The fibering method approach to a singular (p, q)-Laplacian equation
Fereshteh Behboudi
∗Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran
and Abdolrahman Razani
Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran
Abstract. In this paper, the existence of two weak non-negative non-trivial solutions of a nonlinear problem involving the(p, q)-Laplacian operator in a bounded domain with smooth boundary inRN is proved via fibering method.
Keywords: (p, q)-Laplacian equation, Fibering method, Nehari manifold.
AMS Mathematical Subject Classification [2010]: 35J75, 35D30, 35P30.
1. Introduction
The Nehari manifold is closely related to the fibering maps, that is, maps of the formt→I(tu) whereI is the energy functional associated with the problem. Brown [3] proved the existence of solutions for a semilinear elliptic equation involving the sign-changing weight functions via the Nehari manifold and the fibering method. Papageorgioua et al. [6] studied the existence of positive solutions for a weightedp-Laplacian problem
{−div(ξ(x)|∇u|p−2∇u) =a(x)u−γ+λur x∈Ω,
u= 0 x∈∂Ω,
where Ω is a bounded domain with Lipschitz boundary in RN, 0 < γ < 1 and the differential operator is a weightedp-Laplacian with a weightξ∈L∞(Ω),ξ≥0.
Recently, the existence of two weak solutions for a singular(p, q)-Laplacian type equations with the singular terms is proved (see [1]).
In this paper, we are concerned with a quasi-linear problem, that is, a singular (p, q)-Laplacian elliptic problem
(1)
−∆pu−∆qu+θ(x)up−1=β(x)up−1+λa(x)u−γ+b(x)ur−1 x∈Ω,
u≥0 x∈Ω,
u= 0 x∈∂Ω,
whereΩ ⊂RN is a bounded domain with smooth boundary. The real numbers λ, p, q, γ and r are satisfying the assumptions
λ >0, 0< γ <1, 1< q < p < r < p∗,
where p < N and p∗ := NN p−p. Here, ∆mu = div(|∇u|m−2∇u) is the m-Laplacian operator for m∈ {p, q}. Furthermore,θ∈Lm(Ω)is an indefinite function wherem > Np. The weight functions a, b, β∈L∞(Ω) anda(x), b(x)>0 a.e. inΩ.
∗speaker
1
F. Behboudi and A. Razani 2. Preliminaries
In this section, we recall the necessary preliminaries and notations. Firstly, we recall that the norm in Lebesgue spaceLp(Ω)is
∥u∥Lp(Ω)= ( ∫
Ω
|u(x)|pdx )1p
,
and the Sobolev spaceW01,p(Ω)is the closure ofC0∞(Ω)in W1,p(Ω) endowed with the norm
∥u∥p= ( ∫
Ω
|∇u(x)|pdx )1p
,
for every u in W01,p(Ω). Since Ω is bounded and q < p, we have the continuous embedding W01,p(Ω),→W01,q(Ω)such that
∥u∥q ≤C∥u∥p,
whereu∈W01,p(Ω), for some positive constantC:=C(N, p, q,Ω).
For simplicity, we set X := W01,p(Ω) and X∗ := W0−1,p′(Ω) (the dual space of X) where
1
p +p1′ = 1.
The natural space to study (p, q)-Laplacian problems is Sobolev space W01,p(Ω). Notice that (X,∥.∥p)is a uniformly convex reflexive Banach space.
We consider the generalized eigenvalue problem (2)
{−∆pu−∆qu+θ(x)|u|p−2u=λ|u|p−2u x∈Ω,
u= 0 x∈∂Ω,
whereΩ⊂RN is a bounded domain with smooth boundary,λ∈Rand1< q < p. Alsoθ∈Lm(Ω) form > Np. Recently, the fractional form of the problem (2) is studied in [2].
Proposition 2.1. The problem
{−∆pu+θ(x)|u|p−2u=λ|u|p−2u x∈Ω,
u= 0 x∈∂Ω,
whereλ∈R, admits the first eigenvalueλ1,
(3) λ1:= inf
{ ∫
Ω
|∇u|pdx+
∫
Ω
θ(x)|u|pdx:
∫
Ω
|u|pdx= 1 }
.
Definition 2.2. We say thatu̸= 0,u∈X, is an eigenfunction ofλ1, if the following Euler- Lagrange equation holds for all functionsv∈X
∫
Ω
|∇u|p−2∇u∇vdx+
∫
Ω
θ(x)|u|p−2uvdx=λ1
∫
Ω
|u|p−2uvdx.
Moreover, we define
(4) η1:= inf
{∥u∥pp+p q∥u∥qq+
∫
Ω
θ(x)|u|pdx:
∫
Ω
|u|pdx= 1 }
.
The numberη1 is called the first generalized eigenvalue of (2).
Remark 2.3. Notice thatλ1=η1, where the values ofλ1andη1are given by (3) and (4). In addition the infimum ofη1 is not attained.
Proposition 2.4. [5] Assume that the function β ∈L∞(Ω) with β(x) ≤λ1 a.e. in Ω and meas{x: β(x)< λ1}>0, then there existsc >0 such that
∫
Ω
|∇u|pdx+
∫
Ω
θ(x)|u|pdx−
∫
Ω
β(x)|u|pdx≥c
∫
Ω
|∇u|pdx, for eachu∈X.
2
A SINGULAR (p, q)-LAPLACIAN EQUATION 3. Two solutions
The energy functionalJλ:X→Rassociated with (1) is defined as follows Jλ(u) := 1
p
(∥u∥pp+
∫
Ω
θ(x)|u|pdx−
∫
Ω
β(x)|u|pdx )
+1
q∥u∥qq− λ 1−γ
∫
Ω
a(x)|u|1−γdx−1 r
∫
Ω
b(x)|u|rdx,
for everyu∈X. Notice that the functionalJλ is unbounded from below on the space X. Nehari manifold is a good candidate for a subset ofX such that the functionalJλis bounded on it and is as follows
Nλ:={u∈X\ {0}: ⟨Jλ′(u), u⟩= 0}.
Clearly, critical points ofJλmust lie onNλ. The fibering mapϕ:R+→Rfor the functionalJλis defined byϕu(t) :=Jλ(tu). These maps are introduced by Drabek and Pohozaev in [4]. For every u∈X, we have
ϕu(t) =tp p
(∥u∥pp+
∫
Ω
θ(x)|u|pdx−
∫
Ω
β(x)|u|pdx )
+tq q∥u∥qq
−λt1−γ 1−γ
∫
Ω
a(x)|u|1−γdx−tr r
∫
Ω
b(x)|u|rdx.
Notice thattu∈Nλ if and only ifϕ′u(t) = 0and especiallyu∈Nλ if and only ifϕ′u(1) = 0. We decomposeNλ into three disjoint parts (see [7])
Nλ+={u∈Nλ: ϕ′′u(1)>0}, Nλ0={u∈Nλ: ϕ′′u(1) = 0}, Nλ−={u∈Nλ: ϕ′′u(1)<0}. A computation shows that Jλ is coercive and bounded below on Nλ. Thus we can prove the following lemmas.
Lemma 3.1. Supposeu is a maximum or minimum of Jλ on Nλ and u /∈Nλ0. Then u is a critical point ofJλ.
Lemma 3.2. There existsλ0>0 such that for eachλ∈(0, λ0),Nλ0=∅. For applying the fibering method, we define the functionFu:R+→Rby
Fu(t) :=tp+γ−1
(∥u∥pp+
∫
Ω
θ(x)|u|pdx−
∫
Ω
β(x)|u|pdx ) +tq+γ−1∥u∥qq−tr+γ−1
∫
Ω
b(x)|u|rdx.
Clearly,tu∈Nλif and only if tis a solution of equation Fu(t) =λ
∫
Ω
a(x)|u|1−γdx.
Since∫
Ωb(x)|u|rdx >0, it is clear thatFu(t)→ −∞ast→ ∞,Fu′(t)>0 fortsmall enough and Fu′(t)<0 fortlarge enough. We prove that there exists uniquetmax>0such thatFu′(tmax) = 0.
It is worth noting thatFuis increasing in(0, tmax)and decreasing in(tmax,∞). Then, there exist t1< tmaxandt2> tmaxsuch thatFu(t1) =Fu(t2) =λ∫
Ωa(x)|u|1−γdx. That meanst1u, t2u∈Nλ. Also,Fu′(t1)>0 andFu′(t2)<0 leads tot1u∈Nλ+ andt2u∈Nλ−. We set
mλ:= inf
u∈Nλ
Jλ(u), m+λ := inf
u∈Nλ+
Jλ(u), m−λ := inf
u∈Nλ−
Jλ(u).
By using the properties of fibering maps, we can prove the existence of two positive solutions, which one of them is inNλ+ and the other one is inNλ−.
3
F. Behboudi and A. Razani
Proposition 3.3. There existsˆλ∈(0, λ0] andu∗∈Nλ+ such that Jλ(u∗) =m+λ = inf
Nλ+
Jλ,
for everyλ∈(0,λ). Moreover,ˆ u∗(x)≥0 for everyx∈Ω.
Proposition 3.4. If λ∈(0,λ), then the problemˆ (1)admits a weak positive solution u∗∈X such thatu∗>0 in ΩandJλ(u∗)<0.
We can minimizeJλ on the Nehari manifoldNλ− and get the second non-negative solution.
Proposition 3.5. There exists˜λ∈(0, λ0] andv∗∈Nλ− such that Jλ(v∗) =m−λ = inf
Nλ−
Jλ,
for everyλ∈(0,λ).˜
Proposition 3.6. Ifλ∈(0,λ), then˜ v∗ is a weak solution of problem(1)such thatv∗>0in ΩandJλ(v∗)>0.
Now we state the main result of the paper.
Theorem 3.7. Assume that the function β ≤λ1, where the value of λ1 is given by (3) and meas{x: β(x)≤λ1}>0. Then there exists λ∗>0 such that for every λ∈(0, λ∗), the problem (1) admits at least two weak non-negative non-trivial solutions.
By the above propositions, one can prove the existence of two weak solutions. If we set λ∗= min{λ,ˆ ˜λ}and sinceNλ+∩Nλ− =∅, one can conclude thatu∗ andv∗ are distinct.
References
1. F. Behboudi and A. Razani,Two weak Solutions for a singular (p, q)-Laplacian problem, Filomat, 33 (2019), 3399–3407.
2. F. Behboudi, A. Razani and M. Oveisiha, Existence of a mountain pass solution for a nonlocal fractional(p, q)- Laplacian problem,Bound. Value Probl.2020, Article ID 149 (2020). doi:10.1186/s13661-020-01446-w
3. K. J. Brown and Y. Zhang,The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481–499.
4. P. Drabek and S. I. Pohozaev,Positive solutions for thep-Laplacian: application of the fibering method, Proc.
Roy. Soc. Edinburgh Sect. A, 127 (1997), 703–726.
5. D. A. Kandilakis and M. Magiropoulos,Existence of solutions for(p, q)-Laplacian equations with an indefinite potential, Complex Var. Elliptic Equ., (2019). doi:10.1080/17476933.2019.1631289
6. N. S. Papageorgiou and P. Winkert, Positive solutions for weighted singularp-Laplace equations via Nehari manifolds, Appl. Anal., (2019). doi:10.1080/00036811.2019.1688791
7. G. Tarantello,On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 281–304.
E-mail: f.behboudi@edu.ikiu.ac.ir E-mail: razani@sci.ikiu.ac.ir
4
