Superlinear Ambrosetti Prodi problem for the p-laplacian operator

Transkript

1 Nonlinear Differ. Equ. Appl. 17 (010), c 010 Birkhäuser Verlag Basel/Switzerland /10/ published online January 8, 010 DOI /s Nonlinear Differential Equations and Applications NoDEA Superlinear Ambrosetti Prodi problem for the p-laplacian operator T. Junges Miotto Abstract. Based on a new Liouville theorem, we study a superlinear Ambrosetti Prodi problem for the p-laplacian operator, 1 < p < N. For this, we use the sub and supersolution method, blow up technique and the Leray Schauder degree theory. Mathematics Subject Classification (000). 35J5; 35B45; 47H11. Keywords. Leray Schauder degree, Sub-supersolutions, Blow up technique, Multiplicity of solutions. 1. Introduction The main purpose of this work is to investigate the existence of multiple solutions of the problem { Δp u = f(x, u)+tφ + h, in Ω (P u =0, on Ω, t ) where Ω R N is a bounded smooth domain, Δ p = div( u p u) isthe p-laplacian operator and 1 <p<n. We assume that φ, h L (Ω), with φ 0inΩandt R is a parameter, where will be defined later. Denoting λ 1 the first eigenvalue of ( Δ p,w 1,p 0 (Ω)) we consider f : Ω R R a continuous function satisfying the following conditions: f(x, u) (H1) lim sup u u p u = μ<λ 1, uniformly x Ω, (H) There exists σ > 0 such that f(x, s) +σ s p s is increasing in s, f(x, 0) = 0. f(x, u) (H3) lim u + u α = a(x), a C(Ω), a 1, p 1 <α<p N N p, where p = Np N p, ((H3) is known as p-superlinear condition at infinity and simply superlinear when p = ).

2 338 T. Junges Miotto NoDEA This problem belongs to a class of problems known as the Ambrosetti Prodi type. For the case p = with different variants and formulations, it has been extensively studied by several authors. We shall quote here the original work [1], as well as the subsequent developments [3 5, 14] and the references therein. For the superlinear case see for example [6, 9 11]. More recently, Ambrosetti Prodi type results for p-laplacian operator, with p>1 have been studied, but few results are really known. We can cite Perera [19] which used the linking and homotopy theory to study a similar problem with zero Dirichlet condition on the boundary and asymmetric nonlinearities. Mawhin in [18] studied the existence of periodic solutions for the ODE case with φ = 1. Arcoya and Ruiz in [] treated problem (P t ) with f growing as u p u, that is, the p-linear case, which was also studied by Koizumi and Schmitt in [15]. Our main contribution is to study this problem where f has a p-superlinear behavior. Motivated by results in [] we prove the following theorem: Theorem 1.1. Suppose (H1) (H3) hold. There exist <t t < + such that problem (P t ) has: i) at least, two solutions provided that t<t, ii) at least, one solution provided that t t, iii) no solution provided that t>t. We conjecture that t = t, but this question is still open. In order to prove Theorem 1.1, we use the sub and supersolution method and topological arguments. Then we need a priori bounds on the eventual solutions of (P t )to apply the Leray Schauder degree. In order to use the sub and supersolution method we only need the hypothesis (H1), and therefore the first solution is obtained arguing as in []. We can also cite the work of de Figueiredo, Gossez and Ubilla in [7] that use a variational approach to the method of upper lower solutions. However to get the second solution we need the a priori estimates of eventual solutions of (P t ). We remark that Arcoya and Ruiz s proof of this fact does not apply for our case. To overcome this fact, we follow [8], obtaining not only a priori bound on negative part of u, but also on t, such that (P t ) has a solution and in this way we obtain an a priori bound for u. The main difficulty is that the operator studied in [8] is linear, then the a priori bound on t follows easily from the linearity of the operator and from ABP estimate. For our case, since our operator is not linear, we have to adapt a Dong s result [1] thatusea comparison principle. For the boundedness of u we use the blow up technique, which only was possible to use thanks to a recent paper of Lorca [17], where he proves a Liouville type result for a half space. To obtain the second solution we will use the Leray Shauder degree theory, following the results of [] and [5]. This paper is organized as follows. Section is devoted to finding the first solution. In Sect. 3, we obtain a priori estimates for eventual solutions of (P t ) using the blow up method and in Sect. 4 we apply the Leray Shauder degree theory to obtain the second solution.

3 Vol. 17 (010) Superlinear Ambrosetti Prodi problem 339 All along this work we will use the following notations: i) represents the L norm in Ω, ii) C 0 (Ω) = {u C(Ω); u(x) =0, x Ω}. Similarly, we define C0(Ω) 1 and C 1,α 0 (Ω). iii) for any h 1,h L (Ω), we say that h 1 h if, for any compact subset K Ω, there exists ε>0 such that h 1 (x) +ε<h (x) for almost every x K. iv) for any u, v C0(Ω); 1 we will say that u v in Ω if u(x) <v(x) forx Ω and u v (x) < (x) for all x Ω, where η(x) denotes the inward normal to Ω atx.. First solution Our main tool to find the first solution will be the sub and supersolution method. We will find now the sub and supersolutions for (P t ): Proposition.1. For every w C0(Ω), 1 t R, there exists u C 1,α 0 (Ω) such that u w and Δ p u f(x, u)+tφ + h, in Ω. Proof. Let w C0(Ω), 1 t R be fixed, but arbitrary. By (H1) we have that for ε (0,λ 1 μ), there exists constant C>0such that f(x, u) (μ + ε) u p u C, for all u<0. Let u W 1,p 0 (Ω) be solution of Δ p u =(μ + ε) u p u C + tφ + h, in Ω, u =0on Ω. where without loss of generality we take C large enough to get C + tφ + h<0. Claim.. We can take C>0 large enough such that u w. In fact, let {C n } be a sequence such that C n (without loss of generality assume that C n > 1) and {u n } a sequence of solutions of Δ p u n =(μ + ε) u n p u n C n + tφ + h, in Ω, u n =0, on Ω. (.1) u n Thus, if v n =, then we have that v (C n ) 1 n is solution of Δ p v n =(μ + ε) v n p v n 1+ tφ + h, in Ω, v n =0, on Ω. C n By Lemma. in [] we obtain v n C 1,α (Ω), 0 <α<1, and v n C 1,α (Ω) M. Thus by the compact imbedding C 1,α (Ω) C 1,β (Ω), 0 β<αwehave that, up to a subsequence, v n v in C 1,β (Ω). By Lemma.3 in [] wegetv n v which is solution of { Δp v =(μ + ε) v p v 1, in Ω (.) v =0, on Ω. By Maximum Principle it follows that v 0 and by Vazquez Maximum Principle we get v 0.

4 340 T. Junges Miotto NoDEA Note that v n = un v 0, thus u (C 1 n < 0, un un n) < 0and,u n as n. Since u n,w C0(Ω) 1 we have that for n large enough, u n w. Let n 0 such that u n0 w and C n0 >C. Then, defining u = u n0, since u n0 is solution of (.1) wehave, Δ p u =(μ + ε) u p u C n0 + tφ + h < (μ + ε) u p u C + tφ + h f(x, u)+tφ + h, that is, u is subsolution of (P t ) that satisfies u w. Proposition.3. Given a function w C0(Ω), 1 there exists t such that for every t t, there exists a strict supersolution u C0(Ω) 1 for (P t ) with u w. Proof. Fix an open set Ω 0 such that Ω 0 Ω. Define M> max f(x, s) + h. x Ω,s [ 1,0] For each n>0, let u n W 1,p 0 (Ω) be the solution of Δ p u n = g n (x), in Ω, u n =0, on Ω, where { n g n (x) =,x Ω 0 M, x Ω\Ω 0. g Note that n n χ Ω0 in L (Ω), as n goes to, where χ Ω0 is the characteristic function of Ω 0. Define now v n = un n. By Lemma.3 in [], we deduce that the sequence v n v in C 1,α (Ω), 0 <α<1, v solution of { Δp v = χ Ω0, in Ω, v =0, on Ω. By Maximum Principle and Vazquez Maximum Principle we have that v 0. Arguing as Claim. we obtain that for n 0 large enough, u n0 w and u n0 < 0 in Ω. We can assume that u n0 < 1 inω 0. Define ū = u n0 and t < (n 0 + max x Ω,s [min ū, 1] f(x, s) + h ). infū 1 ((, 1]) φ Take into account that, since φ 0, infū 1 ((, 1]) φ>0 (here the infimum is considered in an essential way). For x Ω there are two possibilities: either ū(x) [ 1, 0), or ū(x) < 1. In the first case we have that x Ω\Ω 0, since ū< 1 inω 0, and then f(x, ū)+ tφ(x)+h(x) <M= g n0 = Δ p ū. On the other hand, if ū(x) < 1, then, by definition of t, f(x, ū) + tφ(x) +h(x) f(y, s) + t inf φ + h ū 1 ((, 1]) max y Ω,s [min ū, 1] < n 0 = g n0 = Δ p ū.

5 Vol. 17 (010) Superlinear Ambrosetti Prodi problem 341 Therefore ū is supersolution of (P t) and for all t t, ū is supersolution of (P t ) with ū w, which concludes the proof. The next result is the well known sub and supersolution method, which give us the first solution, through the Schauder fixed point theorem. Theorem.4. Let u, u C 1 (Ω) be the sub and supersolution of (P t ), respectively, such that u u in Ω, with u 0 u on Ω. Then there exists u C 1 (Ω) solution of (P t ) such that u u u in Ω and u =0on Ω. Proof. Define the operators N t : C0(Ω) 1 L (Ω) and T : L (Ω) C0(Ω) 1 by N(v) =f(x, v)+σ v p v + tφ + h, v C0(Ω) 1 and T (v) =w iff Δ p w + σ w p w = v, v L (Ω). Define also Q t : C0(Ω) 1 C0(Ω) 1 by Q t = T N t. Note that u is a fixed point of Q t if and only if u is a solution of (P t ). We will show that Q t is compact: since Q t = T N t and N t is continuous, it is enough to show that T is compact. In fact, let {u n } be a bounded sequence in L (Ω) and w n = T (u n ), thus Δ p w n + σ w n p w n = u n. By Lemma. in [] w n C 1,α (Ω) with w n C 1,α < C. By the compact embedding C 1,α (Ω) C 1,β (Ω) it follows that, up to a subsequence, w n w in C 1,β (Ω), therefore T is compact. Now, define O = {u C0(Ω); 1 u(x) u(x) u(x)}. Note that O is closed and convex. Still, by the comparison principle (see Tolksdorf [1]) we have that Q t (O) O, and by Lemma. in [] Q t (O) is bounded. Thus, by Schauder fixed point theorem there exists u C0(Ω) 1 which is a fixed point of Q t, that is, there exists u C0(Ω) 1 solution of (P t ) such that u u u in Ω. 3. A priori bounds In order to obtain the second solution using the degree theory, we need to get a priori estimates. In this section we start getting a estimate for the negative part of u, and later we show that if u is solution of (P t )fort bigger than a certain negative constant, then t<c(1 + u ). Finally, we prove an a priori bound on u using the blow up technique, as introduced by Gidas Spruck [13]. We remark here that to use the blow up technique we need the Liouville type results for R N and half space, which had been gotten by [0] and [17], respectively. For that matter, in our hypothesis (H3) we need to impose N N p that p 1 <α<p to be able to use those results. For the reader s convenience we enunciate these results:

6 34 T. Junges Miotto NoDEA Theorem 3.1 (Theorem III [0]). Assume g is subcritical and g(u) > 0 for all u>0. Then every bounded solution of Δ p u + g(u) =0, u 0 x R N, p < N, is trivial. Theorem 3. (Theorem 3.1 [17]). Consider the problem u m Δ p u Cu m, x R N +, (3.1) where C 1. Assume that p 1 <m<p N N p. Then, there is no positive solution of (3.1) in C 1 (R N + ). To prove the boundedness of the negative part, we will need the following result due to Ladyzhenskaya and Ural tseva [16]: Theorem 3.3 (Theorem 5.1 [16]). Suppose u W 1,m (Ω) L q (Ω) has bounded essential max Ω u(x) and that for k ˆk essential max Ω u(x) satisfies the inequality ( ) m u m ˆγ u k l l M +ˆγ k αi (mes(a k )) 1 m N +εi, A k A k i=1 where A k = {x Ω; u(x) k}, ˆγ,l,α i and M are positive constants. Suppose also that l Nm N m,ε i > 0, m α i <ε i q + m. Then, essential max Ω u(x) does not exceed some constant that depends on ˆγ,l, α i, ε i (for i =1,..., M), m, N, ˆk, q, mes(ω), and either the norm u L1 (Aˆk) (for q< Nm N m ) or the norm u L q (for q (Aˆk) Nm N m ). Next, arguing as [], we will prove the boundedness of negative part of a eventual solution of (P t ): Theorem 3.4. For any C 0 R +, there exists constant M>0such that for all t C 0,ifu is solution of (P t ) then u M. Proof. Let u be a solution of (P t ). Considering u as a test function we get u p u u = (f(x, u)+tφ + h)u. Ω Ω Now, if Ω = {x Ω; u(x) 0}, since u = u in Ω and u = u in Ω,wehavethat u p = f(x, u)u + tφu + hu. Ω Ω Since f satisfies (H1) we have that for all ε (0,λ 1 μ), there exists C>0 large enough such that f(x, u) (μ + ε) u p u C, for all u<0, that is f(x, u)u (μ + ε) u p Cu, for all u<0. Then, u p (μ + ε) u p Cu + tφu + hu. Ω Ω

7 Vol. 17 (010) Superlinear Ambrosetti Prodi problem 343 Thus, by the variational characterization of λ 1 and by Holder inequality it follows that ( ) ( ) ( ) (μ + ε) 1 u p p C C 0 φ + h p u p 1 p, λ 1 Ω Ω Ω which implies u p M, (3.) where M is independent of t C 0. Now, consider ˆk = 1 and for k ˆk define the set A k = {x Ω; u (x) >k}, then u p = u p u (u k) = u p u (u k) +. A k A k Ω Since u is a solution of (P t ) and again by (H1) we have Ω u p u (u k) + (f(x, u) C 0 φ + h)(u k) A k (μ + ε) u p u (u k)+ (C + C 0 φ + h )(u k) A k A k = I 1 + I. For I we have that I (C + C 0 φ + h ) u k p p. A k In order to estimate these integrals, we will use the following Young s inequality ( ab δa 1 s ) s 1 s s + (1 s)b 1 1 s, a, b 0, s (0, 1), δ > 0. δ Setting γ = C + C 0 φ + h, by Young s inequality with s = 1 p (0, 1) > 0 where γ (p( N p N p ),p)fori, and for I 1 taking s = p > 0 we obtain, and δ = λ1kγ 4 γ (0, 1) and δ = λ1 4(μ+ε) u p I 1 + I A k λ ( ) 1 4(μ + ε)(p 1) u p μ + ε + u k p 4 A k pλ 1 p A k + λ 1k γ ( ) 1 u p k p 4 γ p 1 + γ 4 A k pλ 1 k γ p k p mes(ak ).

8 344 T. Junges Miotto NoDEA By the variational characterization of λ 1, it follows ) ( ) 14 kγ p 4(μ + ε)(p 1) (1 u p μ + ε u k p 4 A k pλ 1 p A }{{}}{{} k ν ν 1 ( 4 γ + γ pλ 1 ) 1 p 1 p } {{ } ν 3 mes(a k )k p γ. It follows from definition of γ that k γ p < 1 for all k>ˆk and then ν 1 > 1. We have also that ν,ν 3 > 0. Take γ = max{ν,ν 3 }, then ( ) u p γ u k p + γk p γ mes(ak ). A k A k Since 1 <p<n, we have by Sobolev imbedding and by (3.) that u L p (Ω) C u W 1,p 0 (Ω) C, where C > 0 is independent of t C 0. By Theorem 3.3 with q = p, m = l = p, M = 1, ε1 = p N, α 1 = p γ ( p, p p N + p),ˆγ = γ and ˆk = 1, we obtain that u M, where M depends only of p, γ, N, 1,mes(Ω), C. Theorem 3.5. For any C 0 R +, there exists constant C 1 such that for all t C 0,ifu is solution of (P t ) then t C 1 (1 + u ). Proof. Let C 0 R + and t C 0. If C 0 t 0, then let C 1 0 be an arbitrary constant, thus t C 1 C 1 (1 + u ) and the theorem is proved. Suppose that t>0. Let u be solution of (P t ). By Theorem 3.4 we have that u M, hence by (H) we have f(x, u)+σ u p u f(x, M)+σ M p ( M). Define k 1 =inff(x, M)+σ M p ( M) Ω (note that k 1 does not depend on t). Let Ω 0 Ω be a compact set and δ>0 such that φ>δin Ω 0, then Δ p u + σ u p u = f(x, u)+σ u p u + tφ + h >k 1 + tδ h, in Ω 0.If t k 1 + h δ, (3.3) then defining C 1 = it follows that k1+ h δ

9 Vol. 17 (010) Superlinear Ambrosetti Prodi problem 345 t C 1 C 1 (1 + u ) and the theorem is proved. Assume t> k 1 + h δ. Thus we have that tδ > k 1 + h k 1 + h and hence k 1 h +tδ > 0. ) 1 Define w 1 = v, where v is the solution of ( k1+tδ h Δ p v + σ v p v =1inΩ 0, v =0on Ω 0. Then we have ( ) Δ p w 1 + σ w 1 p k1 + tδ h w 1 = ( Δ p v + σ v p v) ( ) k1 + tδ h = < Δ p u + σ u p u, in Ω 0. Since u C 1 (Ω), it follows that u inf Ω0 u(x) M. Now we will analyze the two cases: inf Ω0 u(x) 0 and inf Ω0 u(x) < 0. First case: If inf Ω0 u(x) 0, then w 1 =0 inf Ω0 u(x) u on Ω 0 and by the Comparison Principle we obtain u w 1 in Ω 0.Letx 0 Ω 0 such that v(x 0 )= v, thus ( ) 1 k1 + tδ h u u(x 0 ) v(x0 ) that is t u δ v + h k 1 u δ δ v + h k 1 δ < u δ v + t, or t< 4 u δ v. (3.4) Second case: If inf Ω0 u(x)<0, then z =v M, where M = satisfies, in Ω 0, Δ p z + σ z p z = Δ p v + σ v M p (v M) < Δ p v + σ v p v =1, and z = v M = M, on Ω 0. M ( ) 1 k1 +tδ h

10 346 T. Junges Miotto NoDEA ( ) 1 k1+tδ h Thus defining w = z, it follows that ( Δ p w + σ w p k1 + tδ h w = < Δ p u + σ u p u in Ω 0. Still ( ) 1 ( k1 + tδ h k1 + tδ h w = z = = M u, ) ( Δ p z + σ z p z) on Ω 0. By the Comparison Principle we have u w on Ω 0. Let x 0 Ω 0 as above, thus that is, or t Define { C 1 = max u u(x 0 ) w (x 0 ) ( k1 + tδ h = ( k1 + tδ h = ( u + M) δ v 4 δ v, + h k 1 δ ) 1 (v(x0 ) M) ) 1 v M, ) 1 < ( u + M ) δ v M + t, t< p ( u + M ) δ v. (3.5) p δ v, p M δ v, h k 1 δ }. Then by (3.3), (3.4) and (3.5) wehavet C 1 (1 + u ), which concludes the proof. We also need the following theorem: Theorem 3.6. There exists a constant C such that for all t 1 and for all solution u of (P t ) with this t we have u C t 1 α. Proof. Consider g :Ω R R defined by g(x, s) =f(x, s) a(x)(s + ) α. Note that g is continuous and by (H3) we have g(x, s) lim s s α =0. (3.6)

11 Vol. 17 (010) Superlinear Ambrosetti Prodi problem 347 Suppose that the conclusion of theorem is false, then there exists sequences {t n } such that t n 1and{u n } solution of (P tn ) such that u n α >nt n. (3.7) Since u n is solution of (P tn ), it follows from definition of g that Δ p u n = a(x)(u + n ) α + g(x, u n )+t n φ + h. (3.8) By Theorem 3.4 we have that u n M, thus u + n = u n for n n 0 large enough. Define λ n = u n 1 q, where q = p α () > 0. Then it follows that λ n = u n 1 q (ntn ) 1 qα n 1 qα 0 as n. Without loss of generality we can assume that λ q n < 1forn n 0. Consider also, for n n 0 a function v n defined by v n (x) =λ q nu n (λ n x + x n )+M, where u n (x n )= u + n = u n.thus,forn n 0, v n (0) = 1 + M and 0 v n (x) 1+M. On the other hand (3.8) and a direct computation show that v n satisfies Δ p v n (x) =λ q()+p n a(y n )(u + n (y n )) α + λ q()+p n t n φ(y n ) +λ q()+p n [g(y n,u n (y n )) + h(y n )] = λ q()+p qα n a(y n )[v n (x) M + λ q n u n (y n )] α + λ q()+p n t n φ(y n ) +λ q()+p n [g(y n,u n (y n )) + h(y n )], where y n = λ n x + x n for x Ω n = Ω xn λ n and n n 0. Now, note that by definition of q we have q(p 1) + p qα = 0. Still, by (3.7), λ q()+p n t n = u n α t n 0 as n. Thus, since φ L (Ω), we have that λ q()+p n t n φ 0asn.Moreover, lim n λq()+p n [g(y n,u n (y n )) + h(y n )] = lim n g(λ n x + x n,u n (λ n x + x n )) u n α, since h L (Ω). We know that {u n (λ n x + x n )} is bounded from below by Theorem 3.4. If{u n (λ n x + x n )} is bounded from above, for all x Ω n and n N, then since g is continuous, we have that g(λ n x + x n,u n (λ n x + x n )) lim n u n α =0. If {u n (λ n x + x n )} is unbounded from above, then there exists { x n } Ω n such that u n (λ n x n + x n ) as n. Since α>0, by definition of {x n } we have that (u n (λ n x n + x n )) α (u n (x n )) α and then by (3.6), g(λ n x n +x n,u n (λ n x n +x n )) g(λ n x n +x n,u n (λ n x n +x n )) lim n u n α lim n (u n (λ n x n +x n )) α =0. Hence, since Ω is compact, we have that, for a subsequence, x n x 0 Ω. By regularity theorems in [1, ] forp-laplacian operators, one can obtain

12 348 T. Junges Miotto NoDEA estimates on {v n } ensuring that, for a subsequence, v n v locally uniformly, with v C 1 (G) andv satisfies Δ p v = a(x 0 )(v M) α in G, where G = R N,ifx 0 ΩorG = R N +,ifx 0 Ω. Since v n (x) M + λ q n u n (y n )=λ q n u + n (y n ) 0, for all x Ω n,n n 0 and v n ( ) M + λ q n u n (λ n +x n ) v M locally uniformly, we have that v M 0, for all x G. Thus defining w = v M, wehavethatw C 1 (G), w(0) = 1 and w(x) 0, for all x G.Moreover,w satisfies Δ p w = a(x 0 )w α in G. If G = R N, by Theorem 3.1 we have that w 0, which is a contradiction with the fact that w(0) = 1. And if G = R N +, in the same way, we obtain a contradiction with the Theorem 3.. Thus we conclude the proof. Remark 3.7. Since tφ = t + φ t φ, we can consider h = h t φ and suppose t>0, thus repeating the proof of Theorem 3.6 replacing t by max{t, 1}, we obtain: for each C 0 R +, there exists a constant C such that for all t C 0 and for all solution of (P t ) with this t we have u α C max{t, 1}. From Theorems 3.5 and 3.6 we get the following results: Theorem 3.8. Given t 0 R, there exist R, R >0 such that for all t t 0,ifu is a solution of (P t ) then u R and t R. Proof. Let t 0 R and define C 0 = t 0 0. It follows from Theorem 3.5 that for all t t 0 t 0 = C 0,ifu is a solution of (P t ) then t C 1 (1 + u ). By Remark 3.7 we have that t C 1 (1 + u ) C 1 (1 + (C max{1,t}) α ). If t 1, then t C 1 (1 + (C t) α ). Since α < 1 we infer that t is bounded, that is, there exists R 1 such that t R. By Remark 3.7 we obtain that u α C max{1,t} RC, defining R =( RC ) 1 α we obtain the result. Theorem 3.9. Given t 0 R, there exist R, R >0 such that for all t t 0,ifu is a solution of (P t ) then u C 1 (Ω) R and t R. Proof. Given t 0 R, define C 0 = t 0 0. By Theorem 3.8 we have that if u is a solution of (P t ) then u < R and t< R. Now, since u is a solution of (P t ) we have Δ p u = f(x, u)+tφ + h. Note that d(x, u) =f(x, u)+tφ + h is a Caratheodory function that satisfies d(x, u) f(x, u) + t φ + h. Since u [ M, R] ( M given by Theorem 3.4), f(x, u) R 1 for (x, u) Ω [ M, R] and then d(x, u) R 1 + R φ + h = R. Thus by Lemma. in [] u C 1,α (Ω), for 0 α<1and u C 1,α (Ω) R.

13 Vol. 17 (010) Superlinear Ambrosetti Prodi problem Second solution As we said previously, employing Leray Schauder degree Theory, we complete the proof Theorem 1.1 by finding another solution of (P t ). Now we are ready to prove it. Let us denote S = {t R;(P t ) has at least one supersolution}. From Proposition.3, we have that there exists t R such that (P t )hasa supersolution and for all t t, (P t ) has a supersolution, thus S is not empty and if t S then (,t] S. Define S 1 = {t R;(P t ) has at least one solution}. Obviously S 1 S. Moreover, if t S, then (P t ) has one supersolution ū. It follows from Proposition.1 that there exists subsolution u such that u ū. Thus by Theorem.4 there exists u solution of (P t ) with u u ū and therefore t S 1.ThusS 1 = S. By Theorem 3.8 wegetthat(p t ) does not have solution for all t R, thus S is bounded from above and we can define t =supt. S It follows from definition of supremum that for all t t,(p t )doesnothave solution and therefore we prove the item (iii) of Theorem 1.1. Now we will show (ii). From definition of supremum there exists {t n } Ssuch that t n t. Without loss of generality we can assume that t n is increasing. Since t n S, there exists u n solution of (P tn ), that is, for all ϕ W 1,p 0 (Ω), u n p u n ϕ = (f(x, u n )+t n φ + h)ϕ. (4.1) Ω Ω Considering t 0 = t 1 (the first element of the above sequence), we have by Theorem 3.4 and Theorem 3.8 that u n [ M,R], thus f(x, u n ) K. Still by Theorem 3.8 it follows that f(x, u n )+t n φ + h K + R φ + h C thus by Lemma. in [] it follows that {u n } is bounded in C 1,α (Ω) and therefore it has a convergent subsequence u nk u in C 1,β (Ω), 0 β<α.thus, take a limit as n k in (4.1) weget u p u ϕ = (f(x, u )+t φ + h)ϕ, Ω Ω that is, u is solution of (P t ). Hence, t Sand S =(,t ], which concludes the item ii). We will show now the item i). Letu be solution of (P t ) and define T = {t R;(P t ) has at least one supersolution ū u }.

14 350 T. Junges Miotto NoDEA It follows from Proposition.3 that T is a nonempty interval which is unbounded from below. Consider t 0 T. We will show that (P t0 ) has at least two solutions. Since t 0 T, there exists ū supersolution of (P t0 ) such that ū u. Proposition.1 and Theorem.4 provide us a solution u 0 of (P t0 ). Taking a smaller subsolution, if necessary, u 0 verifies u u 0 ū u. Define D = {u C0(Ω); 1 u <u<u u, < u < u, u C 1 (Ω) <R 0}, where R 0 is defined later. Define also f(x, u)+σ u p u, ξ u f(x, ξ) = f(x, ξ)+σ ξ p ξ, u ξ u f(x, u )+σ u p u, ξ u and Q t : C0(Ω) 1 C0(Ω) 1 by Q t (v) =w iff Δ p w + σ w p w = f(x, v)+tφ + h. From Theorem.4 we have that Q t is compact. Since f is bounded we have that the solutions of equation above are uniformly bounded in C 1,α (Ω) and then we can define R 0 > sup{ Q t0 (v) C 1 0 (Ω) ; v C1 0(Ω)}. We will show that Q t0 ( D) D. In fact, let v D and consider w = Q t0 (v). Since f is increasing and t 0 t we have that Δ p w + σ w p w = f(x, v)+t 0 φ + h f(x, u )+t 0 φ + h Δ p u + σ u p u. By Comparison Principle, w u. We will show that w contradiction that w and η is an inward normal vector, we have u w (x)< (x)= lim λ 0 u. Suppose by (x) > u (x) for some x Ω, since w(x) =u (x) =0 w(x + λη) w(x) lim λ which is a contradiction. Hence w = u (x), u. Similarly we obtain u w and u (x + λη) u (x) λ 0 λ u w. By definition of R 0 it follows that w D. Moreover if Q t is given by Theorem.4, we have that Q t = Q t in D, u is a solution of (P t ) if and only if u is a fixed point of Q t and u 0 D.If the boundary of D contains a solution of (P t0 ) then the proof is completed. Suppose that D does not contain any solution of (P t0 ). Let ψ Dand H λ = H :[0, 1] D C 1 0(Ω) defined by H(λ, u) =λq t0 (u)+(1 λ)ψ. Note that D is an open, bounded and convex set thus since Q t0 ( D) D, we have that H(λ, w) D, for all (λ, w) [0, 1) D. Therefore we conclude that 0 (I H(λ, ))( D), for all λ [0, 1], because if this occurred, we would have H(λ 0,v)=v, for some v D and for some λ 0 [0, 1]. Now, H(λ, v) D,for all λ [0, 1), and for λ = 1 we would have that H(1,v)=v, forv D, which

15 Vol. 17 (010) Superlinear Ambrosetti Prodi problem 351 is a contradiction since D does not contain any solution of (P t0 ). Therefore deg(i H λ, D, 0) is well defined. By the invariance homotopy we have deg(i ψ, D, 0) = deg(i H 0, D, 0) = deg(i H 1, D, 0) = deg(i Q t0, D, 0). Now, take y(λ) =λψ and H λ = H :[0, 1] D C0(Ω) 1 defined by H(λ, u) =(1 λ)ψ, again by the invariance homotopy we have deg(i ψ, D, 0) = deg(i H 0, D,y(0)) = deg(i H 1, D,y(1)) = deg(i,d,ψ)=1. Therefore it follows that deg(i Q t0, D, 0) = 1. We will prove that for r R large enough, deg(i Q t0,b r (0), 0) = 0. For t 0,let R, R > 0 be given by Theorem 3.9. Consider r>r, ˆt > R and define H t = H :[t 0, ˆt] B r (0) C0(Ω) 1 by H(t, u) =Q t (u). Note that H is compact and 0 (I H(t, ))( B r (0)), for all t [t 0, ˆt] (because on the contrary case there exists v B r (0) such that Q t (v) = H(t, v) =v, for some t [t 0, ˆt], and then v is a solution of (P t ) with v = r>r,which is a contradiction since for all t t 0, the solutions of (P t ) are bounded from above by R). By the invariance homotopy we have deg(i H t0,b r (0), 0) = deg(i Hˆt,B r(0), 0) = 0, since from Theorem 3.8, fort> R (P t ) has no solution. By the excision property of the degree, 0=deg(I Q t0,b r (0), 0) = deg(i Q t0, D, 0) + deg(i Q t0,b r (0) \D, 0), and hence deg(i Q t0,b r (0) \D, 0) 0, that is, there exists v B r (0) \D solution of (P t0 ) and thus (P t0 ) has two solutions. Thus we have that for t T, there exists at least two solutions for (P t ). Consider t =supt. T Thus (P t ) has at least two solutions for all t<t, concluding the proof of the theorem. Remark 4.1. We would like to prove that t = t, but we do not know how to do it. Indeed, we note that t t,thuswehavetoprovethatt t,thatis, t T.Now,sinceu is solution of (P t ), in particular, u is a supersolution of (P t )andu u. By Proposition.1 and Theorem.4 there is a solution u of (P t ) that satisfies u u u. But we do not get to assure that u u. Thus we can not assure that u D, only u D. Hence the reasoning used in the sequence and the Leray Schauder degree can not be applied.

16 35 T. Junges Miotto NoDEA Acknowledgments This paper was developed in part during my Ph.D. thesis at UNICAMP, under advise of Professor Djairo G. de Figueiredo. The author would like to thank Professor Djairo G. de Figueiredo and Professor Olimpio H. Miyagaki for all discussions and encouragement during the development of this work. The author was supported partially by Capes-Brazil. References [1] Ambrosetti, A., Prodi, G.: On the inversion of some differentiable mappings with singularities between Banach spaces. Ann. Mat. Pure Appl. 93, (197) [] Arcoya, D., Ruiz, D.: The Ambrosetti Prodi problem for the p-laplace operator. Comm. Partial Differ. Equ. 31, (006) [3] Berger, M.S., Podolak, E.: On the solutions of a nonlinear Dirichlet problem. Indiana Univ. Math. J. 4, (1974/1975) [4] Dancer, E.N.: On the ranges of certain weakly nonlinear elliptic partial differential equations. J. Math. Pures Appl. 57, (1978) [5] de Figueiredo, D.G.: Lectures on boundary value problems of Ambrosetti Prodi type. 1th Brazilian Seminar of Analysis, Brasil, 1980 [6] de Figueiredo, D.G.: On the superlinear Ambrosetti Prodi problem. Nonlinear Anal. 8, (1984) [7] de Figueiredo, D.G., Gossez, J.-P., Ubilla, P.: Local superlinearity and sublinearity for the p-laplacian. J. Funct. Anal. 57(3), (009) [8] de Figueiredo, D.G., Sirakov, B.: On the Ambrosetti Prodi problem for nonvariational elliptic systems. J. Differ. Equ. 40, (007) [9] de Figueiredo, D.G., Solimini, S.: A variational approach to superlinear elliptic problems. Comm. Partial Differ. Equ. 9, (1984) [10] de Figueiredo, D.G., Srikanth, P.N., Santra, S.: Non-radially symmetric solutions for a superlinear Ambrosetti Prodi type problem. Comm. Contem. Math. 7(6), (005) [11] de Figueiredo, D.G., Yang, J.: Critical superlinear Ambrosetti Prodi problems. Topol. Methods Nonlinear Anal. 14(1), (1999) [1] Dong, W.: A priori estimates and existence of positive solutions for a quasilinear elliptic equation. J. Lond. Math. Soc. 7, (005) [13] Gidas, B., Spruck, J.: A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differ. Equ. 6, (1981) [14] Hess, P.: On a nonlinear elliptic boundary value problem of the Ambrosetti Prodi type. Boll. Un. Mat. Ital. A 17, (1980)

17 Vol. 17 (010) Superlinear Ambrosetti Prodi problem 353 [15] Koizumi, E., Schmitt, K.: Ambrosetti Prodi-type problems for quasilinear elliptic equations. Diff. Int. Equ. 18, 41 6 (005) [16] Ladyzhenskaya, O.A., Ural tseva, N.N.: Linear and Quasilinear Elliptic Equations. 1st edn. Academic Press, London (1968) [17] Lorca, S.: Nonexistence of positive solution for quasilinear elliptic problems in the half-space. J. Inequal. Appl. 007, 1 4 (007) [18] Mawhin, J.: The periodic Ambrosetti Prodi problem for nonlinear perturbations of the p-laplacian. J. Eur. Math. Soc. 8, (006) [19] Perera, K.: An existence result for a class of quasilinear elliptic boundary value problems with jumping nonlinearities. Topol. Methods Nonlinear Anal. 0, (00) [0] Serrin, J., Zou, H.: Cauchy Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math. 189, (00) [1] Tolksdorf, P.: On the Dirichlet problem for quasilinear equations in domains with conical boundary points. Comm. Partial Differ. Equ. 8, (1983) [] Tolksdorf, P.: Regularity for more general class of quasilinear elliptic equations. J. Differ. Equ. 51, (1984) T. Junges Miotto Departamento de Matemática, Universidade Federal de Santa Maria, Av. Roraima, 1000 CCNE, Santa Maria, RS , Brazil jungesmiotto@gmail.com Received: 5 February 009. Accepted: 13 January 010.