We consider a nonlinear Neumann elliptic equation driven by the $p$-Laplacian and a Carathéodory perturbation. The energy functional of the problem need not be coercive. Using variational methods we prove an existence theorem and a multiplicity theorem, producing two nontrivial smooth solutions. Our formulation incorporates strongly resonant equations.

Gasiński, Leszek

Papageorgiou, Nikolaos S.

Schedae Informaticae

English

Jagiellonian Univeristy Repository

Schedae Informaticae Vol. 21 (2012): 27–40doi: 10.4467/20838476SI.12.002.0812Existence and Multiplicity of Solutionsfor Noncoercive Neumann Problemswith p-Laplacian∗Leszek Gasiński1, Nikolaos S. Papageorgiou21Faculty of Mathematics and Computer Science, Jagiellonian University, Lojasiewicza 6, 30-348 Kraków, Polande-mail: Leszek.Gasinski@ii.uj.edu.pl2Department of Mathematics, National Technical University,Zografou Campus, Athens 15780, Greecee-mail: npapg@math.ntua.grAbstract. We consider a nonlinear Neumann elliptic equation driven bythe p-Laplacian and a Carathéodory perturbation. The energy functional ofthe problem need not be coercive. Using variational methods we prove anexistence theorem and a multiplicity theorem, producing two nontrivial smoothsolutions. Our formulation incorporates strongly resonant equations.Keywords: Palais–Smale condition, noncoercive functional, second deforma-tion theorem.1. IntroductionLet Ω ⊆ RN be a bounded domain with a C2-boundary ∂Ω. We study the followingnonlinear Neumann problem:−∆pu(z) = f(z, u(z))a.e. in Ω,∂u∂n= 0 on ∂Ω.(1)∗This research has been partially supported by the Ministry of Science and Higher Educationof Poland under Grants no. N201 542438 and N201 604640.28Here ∆p denotes the p-Laplace differential operator, defined by∆pu(z) = div(‖∇u(z)‖p−2∇u(z))∀u ∈W 1,p(Ω)(with 1 < p < +∞). Also, f(z, ζ) is a Carathéodory function, i.e., for all ζ ∈ R,the function z 7−→ f(z, ζ) is measurable and for almost all z ∈ Ω, the functionζ 7−→ f(z, ζ) is continuous.The aim of this work is to prove existence and multiplicity results for problem(1), when the energy functional of the problem is noncoercive. In fact, our hypothe-ses on the reaction f incorporate in our framework equations which are stronglyresonant at infinity. Such problems are of special interest, since they exhibit apartial lack of compactness. Recently, there have been some existence and multi-plicity results for Neumann problems driven by the p-Laplacian. We mention theworks of Anello [1], Filippakis–Gasiński–Papageorgiou [3], Motreanu–Papageorgiou[6], O’Regan–Papageorgiou [7], Wu–Tan [8]. In Anello [1] and Wu–Tan [8], thekey hypothesis is that p > N (low dimension problems). This condition impliesthat W 1,p(Ω) is embedded compactly in C(Ω) (Sobolev embedding theorem) andthis is their key mathematical tool. In Filippakis–Gasiński–Papageorgiou [3] andMotreanu–Papageorgiou [6], the potential function is nonsmooth (hemivariationalinequality) and the energy function is coercive. Finally in O’Regan–Papageorgiou[7], the energy function is bounded below but need not be coercive. In fact, thepotential functionF (t, ζ) =ζ∫0f(z, s) dsis unbounded below as ζ → ±∞. The authors prove a multiplicity theorem usingthe notion of homological linking.2. Mathematical backgroundLet X be a Banach space and let X∗ be its topological dual. In what follows, by〈·, ·〉 we denote the duality brackets for the pair (X∗, X). Let ϕ ∈ C1(X) and c ∈ R.We say that ϕ satisfies the Palais–Smale condition at level c ∈ R, if the following istrue:Every sequence {xn}n>1 ⊆ X, such thatϕ(xn) −→ c and ϕ′(xn) −→ 0 inX∗,admits a strongly convergent subsequence.The following result is an easy consequence of the above definition (see Gasiński–Papageorgiou [4, p. 650]).29THEOREM 1. If ϕ ∈ C1(X) is bounded below, c = infXϕ and ϕ satisfies thePalais–Smale condition at level c, then there exists x0 ∈ X, such that c = ϕ(x0), i.e.x0 is a critical point of ϕ.For ϕ ∈ C1(X) and c ∈ R, we introduce the following sets:ϕc ={x ∈ X : ϕ(x) 6 c},Kϕ ={x ∈ X : ϕ′(x) = 0},Kcϕ ={x ∈ Kϕ : ϕ(x) = c}.The next result is a basic tool in the minimax theorems of the critical point theoryand it is known as the “second deformation theorem” (see Gasiński–Papageorgiou[4, p. 628]).THEOREM 2. If ϕ ∈ C1(X), a ∈ R, a < b 6 +∞, ϕ satisfies the Palais–Smale condition for every c ∈ [a, b), ϕ has no critical values in (a, b) and ϕ−1({a})contains at most a finite number of critical points of ϕ, then there exists a homotopyh : [0, 1]× (ϕb \Kbϕ) −→ ϕb, such that(a) h(1, ϕb \Kbϕ)⊆ ϕa;(b) h(t, x) = x for all t ∈ [0, 1], all x ∈ ϕa;(c) ϕ(h(t, x))6 ϕ(h(s, x))for all t, s ∈ [0, 1], s 6 t, all x ∈ ϕb \Kbϕ.Let A : W 1,p(Ω) −→W 1,p(Ω)∗ be the nonlinear map, defined by〈A(u), v〉=∫Ω‖∇u‖p−2(∇u,∇v)RN dz ∀u, v ∈W 1,p(Ω). (2)From Gasiński–Papageorgiou [4, p. 746], we havePROPOSITION 3. The nonlinear map A : W 1,p(Ω) −→W 1,p(Ω)∗ defined by (2)is bounded, continuous, strictly monotone, hence maximal monotone too and of type(S)+, i.e. ifun −→ u weakly in W 1,p(Ω)andlim supn→+∞〈A(un), un − u〉6 0,then un −→ u in W 1,p(Ω).In what follows by ‖ · ‖ we denote the norm of W 1,p(Ω), i.e.‖u‖ =(‖u‖pp + ‖∇u‖pp) 1p ∀u ∈W 1,p(Ω).Also, if g : Ω × R −→ R is a measurable function (for example a Carathéodoryfunction), thenNg(u)(·) = g(·, u(·))∀u ∈W 1,p(Ω).Finally, by | · |N we denote the Lebesgue measure on RN .303. Existence theoremThe existence theorem will be obtained for a more general version of problem than(1). Namely, let h ∈ L∞(Ω) be such that∫Ωh(z) dz = 0.We consider the following nonlinear Neumann problem:−∆pu(z) = f(z, u(z))+ h(z) in Ω,∂u∂n= 0 on ∂Ω.(3)We work with the Sobolev space W 1,p(Ω) and consider the following direct sumdecomposition of this spaceW 1,p(Ω) = R⊕ V,whereV ={u ∈W 1,p(Ω) :∫Ωu dz = 0}.Hence, every u ∈W 1,p(Ω) admits a unique decompositionu = u+ û, with u ∈ R and û ∈ V.Recall that the elements of V satisfy‖û‖p 6 c0(N, p)‖∇û‖p ∀û ∈ V, (4)for some c0(N, p) > 0 (this is the so called Poincaré–Wirtinger inequality; seeGasiński–Papageorgiou [4, p. 841]). In particular, (4) implies thatû 7−→ ‖∇û‖pis an equivalent norm on V .For h ∈ L∞(Ω) with ∫Ωh(z) dz = 0,we consider the following auxiliary Neumann problem:−∆pu(z) = h(z) in Ω,∂u∂n= 0 on ∂Ω.(5)Let ψ : V −→ R be the C1-functional, defined byψ(û) =1p‖∇û‖pp −∫Ωh(z)û(z) dz ∀û ∈ V.31PROPOSITION 4. Problem (5) has a unique solution û0 ∈ V ∩ C1(Ω), which isthe unique minimizer of ψ.Proof. By virtue of the Poincaré–Wirtinger inequality (see (4)), we see that ψ iscoercive. Also, using the Sobolev embedding theorem, we see easily that ψ is se-quentially weakly lower semicontinuous. Hence, by the Weierstrass theorem, we canfind û0 ∈ V , such thatψ(û0) = inf{ψ(û) : û ∈ V},soψ′(û0) = 0 in V∗and thus 〈A(û0), v〉=∫Ωhv dz ∀v ∈ V. (6)Letv(z) = y(z)− 1|Ω|N∫Ωy dz with y ∈W 1,p(Ω).Then v ∈ V and from (6), we have〈A(û0), y〉=∫Ωhy dz(since∫Ωh dz = 0), soA(û0) = h in W1,p(Ω)∗(since y ∈W 1,p(Ω) is arbitrary) and thus−∆pû0(z) = h(z) a.e. in Ω,∂û0∂n= 0 on ∂Ω.Nonlinear regularity theory (see Lieberman [5]) implies thatû0 ∈ V ∩ C1(Ω).The uniqueness of û0 follows from the strict monotonicity of A (see Proposition3).Now let us introduce our hypotheses on the reaction f :H : f : Ω×R −→ R is a Carathéodory function, such that f(z, 0) = 0 for almost allz ∈ Ω and32(i) we have ∣∣f(z, ζ)∣∣ 6 a(z) + c|ζ|r−1 for almost all z ∈ Ω, all ζ ∈ R,with a ∈ L∞(Ω)+, c > 0, p < r < p∗, wherep∗ =NpN − pif p < N,+∞ if p > N ;(ii) ifF (z, ζ) =ζ∫0f(z, s) ds,thenF (z, ζ) 6 ξ(z) for almost all z ∈ Ω, all ζ ∈ R,with ξ ∈ L1(Ω);(iii) there exists c0 ∈ R \ {0}, such that∫ΩF (z, c0) dz > 0.EXAMPLE 5. The following function satisfies hypotheses H (for the sake of sim-plicity we drop the z-dependence):f(ζ) =|ζ|p−2ζ if |ζ| 6 1,− ζ|ζ|p+2+4ζ(1 + ζ2)|ζ|if |ζ| > 1.In this case the potential function F is given byF (ζ) =1p|ζ|p if |ζ| 6 1,1p|ζ|p+ 4 arctan |ζ| − π if |ζ| > 1.Let ϕ : W 1,p(Ω) −→ R be the energy functional for problem (3), given byϕ(u) =1p‖∇u‖pp −∫ΩF(z, u(z))dz −∫Ωh(z)u(z) dz ∀u ∈W 1,p(Ω).Evidently ϕ ∈ C1(W 1,p(Ω)). Recall that for every u ∈W 1,p(Ω), we have in a uniquewayu = u+ û with u ∈ R, û ∈ V.33So, we can writeϕ(u) = ψ(û)−∫ΩF(z, u(z))dz ∀u ∈W 1,p(Ω)(recall that∫Ωh dz = 0).Hypotheses H incorporate in our framework, problems which are strongly reso-nant at infinity. It is well known that such problem exhibit a partial lack of compact-ness (see Bartolo–Benci–Fortunato [2]). This is reflected in the next proposition. Inwhat follows û0 ∈ V ∩ C1(Ω) is the unique solution of problem (5), established inProposition 4. Alsoβ =∫Ωlim sup|ζ|→+∞F (z, ζ) dz.By virtue of hypothesis H(ii), β ∈ [−∞,+∞).PROPOSITION 6. If hypotheses H hold andc < ψ(û0)− β = ξ∗ ∈ (−∞,+∞],then ϕ satisfies the Palais–Smale condition at level c.Proof. Let {un}n>1 ⊆W 1,p(Ω) be a sequence, such thatϕ(un) −→ c < ξ∗ (7)andϕ′(un) −→ 0 in W 1,p(Ω)∗. (8)Recall thatun = un + ûn ∀n > 1,with un ∈ R, ûn ∈ V . On account of (7), we haveϕ(un) 6 M1 ∀n > 1,for some M1 > 0, soM1 >1p‖∇ûn‖pp −∫ΩF (z, un) dz −∫Ωhûn dz>1p‖∇ûn‖pp − c1‖∇ûn‖p − c2 ∀n > 1, (9)for some c1, c2 > 0. Here we have used hypothesis H(ii) and the Poincaré–Wirtingerinequality (see (4)). Since p > 1, from (9), we infer thatthe sequence {ûn}n>1 ⊆W 1,p(Ω) is bounded. (10)So, by passing to a suitable subsequence if necessary, we may assume thatûn −→ û weakly in W 1,p(Ω), (11)ûn −→ û in Lp(Ω), (12)ûn(z) −→ û(z) for almost all z ∈ Ω (13)34and ∣∣ûn(z)∣∣ 6 η̂(z) for almost all z ∈ Ω, all n > 1,with η̂ ∈ Lp(Ω).Claim. The sequence {un}n>1 ⊆W 1,p(Ω) is bounded.Arguing by contradiction and passing to subsequence if necessary, we may sup-pose that‖un‖ −→ +∞.Since‖un‖ 6 ‖un‖+ ‖ûn‖ ∀n > 1,from (10), it follows that |un| −→ +∞ (recall that {un}n>1 ⊆ R). We have∣∣un(z)∣∣ > |un| − ∣∣ûn(z)∣∣ > |un| − η̂(z) for almost all z ∈ Ω, all n > 1(see (11)), so ∣∣un(z)∣∣ −→ +∞ for almost all z ∈ Ω.Since û0 ∈ V is the minimizer of ψ (see Proposition 4), we haveϕ(un) = ψ(ûn)−∫ΩF(z, un(z))> ψ(û0)−∫ΩF(z, un(z))dz.From (7) and the Fatou lemma (see hypothesis H(ii)), we haveξ∗ > c > ψ(û0)−∫Ωlim supn→+∞F (z, un) dz = ψ(û0)− β = ξ∗,a contradiction. This proves the Claim.By virtue of the Claim, passing to a subsequence if necessary, we may assumethatun −→ u weakly in W 1,p(Ω), (14)un −→ u in Lr(Ω). (15)From (8), we have ∣∣〈ϕ′(un), y〉∣∣ 6 εn‖y‖ ∀y ∈W 1,p(Ω),with εn ↘ 0, so∣∣∣∣〈A(un), y〉− ∫Ωf(z, un)y dz −∫Ωhy dz∣∣∣∣ 6 εn‖y‖ ∀n > 1.We choose y = un − u ∈W 1,p(Ω). Then∣∣∣∣〈A(un), un − u〉− ∫Ωf(z, un)(un − u) dz −∫Ωh(un − u) dz)∣∣∣∣35≤ εn‖un − u‖ ∀n > 1. (16)From (14), we have∫Ωf(z, un)(un − u) dz −→ 0 and∫Ωh(un − u) dz −→ 0.Therefore, if in (16) we pass to the limit as n→ +∞, thenlimn→+∞〈A(un), un − u〉= 0,soun −→ u in W 1,p(Ω)(see Proposition 3) and so ϕ satisfies the Palais–Smale condition at any level c <ξ∗.Using this proposition, we can have an existence theorem for problem (1).THEOREM 7. If hypotheses H hold andβ <∫ΩF (z, û0) dz,then problem (3) admits a nontrivial solution u∗ ∈ C1(Ω).Proof. From Proposition 4 and hypothesis H(ii), we haveϕ(u) = ψ(û)−∫ΩF (z, u) dz > ψ(û0)− ‖ξ‖1 ∀u ∈W 1,p(Ω),so ϕ is bounded below.Letmϕ = inf{ϕ(u) : u ∈W 1,p(Ω)}> −∞.Then−∞ < mϕ 6 ϕ(û0) = ψ(û0)−∫ΩF (z, û0) dz < ψ(û0)− β,so ϕ satisfies the Palais–Smale condition at level mϕ (see Proposition 6).Theorem 1 implies that we can find u∗ ∈W 1,p(Ω), such thatϕ(u∗) = mϕ 6 ϕ(c0) < 0 = ϕ(0)(see hypothesis H(iii)), sou∗ 6= 0.Also, we haveϕ′(u∗) = 0,soA(u∗) = Nf (u∗) + hand thus u∗ ∈ C1(Ω) (see (7)) is a nontrivial solution of (3).REMARK 8. A careful inspection of the above proof, reveals that hypothesis H(iii)is needed only if h = 0, to guarantee the nontriviality of u∗.364. Multiplicity theoremIn this section we prove a multiplicity theorem for problem (1) (i.e., now h = 0).For this purpose, we strengthen the hypotheses on f as follows:H ′: f : Ω×R −→ R is a Carathéodory function, such that f(z, 0) = 0 for almost allz ∈ Ω, hypotheses H ′(i)− (iii) are the same as H(i)− (iii) and(iv) we haveβ =∫Ωlim sup|ζ|→+∞F (z, ζ) < 0and there exists ϑ ∈ L∞(Ω)+, ϑ 6= 0, such thatϑ(z) 6 lim infζ→0F (z, ζ)|ζ|puniformly for almost all z ∈ Ω;(v) we haveF (z, ζ) 6λ̂1p|ζ|p for almost all z ∈ Ω, all ζ ∈ R,with λ̂1 > 0 being the first nonzero eigenvalue of the negative Neumann p-Laplacian.EXAMPLE 9. The following function satisfies hypotheses H ′ (as before, for thesake of simplicity, we drop the z-dependence):f(ζ) =λ̂1|ζ|p−2ζ if |ζ| 6 1,(λ̂1 + 1)ζ|ζ|p+2− |ζ|r−2ζ if |ζ| > 1,where p < r < p∗. In this case the potential function F is given byF (ζ) =λ̂1p|ζ|p if |ζ| 6 1,λ̂1 + 1p1|ζ|p− 1r|ζ|r − r − prpif |ζ| > 1.Now the energy functional ϕ̂ : W 1,p(Ω) −→ R is given byϕ̂(u) =1p‖∇u‖pp −∫ΩF(z, u(z))dz ∀u ∈W 1,p(Ω).Evidently ϕ̂ ∈ C1(W 1,p(Ω)).37THEOREM 10. If hypotheses H ′ hold, then problem (1) has at least two nontrivialsmooth solutions u∗, v∗ ∈ C1(Ω).Proof. Since h = 0, we haveû0 = 0.Therefore ∫ΩF(z, û0(z))dz = 0.Since by hypothesis H ′(iv), β < 0, we can apply Theorem 7 and have one nontrivialsmooth solution u∗ ∈ C1(Ω).By virtue of hypothesis H ′(iv), for a given ε > 0 we can find δ = δ(ε) > 0, suchthatF (z, ζ) >(ϑ(z)− ε)|ζ|p for almost all z ∈ Ω, all |ζ| 6 δ.If ĉ ∈ [−δ, δ], thenϕ(ĉ) = −∫ΩF (z, ĉ) dz 6 |ĉ|p(ε|Ω|N −∫Ωϑ dz).Choosing ε ∈(0, 1|Ω|N∫Ωϑ dz), we see that ϕ(ĉ) < 0 and somax{ϕ(v) : v ∈ BR ∩ R}= µR < 0 ∀R ∈(0, δ|Ω|1pN), (17)whereBR ={u ∈W 1,p(Ω) : ‖u‖ 6 R}.We consider the setC(p) ={u ∈W 1,p(Ω) :∫Ω∣∣u(z)∣∣p−2u(z) dz = 0}.Then for every u ∈ C(p), we haveϕ(u) >1p‖∇u‖pp −λ̂1p‖u‖pp(see H ′(v)), soinfC(p)ϕ = 0 (18)(see Gasiński–Papageorgiou [4]).Let us fix r ∈(0, δ|Ω|1pN). LetΓ ={γ ∈ C(Br ∩ R, W 1,p(Ω)): γ|∂Br∩R= id|∂Br∩R}and defineĉr = infγ∈Γmaxv∈Br∩Rϕ(γ(v)). (19)38Note that (∂Br ∩ R)∩ C(p) = ∅.Also, let γ ∈ Γ and defineσ(ξ) =∫Ω∣∣γ(ξ)∣∣p−2γ(ξ) dz ∀ξ ∈ Br ∩ R.Then∂Br ∩ R ={± r0 = ±r|Ω|1pN}and soσ(−r0) < 0 < σ(r0).By virtue of the Bolzano theorem, we can find ξ̂ ∈ Br ∩ R, such thatσ(ξ̂) =∫Ω∣∣γ(ξ̂)∣∣p−2γ(ξ̂) dz = 0,soγ(ξ̂) ∈ C(p),thusγ(Br ∩ R)∩ C(p) 6= ∅and finallycr > 0 (20)(see (18) and (19)). Suppose that {0, u∗} are the only critical points of ϕ. We seta = inf ϕ = ϕ(u∗) < 0 and b = ϕ(0) = 0.By virtue of Proposition 6 and hypothesis H ′(iv), we see that ϕ satisfies the Palais–Smale condition for every level c ∈ [a, b]. Also,ϕ−1({a}) = {u∗}.Therefore, we can apply the second deformation lemma (see Theorem 2) and havea homotopyĥ : [0, 1]× (ϕb \Kbϕ) −→ ϕb,such thatĥ(1, ϕb \Kbϕ)⊆ ϕa = {u∗} (21)andϕ(ĥ(t, u))6 ϕ(ĥ(s, u))∀s, t ∈ [0, 1], s 6 t, all u ∈ ϕb \Kbϕ. (22)We consider the mapγ0 : Br ∩ R −→ W 1,p(Ω),defined byγ0(u) =u∗ if ‖u‖ 6 r2 ,ĥ(2(r − ‖u‖)r,ru‖u‖)if ‖u‖ > r2 .(23)39If u ∈ R and ‖u‖ = r2 , thenĥ(2(r − ‖u‖)r,ru‖u‖)= h(1, 2u) = u∗(see (17) and (21)). Hence, from (23), we see that γ0 is continuous. Also, if u ∈∂Br ∩ R, thenγ0(u) = ĥ(0, u) = u(see (23)). Therefore γ0 ∈ Γ. From (17), (22) and (23), we haveϕ(γ0(u))< 0 ∀u ∈ Br ∩ R,soĉr < 0 (24)(see (19)).Comparing (20) and (24), we reach a contradiction. This means that we can findv∗ ∈ Kϕ, such that v∗ 6∈ {0, u∗}. ThenA(v∗) = Nf (v∗),so −∆pv∗(z) = f(z, v∗(z))a.e. in Ω,∂v∗∂n= 0 on ∂Ω.(see (7)). Nonlinear regularity theory (see Lieberman [5]) implies that v∗ ∈ C1(Ω).This is the desired second nontrivial smooth solution of (1).5. References[1] Anello G.; Existence of infinitely many weak solutions for a Neumann problem, Non-linear Anal. 57, 2004, pp. 199–209.[2] Bartolo P., Benci V., Fortunato D.; Abstract critical point theorems and applicationsto some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal. 7,1983, pp. 981–1012.[3] Filippakis M., Gasiński L., Papageorgiou N.S.; Multiplicity result for nonlinear Neu-mann problems, Canad. J. Math. 58, 2006, pp. 64–92.[4] Gasiński L., Papageorgiou N.S.; Nonlinear Analysis, Chapman and Hall/CRC Press,Boca Raton, FL, 2006.[5] Lieberman G.M.; Boundary regularity for solutions of degenerate elliptic equations,Nonlinear Anal. 12, 1988, pp. 1203–1219.40[6] Motreanu D., Papageorgiou N.S.; Existence and multiplicity of solutions for Neumannproblems, J. Differential Equations 232, 2007, pp. 1–35.[7] O’Regan D., Papageorgiou N.S.; The existence of two nontrivial solutions via homolog-ical local linking for the non-coercive p-Laplacian Neumann problem, Nonlinear Anal.70, 2009, pp. 4386–4392.[8] Wu X.-P., Tan K.K.; On existence and multiplicity of solutions of Neumann boundaryvalue problems for quasi-linear elliptic equations, Nonlinear Anal. 65, 2006, pp. 1334–1347.

Existence and multiplicity of solutions for noncoercive neumann problems with p-Laplacian

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Existence and Multiplicity of Solutions for Noncoercive Neumann Problems with $p$-Laplacian

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Existence and multiplicity of solutions for noncoercive neumann problems with p-Laplacian

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Existence and multiplicity of solutions for noncoercive neumann problems with p-Laplacian

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