This paper concerns singular elliptic Kirchhof’s equations whose nonlinearity has a critical growth and contains an inhomogeneous perturbation in a regular bounded domain of R3. To explore the existence of a ground state solution, we rely on various techniques related to variational methods and the Nehari decomposition

Benmansour, Safia

Matallah, Atika

Yagoub, Nadjet

English

Safia Benmansour

Nadjet Yagoub

Atika Matallah

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A ground state solution for a nonhomogeneous elliptic Kirchhoff type problem involving critical growth and Hardy term

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RAD HAZU. MATEMATIČKE ZNANOSTIVol. 27 = 555 (2023): 133-142DOI: https://doi.org/10.21857/m8vqrtgr09A GROUND STATE SOLUTION FOR ANONHOMOGENEOUS ELLIPTIC KIRCHHOFF TYPEPROBLEM INVOLVING CRITICAL GROWTH AND HARDYTERMSafia Benmansour, Nadjet Yagoub and Atika MatallahAbstract. This paper concerns singular elliptic Kirchhof’s equationswhose nonlinearity has a critical growth and contains an inhomogeneousperturbation in a regular bounded domain of R3. To explore the exis-tence of a ground state solution, we rely on various techniques related tovariational methods and the Nehari decomposition.1. IntroductionIn this work, we explore the existence of a ground state solution for thefollowing Kirchhoff type problem with Dirichlet boundary conditions, a Hardyterm and a critical Sobolev exponent:(Pµ) −(α∫Ω(|∇u|2 − µ u2|x|2)dx+ β)(∆u+ µ u|x|2)= u2∗−1 + g in Ω,u = 0 on ∂Ω,where Ω is a smooth bounded domain of RN (N ≥ 3), µ < µ̄, µ̄ = (N−2)24 ,α, β are positive constants, 2∗ is the critical Sobolev exponent defined by2∗ := 2N/ (N − 2) and g satisfying a suitable condition.This problem is related to the following well known Hardy inequality [3]:(∫Ωu2|x|2dx)1/2≤ 1/µ(∫Ω|∇u|2 dx)1/2for all u ∈ C∞0 (Ω) .2020 Mathematics Subject Classification. 35J20, 35IJ60, 47J30.Key words and phrases. Variational methods, critical growth, Hardy term, singularelliptic problems, Kirchhoff equations.133134 S. BENMANSOUR, N. YAGOUB AND A. MATALLAHWe shall work with the Hilbert space Hµ := Hµ (Ω) for 0 ≤ µ < µequipped with the norm∥u∥2µ :=∫Ω(|∇u|2 − µ u2|x|2)dx,which is equivalent to the norm of H10 (Ω). The space H−1 is the topologicaldual of Hµ endowed with the norm∥f∥− = sup∥u∥µ≤1|f (u)|∥u∥µand ∥u∥6 :=(∫Ω |u|6dx)1/6 is the norm in L6(Ω).Without the Hardy term µ u|x|2, the problem (Pµ) is related to the originalKirchhoff’s equation [8] proposed by Kirchhoff himself in 1883 to describe thetransversal oscillations of a stretched string. He take into account the changesin length of the string produced by transverse vibrations. His model can beconsidered as a generalization of the classical D’Alembert wave equation forfree vibrations of elastic strings. These problems serve also to model otherphysical phenomena as biological systems where u describes a process whichdepends on the average of itself (for example, population density), for moredetails see [1], [2] and the references therein.In the case α = 0 (without nonlocal term) and β = 1, Tarantello [9]established a multiplicity results to (P0) when g satisfies a certain hypothesis.On the other hand, Kang et al. [7] generalized the main result of [9] tothe following singular problem −∆u− µu|x|2= |u|2∗−2 u+ λ u|x|c+ g (x) in Ωu = 0 on ∂Ω,with µ < µ̄ = (N−2)24 , 0 < c < 2, 0 < λ < λ1 (µ) , where λ1 (µ) is the firsteigenvalue of the operator Lµ := −∆ − µ1|x|2. Here, the authors imposed thepresence of the term λ u|x|cwhich provided them with the main tool to obtainthe second solution.A natural and interesting question is whether the results concerning thesolutions in [9] remain valid for a nonlocal operator with a Hardy term. Stim-ulated by [9] and [7], we study our problem and give some positive answer.However, in the case α > 0, the problem becomes more complicated to study,in particular in dimension higher than 3. Furthermore, the main difficultieswhen we investigate our problem are that for this kind of problems with crit-ical exponent, it is not easy to verify that the critical value level is containedNONHOMOGENEOUS CRITICAL SINGULAR KIRCHHOFF EQUATIONS 135in the range where Palais-Smale condition holds, this is the reason why themultiplicity result is not expected. To our best knowledge, the results in thispaper are new.Before stating our result, we give some notation and assumptions. Thebest constant Sµ is defined bySµ := infu∈Hµ\{0}∥u∥2µ(∫Ω |u|6dx)1/3 .By Sobolev inequality, note that Sµ > 0.The weight function g belongs to H−1 and satisfies the following hypoth-esis(gµ)∣∣∣∣∫Ωgudx∣∣∣∣ ≤ Gµ(u), ∀ u ∈ Hµ and ∥u∥6 = 1withGµ(u) :=1105/2[12α2 ∥u∥8µ + 80β ∥u∥2µ + 4α ∥u∥4µ Gµ(u)][3α ∥u∥4µ + Fµ(u)]1/2where Fµ(u) :=√9α2 ∥u∥8µ + 20β ∥u∥2µ.Now, we state our main result.Theorem 1.1. Let N = 3, µ < µ̄, α > 0, β ≥ 0 and g ̸= 0 satisfying(gµ). Then the problem (Pµ) admits a ground state solution.2. Some preliminary resultsWe define the energy functional corresponding to the problem (Pµ) byJµ(u) =α4 ∥u∥4µ +β2 ∥u∥2µ −16 ∥v∥66 −∫Ωgudx, ∀u ∈ Hµ.Note that Jµ ∈ C1( Hµ,R). As the approach is variational so, a weak solutionof problem (Pµ) is a critical point of Jµ i.e. it satisfies:(α ∥u∥2µ + β)(∫Ω(∇u∇v−µ uv|x|2)dx)−∫Ωu5vdx−∫Ωgvdx = 0, ∀v ∈ Hµ.The following definitions and lemmas play crucial roles in the sequel ofthis work.Definition 2.1. A nontrivial solution of the problem (Pµ) is called aground state solution if its energy is less than the energies of all other non-trivial solutions of (Pµ).136 S. BENMANSOUR, N. YAGOUB AND A. MATALLAHDefinition 2.2. A sequence (un) is said to be a Palais-Smale sequenceat level c ((P-S)c in short) for Jµ in Hµ ifJµ(un) = c+ on (1) and J ′µ(un) = on (1) in H−1.We say that Jµ verifies the Palais-Smale condition at level c if any (P-S)csequence for Jµ has a convergent subsequence in Hµ.The functional Jµ is not bounded from below on Hµ but it is on a subsetof Hµ. A good candidate for an appropriate subset of Hµ is the so calledNehari manifold defined byNµ={u ∈ Hµ\{0} :〈J ′µ(u), u〉=0}.So, u belongs to Nµ implies that, α ∥u∥4µ +β ∥u∥2µ = ∥v∥66 +∫Ω gudx. For moredetails about the Nehari decomposition, interested readers can consult [4] and[7]. Let us define its subsetsN +µ := {u ∈ Nµ : h′′u(1) > 0} , N 0µ := {u ∈ Nµ : h′′u(1) = 0}andN −µ := {u ∈ Nµ : h′′u(1) < 0} ,where hu(s) = Jµ(su) for s ∈ R∗ and u ∈ Hµ\{0} and h′′u(s) = −5 ∥u∥66 s4 +3α ∥u∥4µ s2 + β ∥u∥2µ . For more details about these maps see [5].SetHu(s) = h′u(s) +∫Ωgudx = − ∥u∥66 s5 + α ∥u∥4µ s3 + β ∥u∥2µ s.The function Hu(s) attains its maximum G̃µ(u) at the point sumax whereG̃µ(u) := 10−5/2 ∥u∥−96 [12α2 ∥u∥8µ + 80β ∥u∥66 ∥u∥2µ+ 4α ∥u∥4µ F̃µ(u)][3α ∥u∥4µ + F̃µ(u)]1/2andsumax = 101/2 ∥u∥36(3α ∥u∥4µ + F̃µ(u))1/4,withF̃µ(u) := ∥u∥µ(9α2 ∥u∥6µ + 20β ∥u∥66)1/2.Let, for α ≥ 0,ξ̃g := infv∈Hµ\{0}{G̃µ(v) −∣∣∣∣∫Ωgvdx∣∣∣∣} , ξg := inf∥v∥6=1{Gµ(v) −∫Ωgvdx}.To prove that the subsets N +µ and N −µ are not empty, we need the followinglemma.NONHOMOGENEOUS CRITICAL SINGULAR KIRCHHOFF EQUATIONS 137Lemma 2.3. Suppose that the hypothesis (gµ) holds. Then, for any u ∈Hµ\{0}, there exist three unique values s+1 = s+1 (u), s− = s−(u) ̸= 0 ands+2 = s+2 (u) such that:i) s+1 < −sumax, s+1 u ∈ N −µ , and Jµ(s+1 u) = maxs≤−sumaxJµ(su),ii) −sumax < s− < sumax, s−u ∈ N +µ and Jµ(s−u) = min|s|≤sumaxJµ(su)iii) s+2 > sumax, s+2 u ∈ N −µ and Jµ(s+2 u) = maxs≥sumaxJµ(su).The proof of this lemma follows from the fact that Hu(s) is concave.We have for t > 0,Ψ(tu) = tΨ(u), where Ψ(u) = G̃µ(u) −∣∣∣∣∫Ωgudx∣∣∣∣ ,and for a given γ > 0, we derive that(2.1) inf∥v∥6≥γΨ(u) ≥ γξ̃g.In particular, if g satisfies (gµ), this infimum is bounded away from zero.Lemma 2.4. If g satisfies (gµ), then N 0µ = ∅.Proof. If N 0µ ̸= ∅. then, for u ∈ N 0µ we have that(2.2) 3α ∥u∥4µ + β ∥u∥2µ = 5 ∥u∥66 .Thus, we obtainF̃µ(u) = 3α ∥u∥4µ + 2β ∥u∥2µ and (sumax)2 = 1Consequently, we get(2.3)Ψ(u) = G̃µ(u) −∣∣∣∣∫Ωgudx∣∣∣∣ ≤ G̃µ(u) − ∫Ωgudx= Hu(1) −∫Ωgudx = h′u(1) = 0.Condition (2.2) implies that∥v∥6 ≥(β5Sµ)1/4:= γ.From (2.1) and (2.3) we obtain0 < γξ̃g ≤ Ψ(u) = 0,which yields a contradiction.138 S. BENMANSOUR, N. YAGOUB AND A. MATALLAHLemma 2.5. Let u ∈ Nµ. There exist ε > 0 and a differentiable functions : B(0, ε) ⊂ Hµ −→ R+ such that s(0) = 1, s(v)(u−v) ∈ Nµ for ∥v∥ < ϵ and(2.4)⟨s′(0), v⟩ =2(2α ∥u∥2µ + β)(∫Ω(∇u∇v−µuv|x|2 )dx)− 6β∫Ω u5vdx−∫Ω gvdx3α ∥u∥4µ + β ∥u∥2µ − 5 ∥u∥66.Proof. Let F : R ×H → R, defined byF (t, w) = αt3||u− w||4µ + βt||u− w||2µ − t5 ∥u− w∥66 −∫Ωg(u− w)dx.As F (1, 0) = 0, ∂F∂t (1, 0) = 3α ∥u∥4µ + β ∥u∥2µ − 5 ∥u∥66 ̸= 0 then, the proofis obtained by the implicit function theorem at the point (1, 0).Define(2.5) c0 = infv∈N +µJµ (v) and c1 = infv∈N −µJµ (v) .From Lemma 2, we easily deduce that c0 = infu∈NµJµ (u) .Lemma 2.6. The functional Jµ is coercive and bounded from below on Nµand −2548β ∥g∥2_ ≤ c0 < 0.Proof. Let u ∈ Nµ, so α ∥u∥4µ +β ∥u∥2µ = ∥u∥66 +∫Ω gudx. Consequently,we obtainJµ(u) =α12 ∥u∥4µ +β3 ∥u∥2µ −56∫Ωgudx≥ β3 ∥u∥2µ −56 ∥g∥_ ∥u∥µ ,≥ −2548β ∥g∥2_ ,Then, Jµ is coercive and bounded from below on Nµ and c0 ≥ −2548β ∥g∥2_ .Let v ∈ Hµ such that−∆v − µ v|x|2= g.As g ̸≡ 0, thus∫Ω gvdx = ∥v∥2µ = ∥g∥2_ .Let s0 = s−(v), v ∈ Hµ\{0} defined as in Lemma 1. So, s0v ∈ N +µ andconsequently, we have thatJµ(s0v) = −3α4 s40 ∥v∥4µ −β2 s20 ∥v∥2µ +56s60 ∥v∥66≤ −α4 s40 ∥v∥4µ −β3 s20 ∥v∥2µ < 0,thus, c0 < 0.NONHOMOGENEOUS CRITICAL SINGULAR KIRCHHOFF EQUATIONS 139Lemma 2.7. There exist (un) ⊂ N +µ and (vn) ⊂ N −µ verifyingi) Jµ (un) < c0 + 1n and Jµ (w) ≥ Jµ (un) −1n ∥w − un∥µ ∀ w ∈ N+µ .ii) Jµ (vn) < c1 + 1n and Jµ (w) ≥ Jµ (vn) −1n ∥w − vn∥µ ∀ w ∈ N−µ .Proof. From Lemma 4, we deduce that Jµ is coercive and bounded frombelow on Nµ.Applying the Ekeland Variational Principle to the minimizing problem(2.5) (see [6]), we obtain two minimizing sequences (un) ⊂ N +µ and (vn) ⊂ N −µsatisfying i) and ii).Let (un) ⊂ N +µ then, we have thatJµ (un) =α12 ∥un∥4µ +β3 ∥un∥2µ −56∫Ωgundx < c0 +1n≤ −β3 s20 ∥g∥2_ ,which implies that(2.6)∫Ωgundx ≥25βs20 ∥g∥2_ > 0,and consequently, we get(2.7) 25βs20 ∥g∥_ ≤ ∥un∥µ ≤52β ∥g∥_ .Therefore, (un) is bounded in Hµ.Lemma 2.8. Let g satisies (gµ). Then,∥∥J ′µ (un)∥∥µ tends to 0 as n goesto +∞.Proof. If∥∥J ′µ (un)∥∥µ > 0 for n large then, by Lemma 3 with u = unand w = δ J′µ(un)∥J′µ(un)∥µ, δ > 0 small, and we find sn(δ) := s[δJ′µ(un)∥J′µ(un)∥µ], suchthatwδ = sn(δ)[un − δJ ′µ (un)∥∥J ′µ (un)∥∥µ]∈ Nµ.Applying the Ekeland Variational Principle [6], we get1n∥wδ − un∥µ ≥ Jµ (un) − Jµ (wδ) = (1 − sn(δ)) ⟨Jµ(wδ), un⟩+ δsn(δ)〈J ′µ(wδ),J ′µ (un)∥∥J ′µ (un)∥∥〉+ on (δ) .This implies that1n(1 + |s′n(0)| ∥un∥µ) ≥ −s′n(0)〈J ′µ(un), un〉+∥∥J ′µ (un)∥∥µ = ∥∥J ′µ (un)∥∥µ ,140 S. BENMANSOUR, N. YAGOUB AND A. MATALLAHwhere s′n(0) =〈s′(0), J′µ(un)∥J′µ(un)∥µ〉. Thus, from (2.7), we conclude that∥∥J ′µ (un)∥∥µ ≤ Cn (1 + |s′n(0)|) .On the other hand, by (2.4) and (2.7), we have that|s′n(0)| ≤C∣∣∣3α ∥un∥4µ + β ∥un∥2µ − 5 ∥un∥66∣∣∣ .Now, we assume that(2.8) 3α ∥un∥4µ + β ∥un∥2µ − 5 ∥un∥66 = on(1).From (2.7) and (2.8), we derive that∥un∥6 ≥ γ, for some γ > 0.By (2.8) and the fact that un ∈ Nµ, we get∫Ωgundx = −2α ∥un∥4µ + 4 ∥un∥66 + on(1).Considering the definition of ξ̃g and the above equality, we lead to0 < γξ̃g ≤ γ(G̃µ(un) −∫Ωgundx)+ on(1)= γh′un(1) + on(1)= on(1).which is absurd. Thus,∥∥J ′µ (un)∥∥µ tends to 0 as n → ∞.3. Existence of a ground state solutionLet (un) be the minimizing sequence obtained in the Lemma 5, (un) isbounded in Hµ, then, un ⇀ u0 weakly in Hµ. Consequently, we get〈J ′µ (u0) , w〉= 0, ∀w ∈ Hµ.Thus, u0 is a critical point for Jµ.By (2.6), we have that ∫Ωgu0dx > 0.Then, u0 ̸= 0 and u0 ∈ Nµ.Hence, we getc0 ≤ Jµ (u0) =α12 ∥u0∥4µ +β3 ∥u0∥2µ −56∫Ωgu0dx ≤ limn→∞Jµ(un) = c0.Then, c0 = Jµ (u0) and un → u0 in Hµ and necessarily u0 ∈ N +µ .NONHOMOGENEOUS CRITICAL SINGULAR KIRCHHOFF EQUATIONS 141Acknowledgements.This research work is supported by the General Direction of ScientificResearch and Technological Development (DGRSDT), Algeria.References[1] C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinearelliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005) 85–93.[2] P. D’Ancona and Y. Shibata, On global solvability of nonlinear viscoelastic equationsin the analytic category, Math. Methods Appl. Sci. 17 (1994) 477–489.[3] K. S. Chou and C. W. Chu, On the best constant for a weighted Sobolev-Hardy in-equality, J. London Math. Soc. (2) 48 (1993), 137–151.[4] C. Y. Chen, Y. C. Kuo and T. F. Wu, The Nehari manifold for a Kirchhoff typeproblems with critical exponent functions, J. Differential Equations 250 (2011), 1876–1908.[5] P. Drábek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application ofthe fibering method, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 703–726.[6] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353.[7] D. Kang and Y. Deng, Multiple solutions for inhomogeneous elliptic problems involvingcritical Sobolev-Hardy, Nonlinear Anal. 60 (2005), 729–753.[8] G. Kirchhoff, Mechanik, Teubner, Leipzig 1883.[9] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev expo-nent, Ann. Inst. H. Poincaré C Anal. Non Linéaire 9 (1992) 281–304.Rješenje osnovnog stanja za nehomogen eliptički problemKirchhoffovog tipa uključujući kritični rast i Hardyjev članNadjet Yagoub, Safia Benmansour i Atika MatallahSažetak. Ovaj se članak bavi singularnim eliptičkim Kirch-hofovim jednadžbama čija nelinearnost ima kritični rast i sadržinehomogenu perturbaciju u regularnoj ograničenoj domeni od R3.Da bismo istražili postojanje rješenja temeljnog stanja, oslanjamose na razne tehnike povezane s varijacijskim metodama i Nehari-jevom dekompozicijom.142 S. BENMANSOUR, N. YAGOUB AND A. MATALLAHSafia BenmansouEcole supérieure de management de TlemcenLaboratoire d’analyse et contrôle des équations aux dérivées partiellesUniversité Djilali Liabes Sidi Bel Abbès, AlgérieE-mail: safiabenmansour@hotmail.frNadjet YagoubLaboratoire d’analyse et contrôle des équations aux dérivées partiellesUniversité Djilali Liabes Sidi Bel Abbès, AlgérieE-mail: manadjet222@gmail.comAtika MatallahEcole supérieure de management de TlemcenLaboratoire d’analyse et contrôle des équations aux dérivées partiellesUniversité Djilali Liabes Sidi Bel Abbès, AlgérieE-mail: atika_matallah@yahoo.frReceived: 11.10.2022.Revised: 7.1.2023.; 8.2.2023.Accepted: 14.3.2023.

A ground state solution for a nonhomogeneous elliptic Kirchhoff type problem involving critical growth and Hardy term

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