The bifurcation diagram of an elliptic Kirchhoff-type equation with respect to the stiffness of the material
aa r X i v : . [ m a t h . A P ] A p r THE BIFURCATION DIAGRAM OF AN ELLIPTIC KIRCHHOFF-TYPEEQUATION WITH RESPECT TO THE STIFFNESS OF THE MATERIAL
KAYE SILVA
Abstract.
We study a superlinear and subcritical Kirchhoff type equation which isvariational and depends upon a real parameter λ . The nonlocal term forces some of thefiber maps associated with the energy functional to have two critical points. This suggestmultiplicity of solutions and indeed we show the existence of a local minimum and a mountainpass type solution. We characterize the first parameter λ ∗ for which the local minimum hasnon-negative energy when λ ≥ λ ∗ . Moreover we characterize the extremal parameter λ ∗ forwhich if λ > λ ∗ , then the only solution to the Kirchhoff equation is the zero function. In fact, λ ∗ can be characterized in terms of the best constant of Sobolev embeddings. We also studythe asymptotic behavior of the solutions when λ ↓ Introduction
In this work we study the following Kirchhoff type equation(1.1) − (cid:18) a + λ Z |∇ u | (cid:19) ∆ u = | u | γ − u in Ω ,u = 0 on ∂ Ω , where a > λ > ⊂ R is a boundedregular domain. Let H (Ω) denote the standard Sobolev space and Φ λ : H (Ω) → R the energyfunctional associated with (1.1), that is(1.2) Φ λ ( u ) = a Z |∇ u | + λ (cid:18)Z |∇ u | (cid:19) − γ Z | u | γ . We observe that Φ λ is a C functional. By definition a solution to equation (1.1) is a criticalpoint of Φ λ . Our main result is the following Theorem 1.1.
Suppose γ ∈ (2 , . Then there exist parameters < λ ∗ < λ ∗ and ε > suchthat:1) For each λ ∈ (0 , λ ∗ ] problem (1.1) has a positive solution u λ which is a global minimizerfor Φ λ when λ ∈ (0 , λ ∗ ] , while u λ is a local minimizer for Φ λ when λ ∈ ( λ ∗ , λ ∗ ) .Moreover Φ ′′ λ ( u λ )( u λ , u λ ) > for λ ∈ (0 , λ ∗ ) and Φ ′′ λ ∗ ( u λ ∗ )( u λ ∗ , u λ ∗ ) = 0 .2) For each λ ∈ (0 , λ ∗ + ε ) problem (1.1) has a positive solution w λ which is a mountainpass critical point for Φ λ .3) If λ ∈ (0 , λ ∗ ) then Φ λ ( u λ ) < while Φ λ ∗ ( u λ ∗ ) = 0 and if λ ∈ ( λ ∗ , λ ∗ ] then Φ λ ( u λ ) > .4) Φ λ ( w λ ) > and Φ λ ( w λ ) > Φ λ ( u λ ) for each λ ∈ (0 , λ ∗ + ε ) .5) If λ > λ ∗ then the only solution u ∈ H (Ω) to the problem (1.1) is the zero function u = 0 . Mathematics Subject Classification.
Primary 35A02, 35A15, 35B32,
Key words and phrases.
Nehari Manifold, Variational Methods, Extremal Parameter, Kirchhoff.
Kirchhoff type equations have been extensively studied in the literature. It was proposed byKirchhoff in [5] as an model to study some physical problems related to elastic string vibrationsand since then it has been studied by many author, see for example the works of Lions [6],Alves et al. [1], Wu et al. [2], Zhang and Perera [14] and the references therein. Physicallyspeaking if one wants to study string or membrane vibrations, one is led to the equation (1.1),where u represents the displacement of the membrane, | u | p − u is an external force, a and λ arerelated to some intrinsic properties of the membrane. In particular, λ is related to the Youngmodulus of the material and it measures its stiffness.Our main interest here is to analyze equation (1.1) with respect to the parameter λ (stiffness)and provide a description of the bifurcation diagram. To this end, we will use the fiberingmethod of Pohozaev [9] to analyse how the Nehari set (see Nehari [7, 8]) change with respectto the parameter λ and then apply this analysis to study bifurcation properties of the problem(1.1) (see Chen et al. [2] and Zhang et al. [13]). In fact, the extremal parameter λ ∗ (seeIl’yasov [3]) which appears in the Theorem 1.1 can be characterized variationally by λ ∗ = C a,γ sup (cid:0)R | u | γ (cid:1) γ (cid:0)R |∇ u | (cid:1) γγ − : u ∈ H (Ω) \ { } , where C a,γ is some positive constant. One can easily see from the last expression that λ ∗ = C a,γ S γ − γ γ , where S γ is best Sobolev constant for the embedding H (Ω) ֒ → L γ (Ω).In this work the extremal parameter λ ∗ has the important role that if λ > λ ∗ then theNehari set is empty while if λ ∈ (0 , λ ∗ ) then the Nehari set is not empty. Another interestingparamenter is λ ∗ < λ ∗ which is characterized by the property that if λ ∈ (0 , λ ∗ ), theninf u ∈ H (Ω) Φ λ ( u ) < λ ≥ λ ∗ the infimum is zero. When λ ∈ (0 , λ ∗ ] one can easilyprovide a Mountain Pass Geometry and a global minimizer for the functional Φ λ . Althoughhere we characterize λ ∗ variationally, one can see that the parameter a ∗ defined in Theorem 1.3(ii) of Sun and Wu [12] serves to the same purpose as λ ∗ and hence our result for λ ∈ (0 , λ ∗ ) isnot new, however, when λ > λ ∗ we could not find this result in the literature and in this casewe need to provide some finer estimates on the Nehari sets in order to solve some technicalissues to obtain again a Mountain Pass Geometry and a local minimizer for the functional Φ λ .The hypothesis γ ∈ (2 ,
4) has the fundamental role that it forces the problem to besuperlinear, subcritical and it allows the existence of fiber maps with two critical points. Theexistence of these kinds of fiber maps implies multiplicity of solutions (at least two solutions)and once for λ > λ ∗ there is no solution at all, the parameter λ ∗ is a bifurcation point wherethese solutions collapses. We refer the reader to the recently works of Siciliano and Silva [10],Il’yasov and Silva [4], Silva and Macedo [11], where the extremal parameters of some indefinitenonlinear elliptic problems were analyzed.Concerning the asymptotic behavior of the solutions when λ ↓ Theorem 1.2.
There holdsi) Φ λ ( u λ ) → −∞ and k u λ k → ∞ as λ ↓ .ii) w λ → w in H (Ω) where w ∈ H (Ω) is a mountain pass critical point associated tothe equation − a ∆ w = | w | p − w . This work is organized as follows: In Section 2 we provide some definitions and provetechnical results which will be used in the next sections. In Section 3 we show the existence oflocal minimizers for the functional Φ λ . In Section 4 we prove the existence of a mountain passcritical point for the functional Φ λ . In Section 5 we prove Theorem 1.1. In Section 6 we prove XTREMAL PARAMETERS OF A KIRCHHOFF TYPE EQUATION 3
Theorem 1.2. In Section 7 we provide a picture detailing the bifurcation diagram with respectto the energy and make some conjectures and in the Appendix we prove some auxiliary results.2.
Technical Results
We denote by k u k the standard Sobolev norm on H (Ω) and k u k γ the L γ (Ω) norm. It followsfrom (1.2) that Φ λ ( u ) = a k u k + λ k u k − γ k u k γγ , ∀ u ∈ H (Ω) . For each λ > N λ = { u ∈ H (Ω) \ { } : Φ ′ λ ( u ) u = 0 } . To study the Nehari set we will make use of the fiber maps: for each λ > u ∈ H (Ω) \ { } define ψ λ,u : (0 , ∞ ) → R by ψ λ,u ( t ) = Φ λ ( tu ) . It follows that N λ = { u ∈ H (Ω) \ { } : ψ ′ λ,u (1) = 0 } . We divide the Nehari set into three disjoint sets as follows: N λ = N + λ ∪ N λ ∪ N − λ , where N + λ = { u ∈ H (Ω) \ { } : ψ ′ λ,u (1) = 0 , ψ ′′ λ,u (1) > } , N λ = { u ∈ H (Ω) \ { } : ψ ′ λ,u (1) = 0 , ψ ′′ λ,u (1) = 0 } , and N − λ = { u ∈ H (Ω) \ { } : ψ ′ λ,u (1) = 0 , ψ ′′ λ,u (1) < } . By using the Implicit Function Theorem one can prove the following
Lemma 2.1. If N + λ , N − λ are non empty then N + λ , N − λ are C manifolds of codimension in H (Ω) . Moreover if u ∈ N + λ ∪ N − λ is a critical point of (Φ λ ) |N + λ ∪N − λ , then u is a critical pointof Φ λ . In order to understand the Nehari set N λ we study the fiber maps ψ λ,u . Simple Analysisarguments show that Proposition 2.2.
For each λ > and u ∈ H (Ω) \ { } , there are only three possibilities forthe graph of ψ λ,u I) The function ψ λ,u has only two critical points, to wit, < t − λ ( u ) < t + λ ( u ) . Moreover, t − λ ( u ) is a local maximum with ψ ′′ λ,u ( t − λ ( u )) < and t + λ ( u ) is a local minimum with ψ ′′ λ,u ( t + λ ( u )) > ;II) The function ψ λ,u has only one critical point when t > at the value t λ ( u ) . Moreover, ψ ′′ λ,u ( t λ ( u )) = 0 and ψ λ,u is increasing;III) The function ψ λ,u is increasing and has no critical points. It follows from Proposition 2.2 that N + λ , N − λ are non empty if and only if the item I ) issatisfied. Therefore, it remains to show whether I ) is satisfied or not. For this purpose westudy for what values of λ there holds N λ = ∅ . Note that tu ∈ N λ for t > u ∈ H (Ω) \{ } if and only if ( ψ ′ λ,u ( t ) = 0 ,ψ ′′ λ,u ( t ) = 0 , K. SILVA or equivalently(2.1) ( a k u k + λ k u k t − k u k γγ t γ − = 0 ,a k u k + 3 λ k u k t − ( γ − k u k γγ t γ − = 0 . We solve the system (2.1) with respect to the variable ( t, λ ) to obtain for each u ∈ H (Ω) \ { } a unique pair ( t ( u ) , λ ( u )) such that(2.2) t ( u ) = (cid:18) a − γ k u k k u k γγ (cid:19) γ − , (2.3) λ ( u ) = C a,γ (cid:18) k u k γ k u k (cid:19) γγ − , where C a,γ = a (cid:18) γ − − γ (cid:19) (cid:18) − γ a (cid:19) γ − . We define the extremal parameter (see Il’yasov [3])(2.4) λ ∗ = sup u ∈ H (Ω) \{ } λ ( u ) . We also consider another parameter which is defined as a solution of the system ( ψ λ,u ( t ) = 0 ,ψ ′ λ,u ( t ) = 0 , or equivalently(2.5) a k u k + λ t k u k − γ t γ − k u k γγ = 0 ,a k u k + λt k u k − t γ − k u k γγ = 0 . Similar to the system (2.1) we can solve the system (2.5) with respect to the variable ( t, λ ) tofind a unique pair ( t ( u ) , λ ( u )). Moreover, one can easily see that λ ( u ) = C ,a,γ λ ( u ) , ∀ u ∈ H (Ω) \ { } , where C ,a,γ = 2 (cid:18) γ (cid:19) γ − . Observe that C ,a,γ <
1. We define(2.6) λ ∗ = sup u ∈ H (Ω) \{ } λ ( u ) . The functions λ ( u ) and λ ( u ) has the following geometrical interpretation Proposition 2.3.
For each u ∈ H (Ω) \ { } there holdsi) λ ( u ) is the unique parameter λ > for which the fiber map ψ λ,u has a critical pointwith second derivative zero at t ( u ) . Moreover, if < λ < λ ( u ) , then ψ λ,u satisfies I) ofProposition 2.2 while if λ > λ ( u ) , then ψ λ,u satisfies III) of Proposition 2.2.ii) λ ( u ) is the unique parameter λ > for which the fiber map ψ λ,u has a critical pointwith zero energy at t ( u ) . Moreover, if < λ < λ ( u ) , then inf t> ψ λ,u ( t ) < while if λ > λ ( u ) , then inf t> ψ λ,u ( t ) = 0 . XTREMAL PARAMETERS OF A KIRCHHOFF TYPE EQUATION 5
Proof. i ) The uniqueness of λ ( u ) comes from equation (2.1). Assume that λ ∈ (0 , λ ( u )), then ψ λ,u must satisfies I ) or III ) of Proposition 2.2. We claim that it must satisfies I ). Indeed,suppose on the contrary that it satisfies III ). Once ψ ′ λ ( u ) ,u ( t ) > ψ ′ λ,u ( t ) > , ∀ t > , we reach a contradiction since ψ ′ λ ( u ) ,u ( t ( u )) = 0 where t ( u ) is given by (2.2), therefore ψ λ,u must satisfies I ). Now suppose that λ > λ ( u ), then ψ ′ λ,u ( t ) > ψ ′ λ ( u ) ,u ( t ) ≥ , ∀ t > , and hence ψ λ,u must satisfies III ). ii ) The uniqueness of λ ( u ) comes from equation (2.5). If 0 < λ < λ ( u ) then from thedefinition we have ψ λ,u ( t ( u )) < ψ λ ( u ) ,u ( t ( u )) = 0 , which implies that inf t> ψ λ,u ( t ) <
0. If λ > λ ( u ) then ψ λ,u ( t )) > ψ λ ( u ) ,u ( t ) ≥ , ∀ t > , and therefore inf t> ψ λ,u ( t ) = ψ λ,u (0) = 0. (cid:3) Now we turn our attention to the parameters λ ∗ and λ ∗ . Proposition 2.4.
There holds λ ∗ < λ ∗ < ∞ . Moreover, there exists u ∈ H (Ω) \ { } suchthat λ ( u ) = λ ∗ and λ ( u ) = λ ∗ .Proof. Indeed, from the Sobolev embedding it follows that λ , λ ∗ < ∞ . Now observe that λ ( u )is 0-homogeneous, that is λ ( tu ) = λ ( u ) for each t >
0. It follows that there exists a sequence u n ∈ H (Ω) \ { } such that k u n k = 1 and λ ( u n ) → λ ∗ as n → ∞ . We can assume that u n ⇀ u in H (Ω) and u n → u in L γ (Ω). Moreover, from (2.3) it follows that u = 0. We conclude that λ (cid:18) u k u k (cid:19) = λ ( u ) ≥ C a,γ (cid:18) lim n →∞ k u n k γ lim inf n →∞ k u n k (cid:19) γγ − ≥ lim sup n →∞ λ ( u n ) = λ ∗ , and hence u n → u in H (Ω) and u satisfies λ ( u ) = λ ∗ . Once λ ( u ) is a mulitple of λ ( u ) itfollows also that λ ( u ) = λ ∗ and from C ,a,γ <
1, we conclude that λ ∗ < λ ∗ . (cid:3) As a consequence of Proposition 2.4 we have the following
Proposition 2.5.
There holdsi) For each λ ∈ (0 , λ ∗ ) we have that N + λ and N − λ are non empty. Moreover, if λ > λ ∗ then N λ = ∅ .ii) For each λ ∈ (0 , λ ∗ ) there exists u ∈ H (Ω) \ { } such that Φ λ ( u ) < . Moreover, if λ ≥ λ ∗ then inf t> ψ λ,u ( t ) = 0 for each u ∈ H (Ω) \ { } .Proof. i ) From Proposition 2.4, there exists u ∈ H (Ω) \ { } such that λ ( u ) = λ ∗ . It followsfrom Proposition 2.3 that for each λ ∈ (0 , λ ∗ ) the fiber map ψ λ,u satisfies I ) of Proposition 2.2and hence t − λ ( u ) u ∈ N − λ and t + λ ( u ) u ∈ N + λ . Now suppose that λ > λ ∗ , then it follows that λ > λ ∗ ≥ λ ( u ) for each u ∈ H (Ω) \ { } , which implies from Proposition 2.3 that ψ λ,u satisfies III ) of Proposition 2.2 and hence N λ = ∅ . ii ) From Proposition 2.4, there exists u ∈ H (Ω) \ { } such that λ ( u ) = λ ∗ . It followsfrom Proposition 2.3 that for each λ ∈ (0 , λ ∗ ), there exists t > λ ( tu ) <
0. Nowassume that λ ≥ λ ∗ . Therefore λ > λ ∗ ≥ λ ( u ) for each u ∈ H (Ω) \ { } , which implies fromProposition 2.3 that inf t> ψ λ,u ( t ) = 0. (cid:3) From Proposition 2.5 we obtain the following nonexistence result.
K. SILVA
Corollary 2.6.
For each λ > λ ∗ the functional Φ λ does not have critical points other than u = 0 .Proof. Indeed, observe that for each λ > λ ∗ there holds N λ = ∅ . (cid:3) Now we turn our attention to some estimates which will prove to be useful on the nextsection. We start with:
Proposition 2.7.
Suppose that λ ∈ (0 , λ ∗ ] , then there exists r λ > such that k u k ≥ r λ foreach u ∈ N λ .Proof. The existence of r λ is straightforward from a k u k + λ k u k − C k u k γ ≤ a k u k + λ k u k − k u k γγ = 0 , ∀ u ∈ N λ , where C > (cid:3)
Proposition 2.8.
For each λ ∈ (0 , λ ∗ ] , there holds Φ λ ( u ) = ( γ − γ (4 − γ ) a λ , ∀ u ∈ N λ . Proof.
In fact, if u ∈ N λ , then(2.7) ( a k u k + λ k u k − k u k γγ = 0 , a k u k + 4 λ k u k − γ k u k γγ = 0 . It follows from (2.7) that(2.8) k u k = γ − − γ aλ . Moreover, from (2.7) we also have that(2.9) Φ λ ( u ) = γ − γ a k u k − − γ γ λ k u k , ∀ u ∈ N λ . We combine (2.8) with (2.9) to prove the proposition. (cid:3)
We conclude this Section with some variational properties related to the functional Φ λ . Lemma 2.9.
For each λ ∈ (0 , λ ∗ ) there holdsi) The functional Φ λ is weakly lower semi-continuous and coercive.ii) Suppose that u n is a Palais-Smale sequence at the level c ∈ R , that is Φ λ ( u n ) → c and Φ ′ λ ( u n ) → as n → ∞ , then u n converge strongly to some u .iii) There exist C λ > and ρ λ > satisfying Φ λ ( u ) ≥ C λ , ∀ u ∈ H (Ω) , k u k = ρ λ , and lim C λ → ρ λ = 0 . Proof. i ) is obvious. To prove ii ), observe from i ) that u n is bounded and therefore we canassume that u n ⇀ u in H (Ω) and u n → u in L γ (Ω). From the limit Φ ′ λ ( u n ) → n → ∞ we infer thatlim sup n →∞ [ − ( a + λ k u n k )∆ u n ( u n − u )] = lim sup n →∞ | u n | γ − u n ( u n − u ) = 0 , which easily implies that u n → u in H (Ω). XTREMAL PARAMETERS OF A KIRCHHOFF TYPE EQUATION 7 iii ) It follows from the inequalityΦ λ ( u ) ≥ a k u k + λ k u k − Cγ k u k γ , ∀ H (Ω) , where the constant C is positive. (cid:3) Local Minimizers for Φ λ In this section we prove the following
Proposition 3.1.
For each λ ∈ (0 , λ ∗ ) the functional Φ λ has a local minimizer u λ ∈ H (Ω) \ { } . Moreover, if λ ∈ (0 , λ ∗ ) then Φ λ ( u λ ) < while Φ λ ∗ ( u λ ∗ ) = 0 and if λ ∈ ( λ ∗ , λ ∗ ) then Φ λ ( u λ ) > . Remark 1.
In fact if λ ∈ (0 , λ ∗ ] then the local minimizer given by the Lemma 3.2 is a globalminimizer. We divide the proof of Proposition 3.1 in some Lemmas.
Lemma 3.2.
For each λ ∈ (0 , λ ∗ ) the functional Φ λ has a global minimizer u λ with negativeenergy.Proof. It is a consequence of Lemma 2.9 and Proposition 2.5. (cid:3)
Lemma 3.3.
The functional Φ λ ∗ has a global minimizer u λ ∗ = 0 with zero energy.Proof. Suppose that λ n ↑ λ ∗ as n → ∞ and for each n choose u n ≡ u λ n , where u λ n is givenby Lemma 3.2. From the inequality Φ λ n ( u n ) < n and Lemma 2.9 we obtain that u n is bounded. Therefore we can assume that u n ⇀ u in H (Ω) and u n → u in L γ (Ω). FromLemma 2.9 we have that Φ λ ∗ ( u ) ≤ lim inf n →∞ Φ λ n ( u n ) ≤ . From Proposition 2.5 we conclude that Φ λ ∗ ( u ) = 0 and hence Φ λ ∗ ( u ) = lim n →∞ Φ λ n ( u n ).Therefore u n → u in H (Ω) and from Proposition 2.7 we obtain that u = 0. If u λ ∗ ≡ u theproof is complete. (cid:3) Remark 2.
Observe that λ ∗ ( u λ ∗ ) = λ ∗ and hence λ ∗ ( u λ ∗ ) = λ ∗ . In order to show the existence of local minimizers when λ > λ ∗ we need the followingdefinition: for λ ∈ (0 , λ ∗ ) define(3.1) ˆΦ λ = inf { Φ λ ( u ) : u ∈ N + λ ∪ N λ } . Remark 3.
From the definitions, Proposition 2.2 and Proposition 2.5 we conclude that ˆΦ λ = inf u ∈ H (Ω) Φ λ ( u ) , ∀ λ ∈ (0 , λ ∗ ] . Proposition 3.4.
For each λ ∈ ( λ ∗ , λ ∗ ) there holds ˆΦ λ < ( γ − γ (4 − γ ) a λ . Proof.
Indeed, first observe from Remark 2 that t + λ ( u λ ∗ ) is defined for each λ ∈ ( λ ∗ , λ ∗ ).From Proposition A.2 in the Appendix we know that t − λ ( u λ ∗ ) < t λ ∗ ( u λ ∗ ) < t + λ ( u λ ∗ ) for each K. SILVA λ ∈ ( λ ∗ , λ ∗ ) and therefore ˆΦ λ ≤ Φ λ ( t + λ ( u λ ∗ ) u λ ∗ ) < Φ λ ( t λ ∗ ( u λ ∗ ) u λ ∗ ) < Φ λ ∗ ( t λ ∗ ( u λ ∗ ) u λ ∗ )= ( γ − γ (4 − γ ) a λ ∗ , ∀ λ ∈ ( λ ∗ , λ ∗ ) , (3.2)where the equality comes from Proposition 2.8. We combine (3.2) with λ < λ ∗ to complete theproof. (cid:3) Lemma 3.5.
For each λ ∈ ( λ ∗ , λ ∗ ) there exists u λ ∈ N + λ such that Φ λ ( u λ ) = ˆΦ λ .Proof. Indeed, suppose that u n ∈ N + λ ∪ N λ satisfies Φ λ ( u n ) → ˆΦ λ . From Lemma 2.9 we havethat u n is bounded and therefore we can assume that u n ⇀ u in H (Ω) and u n → u in L γ (Ω).From a k u n k + λ k u n k − k u n k γγ = 0 for all n and Proposition 2.7 we conclude that u = 0. Weclaim that u n → u in H (Ω). In fact, suppose on the contrary that this is false. It follows that ψ ′ λ,u (1) = a k u k + λ k u k − k u k γγ < lim inf n →∞ ( a k u n k + λ k u n k − k u n k γγ ) = 0 , and hence we conclude that the fiber map ψ λ,u satisfies I ) of Proposition 2.2 and t − λ ( u ) < The Lemmas 3.2 and 3.3 guarantee the existence of a globalminimizer u λ for the functional Φ λ satisfying: if λ ∈ (0 , λ ∗ ) then Φ λ ( u λ ) < λ ∗ ( u λ ∗ ) = 0. For λ ∈ ( λ ∗ , λ ∗ ) we use Lemma 3.5 in order to obtain a local minimizerfor the functional Φ λ . It remains to show that Φ λ ( u λ ) > λ ∈ ( λ ∗ , λ ∗ ), however, onceˆΦ λ ∗ = 0 this is a consequence of Proposition A.1. (cid:3) Mountain Pass Solution for Φ λ In this Section we show the exsitence of a mountain pass type solution to equation (1.1). Inorder to formulate our result we need to introduce some notation. For each λ ∈ (0 , λ ∗ ) define(4.1) c λ = inf ϕ ∈ Γ λ max t ∈ [0 , Φ λ ( ϕ ( t )) , where Γ λ = { ϕ ∈ C ([0 , , H (Ω)) : ϕ (0) = 0 , ϕ (1) = ¯ u λ } with ¯ u λ = u λ ∗ if λ ∈ (0 , λ ∗ ] and¯ u λ = u λ for λ ∈ ( λ ∗ , λ ∗ ). Proposition 4.1. There exists ε > such that for each λ ∈ (0 , λ ∗ + ε ) one can find w λ ∈ H (Ω) satisfying Φ λ ( w λ ) = c λ and Φ ′ λ ( w λ ) = 0 . Moreover c λ > and c λ > ˆΦ λ . To prove Proposotion 4.1 we need some auxiliary results. Lemma 4.2. Given δ > , there exists ε δ > such that < ˆΦ λ ≤ δ, ∀ λ ∈ ( λ ∗ , λ ∗ + ε δ ) . XTREMAL PARAMETERS OF A KIRCHHOFF TYPE EQUATION 9 Proof. The inequality ˆΦ λ > u λ ∗ be given as in Proposition3.3. Observe that if λ ↓ λ ∗ , then Φ λ ( u λ ∗ ) → Φ λ ∗ ( u λ ∗ ) = 0. Moreover, since from Remark 2 thefiber map ψ λ ∗ ,u λ ∗ satisfies I ) of Proposition 2.2, we have from Proposition 2.3 that λ ∗ < λ ( u λ ∗ ).It follows that there exists ε > λ ∗ + ε < λ ( u λ ∗ ). From Propositions 2.2 and 2.3,for each λ ∈ ( λ ∗ , λ ∗ + ε ), there exists t + λ ( u λ ∗ ) > t + λ ( u λ ∗ ) u λ ∗ ∈ N + λ . Note that t + λ ( u λ ∗ ) → λ ↓ λ ∗ and thereforeˆΦ λ ≤ Φ λ ( t + λ ( u λ ∗ ) u λ ∗ ) → Φ λ ∗ ( u λ ∗ ) = 0 , λ ↓ λ ∗ . If ε ,δ > λ ( t + λ ( u λ ∗ ) u λ ∗ ) < δ for each λ ∈ ( λ ∗ , λ ∗ + ε ,δ ), thenwe set ε δ = min { ε , ε ,δ } and the proof is complete. (cid:3) Definition 1. For λ ∈ (0 , λ ∗ ) denote (4.2) M λ = min (cid:26) C λ , ( γ − γ (4 − γ ) a λ (cid:27) , where C λ is given by Lemma 2.9 and ( γ − γ (4 − γ ) a λ is given by Proposition 2.8. We assume that ρ λ < r λ where both numbers are given by Lemma 2.9 and Proposition 2.7 respectively. Choose < δ < M λ and from Proposition 3.4 we take the corresponding ε δ . Now we are in position to prove Proposition 4.1 Proof of Proposition 4.1. The proof will be done once we show that the functional Φ λ has amountain pass geometry (remember that u λ is a local minimizer for Φ λ ), however, one can seefrom Definition 1 that(4.3) inf k u k = ρ λ Φ λ ( u ) ≥ M λ > max { Φ λ (0) , Φ λ (¯ u λ ) } , which is the desired mountain pass geometry. It follows that c λ ≥ M λ > Φ λ (¯ u λ ) andΦ λ (¯ u λ ) ≥ ˆΦ λ if λ ∈ (0 , λ ∗ ] and Φ λ (¯ u λ ) = ˆΦ λ otherwise.We infer that there exists a Palais-Smale sequence for the functional Φ λ at the level c λ , thatis, there exists w n ∈ H (Ω) such that Φ λ ( w n ) → c λ and Φ λ ( w n ) → 0. From Lemma 2.9 wehave that w n → w in H (Ω) and hence Φ λ ( w λ ) = c λ and Φ ′ λ ( w λ ) = 0. (cid:3) Proof of Theorem 1.1 In this Section we prove our main result Proof of the Theorem 1.1. The existence of u λ and w λ are given by Propositions 3.1 and 4.1.Observe that u λ being a global minimizer for Φ λ when λ ∈ (0 , λ ∗ ] it is obviously a criticalpoint for Φ λ and hence a solution to (1.1). If λ ∈ ( λ ∗ , λ ∗ ) we saw in Lemma 3.5 that u λ ∈ N + λ and hence from Lemma 2.1 it is a critical point for the functional Φ λ . The case λ = λ ∗ goesas following. Choose a sequence λ ↑ λ ∗ and a corresponding sequence u n ≡ u λ n such thatΦ λ n ( w n ) = ˆΦ λ n and Φ ′ λ n ( u n ) = 0 for each n ∈ N . Observe from the proof of Proposition 3.4that ˆΦ λ n < ( γ − γ (4 − γ ) a λ ∗ , ∀ n ∈ N , and therefore from Lemma 2.9 we conclude that u n → u in H (Ω). From Proposition A.1 weobtain that Φ λ ∗ ( u ) = lim n →∞ Φ λ n ( w n ) = lim n →∞ ˆΦ λ n > , and hence u = 0. By passing the limit it follows that Φ ′ λ ∗ ( u ) = 0. Moreover from the definitionof λ ∗ we also obtain that Φ ′′ λ ∗ ( u )( u, u )=0. If we set u λ ∗ ≡ u the proof of Theorem 1.1 items 1),2) and 3) is complete.The item 4) is a consequence of Proposition 4.1. Item 5) is proved by using the fact thatevery critical point of Φ λ lies in N λ and Proposition 2.5. To conclude we observe that standardarguments using the fact that Φ λ ( u ) = Φ λ ( | u | ) provide positive solutions. (cid:3) Asymptotic Behavior of u λ and w λ as λ ↓ : H (Ω) → R by Φ ( u ) = a k u k − γ k u k γγ , and observe that Φ ( u λ ∗ ) < Φ λ ∗ ( u λ ∗ ) = 0, where u λ ∗ is given by Theorem 1.1. Define c = inf ϕ ∈ Γ max t ∈ [0 , Φ ( ϕ ( t )) , where Γ = { ϕ ∈ C ([0 , 1] : H (Ω)) : ϕ (0) = 0 , ϕ (1) = u λ ∗ } . Standard arguments provide afunction w ∈ H (Ω) such that Φ ( w ) = c > ′ ( w ) = 0. For λ ∈ (0 , λ ∗ ), let usassume that u λ , w λ are given by Theorem 1.1. In this section we prove the following Proposition 6.1. There holdsi) Φ λ ( u λ ) → −∞ and k u λ k → ∞ as λ ↓ .ii) w λ → w in H (Ω) where w ∈ H (Ω) satisfies Φ ( w ) = c and Φ ′ ( w ) = 0 .Proof. i ) Indeed, choose any u ∈ H (Ω) and suppose without loss of generality that λ ∈ (0 , λ ( u )). It follows from Proposition 2.2 that ψ λ,u ( t ) ≥ ψ λ,u ( t + λ ( u )) ≥ inf u ∈ H (Ω) Φ λ ( u ) = ˆΦ λ .Now observe that for fixed t > ψ λ,u ( t ) → a k u k t − γ k u k γγ t γ , as λ ↓ . Once lim t →∞ (cid:18) a k u k t − γ k u k γγ t γ (cid:19) = −∞ , it follows from (6.1) that given M < t > δ > λ ∈ (0 , δ ), then ψ λ,u ( t ) < M and hence ˆΦ λ < M , which proves that Φ λ ( u λ ) → −∞ as λ ↓ 0. One can easilyinfer from the last convergence that k u λ k → ∞ as λ ↓ (cid:3) To prove the item ii ) of Proposition 6.1 we need to establish some results. Lemma 6.2. The function [0 , λ ∗ ) ∋ λ c λ = Φ λ ( w λ ) is non-decreasing. Moreover c λ → c as λ ↓ .Proof. First observe that Γ λ = Γ for each λ ∈ (0 , λ ∗ ]. Suppose that 0 ≤ λ < λ ′ < λ ∗ and fixany ϕ ∈ Γ. It follows that max t ∈ [0 , Φ λ ( ϕ ( t )) < max t ∈ [0 , Φ λ ′ ( ϕ ( t )) and by taking the infimumin both sides we conclude that c λ ≤ c λ ′ .Once c λ is non-decreasing, we can assume that c λ → c ≥ c as λ ↓ 0. Suppose on the contrarythat c > c . Given δ > c + ε < c choose ϕ ∈ Γ such that c ≤ max t ∈ [0 , Φ ( ϕ ( t )) XTREMAL PARAMETERS OF A KIRCHHOFF TYPE EQUATION 11 Proof of ii ) of Proposition 6.1. Indeed, suppose that λ n ↓ n ∈ N choose w n ≡ w λ n such that Φ λ n ( w n ) = c λ n and Φ ′ λ n ( w n ) = 0. We claim that λ n k w n k → n → ∞ . In fact, for each n we can find a path ϕ n ∈ Γ λ n = Γ and a function v n such thatΦ λ n ( v n ) = max t ∈ [0 , Φ λ n ( ϕ ( t )) and(6.2) 0 < Φ λ n ( v n ) − c λ n → , k v n − w n k → , k v n − w n k γ → , as n → ∞ . Now observe from the definition of c , Lemma 6.2 and (6.2) that(6.3) 0 < lim n →∞ Φ ( v n ) − c ≤ lim n →∞ Φ λ n ( v n ) − c = lim n →∞ (Φ λ n ( v n ) − c λ n ) = 0 . It follows from (6.2) and (6.3) that a k v n k − p k v n k γγ → a k v n k + λ n k v n k − p k v n k γγ → , as n → ∞ , which implies that λ n k v n k → n → ∞ . From (6.2) we conclude that | λ n k w n k − λ n k v n k | → , as n → ∞ , and hence λ n k w n k → n → ∞ as we desired. Now note from the equations Φ λ n ( w n ) = c λ n and Φ ′ λ n ( w n ) = 0, n ∈ N that(6.4) a k w n k + λ n k w n k − γ k w n k γγ = c λ n ,a k w n k + λ n k w n k − k w n k γγ = 0 , which combined with the limit λ n k w n k → n → ∞ and the Lemma 6.2 implies that a λ n k w n k − λ n γ k w n k γγ = o (1) ,aλ n k w n k − λ n k w n k γγ = 0 . We multiply the first equation by − γ and sum with the second equation to obtain that (cid:16) − γ (cid:17) aλ n k w n k = o (1) , which implies that λ n k w n k → n → ∞ . Now we claim that k w n k is bounded. In fact,suppose on the contrary that up to a subsequence k w n k → ∞ as n → ∞ . From (6.4) we obtainthat a λ n k w n k − γ k w n k γγ k w n k = o (1) ,a + λ n k w n k − k w n k γγ k w n k = 0 . Once λ n k w n k → n → ∞ we conclude that γ = 2 which is a contradiction. Since k w n k is bounded we obtain that Φ ( w n ) → c and Φ ′ ( w n ) → n → ∞ and hence w n → w as n → ∞ , where w satisfies Φ ( w ) = c and Φ ′ ( w ) = 0. (cid:3) Proof of the Theorem 1.2. It is a consequence of Proposition 6.1. (cid:3) λ Energy λ ∗ λ ∗ + ε λ ∗ −∞ Φ λ ( w λ ) Φ λ ( u λ ) Figure 1. Energy depending on λ Some Conclusions and Remarks If we plot the energy of the two solutions as a function of λ we obtain the following picture:Observe from Proposition A.1 that the energy of the local minimum depending on λ iscontinuous and increasing (red plot) and although we could not prove it, we believe that thesame holds true for the energy of the mountain pass solution (blue plot). We also believe that λ ∗ is a bifurcation turning point, that is, the two types of solutions must coincide at λ ∗ asFigure 1 suggests. Appendix A. Proposition A.1. The function (0 , λ ∗ ) ∋ λ ˆΦ λ is continuous and increasing.Proof. First we prove that (0 , λ ∗ ) ∋ λ ˆΦ λ is decreasing. Indeed, suppose that λ < λ ′ . FromLemmas 3.2, 3.3 and 3.5, there exists u λ ′ such that ˆΦ λ ′ = Φ λ ′ ( u λ ′ ). Since the fiber map ψ λ ′ ,u λ ′ obvioulsy satisfies I) of Proposition 2.2 it follows from Proposition 2.3 that ψ λ,u λ ′ also satisfiesI) of Proposition 2.2 and thenˆΦ λ ≤ Φ λ ( t + λ ( u λ ′ ) u λ ′ ) < Φ λ ( t + λ ′ ( u λ ′ ) u λ ′ ) = Φ λ ′ ( u λ ′ ) = ˆΦ λ ′ . Now we prove that (0 , λ ∗ ) ∋ λ ˆΦ λ is continuous. In fact, suppose that λ n ↑ λ ∈ (0 , λ ∗ )and choose u n ≡ u λ n such that ˆΦ λ n = Φ λ n ( u n ) for all n . Similar to the proof of Lemma3.5 we may assume that u n → u ∈ N + λ . We claim that ˆΦ λ n → ˆΦ λ as n → ∞ . Indeed, once(0 , λ ∗ ) ∋ λ ˆΦ λ is increasing, we can assume that ˆΦ λ n < ˆΦ λ for each n and ˆΦ λ n → Φ λ ( u ) ≤ ˆΦ λ as n → ∞ , wich implies that Φ λ ( u ) = ˆΦ λ .Now suppose that λ n ↓ λ ∈ (0 , λ ∗ ). Once (0 , λ ∗ ) ∋ λ ˆΦ λ is increasing, we can assume thatˆΦ λ n > ˆΦ λ for each n and lim n →∞ ˆΦ λ n ≥ ˆΦ λ . Choose u λ such that ˆΦ λ = Φ λ ( u λ ) and observethat ˆΦ λ ≤ lim n →∞ ˆΦ λ n ≤ lim n →∞ Φ λ n ( t + λ n ( u λ ) u λ ) = ˆΦ λ . (cid:3) For the next proposition we assume that u λ ∗ is given as in Lemma 3.3 and t ( u λ ∗ ) is definedin (2.2). Observe from Remark 2 that t + λ ( u λ ∗ ) is well defined for each λ ∈ (0 , λ ∗ ). Proposition A.2. There holdsi) The function (0 , λ ∗ ) ∋ λ t + λ ( u λ ∗ ) is decreasing and continuous.ii) The function (0 , λ ∗ ) ∋ λ t − λ ( u λ ∗ ) is increasing and continuous.Moreover lim λ ↑ λ ∗ t + λ ( u λ ∗ ) = lim λ ↑ λ ∗ t − λ ( u λ ∗ ) = t ( u λ ∗ ) . XTREMAL PARAMETERS OF A KIRCHHOFF TYPE EQUATION 13 Proof. Indeed, let t λ ≡ t + λ ( u λ ∗ ) and note that t λ satisfies ψ ′ λ ( t λ ) = 0 for each λ ∈ (0 , λ ∗ ). Byimplicit differentiation and the fact that ψ ′′ λ ( t λ ) > 0, we conclude that (0 , λ ∗ ) ∋ λ t + λ ( u λ ∗ ) isdecreasing and continuous, which proves i ) The proof of ii ) is similar and the limitslim λ ↑ λ ∗ t + λ ( u λ ∗ ) = lim λ ↑ λ ∗ t − λ ( u λ ∗ ) = t ( u λ ∗ ) , are straightforward from the definitions. (cid:3) References 1. C. O. Alves, F. J. S. A. Corrˆea, and T. F. Ma, Positive solutions for a quasilinear elliptic equation ofKirchhoff type , Comput. Math. Appl. (2005), no. 1, 85–93. MR 2123187 22. Ching-yu Chen, Yueh-cheng Kuo, and Tsung-fang Wu, The Nehari manifold for a Kirchhoff typeproblem involving sign-changing weight functions , J. Differential Equations (2011), no. 4, 1876–1908.MR 2763559 23. Yavdat Ilyasov, On extreme values of Nehari manifold method via nonlinear Rayleigh’s quotient , Topol.Methods Nonlinear Anal. (2017), no. 2, 683–714. MR 3670482 2, 44. 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Silva) Instituto de Matem´atica e Estat´ıstica.Universidade Federal de Goi´as,74001-970, Goiˆania, GO, Brazil E-mail address ::