Notations | Used For |
In this paper, we consider the following nonlinear Kirchhoff-type problem with sublinear perturbation and steep potential well
{−(a+b∫R3|∇u|2dx)Δu+λV(x)u=f(x,u)+g(x)|u|q−2u in R3,u∈H1(R3),
where a and b are positive constants, λ>0 is a parameter, 1<q<2, the potential V∈C(R3,R) and V−1(0) has a nonempty interior. The functions f and g are assumed to obey a certain set of conditions. The existence of two nontrivial solutions are obtained by using variational methods. Furthermore, the concentration behavior of solutions as λ→∞ is also explored.
Citation: Shuwen He, Xiaobo Wen. Existence and concentration of solutions for a Kirchhoff-type problem with sublinear perturbation and steep potential well[J]. AIMS Mathematics, 2023, 8(3): 6432-6446. doi: 10.3934/math.2023325
[1] | Marwa M. Tharwat, Marwa M. Ahmed, Ammara Nosheen, Khuram Ali Khan, Iram Shahzadi, Dumitru Baleanu, Ahmed A. El-Deeb . Dynamic inequalities of Grüss, Ostrowski and Trapezoid type via diamond-α integrals and Montgomery identity. AIMS Mathematics, 2024, 9(5): 12778-12799. doi: 10.3934/math.2024624 |
[2] | Andrea Aglić Aljinović, Domagoj Kovačević, Mehmet Kunt, Mate Puljiz . Correction: Quantum Montgomery identity and quantum estimates of Ostrowski type inequalities. AIMS Mathematics, 2021, 6(2): 1880-1888. doi: 10.3934/math.2021114 |
[3] | Mehmet Kunt, Artion Kashuri, Tingsong Du, Abdul Wakil Baidar . Quantum Montgomery identity and quantum estimates of Ostrowski type inequalities. AIMS Mathematics, 2020, 5(6): 5439-5457. doi: 10.3934/math.2020349 |
[4] | Hüseyin Budak, Ebru Pehlivan . Weighted Ostrowski, trapezoid and midpoint type inequalities for RiemannLiouville fractional integrals. AIMS Mathematics, 2020, 5(3): 1960-1984. doi: 10.3934/math.2020131 |
[5] | Erhan Set, Ahmet Ocak Akdemir, Abdurrahman Gözpınar, Fahd Jarad . Ostrowski type inequalities via new fractional conformable integrals. AIMS Mathematics, 2019, 4(6): 1684-1697. doi: 10.3934/math.2019.6.1684 |
[6] | Mustafa Gürbüz, Yakup Taşdan, Erhan Set . Ostrowski type inequalities via the Katugampola fractional integrals. AIMS Mathematics, 2020, 5(1): 42-53. doi: 10.3934/math.2020004 |
[7] | Serap Özcan, Saad Ihsan Butt, Sanja Tipurić-Spužević, Bandar Bin Mohsin . Construction of new fractional inequalities via generalized n-fractional polynomial s-type convexity. AIMS Mathematics, 2024, 9(9): 23924-23944. doi: 10.3934/math.20241163 |
[8] | Humaira Kalsoom, Muhammad Idrees, Artion Kashuri, Muhammad Uzair Awan, Yu-Ming Chu . Some New (p1p2,q1q2)-Estimates of Ostrowski-type integral inequalities via n-polynomials s-type convexity. AIMS Mathematics, 2020, 5(6): 7122-7144. doi: 10.3934/math.2020456 |
[9] | Anjum Mustafa Khan Abbasi, Matloob Anwar . Ostrowski type inequalities for exponentially s-convex functions on time scale. AIMS Mathematics, 2022, 7(3): 4700-4710. doi: 10.3934/math.2022261 |
[10] | Gültekin Tınaztepe, Sevda Sezer, Zeynep Eken, Sinem Sezer Evcan . The Ostrowski inequality for s-convex functions in the third sense. AIMS Mathematics, 2022, 7(4): 5605-5615. doi: 10.3934/math.2022310 |
In this paper, we consider the following nonlinear Kirchhoff-type problem with sublinear perturbation and steep potential well
\begin{eqnarray*} \left \{\begin{array}{ll} -\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\Big)\Delta u+\lambda V(x)u = f(x,u)+g(x)|u|^{q-2}u\ \ \mbox{in}\ \mathbb{R}^3,\\ \\ u\in H^1(\mathbb{R}^3), \\ \end{array} \right. \label{1} \end{eqnarray*}
where a and b are positive constants, λ>0 is a parameter, 1<q<2, the potential V∈C(R3,R) and V−1(0) has a nonempty interior. The functions f and g are assumed to obey a certain set of conditions. The existence of two nontrivial solutions are obtained by using variational methods. Furthermore, the concentration behavior of solutions as λ→∞ is also explored.
In 1938, A. M. Ostrowski introduced the subsequent appealing integral-based inequality to relate the value of a function Ψ with its integral from α to β:
Consider Ψ:[α,β]→R is continuous on [α,β] and differentiable in (α,β), such that Ψ′:(α,β)→R is bounded in (α,β), i.e. ‖Ψ′‖∞:=supˆa∈(α,β)|Ψ′(ˆa)|<∞. Subsequently for any ˆa∈[α,β], we have:
|(β−α)Ψ(ˆa)−β∫αΨ(s)ds|≤[(β−α)24+(ˆa−α+β2)2]‖Ψ′‖∞. | (1.1) |
If the fraction 14 cannot be substituted by any smaller value then it leads to sharpness of the above inequality.
Afterward, several researchers have brought in few findings through the extensions and generalizations of Inequality (1.1). Such inequalities can be employed to guesstimate the inaccuracy of approximation in integration while investigating the steadiness and consistency of statistical calculation [1].
Stefan Hilger [2] developed the calculus of measure chains in 1988. His Ph.D. supervisor Bernd Aulbach described core contributions of this theory of Unification, Extension and Discretization. The concept of time scales is a novelty in applied sciences as well as in mathematics as it enlightens a number of indefinite points about differential equations and solutions of some fractional order differential equations, which have been proved to be inadequate for their solution. Additionally, it has enlarged its contribution to the literature with its applications in areas such as engineering, biostatistics, mathematical biology, functional spaces, optimization theory and dynamic inequalities. The theory of time scales has attracted a great attention of researchers for resolving many problems in analysis. Some dynamical inequalities on time scales can be found in [3,4,5], where researchers established convex function-based inequalities, new Hardy-type inequalities and inequalities for product of different kinds of convex functions respectively using various analytical and theoretical approach on time scales. Furthermore Hu & Wang [6] discussed dynamic inequalities on time scales with applications in permanence of predator-prey system and Saker [7] employed some dynamic inequalities of Opial-type on time scales to prove numerous results related to the spacing between successive zeros of a solution of a second order dynamic equation with a damping term. Bohner and Matthews ([8,Theorem 3.5]) characterized the following relation on Γ as a generalization of the Ostrowski Inequality (1.1) :
Let α,β,ˆu,ˆv∈Γ,α<β, and Ψ:[α,β]∩Γ=[α,β]Γ→R is differentiable. Then we have the inequality:
|Ψ(ˆv)−1β−αβ∫αΨσ(ˆu)Δ(ˆu)|≤Mβ−α(Ω2(ˆv,α)+Ω2(ˆv,β)), |
where M=supα<ˆv<β|ΨΔ(ˆv)|<∞ and [α,β]Γ is a closed interval under Γ. If its R.H.S of Inequality (1.1) cannot be substituted by any smaller value then it leads to sharpness of the above inequality.
Liu & Ngô ([9,Lemma 3.2]) proved the following identity: Let α,β,ˆu,ˆv∈Γ,α<β and Ψ:[α,β]Γ→R is differentiable. Then the following relation for all η∈[0,1] holds.
(1−η)Ψ(ˆv)+ηΨ(α)+Ψ(β)2=1β−αβ∫αΨσ(ˆu)Δˆu+1β−αβ∫αK(ˆv,ˆu)ΨΔ(ˆv)Δˆu, | (1.2) |
where,
K(ˆu,ˆv)={ˆu−α+ηβ−α2,ˆu∈[α,ˆv),ˆu−β−ηβ−α2,ˆu∈[ˆv,β]. |
By making use of (1.2), they have expanded the Inequality (1.1) by considering parameter η and developed the following Ostrowski type inequality involving parameter η ([9,Theorem 3.1]):
Let α,β,ˆu,ˆv∈Γ,α<β, and that Ψ:[α,β]Γ→R is differentiable function. Then the following relation holds true:
|(1−η)Ψ(ˆv)+ηΨ(α)+Ψ(β)2−1β−αβ∫αΨσ(ˆu)Δˆu|≤Mβ−α[Ω2(α,α+ηβ−α2)+Ω2(ˆv,α+ηβ−α2)+Ω2(ˆv,β−ηβ−α2)+Ω2(β,β−ηβ−α2)] | (1.3) |
for all η∈[0,1], such that α+ηβ−α2 and β−ηβ−α2 are in Γ where ˆv∈[α+ηβ−α2,β−ηβ−α2]∩Γ, and M=supα<ˆv<β|ΨΔ(ˆv)|<∞. Its sharpness is conditioned with
η2α(β−α)+η24(β−α)2≤α+ηβ−α2∫αˆuΔˆu. |
Here [α,β]Γ represents a closed interval on Γ.
Xu & Fang [10,Lemma 1] investigated the following identity: Let α,β,ˆu,ˆv∈Γ,α<β,Ψ:[α,β]Γ→R is differentiable, and ζ:[0,1]→[0,1] is given, then we have
H−ζ(η)2Ψ(ˆv)+ζ(η)Ψ(α)+GΨ(β)2=1β−αβ∫αΨσ(ˆu)Δˆu+1β−αβ∫αK(ˆu,ˆv)ΨΔ(ˆu)Δˆu, | (1.4) |
where
K(ˆu,ˆv)={ˆu−α1,ˆu∈[α,ˆv),ˆu−α2,ˆu∈[ˆv,β]. |
By making use of (1.4), they proved Ostrowski Inequality [10,Theorem 1] by using a parametric function and developed the result:
Let α,β,ˆu,ˆv∈Γ,α<β,Ψ:[α,β]Γ→R is differentiable, and ζ:[0,1]→[0,1], then
|G−ζ(η)2Ψ(ˆv)+ζ(η)Ψ(α)+HΨ(β)2−1β−ββ∫αΨΔ(ˆu)Δˆu|≤Mβ−α(Ω2(α,α1)+Ω2(ˆv,α1)+Ω2(ˆv,α2)+Ω2(β,α2)) | (1.5) |
for all η∈[0,1] such that α1 and α2 are in Γ, and t∈[α1,α2]∩Γ, where M=supα<ˆv<β|ΨΔ(ˆv)|<∞. Its sharpness is conditioned with
ζ2(η)−2ζ(η)2α−ζ2(η)2β≥α∫α1ˆuΔˆu. |
We refer [11,12,13] for detail textual-cum-mathematical description on the weighted Ostrowski type inequalities (wOTIs) via time scales. Such as Liu, Tuna and Jiang [11] developed few wOTIs by using weighted Montgomery identity on Γ. Liu and Tuna [13] characterized wOTIs on Γ by utilizing the concept of combined dynamic derivatives on Γ. Several scholars investigated multivariate OTIs on Γ in [14,15,16]. Few authors [17,18,19,20,21,22,23,24,25,26,27,28] utilized various methods to develop OTIs on Γ for functions of two variables. Motivated by the mentioned work, we extend OTIs for bivariate functions, which can be considered as generalizations of OTIs proved by Liu & Ngô, Xu & Fang and by Dragomir et al. [29].
(1) Time scale: It is a closed subset of the real numbers. In present study, it is denoted by the symbol Γ. Examples of time scales include Cantor set, closed intervals, Z.
(2) Forward jump operator: For t∈Γ, the Forward jump operator ρ:Γ→Γ is defined as
σ(ˆv):=inf{ˆu∈Γ:ˆu>ˆv}. |
(3) Γk notation:
Γk={Γ∖(ρ(supΓ),supΓ],supΓ<∞,Γ,supΓ=∞. |
(4) Ψσ notation: If Ψ:Γ→R is a function, then we define a function Ψσ:Γ→R by
Ψσ(ˆv)=Ψ(σ(ˆv)),∀ˆv∈Γ, |
i.e., Ψσ=Ψ∘σ.
(5) rd-continuous function: A function ζ:Γ→R is stated as rd-continuous if it is continuous and its left-sided limits exist at right-dense points and left dense points respectively in Γ. The symbol Crd stands for the family of all such functions.
(6) The delta derivative: Let Ψ:Γ→R with ˆv∈Γk. The term ΨΔ(ˆv) is said to be a number (if exists) if for any ϵ>0, ∃ a neighbourhood U of ˆv (i.e., U=(ˆv−δ,ˆv+δ)∩Γ for any δ>0) such that
|[Ψ(σ(ˆv))−Ψ(ˆu)]−ΨΔ(ˆv)[σ(ˆv)−ˆu]|≤ε|σ(ˆv)−ˆu|forallˆu∈U. |
The value ΨΔ(ˆv) is known as the delta (or Hilger) derivative of Ψ at ˆv. Furthermore, the function Ψ is said to be delta differentiable on Γk if ΨΔ(ˆv) exists ∀ˆv∈Γk.
(7) Integration by parts: If α,β∈Γ and Ψ,ζ∈Crd then
β∫αΨ(σ(ˆv))ζΔ(ˆv)Δˆv=(Ψζ)(β)−(Ψζ)(α)−β∫αΨΔ(ˆv)ζ(ˆv)Δˆv, | (1.6) |
or
β∫αΨ(ˆv)ζΔ(ˆv)Δˆv=(Ψζ)(β)−(Ψζ)(α)−β∫αΨΔ(ˆv)ζ(σ(ˆv))Δˆv. | (1.7) |
(8) Polynomials on time scales: The generalized polynomials are the functions ζk,Ωk:Γ2→R,k∈N0, defined recursively as follow: the functions Ω0 and ζ0 are ζ0(ˆv,ˆu)=Ω0(ˆv,ˆu)=1 for all ˆu,ˆv∈Γ, and, for given ζk and Ωk with k∈N0, the functions ζk+1 and Ωk+1 are ζk+1(ˆv,ˆu)=ˆv∫ˆuζk(σ(τ),ˆu)Δτ, for all ˆu,ˆv∈Γ and Ωk+1(ˆv,ˆu)=ˆv∫ˆuΩk(τ,ˆu)Δτ for all ˆu,ˆv∈Γ. If ΩΔk(ˆv,ˆu) represents the derivative of Ωk(ˆv,ˆu) w.r.t. ˆv for any ˆu, then ΩΔk(ˆv,ˆu)=Ωk−1(ˆv,ˆu),∀k∈N,ˆv∈Γk. Similarly ζΔk(ˆv,ˆu)=ζk−1(σ(ˆv),ˆu)∀k∈N,ˆv∈Γk.
For further study of time scale calculus, readers are referred to [30,31].
Note: In this article, several abbreviations are used in order to lighten the notation and shorten the proofs; we refer to the table at the end just after the section of conclusions. Moreover throughout the paper we consider [a,b]Γ=[a,b]∩Γ.
This section presents the characterization of novel wOTIs on Γ through the generalization of Montgomery identity with parameter functions.
Lemma 2.1. Let α,β,γ,θ,ˆu,ˆv∈Γ with α<β,γ<θ,Ψ:I=[α,β]Γ1×[γ,θ]Γ2→R is differentiable. Assume that the delta derivatives ΨΔΔ(ˆv,ˆu) exist on I and ζ:[0,1]→[0,1]. Then we have the following identity
{Jθ−γ2}β∫αΨσ(ˆv,ˆy)Δˆv +{ζ(η)θ−γ2}β∫αΨσ(ˆv,γ)Δˆv+{Hθ−γ2}β∫αΨσ(ˆv,θ)Δˆv+β∫αk1(ˆx,ˆv){θ∫γΨΔσ(ˆv,ˆu)Δˆu}Δˆv+β∫αθ∫γk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv={J2.(θ−γ)(β−α)4}Ψ(ˆx,ˆy)+{ζ(η)J(θ−γ)(β−α)4}{Ψ(α,ˆy)+Ψ(ˆx,γ)}+{HJ.(θ−γ)(β−α)4}{Ψ(β,ˆy)+Ψ(ˆx,θ)}+{ζ2(η).(θ−γ)(β−α)4}Ψ(α,γ)+{ζ(η)H.(θ−γ)(β−α)4}{Ψ(β,γ)+Ψ(α,θ)}+{H2.(θ−γ)(β−α)4}Ψ(β,θ), | (2.1) |
where
k1(ˆx,ˆv)={ˆv−α1,ˆv∈[α,ˆx);ˆv−α2,ˆv∈[ˆx,β],&k2(ˆy,ˆu)={ˆu−γ1,ˆu∈[γ,ˆy);ˆu−γ2,ˆu∈[ˆy,θ]. |
Proof. Since
β∫αθ∫γk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv=β∫αk1(ˆx,ˆv)[θ∫γk2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)Δˆu]Δˆv. | (2.2) |
Denote
I1=θ∫γk2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)Δˆu=ˆy∫γ[ˆu−γ1]ΨΔΔ(ˆv,ˆu)Δˆu+θ∫ˆy[ˆu−γ2]ΨΔΔ(ˆv,ˆu)Δˆu≐I2+I3. | (2.3) |
By using (1.6), we integrate I2 to get
I2=[ˆy−γ1]ΨΔ(ˆv,ˆy)+ζ(η)θ−γ2)ΨΔ(ˆv,γ)−ˆy∫γΨΔσ(ˆv,ˆu)Δˆu. |
Similarly
I3=[θ−γ2]ΨΔ(ˆv,θ)−[ˆy−γ2]ΨΔ(ˆv,ˆy)−θ∫ˆyΨΔσ(ˆv,ˆu)Δˆu. |
By using I2 and I3 in (2.3), we have
I1=[ˆy−γ−ζ(η)θ−γ2−ˆy+γ+Gθ−γ2]ΨΔ(ˆv,ˆy)+ζ(η)θ−γ2)ΨΔ(ˆv,γ)+[{2−H}θ−γ2]ΨΔ(ˆv,θ)−θ∫γΨΔσ(ˆv,ˆu)Δˆu=[Jθ−γ2]ΨΔ(ˆv,ˆy)+ζ(η)θ−γ2)ΨΔ(ˆv,γ)+[Hθ−γ2]ΨΔ(ˆv,θ)−θ∫γΨΔσ(ˆv,ˆu)Δˆu. |
Use I1 in (2.2) to find
β∫αθ∫γk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv=[Jθ−γ2]I4+{ζ(η)θ−γ2}I5+[Hθ−γ2]I6−β∫αk1(ˆx,ˆv){θ∫γΨΔσ(ˆv,ˆu)Δˆu}Δˆv, | (2.4) |
where I4=β∫αk1(ˆx,ˆv)ΨΔ(ˆv,ˆy)Δˆv, I5=β∫αk1(ˆx,ˆv)ΨΔ(ˆv,γ)Δˆv and I6=β∫αk1(ˆx,ˆv)ΨΔ(ˆv,θ)Δˆv.
Now,
I4=ˆx∫α[ˆv−α1]ΨΔ(ˆv,ˆy)Δˆv+β∫ˆx[ˆv−α2]ΨΔ(ˆv,ˆy)Δˆv≐I7+I8. | (2.5) |
Use (1.6) to find
I7={ˆx−α1}Ψ(ˆx,ˆy)+ζ(η)β−α2)Ψ(α,ˆy)−ˆx∫αΨσ(ˆv,ˆy)Δˆv. |
In similar fashion,
I8={Hβ−α2}Ψ(β,ˆy)−{ˆx−α2}Ψ(ˆx,ˆy)−β∫ˆxΨσ(ˆv,ˆy)Δˆv. |
By adding I7 and I8, we have
I4=[{J}β−α2]Ψ(ˆx,ˆy)+{ζ(η)β−α2}Ψ(α,ˆy)+{Hβ−α2}Ψ(β,ˆy)−β∫αΨσ(ˆv,ˆy)Δˆv. |
Similar calculations for I5 and I6 give
I5=[{J}β−α2]Ψ(ˆx,γ)+{ζ(η)β−α2}Ψ(α,γ)+{Hβ−α2}Ψ(β,γ)−β∫αΨσ(ˆv,γ)Δˆv |
and
I6={Jβ−α2}Ψ(ˆx,θ)+{ζ(η)β−α2}Ψ(α,θ)+{Hβ−α2}Ψ(β,θ)−β∫αΨσ(ˆv,θ)Δˆv. |
Using I4, I5 and I6 in (2.4), we have
β∫αθ∫γk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv=[Jθ−γ2][{Jβ−α2}Ψ(ˆx,ˆy)+{ζ(η)β−α2}Ψ(α,ˆy)+{Hβ−α2}Ψ(β,ˆy)−β∫αΨσ(ˆv,ˆy)Δˆv}]+{ζ(η)θ−γ2}[{Jβ−α2}Ψ(ˆx,γ)+{ζ(η)β−α2}Ψ(α,γ)+{Hβ−α2}Ψ(β,γ)−β∫αΨσ(ˆv,γ)Δˆv]+[Hθ−γ2][{(1−ζ(η)+ζ(1−η))β−α2}Ψ(ˆx,θ)+ζ(η)β−α2Ψ(α,θ)+{Hβ−α2}Ψ(β,θ)−β∫αΨσ(ˆv,θ)Δˆv]−β∫αk1(ˆx,ˆv){θ∫γΨΔσ(ˆv,ˆu)Δˆu}Δˆv. |
Simplification yields
{Jθ−γ2}β∫αΨσ(ˆv,ˆy)Δˆv+{ζ(η)θ−γ2}β∫αΨσ(ˆv,γ)Δˆv+{Hθ−γ2}β∫αΨσ(ˆv,θ)Δˆv+β∫αk1(ˆx,ˆv){θ∫γΨΔσ(ˆv,ˆu)Δˆu}Δˆv+β∫αθ∫γk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv={J2.(θ−γ)(β−α)4}Ψ(ˆx,ˆy)+{ζ(η)J(θ−γ)(β−α)4}{Ψ(α,ˆy)+Ψ(ˆx,γ)}+{HJ.(θ−γ)(β−α)4}{Ψ(β,ˆy)+Ψ(ˆx,θ)}+{ζ2(η).(θ−γ)(β−α)4}Ψ(α,γ)+{ζ(η)H.(θ−γ)(β−α)4}{Ψ(β,γ)+Ψ(α,θ)}+{H2.(θ−γ)(β−α)4}Ψ(β,θ). |
Remark 2.1. If Ψ is single valued function then Eq (2.1) coincides with [10,Lemma 1].
Corollary 2.1. Let α,β,γ,θ,ˆu,ˆv∈Γ with α<β,γ<θ,Ψ:I=[α,β]Γ1×[γ,θ]Γ2→R is differentiable. Assume that the delta derivatives ΨΔΔ(ˆv,ˆu) exist on I and η∈[0,1]. We then have the equation
{(1−η)2(θ−γ)(β−α)}Ψ(ˆx,ˆy)+{η(1−η)(θ−γ)β−α2}[Ψ(α,ˆy)+Ψ(β,ˆy)+Ψ(ˆx,γ)+Ψ(ˆx,θ)]+{η2.(θ−γ)(β−α)4}[Ψ(α,γ)+Ψ(β,γ)+Ψ(α,θ)+Ψ(β,θ)]=(θ−γ)β∫αΨσ(ˆv,θ)Δˆv+β∫αk1(ˆx,ˆv)[θ∫γΨΔσ(ˆv,ˆu)Δˆu]Δˆv+β∫αθ∫γk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv, | (2.6) |
where
k1(ˆx,ˆv)={ˆv−(α+(η)β−α2),ˆv∈[α,ˆx),ˆv−(α+(2−η)β−α2),ˆv∈[ˆx,β],k2(ˆy,ˆu)={ˆu−(γ+(η)θ−γ2),ˆu∈[γ,ˆy),ˆu−(γ+(2−η)θ−γ2),ˆu∈[ˆy,θ]. |
Proof. If we choose ζ(η)=η in Lemma 2.1, we get the required estimate.
Remark 2.2. If Ψ is single valued function then Eq (2.6) coincides with [9,Lemma 3.2].
Theorem 2.1. Suppose that α,β,γ,θ,ˆu,ˆv∈Γ with α<β,γ<θ,Ψ:I=[α,β]Γ1×[γ,θ]Γ2→R is differentiable. Assume that the delta derivatives ΨΔΔ(ˆv,ˆu) exist on I and ζ:[0,1]→[0,1]. We then have the inequality
|J2Ψ(ˆx,ˆy)+ζ(η)J{Ψ(α,ˆy)+Ψ(ˆx,γ)}+HJ{Ψ(β,ˆy)+Ψ(ˆx,θ)}+ζ2(η)Ψ(α,γ)+ζ(η)H{Ψ(β,γ)+Ψ(α,θ)}+{H2Ψ(β,θ)−{J2β−α}β∫αΨσ(ˆv,ˆy)Δˆv}−{ζ(η)2β−α}β∫αΨσ(ˆv,γ)Δˆv−{H2β−α}β∫αΨσ(ˆv,θ)Δˆv|≤4M(β−α)(θ−γ)H2(α,β,ˆx,α1,α2)((θ−γ)+H2(γ,θ,ˆy,γ1,γ2)), | (2.7) |
for all η∈[0,1] such that α1 and α2 are in Γ1, and ˆx∈[α,β]∩Γ1, γ1 and γ2 are in Γ2, ˆy∈[γ,θ]∩Γ2, where M1=Supα<ˆv<βγ<ˆu<θ|ΨΔΔ(ˆv,ˆu)|<∞, M2=Supα<ˆv<βγ<ˆu<θ|ΨΔσ(ˆv,ˆu)|<∞ and M:=Max{M1,M2}.
Proof. By taking absolute value on both sides of (2.1), one yields
|J2Ψ(ˆx,ˆy)+ζ(η)J{Ψ(α,ˆy)+Ψ(ˆx,γ)}+HJ{Ψ(β,ˆy)+Ψ(ˆx,θ)}+ζ2(η)Ψ(α,γ)+ζ(η)H{Ψ(β,γ)+Ψ(α,θ)}+{H2Ψ(β,θ)−{J2β−α}β∫αΨσ(ˆv,ˆy)Δˆv}−{ζ(η)2β−α}β∫αΨσ(ˆv,γ)Δˆv−{H2β−α}β∫αΨσ(ˆv,θ)Δˆv|=4(β−α)(θ−γ)|β∫αk1(ˆx,ˆv){θ∫γΨΔσ(ˆv,ˆu)Δˆu}Δˆv+β∫αθ∫γk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv|. |
Further we have to use |Ψ+ζ|≤|Ψ|+|ζ|, |β∫αΨdˆx|≤β∫α|Ψ|dˆx & |ΨΔσ(ˆv,ˆu)|≤M;|ΨΔΔ(ˆv,ˆu)|≤M;|Ψ.ζ|=|Ψ||ζ| toestimate as following:
|J2Ψ(ˆx,ˆy)+ζ(η)J{Ψ(α,ˆy)+Ψ(ˆx,γ)}+HJ{Ψ(β,ˆy)+Ψ(ˆx,θ)}+ζ2(η)Ψ(α,γ)+ζ(η)H{Ψ(β,γ)+Ψ(α,θ)}+{H2Ψ(β,θ)−{J2β−α}β∫αΨσ(ˆv,ˆy)Δˆv}−{ζ(η)2β−α}β∫αΨσ(ˆv,γ)Δˆv−{H2β−α}β∫αΨσ(ˆv,θ)Δˆv|≤4M(β−α)(θ−γ)((θ−γ){ˆx∫α|k1(ˆx,ˆv)|Δˆv+β∫ˆx|k1(ˆx,ˆv)|Δˆv}+β∫α|k1(ˆx,ˆv)|{ˆy∫γ|k2(ˆy,ˆu)|Δˆu+θ∫ˆy|k2(ˆy,ˆu)|Δˆu}Δˆv)≤4M(β−α)(θ−γ)((θ−γ)H2(α,β,x,α1,α2)+β∫α|k1(ˆx,ˆv)|H2(γ,θ,y,γ1,γ2)Δˆv). |
Simplifications give the required result.
Corollary 2.2. If all the assumptions of Corollary 2.1 hold, then we find the following Ostrowski type inequality for parameter η
|4(1−η)2Ψ(ˆx,ˆy)+2η(1−η)[Ψ(α,ˆy)+Ψ(β,ˆy)+Ψ(ˆx,γ)+Ψ(ˆx,θ)]+η2[Ψ(α,γ)+Ψ(β,γ)+Ψ(α,θ)+Ψ(β,θ)]−4β−αβ∫αΨσ(ˆv,θ)Δˆv|≤4M(β−α)(θ−γ)(Ω2(α,α+ηβ−α2)+Ω2(ˆx,α+ηβ−α2)+Ω2(ˆx,α+(2−η)β−α2)+Ω2(β,α+(2−η)β−α2))((θ−γ)+Ω2(γ,γ+ηθ−γ2)+Ω2(ˆy,γ+ηθ−γ2)+Ω2(ˆy,γ+(2−η)θ−γ2)+Ω2(θ,(γ+(2−η)θ−γ2)). | (2.8) |
Proof. The proof is similar to proof of Theorem 2.1.
Remark 2.3. The inequality (2.8) can be considered as extension of [9,Theorem 3.1]. Since if Ψ is single valued in (2.8), we get [9,Theorem 3.1].
Corollary 2.3. Under the assumptions of Theorem 2.1, we have the following Ostrowski type inequality:
|4(1−η)2Ψ(ˆx,ˆy)+2η2(1−η){Ψ(α,ˆy)+Ψ(ˆx,γ)}+2η(2−η)(1−η){Ψ(β,ˆy)+Ψ(ˆx,θ)}+η4Ψ(α,γ)+η3(2−η){Ψ(β,γ)+Ψ(α,θ)}+η2(2−η)2Ψ(β,θ)−4β−αβ∫α{Ψσ(ˆv,ˆy)+Ψσ(ˆv,γ)+Ψσ(ˆv,θ)}Δˆv|≤4M(β−α)(θ−γ)(Ω2(α,α+η2β−α2)+Ω2(ˆx,α+η2β−α2)+Ω2(ˆx,α+(1+(1−η)2)β−α2)+Ω2(β,α+(1+(1−η)2)β−α2))((θ−γ)+Ω2(γ,γ+η2θ−γ2)+Ω2(ˆy,γ+η2θ−γ2)+Ω2(ˆy,γ+(1+(1−η)2)θ−γ2)+Ω2(θ,(γ+(1+(1−η)2)θ−γ2)). | (2.9) |
Proof. Take ζ(η)=η2 in Theorem 2.1 to meet the requirement.
Remark 2.4. The following Ostrowski type inequalities are obtained by choosing η=0,η=12 and η=1 in Corollary 2.3 respectively:
(a) |Ψ(ˆx,ˆy)−1β−αβ∫α{Ψσ(ˆv,ˆy)+Ψσ(ˆv,γ)+Ψσ(ˆv,θ)}Δˆv|≤ M(β−α)(θ−γ){Ω2(ˆx,α)+Ω2(ˆx,β)}[(θ−γ)+Ω2(ˆy,γ)+Ω2(ˆy,θ)].
(b) |Ψ(ˆx,ˆy)+14{Ψ(α,ˆy)+Ψ(ˆx,γ)}+34{Ψ(β,ˆy)+Ψ(ˆx,θ)}+116Ψ(α,γ)+316{Ψ(β,γ)+Ψ(α,θ)}+916Ψ(β,θ)−4β−αβ∫α{Ψσ(ˆv,ˆy)+Ψσ(ˆv,γ)+Ψσ(ˆv,θ)}Δˆv|≤4M(β−α)(θ−γ)
(Ω2(α,7α+β8)+Ω2(ˆx,7α+β8)+Ω2(ˆx,3α+5β8)+Ω2(β,3α+5β8))((θ−γ)+Ω2(γ,7γ+θ8)+Ω2(ˆy,7γ+θ8)+Ω2(ˆy,3γ+5θ8)+Ω2(θ,3γ+5θ8)). |
(c) |Ψ(α,γ)+Ψ(β,γ)+Ψ(α,θ)+Ψ(β,θ)−4β−αβ∫α{Ψσ(ˆv,ˆy)+Ψσ(ˆv,γ)+Ψσ(ˆv,θ)}Δˆv|≤4M(β−α)(θ−γ)
(Ω2(α,α+β2)+Ω2(ˆx,α+β2)+Ω2(ˆx,α+β2)+Ω2(β,α+β2))((θ−γ)+Ω2(γ,γ+θ2)+Ω2(ˆy,γ+θ2)+Ω2(ˆy,γ+θ2)+Ω2(θ,γ+θ2)). |
Lemma 2.2. Let α,β,γ,θ,ˆu,ˆv∈Γ with α<β,γ<θ,Ψ:I=[α,β]Γ1×[γ,θ]Γ2→R is differentiable. Assume that the delta derivatives ΨΔΔ(ˆv,ˆu) exist on I and ζ1,ζ2:[0,1]→[0,1]. We then have the equation
(J1.J2)Ψ(ˆx,ˆy)+(ζ1(η)J2)Ψ(α,ˆy)+(J2.H1)Ψ(β,ˆy)+(ζ2(μ).J1)Ψ(ˆx,γ)+(ζ1(η).ζ2(μ))Ψ(α,γ)+(ζ2(μ)H1)Ψ(β,γ)+(H2J1)Ψ(ˆx,θ)+(ζ1(η)H2)Ψ(α,θ)+(H1H2)Ψ(β,θ)−(J22β−α).β∫αΨσ(ˆv,ˆy)Δˆv−(ζ2(μ)2β−α).β∫αΨσ(ˆv,γ)Δˆv−(H22β−α)β∫αΨσ(ˆv,θ)Δˆv=4(β−α)(θ−γ)[β∫αk1(ˆx,ˆv)(θ∫γΨΔσ(ˆv,ˆu)Δˆu)Δˆv+β∫αθ∫γk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv], | (2.10) |
where
k1(ˆx,ˆv)={ˆv−e1,ˆv∈[α,ˆx);ˆv−e2,ˆv∈[ˆx,β].k2(ˆy,ˆu)={ˆu−e3,ˆu∈[γ,ˆy);ˆu−e4,ˆu∈[ˆy,θ]. |
Proof. It can easily be proved by following the steps of Lemma 2.1.
Remark 2.5. If ζ1(η) = ζ2(μ) in Lemma 2.2, it becomes Lemma 2.1.
Theorem 2.2. Suppose that α,β,γ,θ,ˆu,ˆv∈Γ with α<β,γ<θ,Ψ:I=[α,β]Γ1×[γ,θ]Γ2→R is differentiable. Assume that the delta derivatives ΨΔΔ(ˆv,ˆu) exist on I and ζ1,ζ2:[0,1]→[0,1]. We then have the inequality
|{J1.J2}Ψ(ˆx,ˆy)+{ζ1(η)J2}Ψ(α,ˆy)+{J2.H1}Ψ(β,ˆy)+{ζ2(μ).J1}Ψ(ˆx,γ)+{ζ1(η).ζ2(μ)}Ψ(α,γ)+{ζ2(μ)H1}Ψ(β,γ)+{H2J1}Ψ(ˆx,θ)+{ζ1(η)H2}Ψ(α,θ)+{H1H2}Ψ(β,θ)−{(1−ζ2(μ)+ζ2(1−μ))2β−α}β∫αΨσ(ˆv,ˆy)Δˆv}−{ζ2(μ)2β−α}β∫αΨσ(ˆv,γ)Δˆv−{H22β−α}β∫αΨσ(ˆv,θ)Δˆv|≤4M(β−α)(θ−γ)H2(α,β,x,e1,e2)((θ−γ)+H2(γ,θ,y,e3,e4)). | (2.11) |
Proof. By using Lemma 2.2 and adopting the technique of proof of Theorem 2.1, we get the desired result.
Remark 2.6. In similar fashion, remaining results of Section 2.2 can be extended for (2.11).
Some important examples of time scales include continuous time scale Γ=R (set of all real numbers, which gives rise to differential equations), discrete time scale Z (set of integers, which gives rise to difference equations) and quantum time Scale qN0,q>1. In this section we have discussed Ostrowski type Inequality (2.8) for these special time scales.
Example 3.1. If we take Γ1=Γ2=R, then the delta integral is the usual Riemann integral i.e. β∫αΨσ(ˆv)Δˆv=β∫αΨ(ˆv)dˆv as σ(ˆv)=ˆv. In this case, the generalized polynomial Ω2 is
Ω2(ˆv,ˆu)=(ˆv−ˆu)22forallˆu,ˆv∈R, |
which implies the following relations:
Ω2(α,α+ηβ−α2)=(α−α−ηβ−α2)22=η2(β−α)28, |
Ω2(ˆx,α+ηβ−α2)=4(ˆx−α)2+η2(β−α)2−4η(ˆx−α)(β−α)8, |
Ω2(ˆx,α+(2−η)β−α2)=4(ˆx−α)2+(2−η)2(β−α)2−4(ˆx−α)(2−η)(β−α)8, |
Ω2(β,α+(2−η)β−α2)=(β−α)2η28, |
Ω2(γ,γ+ηθ−γ2)=η2(θ−γ)28, |
Ω2(ˆy,γ+ηθ−γ2)=4(ˆy−γ)2+η2(θ−γ)2−4η(ˆy−γ)(θ−γ)8, |
Ω2(ˆy,γ+(2−η)θ−γ2)=4(ˆy−γ)2+(2−η)2(θ−γ)2−4(ˆy−γ)(2−η)(θ−γ)8, |
Ω2(θ,γ+(2−η)θ−γ2)=(θ−γ)2η28. |
The Eq (2.8) takes the following form
|4(1−η)2Ψ(ˆx,ˆy)+2η(1−η)[Ψ(α,ˆy)+Ψ(β,ˆy)+Ψ(ˆx,γ)+Ψ(ˆx,θ)]+η2[Ψ(α,γ)+Ψ(β,γ)+Ψ(α,θ)+Ψ(β,θ)]−4β−αβ∫αΨ(ˆv,θ)dˆv|≤M(β−α)(θ−γ)((β−α)2(η2−η)+(α−ˆx)2+(β−ˆx)2)(2(θ−γ)+(θ−γ)2(η2−η)+(γ−ˆy)2+(θ−ˆy)2). | (3.1) |
Remark 3.1. If Ψ is single valued function then assumptions and calculations made in Example 3.1 coincide with [29,Theorem 2].
Example 3.2. Using Γ1=Γ2=Z,α=0=γ,β=n,θ=m,s=j,t=i,Ψ(p,q)=ˆxpˆyq,p=k,q=l and Ψσ(p,q)=σ(ˆxpˆyq)=ˆxp+1ˆyq+1; β∫αΨσ(ˆv,θ)Δˆv=4nn−1∑t=0ˆxt+1ˆym in Eq (2.8), with the known result
Ω2(ˆv,ˆu)=(ˆv−ˆu)!2!(ˆv−s−2)!=(ˆv−ˆu)(ˆv−s−1)2=(ˆv−ˆu2),forallˆu,ˆv∈Z, |
we have
|4(1−η)2ˆxkˆyl+2η(1−η)[ˆx0ˆyl+ˆxnˆyl+ˆxkˆy0+ˆxkˆym]+η2[ˆx0ˆy0+ˆxnˆy0+ˆx0ˆym+ˆxnˆym]−4nn−1∑i=0ˆxi+1ˆym|≤M4mn(n2(η2−η+1)+2k(k−n−1)+n)(5m+4m2(η2−η+1)+8l(l−m−1)). |
Example 3.3. If we take Γ1=Γ2=qN0,q>1,α=γ=qm,β=θ=qn,m<n,η=1 in Eq (2.8), we have
|Ψ(qm,qm)+Ψ(qn,qm)+Ψ(qm,qn)+Ψ(qn,qn)−4qn−qmqn∫qmΨσ(ˆv,qn)Δˆv|≤M(qn−qm)2.14(1+q)2((qm−qn)(qm(2−q)−qn+1)+2(2ˆx−qn−qm)(2ˆx−qn+1−qm+1)+(qn−qm)(qn(2−q)−qm+1))(4(qn−qm)(1+q)+(qm−qn)(qm(2−q)−qn+1)+2(2ˆy−qn−qm)(2ˆy−qn+1−qm+1)+(qn−qm)(qn(2−q)−qm+1)). |
Remark 3.2. If Ψ is single valued then Example 3.1 to Example 3.3 coincide with [9,Corollaries 3.6–3.8]. Furthermore, it is also possible to reset Eq (2.7) instead of Eq (2.8) for these particular time scales, which will be extensions of [10,Corollary 1,2] in case of continuous and discrete time scales.
In this study, a novel approach is employed for the establishment of Ostrowski type integral inequalities for double integrals via Montgomery identity under the setting of time scales calculus. In addition, certain generalizations are made for some weighted and parameterized functions. Moreover, some particular cases, applications and examples are discussed for some specific time scales. It is also worth mentioning that the results of the paper extend the results of [9,10,29]. Further extensions can be sought by the expansion of this proposed study for multiple integrals.
Following notations have been used in the paper for vivid understanding of the concept:
Notations | Used For |
This research received funding support from the NSRF via the Program Management Unit for Human Resources and Institutional Development, Research and Innovation (grant number B05F640088).
The authors declare that there is no conflict of interests regarding the publication of this paper.
[1] |
T. Bartsch, Z. Q. Wang, Existence and multiplicity results for superlinear elliptic problems on RN, Commun. Part. Differ. Eq., 20 (1995), 1725–1741. https://doi.org/10.1080/03605309508821149 doi: 10.1080/03605309508821149
![]() |
[2] |
T. Bartsch, A. Pankov, Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549–569. https://doi.org/10.1142/S0219199701000494 doi: 10.1142/S0219199701000494
![]() |
[3] |
Y. H. Ding, A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Dif., 29 (2007), 397–419. https://doi.org/10.1007/s00526-006-0071-8 doi: 10.1007/s00526-006-0071-8
![]() |
[4] | G. Kirchhoff, Mechanik, Leipzig: Teubner, 1883. |
[5] | A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Am. Math. Soc., 348 (1996), 305–330. |
[6] |
M. Chipot, B. Lovat, Some remarks on non local elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619–4627. https://doi.org/10.1016/S0362-546X(97)00169-7 doi: 10.1016/S0362-546X(97)00169-7
![]() |
[7] |
Y. H. Li, F. Y. Li, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differ. Equ., 253 (2012), 2285–2294. https://doi.org/10.1016/j.jde.2012.05.017 doi: 10.1016/j.jde.2012.05.017
![]() |
[8] |
G. B. Li, H. Y. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3, J. Differ. Equ., 257 (2014), 566–600. https://doi.org/10.1016/j.jde.2014.04.011 doi: 10.1016/j.jde.2014.04.011
![]() |
[9] |
Z. P. Wang, H. S. Zhou, Positive solutions for nonlinear Schrödinger equations with deepening potential well, J. Eur. Math. Soc., 11 (2009), 545–573. https://doi.org/10.4171/JEMS/160 doi: 10.4171/JEMS/160
![]() |
[10] |
H. Y. Ye, Positive high energy solution for Kirchhoff equation in R3 with superlinear nonlinearities via Nehari-Pohozaev manifold, Discret. Contin. Dyn. Syst., 35 (2015), 3857–3877. https://doi.org/10.3934/dcds.2015.35.3857 doi: 10.3934/dcds.2015.35.3857
![]() |
[11] |
Z. S. Liu, S. J. Guo, Positive solutions for asymptotically linear Schrödinger-Kirchhoff-type equations, Math. Meth. Appl. Sci., 37 (2014), 571–580. https://doi.org/10.1002/mma.2815 doi: 10.1002/mma.2815
![]() |
[12] |
Y. S. Jiang, H. S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differ. Equ., 251 (2011), 582–608. http://dx.doi.org/10.1016/j.jde.2011.05.006 doi: 10.1016/j.jde.2011.05.006
![]() |
[13] |
L. G. Zhao, H. D. Liu, F. K. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differ. Equ., 255 (2013), 1–23. https://doi.org/10.1016/j.jde.2013.03.005 doi: 10.1016/j.jde.2013.03.005
![]() |
[14] |
H. Zhang, F. B. Zhang, Ground states for the nonlinear Kirchhoff type problems, J. Math. Anal. Appl., 423 (2015), 1671–1692. http://dx.doi.org/10.1016/j.jmaa.2014.10.062 doi: 10.1016/j.jmaa.2014.10.062
![]() |
[15] |
Z. J. Guo, Ground states for Kirchhoff equations without compact condition, J. Differ. Equ., 259 (2015), 2884–2902. http://dx.doi.org/10.1016/j.jde.2015.04.005 doi: 10.1016/j.jde.2015.04.005
![]() |
[16] |
C. T. Ledesma, Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well, Commun. Pur. Appl. Anal., 15 (2016), 535–547. https://doi.org/10.3934/cpaa.2016.15.535 doi: 10.3934/cpaa.2016.15.535
![]() |
[17] |
Y. H. Li, Q. Geng, The existence of nontrivial solution to a class of nonlinear Kirchhoff equations without any growth and Ambrosetti-Rabinowitz, Appl. Math. Lett., 96 (2019), 153–158. http://dx.doi.org/10.1016/j.aml.2019.04.027 doi: 10.1016/j.aml.2019.04.027
![]() |
[18] |
M. Du, L. X. Tian, J. Wang, F. B. Zhang, Existence of ground state solutions for a super-biquadratic Kirchhoff-type equation with steep potential well, Appl. Anal., 95 (2016), 627–645. https://doi.org/10.1080/00036811.2015.1022312 doi: 10.1080/00036811.2015.1022312
![]() |
[19] |
D. Q. Zhang, G. Q. Chai, W. M. Liu, Existence and concentration of solutions for the nonlinear Kirchhoff type equations with steep well potential, Bound. Value Probl., 2017 (2017), 142. https://doi.org/10.1186/s13661-017-0875-9 doi: 10.1186/s13661-017-0875-9
![]() |
[20] |
F. B. Zhang, M. Du, Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well, J. Differ. Equ., 269 (2020), 10085–10106. https://doi.org/10.1016/j.jde.2020.07.013 doi: 10.1016/j.jde.2020.07.013
![]() |
[21] |
J. T. Sun, T. F. Wu, Steep potential well may help Kirchhoff type equations to generate multiple solutions, Nonlinear Anal., 190 (2020), 111609. https://doi.org/10.1016/j.na.2019.111609 doi: 10.1016/j.na.2019.111609
![]() |
[22] |
L. Zhou, C. X. Zhu, Existence and asymptotic behavior of ground state solutions to Kirchhoff-type equations of general convolution nonlinearity with a steep potential well, Mathematics, 10 (2022), 812. https://doi.org/10.3390/math10050812 doi: 10.3390/math10050812
![]() |
[23] |
D. Choudhuri, Existence and Hölder regularity of infinitely many solutions to a p-Kirchhoff type problem involving a singular nonlinearity without the Ambrosetti-Rabinowitz (AR) condition, Z. Angew. Math. Phys., 72 (2021), 36. https://doi.org/10.1007/s00033-020-01464-9 doi: 10.1007/s00033-020-01464-9
![]() |
[24] | M. Schechter, A variation of the mountain pass lemma and applications, J. London Math. Soc., 44 (1991), 491–502. https://doi.org/101112/jlms/s2-44.3.491 |
[25] |
J. T. Sun, T. F. Wu, Multiplicity and concentration of homoclinic solutions for some second order Hamiltonian systems, Nonlinear Anal., 114 (2015), 105–115. https://doi.org/10.1016/j.na.2014.11.009 doi: 10.1016/j.na.2014.11.009
![]() |
[26] | M. Willem, Minimax theorems, Boston: Birkhäuser, 1996. |
[27] | R. K. Giri, D. Choudhuri, S. Pradhan, Existence and concentration of solutions for a class of elliptic PDEs involving p-biharmonic operator, Mat. Vestn., 70 (2018), 147–154. |
Notations | Used For |
Notations | Used For |