Neumann problems of superlinear elliptic systems at resonance

Neumann problems of superlinear elliptic systems at resonance

Ruyun Ma

, Zhongzi Zhao

and Mantang Ma

1School of Mathematics and Statistics, Xidian University, Xi’an 710071, P. R. China

2Department of Mathematics, Northwest Normal University, Lanzhou 730070, P. R. China

Received 12 October 2021, appeared 2 February 2022 Communicated by Dimitri Mugnai

Abstract. We prove existence of weak solutions of Neumann problem of nonhomo- geneous elliptic system with asymmetric nonlinearities that may resonant at −∞and superlinear at+∞. The proof is based on Mawhin’s coincidence theory and the product formula of Brouwer degree.

Keywords: elliptic equation, Neumann problem, weak solution, continuation methods.

2020 Mathematics Subject Classification: 35J25, 35J60, 47H11.

1 Introduction

Let Ω ⊂ _R^N (N ≥ 2) be a smooth bounded connected domain in real N-dimensional Eu- clidean space. We are concerned with the existence of weak solutions of the following Neu- mann problem of semilinear elliptic systems

∆u+ f(v) =h₁(x), in Ω,

∆v+g(u) =h₂(x), in Ω,

∂u

∂ν = ^∂v

∂ν =0, on ∂Ω,

(1.1)

where f,g : R → _R are continuous functions, _∂ν^∂ denotes the outward normal derivative on

∂Ω, the boundary ofΩ, andh₁,h₂ ∈ L¹(_Ω).

The motivation for this work is the paper F. O. de Paiva, W. Rosa [12], in which the authors showed the following resonant Neumann problems

−_∆u= (v⁺)^p+h₁(x), inΩ,

−_∆v= (u⁺)^q+h₂(x)_{, inΩ,}

∂u

∂ν = ^∂v

∂ν =0, on∂Ω

(1.2)

BCorresponding author. Email: [email protected]

has at least one solution (u,v) in H¹(_Ω)×H¹(_Ω) under the assumptions h₁,h₂ ∈ L^r(_Ω), r> ^N₂, 1< p,q< _N^N₋₂ and

Z

Ωh_i(x)dx<0, i=1, 2. (1.3) We first define the bilinear form associated with the Laplacian operator. For u,v ∈ W^1,1(_Ω),φ,ψ∈W^1,∞(_Ω), we defineB₁(u,φ)andB₂(v,ψ)by

B₁(u,φ) =−

∑

Z

Ω

∂u

∂x_i

∂φ

∂x_idx,

B₂(v,ψ) =−

∑

Z

Ω

∂v

∂x_i

∂ψ

∂x_idx,

where the derivatives are taken in the distributional sense. By aweak solutionof (1.1), we mean a pair(u,v)∈W^1,1(_Ω)×W^1,1(_Ω), such that f(v(·))∈L¹(_Ω),g(u(·))∈ L¹(_Ω)and

B₁(u,φ) +

Z

Ω f(v)φdx =

Z

Ωh₁(x)φdx, ∀ φ∈W^1,∞(_Ω), B2(v,ψ) +

Z

Ωg(u)ψdx =

Z

Ωh2(x)ψdx, ∀ψ∈W^1,∞(_Ω). Denote

f^−∞ =lim sup

s→−_∞

f(s), g^−∞ =lim sup

s→−_∞

g(s), f+_∞ =lim inf

s→+_∞ f(s), g+_∞ =lim inf

s→+_∞ g(s). We will make the following assumptions.

(C0) h₁,h₂ ∈ L¹(_Ω).

(Cl) There are the nonnegative constantsC₁,C₂ ∈(0,∞)such that f(t)≥ −C₁, g(t)≥ −C₂, t∈ _R and for allt ≤0 we have also|f(t)| ≤C₁,|g(t)| ≤C₂.

(C2) There are the constants a,b ∈ _R and p with 1 ≤ p < N/(N−2) for N ≥ 3 and 1≤ p<_∞forN=2 such that for allt ≥0 the inequality

|f(t)|,|g(t)| ≤at^p+b a.e. onΩ.

(C3) We assume f,gtends to be nondecreasing in that there is aγ∈_Rand a numberM ≥0 such that the inequalities

f(t₁)≤ f(t₂) +γ, g(t₁)≤g(t₂) +γ hold a.e. onΩwhenevert₂−t₁≥ M.

(C4)

Z

Ω f^−∞ <

Z

Ωh₁(x)dx<

Z

Ω f+_∞, Z

Ωg^−∞ <

Z

Ωh₂(x)dx<

Z

Ωg+_∞.

Our main result is the following

Theorem 1.1. Under assumptions (C0)–(C4) the Neumann problem(1.1)has a weak solution(u,v)∈ W^1,1(_Ω)×W^1,1(_Ω). Moreover the solution(u,v)∈W^1,q(_Ω)×W^1,q(_Ω)for all1≤q<N/(N−1). Remark 1.2. Obviously, (1.3) in F. O. de Paiva, W. Rosa [12] are the special case of (C0) and (C4).

Remark 1.3. Our proof is based upon ideas found in Ward Jr [16]. He used the well-known Mawhin’s continuation theorem to get a weak solution of the scale elliptic equation

∆u+ f(x,u) =k(x), inΩ,

∂u

∂ν =0, on∂Ω (1.4)

under the conditionsk∈ L¹(_Ω),

|f(x,t)| ≤α(x)|t|^p+β(x), x∈ _Ω,

where p ∈[1,_N^N₋₂),α∈ L^∞(_Ω),β∈ L¹(_Ω), and Landesman–Lazer condition Z

Ω f^−∞ <

Z

Ωk(x)dx<

Z

Ω f+_∞.

Remark 1.4. Similar problems, under Dirichlet and Neumann boundary condition, can be found in D. Arcoya and S. Villegas [2], M. Cuesta and C. De Coster [3], F. M. Ferreira, F. O. de Paiva [4], R. Kannan and R. Ortega [6,7], S. Kyritsi and N. S. Papageorgiou [8], D. Motreanu, V. Motreanu, N. S. Papageorgiou [10], K. Perera [14], N. S. Papageorgiou and V. D. R˘adulescu [13], F. O. de Paiva and A. E. Presoto [11], L. Recova and A. Rumbos [15], J. R. Ward [16].

2 The preliminaries

Before proving Theorem 1.1 we will need a lemma. In the following we will write L^p for L^p(_Ω) andW^1,p forW^1,p(_Ω). We denote the norm in L^p by| · |_p, that of W^1,p by| · |_1,p. For h∈ L¹. LetQhbe the projection

Qh =|_Ω|⁻¹

Z

Ωhdx.

Lemma 2.1([16]). For each h∈ L¹(_Ω)with Qh=0. There is a unique w∈W^1,1(_Ω)with Qw=0 such that

B(w,φ) =

Z

Ωh(x)φdx, for all φ ∈ W^1,∞, where B(_w,φ) = −_∑^N_i=1R

Ω ∂w

∂x_i

∂φ

∂x_idx. Moreover w ∈ W^1,q for each q satisfying 1≤q< N/(N−1)and there is a constant C(q)such that

|w|_1,q≤C(q)|h|₁.

By the Rellich–Kondrachov theorem W^1,q is compactly imbedded in L^p for 1 ≤ p < _N^Nq_−q since q < N/(N−1)≤ N for all N ≥ 2. (e.g. see [1, p. 144]). Assume that the number pin condition (C2) is fixed hereafter, satisfying 1 ≤ p < N/(N−2)if N ≥ 3 and 1 ≤ p < _∞ if N=_2.

Chooseqso that

p< ^Nq

N−q and 1<q< ^N N−1. We have thatW^1,qis compactly imbedded in L^p.

LetX₁ denote the closed subspace ofL¹defined by h∈X₁ if and only if Qh=_0.

LetTdenotes the operator mappingX₁intoW^1,q∩X₁given byh →wwherewis the unique weak solution to

∆w=h inΩ, Qu=0,

∂w

∂ν

=_{0 on}∂Ω.

Note that W^1,q = (W^1,q∩X₁)⊕_R. T maps X₁ intoW^1,q and we see thatΨ◦T maps X₁ compactly into L^pifΨ is the imbedding ofW^1,q intoL^p. Let

K= _Ψ◦T,

and define an operatorL: L^p→ L¹. BecauseL¹is not the dual space to L^∞, we do not use the usual method of definingL. Instead, we let

D(L) =RangeK⊕_R and

L(w₁+α˜) =h,

whereh∈X₁andKh= w₁, forw₁ ∈RangeKand ˜α∈R. It is easy to see thatLis a Fredholm operator: it has closed rangeX₁and since ker(L) =_Rand the dimension of L¹\X₁is clearly 1, the index ofLis 0,

index(L) =dim kerL−dim cokerL.

We now define the substitution operatorsN₁,N₂: L^p→ L¹ by N₁v(x) = f(v(x))−h₁(x), v∈ L^p andx∈ _Ω.

N₂u(x) =g(u(x))−h₂(x), u∈L^pandx∈ _Ω.

It is well known that the conditions on f andgimply thatN_jmapsL^pintoL¹continuously andN_j obviously takes sets bounded inL^p into sets bounded in L¹for j=1, 2.

A function (u,v) ∈ W^1,1×W^1,1 is a weak solution of (1.1) if and only if (u,v) ∈ D(L)× D(L)and

Lu+N₁v=0,

Lv+N₂u=0. (2.1)

Recalling that for u ∈ L¹ we have defined Qu to be the mean value of u, we have from our remarks above that K(I −Q)N_j : L^p → L^p is compact and continuous, clearly QN_j is also compact and continuous for j = 1, 2. Thus N_j is L-compact (see [5]) on ¯G for any open bounded set ¯G in L^p forj= 1, 2. We will use a well known continuation theorem of Mawhin (see [5] and [9]).

3 Proof of the main result

We are in the position to prove our main result.

Proof of Theorem1.1. By one of Mawhin’s continuation theorems (see [5, p. 40] or [9, The- orem 7.2]) and our remarks above, if we can show the existence of a bounded open set G := G^¯ ×G^¯ in L^p×L^p such that conditions (i) and (ii) below hold, then (2.1) has a solu- tion. The conditions are:

(i) For eachλ∈(0, 1)and each(u,v)∈(D(L)×D(L))∩∂G, Lu+λN₁v̸=_0,

Lv+λN₂u̸=0. (3.1)

(ii) QN_jw̸=0 for eachj=1, 2,w∈kerL∩∂G¯ and

d(_Γ,G∩(kerL×kerL), 0)̸=0,

where Γ := (JQN₁,JQN₂), J : ImQ → kerL is an isomorphism, andd is the Brouwer topological degree.

We first verify (i). We consider

Lu+λN₁v=0,

Lv+λN₁u=0 (3.2)

for 0<λ<1. If((u,v),λ)is a solution of (3.2) then B₁(u,φ) +λ

Z

Ω f(v)φ=λ Z

Ωh₁φ, ∀ φ∈W^1,∞, B₂(v,ψ) +λ

Z

Ωg(u)ψ=λ Z

Ωh₂ψ, ∀ ψ∈W^1,∞. In particular by taking φ=ψ=1, then

Z

Ω f(v) =

Z

Ωh₁, Z

Ωg(u) =

Z

Ωh₂. It follows from (Cl) that for eacht ∈_R

|f(t)| ≤ f(t) +2C₁, |g(t)| ≤g(t) +2C₂. Thus

|N₁v|₁=

Z

Ω|f(v)−h₁(x)|dx

≤

Z

Ω

f(v) +2C₁+|h₁(x)|dx

≤

Z

Ωh₁dx+2|C₁| · |_Ω|+

Z

Ω|h₁(x)|dx=:d₁,

|N₂u|₁ =

Z

Ω|g(u)−h₂(x)|dx

≤

Z

Ω

g(u) +2C₂+|h₂(x)|dx

≤

Z

Ωh₂dx+₂|C₂| · |_Ω|+

Z

Ω|h₂(x)|dx =_:d₂.

Writingu=u₁+α, v=v₁+βwithu₁,v₁ ∈RangeKandα,β∈ _Rby Lemma2.1we have

|u₁|_1,q≤C(q)d₁ =:m₁,

|v₁|_1,q≤C(q)d₂=:m₂,

wherem₁andm₂are independently ofλ∈ (0, 1). By the Sobolev imbedding theorem

|u₁|_p ≤m3, |v₁|_p ≤m₄ for some constantsm₃ andm₄.

We now show that for solutions ((u,v),λ) = (u₁+α,v₁+β),λ

that α and β are also bounded independently ofλ∈ (0, 1).

Suppose this is not the case. Then there is a sequence ((u_n,v_n),λ_n) of solutions to (3.2) with

un= u_1n+αn, vn=v_1n+βn

and

|α_n|+|β_n| →_∞, asn→_∞.

Suppose first that a subsequence of {α_n}, relabeled as {α_n}, tends to +∞. Then using

|u_1n|_1,q≤ m₁is easy to show that

nlim→_∞u_n(x) = +_∞ a.e. (3.3) For otherwise there is a constantk₁>0 and sets Ω(n)inΩfor infinitely manyn(without loss of generality we assume for alln) such that|_Ω(n)| ≥δ> 0 andu_n(x)≤k₁ forx∈_Ω(n). We haveu_1n+αn≤k₁implies

k₁|_Ω| ≥

Z

Ω(n)k₁dx≥

Z

Ω(n)u1n+αndx

≥α_n|_Ω(n)| −

Z

Ω|u_1n|

≥αnδ−C

forC, a constant, which contradictsα_n→∞. Thus (3.3) holds and lim inf

n→_∞ g(un(x)) = g+_∞ a.e.

Sinceg(u_n(x))≥ −C₂for all nandC₂ ∈_Rwe have by Fatou’s lemma Z

Ωh₂=lim inf

n→_∞ Z

Ωg(u_n(x))dx ≥

Z

Ωg+_∞dx which contradicts (C4). Thus the{α_n}must be bounded above.

Supposeα_n→ −∞. It follows as for (3.3) that

nlim→_∞u_n(x) =−_∞ a.e.

Because g(t)is not everywhere bounded above by an L¹ function, we cannot use the simple Fatou’s lemma argument as in the case ofα_n → −_∞.

We proceed as follows. Since|u_1n|_1,q≤ m₁, we may without loss of generality assume the existence of ˜u₁ ∈ L^p such thatu_1n→u˜₁ inL^p.

Let 0 < ϵ < |_Ω|be given. Then ˜u₁ ∈ L^p implies that there exists an integer n(ϵ) and a measurable set E⊆_Ωsuch that ifF=_Ω−Ethen |F|<ϵand

u_n(x)≤0, x ∈E, n≥n(ϵ), hence

g(un(x))≤ C₂, x ∈E, n≥n(ϵ).

Moreover there exists another integermsuch that forn≥ mwe have α_n≤ −M, where M is a positive constant.

Thus, forn≥max{n(ϵ),m}, Z

Ωh₂ =

Z

Eg(u_1n+αn) +

Z

Fg(u_1n+αn)

≤

Z

Eg(u_1n+α_n) +

Z

Fg(u_1n) +

Z

Fγ

and

Z

Ωh2 ≤lim sup

n→_∞

Z

Eg(_un) +

Z

Fg(_u1n)

+

Z

F

γ

≤

Z

Eg^−∞dx+

Z

Fg(u˜₁)dx+

Z

Fγ

(3.4)

by Fatou’s lemma for the integral overEand by convergence inL¹for the integral over F.

Now chooseη>0 such that Z

Ωg^−∞dx+η<

Z

Ωh₂dx. (3.5)

We may chooseϵ>0 so small that, since|F|< ϵ,

Z

Fg^−∞dx

< ^η 3,

Z

Fg(u˜₁)dx

< ^η 3,

Z

Fγ

< ^η 3. For such asϵwe have from (3.4) and (3.5)

Z

Ωh₂ ≤

Z

Ωg^−∞dx−

Z

F

g^−∞dx+

Z

F

g(u˜₁)dx+

Z

F

γ≤

Z

Ωg^−∞dx+η<

Z

Ωh₂. (3.6) Therefore we cannot have α_n →+_∞or α_n→ −_∞ and this, combined with|u₁|_p ≤m₃ shows that if ((u,v)_,λ) is a solution of (3.2) then |u|_p = |u₁+α|_p ≤ m₃+C₃ for some constant C₃. Similarly, We can obtain|v|_p=|v₁+α|_p≤ m₄+C₄ for some constantC₄.

This verifies condition (i) above for any ballG in L¹×L¹, centered at the origin and with radius larger thanρ₁=_max{m₃+C₃,m₄+C₄}_.

The verification of condition (ii) is now straightforward. Both the range of Q and the kernel ofL may be identified withR, so that the isomorphism J in (ii) we may take to be the identity onR. Now for α,β∈_R,

QN₁β= |_Ω|⁻¹

Z

Ω

h

f(β)−h₁(x)ⁱdx, QN₂α=|_Ω|⁻¹

Z

Ω

h

g(α)−h₂(x)ⁱdx.

We may now make two simple applications of Fatou’s lemma using (Cl) to show, using (C4), that there exists anr>0 such that

QN₁(β)>0, QN₁(−β)<0, forα>r, QN₂(α)>0, QN₂(−α)<0, for β> r.

Thus for ¯r ≥rmax{1,|_Ω|},

d(QN_j,[−r, ¯¯ r]∩kerL, 0)̸=0, j=1, 2.

By the product formula of Brouwer degree, we obtain

d(_Γ,[−r, ¯¯ r]²∩(kerL×kerL), 0)̸=0.

Now let ρ := max{ρ₁,r·max{1,|_Ω|}}. Then we have that both (i) and (ii) are satisfied on [Bρ]², whereBρ is the ball inL^pwith radius ρcentered at the origin. Thus (2.1) has a solution (u,v)∈ D(L)×D(L)with

|u|_p ≤ρ, |v|_p≤ρ,

and(u,v) ∈ W^1,p×W^1,p and is a weak solution of (1.1). This completes the proof of Theo- rem1.1.

Acknowledgements

The authors are very grateful to the anonymous referees for their valuable suggestions.

References

[1] R. A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York, 1975.MR0450957;Zbl 0314.46030

[2] D. Arcoya, S. Villegas, Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at−_∞and superlinear at+_∞,Math. Z.219(1995), No. 4, 499–

513.https://doi.org/10.1007/BF02572378;MR1343659;Zbl 0834.35048

[3] M. Cuesta, C. De Coster, Superlinear critical resonant problems with small forcing term, Calc. Var. Partial Differential Equations54(2015), No. 1, 349–363. https://doi.org/

10.1007/s00526-014-0788-8;MR3385163;Zbl 1323.35022

[4] F. M. Ferreira, F. O. de Paiva, On a resonant and superlinear elliptic system, Discrete Contin. Dyn. Syst.39(2019), No. 10, 5775–5784.https://doi.org/10.3934/dcds.2019253;

MR4027011;Zbl 1425.35036

[5] R. E. Gaines, J. L. Mawhin,Coincidence degree, and nonlinear differential equations, Lecture Notes in Mathematics, Vol. 568, Springer-Verlag, Berlin–New York, 1977. MR0637067;

Zbl 0339.47031

[6] R. Kannan, R. Ortega, Superlinear elliptic boundary value problems, Czech. Math. J.

37(1987), No. 3, 386–399.MR904766;Zbl 0668.35032

[7] R. Kannan, R. Ortega, Landesman–Lazer conditions for problems with one-side un- bounded nonlinearities,Nonlinear. Anal.9(1985), No. 2, 1313–1317.https://doi.org/10.

1016/0362-546X(85)90090-2;MR820641;Zbl 0624.35033

[8] S. Kyritsi, N. S. Papageorgiou, Multiple solutions for superlinear Dirichlet problems with an indefinite potential,Ann. Math. Pura Appl. (4) 192(2013), No. 2, 297–315. https:

//doi.org/10.1007/s10231-011-0224-z;MR3035141;Zbl 1268.35061

[9] J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, J. Dif- ferential Equations12(1972), 610–636. https://doi.org/10.1016/0022-0396(72)90028-9;

MR328703;Zbl 0244.47049

[10] D. Motreanu, V. Motreanu, N. S. Papageorgiou, Multiple solutions for Dirichlet prob- lems which are superlinear at +_∞ and sublinear at −_∞, Commun. Appl. Anal. 13(2009), No. 3, 341–358.MR2562309;Zbl 1181.35053

[11] F. O. dePaiva, A. E. Presoto, Semilinear elliptic problems with asymmetric nonlinear- ities, J. Math. Anal. Appl. 409(2014), No. 1, 254–262. https://doi.org/10.1016/j.jmaa.

2013.06.042;MR3095036;Zbl 1310.35122

[12] F. O.dePaiva, W. Rosa, Neumann problems with resonance in the first eigenvalue,Math.

Nachr. 290(2017), No. 14–15, 2198–2206. https://doi.org/10.1002/mana.201600139;

MR3711779;Zbl 1377.35071

[13] N. S. Papageorgiou, V. D. Radulescu˘ , Semilinear Neumann problems with indefinite and unbounded potential and crossing nonlinearity,Contemp. Math. 595(2013), 293–315.

https://doi.org/10.1090/conm/595/11801;MR3156380;Zbl 1301.35052

[14] K. Perera, Existence and multiplicity results for a Sturm–Liouville equation asymptot- ically linear at −_∞ and superlinear at +_∞, Nonlinear Anal. 39(2000), No. 6, 669–684.

https://doi.org/10.1016/S0362-546X(98)00228-4;MR1733121;Zbl 0942.34021

[15] L. Recova, A. Rumbos, An asymmetric superlinear elliptic problem at resonance,Nonlin- ear Anal.112(2015), 181–198.https://doi.org/10.1016/j.na.2014.09.019;MR3274292;

Zbl 1303.35013

[16] J. R. Ward Jr., Perturbations with some superlinear growth for a class of second order elliptic boundary value problems,Nonlinear Anal.6(1982), No. 4, 367–374.https://doi.

org/10.1016/0362-546X(82)90022-0;MR654812;Zbl 0533.35033