this paper is to establish the existence of the principal eigencurve of the p−Laplacian operator with the nonconstant weight subject to Neumann boundary conditions. We then study the nonresonce phenomena under the first eigenvalue and under the principal eigencurve, thus we obtain existence results for some nonautonomous Neumann elliptic problems involving the p−Laplacian operator.


Introduction
In this paper we are concerned with the following class of problems where Ω is a bounded domain in IR N (N ≥ 2) with smooth boundary ∂Ω, −∆ p u = −div (|∇u| p−2 ∇u) denotes the p-Laplacian operator with 1 < p < ∞, h is taken Since the 80's, several works have been devoted to questions of nonresonance for this kind of problem, in the semilinear and autonomous case (p = 2, m 1 = 0 and m 2 = 1) has been discussed by many authors (see e.g., [8], [9], [12], [13], [14],• • • ) in connection with various qualitative assumptions on the function g and its potential G.In the nonautonomous case A. Anane and A. Dakkak considered problem (P α ) in the following particular case α = λ 1 (m 1 ) and m 2 = 1, where λ 1 is the first eigenvalue of the p-Laplacian operator with weight and the Neumann boundary condition (see [6]), they showed the existence of the weak solution of the problem (P α ) with conditions of nonresonance below the first eigenvalue of the −∆ p .
Motivated by the papers ( [6], [7]) and some ideas introduced in ( [6]), the goal of this work is to study the existence of solutions in the sense weak for problem (P α ).In other words, assuming that (A1), (A2), (H1) and (H2) hold, our purpose is to show that problem (P α ) has at least one solution that verifies for any v ∈ W 1,p (Ω) and for all h ∈ L ∞ (Ω).The proof of the main result is based on the Leray-Schauder degree method and exploits some techniques introduced in [6].
The remaining part of this paper is organized as follows: In section 2, we recall some results which are necessary in what follows.In section 3, we show (see theorem 3.2) the existence of principal eigencurve of the p-Laplacian operator with Neumann boundary conditions.In section 4, we show a theorem of nonresonance under the first eigenvalue (see theorem 4.4).Finally the results of sections 3 and 4 are then employed in section 5 in order to obtain the main result the existence of solution for problem (P α ).

Preliminary
Throughout this paper, Ω will be a smooth bonded domain of IR N , W 1,p (Ω) will denote the usual Sobolev space equipped with the norm .1,p = ( .p p + ∇(.) p p ) 1 p , where .p is the L p (Ω)-norm.In this preliminary section we collect some results relative to usual nonlinear eigenvalue problem where 2) λ is called an eigenvalue of problem (2.1) if there exists u ∈ W 1,p (Ω) \ {0} such that (u, λ) is a solution of problem (2.1).In this case u is called an eigenfunction associated to λ.
The Lusternik-Schnirelman theory asserts that the spectrum of the p-Laplacian operator contains at least an unbounded sequence of positive eigenvalues, say Unfortunately, to our best knowledge, nothing is known in general about the possible existence of other eigenvalues in ]λ 1 (m), +∞[.Clearly 0 is a principal eigenvalue of problem (2.1), with the constants as eigenfunctions.The search for another principal eigenvalue involves the following quantity: Proposition 2.2.( [6], [11]).1) If m changes sign on Ω and Ω mdx < 0. Then λ 1 (m) > 0 and λ 1 (m) is the unique nonzero principal eigenvalue; this eigenvalue is simple and the corresponding eigenfunction u can be chosen such that u(x) > 0 in Ω, moreover λ 1 (m) is isolated, namely, there exist b > λ 1 (m) such that σ(−∆ p ) ]0, b[= {λ 1 }, where σ(−∆ p ) represents the set of all eigenvalues associated to the problem (2.1).
3. Existence of the principal eigencurve of the −∆ p with weights in the Neumann case In this section, we study the following problem: Find all the real numbers α and β such that λ 1 (αm 1 + βm 2 ) = 1.Precisely, we seek to find all the pairs (α, β) ∈ IR 2 such that the following problem has at least one nontrivial solution u ∈ W 1,p (Ω) the set of pairs (α, β) has the structure of a continuous curve called the principal eigencurve of the −∆ p with weights in the Neumann case.
We define the graph of the first eigencurve of the p-Laplacian with weight subject to Neumann boundary conditions by: Theorem 3.2.Let m 1 and m 2 be two weight functions.Assume that m 1 , m 2 ∈ M + (Ω) and satisfying assumptions(A1) and (A2) respectively.Then for all α ∈ IR there exist a unique real number t α which satisfies.
Proof: Let α ∈ IR.We consider the function f α : t −→ λ 1 (αm 1 + tm 2 ).In view of propositon 2.3 and propsition 2.4, we affirm that f α is decreasing continuous.It follows that f α is injective.In order to show that the equation f α (t) = 1 has a solution we distinguish three cases.

Nonresonance under the first eigenvalue
This section is devoted to the study of the problem (P α ) in the particular case where (α = λ 1 (m 1 )) and the function g and its primitive G satisfying the following conditions That is to say, we show that the following problem admits at least one weak solution The main result of this section lies in the ( [6]).The improvement of this work is due to the insertion of a second weight m 2 in the right side of the problem (P λ 1 ).By using the theory of the Leray-Schauder degree, the hypotheses (g ± ) and (G ± ) are not sufficient to obtain the result of existence (see theorem4.4),for this we are compelled to treat the following possible cases.

Possible cases
. In order to study the problem (P λ1 ) we will suggest four cases.
. There are all possible cases and by a classical method of lower and upper solutions (cf.[1], [12]) one can show that if the case 1 holds, then (P λ 1 ) is solvable for every h ∈ L ∞ (Ω), so it remains to consider only the other cases.
. Therefore we will keep the hypotheses (g + 0 ) and (g − 0 ) in order to be used in the next, and (g + −1 ), (g − −1 ) will be used in the proof of the following proposition and used also in the truncated function f i which will be defined just after.Proposition 4.1.i) If g satisfy the hypothesis (g − −1 ) then, for any given h ∈ L ∞ (Ω) there exists A = A h < 0 such that: ii) If g satisfy the hypothesis (g + −1 ) then, for any given h ∈ L ∞ (Ω) there exists B = B h > 0 such that: Proof: We only show the first assertion since the proof of the second one proceeds in the same way.According to the hypothesis (g − −1 ), let us fix ε > 0, such that lim inf for this A h , there exists A < A h such that Indeed, assume by contradiction, that for all s < A h , we have which gives a contradiction.According to (4.4), we have It is easy to see that Therefore, using (4.3) we conclude that This concludes the proof of (4.1).✷

Homotopic Problems.
Let θ < 0 be fixed, and let µ ∈ [0, 1] and consider for i ∈ {2, 3, 4} the following problem where f i (., .) is defined for every s ∈ IR and a.e x ∈ Ω by where T + A (s) = max(s, A), T − B (s) = min(s, B), A and B comes from the Proposition 4.1.Proposition 4.2.If u is a solution of (P i,µ ) for i = 2 or i = 3, then we have 1) If i = 2, u(x) ≥ A a.e.x ∈ Ω and u is also a solution of (P 4,µ ) 2) If i = 3, u(x) ≤ B a.e.x ∈ Ω and u is also a solution of (P 4,µ ), where A and B comes from the proposition 4.1.
Proof: 1) Since u is a solution of (P 2,µ ), then where since A < 0 and θ < 0, it is easy to see that x ∈ Ω, according to the fact that u is a solution of (P 2,µ ) we get which gives a contradiction, so (u − A) − ≡ 0. This completes the proof.For 2), using similar arguments as in the proof of 1).✷ Corollary 4.3.1) If g satisfy (g − −1 ) and u is a solution of (P 2,µ ), then u is also a solution of (P λ 1 ).
2) If g satisfy (g + −1 ) and u is a solution of (P 3,µ ), then u is also a solution of (P λ 1 ).We are now in position to give the following result.
The proof needs some technical lemmas, the two next lemmas concern an apriori estimates on the possible solutions of the homotopic problem (P ).Lemma 4.5.We assume that g satisfy (g ± ) and the hypotheses of the case i, where i ∈ {2, 3, 4}.Let be (u n , µ n ) be a sequence of solutions of (P i,µ n ), then we have and ϕ is a normed positive eigenfunction associated to the first eigenvalue λ 1 of −∆ p with weight m 1 .Moreover, we have
According to the proposition 4.2, we have u n ≥ A a.e. in Ω, so by using (4.4) and (4.10) it is easy to see that On the other hand, since (u n ) n ⊂ L ∞ (Ω) and q is a continuous function, it follows that q(u n ) is bonded in L ∞ (Ω), then for a subsequence we get Nonresonance Conditions on the Potential

A. Sanhaji and A. Dakkak
The other cases follows directly by the same proceedings.
According to (4.10) and (4.11), we get dx by passage to the limit in the above inequality, we find (4.8).✷ Lemma 4.6.Let u n be a solution of (P i,µ n ) for some i ∈ {2, 3, 4} and for all n, such that u n ∞ → ∞ when n → +∞, and let us fix a ∈ Ω and η > 0 such that B(a, η) ⊂ Ω.So, if g satisfy (g ± ) and the case i holds, then by putting σ x (t) = a + t(x − a), we have where B(a, η) is the ball of a center and radius η.
Proof: For simplicity of the task we prove the lemma only in the case i = 2 and other cases can be treated in a similar way.Using relation (4.8), we deduce that By using the spherical coordinates, we obtain (sin θ j ) N −1−j dθ j dθ N −1 dt = 0 (4.18) where, ω = x−a η ∈ ∂B(0, 1).The above equality imply By using (g ± ) we can see that g satisfy the following growth condition: are bounded in L ∞ ([0, 1]).By using the Lebesgue dominated convergence theorem, we conclude this proof.✷ Lemma 4.7.Let r ∈]0, 1[, and assume that there exists d < 0 such that then there is an equivalence between (G + ) and (G + r ), where The same conclusion holds if we replace the sign + by the sign −.
Proof: Assume that (G + ) hold, then there exists a sequence (s n ) n with lim n→∞ According to (4.19), we get d < l < 0 and ( G(rsn) |rsn| p ) n is bounded, so there exists k ∈ [l, 0] such that lim n→∞ G(rsn) |rsn| p = k for some subsequence, thus we get Reciprocally, let us assume that (G + r ), then there exists a sequence ( s n ) n with lim n→∞ it is easy to see that ( G( sn)  Let us fix h ∈ L ∞ (Ω), so we will distinguish three cases (i.e. 2, 3 and 4), thus for any fixed i ∈ {2, 3, 4} we will assume that the case i holds and we will show the existence of solutions of (P i,µ ), so according to the corollary 4.
where, S and T satisfy T < 0 < S.
In order to simplify this proof we will assume that the case 2 holds, since the proof with the other cases is hardly the same.We take T = 2A, where A < 0 with A coming from proposition 4.1.By using the hypothesis (G + ) and lemma 4. where, r = minϕ maxϕ and ϕ coming from lemma 4.5.The proof is carried out by contradiction.precisely, one assumes the existence of a sequence (u n ) n of solutions to (P i,µ n ) , with µ n ∈ [0, 1] and u n ∈ ∂O Sn,T .then by proposition 4.2, we get u n ≥ A > 2A.It follows that max(u n ) = s n .
Let x n , y n ∈ Ω such that max Ω (u n ) = u n (x n ) and min Ω (u n ) = u n (y n ), we clearly can suppose that x n → x 0 in Ω and y n → y 0 in Ω, where x 0 and y 0 are two points where ϕ attains its maximum and minimum respectively.By character C 1 of Ω, we obtain the existence of a sequence (z k ) k=1,...,m ⊂ Ω such that We write where, m k=1 σ k is a smashed line.Join x n to x 0 by a C 1 path δ 0,n having range in Ω, and join y 0 to y n by a C 1 path γ n,0 having range in Ω.Then, C n =δ 0,n ( max(un) , by using lemma 4.5, it is clear to see that The proof is achieved if we obtain a contradiction from the formula (4.21), thus we will proceed in two claims: Claim 1.We will show that lim n→∞ , with Cn denotes a line integral.So by using lemma 4.6, we obtain Furthermore, we have for all ε > 0, there exists where, σ Similarly, for all ε > 0, there exists By the C 1 character of ∂Ω, δ 0,n and γ n,0 can be taken such that where, ℓ(.) denotes the length of the corresponding path.By combining (4.24), (4.25) and (4.26), we deduce that there exists c > 0 such that for all ε > 0 and for all n sufficiently large, we get ) n is bounded.On the other hand, we have According to the first claim and the fact lim n→∞ r n = r, we conclude the proof of the second claim.Finally, we get a contradiction from (4.21).This concludes the proof of theorem 4.4.

Nonresonance under the principal eigencurve
In this section, we turn to the problem (P α ) and we show that it has at least weak solution.For this purpose, we will apply the main results the two of sections previous.Consequently, as (P α,β ) and (P α ) are equivalent, which gives that the problem (P α ) has a solution u ∈ W 1,p (Ω), for every h ∈ L ∞ (Ω).✷

✷ 4 . 4 .
Proof of main result 7, we can get the existence of a sequence of positive real numbers (s n ) n which satisfy lim n→∞

m
k=1 σ k ) γ n,0 is a C 1 with morsels line which connects the Nonresonance Conditions on the Potential 93 extremity x n and y n .By lemma 4.6, we can rectify the sequence (z k ) k=1,...,m such that lim

94A..
Sanhaji and A. Dakkak    So, this concludes the proof of the first claim by replacing in (4.23).Claim 2. We will show that lim n→∞ |G(sn)−G(rsn)| |sn| p = 0.It is easy to see that for all n ∈ IN * , there exists c n which lives between r n s n and rs n such that |G(r n s n ) − G(rs n )| s p n = (r n − r)g(c n ) s p−1 n Since g satisfy the following growth condition |g(s)| ≤ a|s| p−1 + b, for some positive reals a, b and for all s ∈ [A, +∞[ then, the sequence ( g(cn) s p−1 n