Existence of S-shaped type bifurcation curve with dualcusp catastrophe via variational methods

Marcos Leandro Carvalho

Universidade Federal de Goias, Instituto de Matematica

74690-900, Goiania - GO - Brazil

Yavdat Il’yasov∗

Institute of Mathematics, Ufa Federal Research Centre, RASChernyshevsky str. 112, 450008 Ufa, Russia

Universidade Federal de Goias, Instituto de Matematica

74690-900, Goiania - GO - Brazil

Carlos Alberto Santos

Universidade de Brasılia, Departamento de Matematica70910-900, Brasılia - DF - Brazil

Abstract

We discuss the existence of multiple positive solutions leading to the occurrenceof an S-shaped bifurcation curve to the equations of the form −∆pu = f(µ, λ, u)where ∆p is a p-Laplacian, p > 1, µ, λ ∈ R. We deal with relatively unexploredcases when f(µ, λ, u) is non-Lipschitz at u = 0, f(µ, λ, 0) = 0 and f(µ, λ, u) < 0,u ∈ (0, r), for some r < +∞. We develop the nonlinear generalized Rayleighquotients method to find a range of parameters where the equation may havedistinct branches of positive solutions. As a consequence, applying the Neharimanifold method and the mountain pass theorem, we prove that the equationfor some range of values µ, λ, has at least three positive solutions with twolinearly unstable solutions and one linearly stable. The results evidence that thebifurcation curve is S-shaped and exhibits the so-called dual cusp catastrophewhich is characterized by the fact that the corresponding dynamic equationhas stable states only within the cusp-shaped region in the control plane ofparameters. Our results are new even in the one-dimensional case and p = 2.

Keywords: bifurcation curve; Nehari manifold; nonliear generalized Rayleighquotient; dual cusp catastrophe2020 MSC: 35J60, 35J20, 35B30.

∗Corresponding authorEmail addresses: marcos$_$leandro$_$carvalho@ufg.br (Marcos Leandro Carvalho),

ilyasov02@gmail.com (Yavdat Il’yasov), csantos@unb.br (Carlos Alberto Santos)

Preprint submitted to Elsevier

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1. Introduction

Let Ω be a smooth bounded domain in RN , N ≥ 1. We consider the followingboundary value problem

−∆pu = uγ−1 + µuα−1 − λuq−1 := f(µ, λ, u) in Ω,

u ≥ 0, in Ω,

u = 0 on ∂Ω.

(1)

Here 1 < q < α < p < γ < p∗, p∗ = pNN−p if p < N , and p∗ = +∞ if p ≥ N , and

λ, µ ∈ R.We are interested on the existence of branches of solutions with S-

shaped type bifurcation curve. The S-shaped bifurcation curve arises in thestudy of many problems such as the Liouville–Bratu–Gelfand equation [16],the Kolmogorov-Petrovsky-Piscounov equation [23], the Minkowski-curvatureproblem (see, e.g., [5, 9, 19]), in the reaction-diffusion models which describedvarious spatio-temporal phenomena in biology, physics, chemistry, epidemiology,and ecology (see e.g. [6, 28]). This type of bifurcation curve is characterizedby complex and fascinating behavior of systems [3, 17, 37]. The associateddynamic equations could include rarefaction and shock waves (see, e.g.,[33])and exhibit hysteresis, which is widespread and manifests itself in many physicalphenomena, such as dielectric hysteresis, magnetic hysteresis, elastic hysteresis,and some others [17, 31, 34].

The ”S-shaped bifurcation problem” has attracted a considerable attentionin recent decades beginning with the celebrated papers by Cohen [11] in 1971,Crandall & Rabinowitz [10] in 1973 and Amann [1] in 1976. The study ofthis problem, which usually contains the search for conditions under which theequations have at least three solutions, use different approaches: the bifurcationand continuation methods [10, 12], the method of sub-super-solutions [1, 8, 25],the time-map method and the quadrature technique [8, 19, 36]. In [7, 27], theexistence of three positive solution for some a range of parameter of a problemhas been obtained by using variational approaches including the mountain passtheorem, the Nehari manifold and fibering methods [29].

Most of the results on the existence of an S-shaped bifurcation curve dealwith a nonlinearity f that satisfies f(u) > 0 on (0, r) for some r ∈ (0,+∞],and f ∈ C2[0,+∞) (see e.g. [1, 8, 10, 25]). Note that in (1) we are facing theopposite case, namely, for any µ > 0, λ ∈ R, there exists rµ,λ > 0 such thatf(µ, λ, s) < 0, s ∈ (0, rµ,λ). Moreover, the nonlinearity in (1) is non-Lipschitz atu = 0 if p ≤ 2 and µ 6= 0 or λ 6= 0. An additional feature of (1), which impliesour results is that the S-shaped bifurcation curve of (1) exhibits the so-calleddual cusp catastrophe [17].

Let us state our main results. The problem (1) has a variational structurewith the energy functional Φλ,µ ∈ C1(W 1,p

0 ) given by

Φλ,µ(u) =1

p

∫|∇u|p +

λ

q

∫|u|q − µ

α

∫|u|α − 1

γ

∫|u|γ .

2

By a weak solution of (1) we mean a critical point u of Φλ,µ(u) on W 1,p0 , i.e.,

DΦλ,µ(u) = 0. Here and subsequently, DΦλ(u) denotes the Frechet derivative

of Φλ,µ at u ∈ W 1,p0 and DΦλ,µ(u)(v) denotes its the directional derivative in

direction v ∈ W 1,p0 . Hereinafter, for F ∈ C1(W 1,p

0 ) we use the abbreviatednotations F ′(tu) := d

dtF (tu), t > 0, u ∈W 1,p0 .

A nonzero weak solution u of (1) is said to be ground state if Φλ,µ(u) ≤Φλ,µ(w), for any non-zero weak solution w ∈ W 1,p

0 of (1). We say that weak

solution u of (1) is linearly stable if u is a local minimizer of Φλ,µ(u) in W 1,p0

and linearly unstable otherwise.Our approach is based on the use of so-called nonlinear generalized Rayleigh

quotient(NG-Rayleigh quotient) whose critical values correspond to the extremevalues of the Nehari manifold method [20]. Following it we introduce the so-called NG-Rayleigh extremal values (cf. [4])

µe,+λ := infu∈W 1,p

0 \0Reλ(u) : (Reλ)′(u) = 0, (Reλ)′′(u) > 0, (2)

µe,−λ := infu∈W 1,p

0 \0Reλ(u) : (Reλ)′(u) = 0, (Reλ)′′(u) < 0, (3)

µn,+λ := infu∈W 1,p

0 \0Rnλ(u) : (Rnλ)′(u) = 0, (Rnλ)′′(u) > 0, (4)

µn,−λ := infu∈W 1,p

0 \0Rnλ(u) : (Rnλ)′(u) = 0, (Rnλ)′′(u) < 0. (5)

Here Rnλ,Reλ : W 1,p0 \ 0→ R are the Rayleigh quotients given by

Rnλ(u) :=

∫|∇u|2 + λ

∫|u|q −

∫|u|γ∫

|u|α , u ∈W 1,p0 \ 0, λ ∈ R,

Reλ(u) :=

12

∫|∇u|2 + λ

q

∫|u|q − 1

γ

∫|u|γ

1α

∫|u|α , u ∈W 1,p

0 \ 0, λ ∈ R.(6)

We introduce also the so-called NG-Rayleigh λ-extremal value:

λ = infu∈W 1,p

0 \0

(∫|∇u|p) γ−qγ−p

(∫|u|q)(

∫|u|γ)

p−qγ−p

,

and defineλe = ceq,γ λ and λn = cnq,γ λ, (7)

where

ceq,γ =qγ

2−qγ−2

2γ−qγ−2

cnq,γ , cnq,γ =(p− α)

γ−qγ−p (p− q) p−qγ−q (γ − p)

(α− q)(γ − α)p−qγ−p (γ − q) γ−qγ−p

.

Lemma 1.1. Assume that 1 < q < α < p < γ < p∗. Then

(1o) 0 < λe < λn < +∞,

(2o) 0 < µn,+λ < µe,+λ < µe,−λ < µn,−λ < +∞, for λ ∈ (0, λe).

3

We deal with the Nehari manifold

Nλ,µ = u ∈W 1,p0 \ 0 : Φ′λ,µ(u) = 0.

A local minimum or maximum point of the function Φλ,µ(u) subject to Nλ,µ iscalled the extremal point of Φλ,µ(u) on the Nehari manifold. We denote

Φλ,µ = minu∈Nλ,µ

Φλ,µ(u) and Mλ,µ := u ∈ Nλ,µ : Φλ,µ = Φλ,µ(u).

Theorem 1.2. Assume that 1 < q < α < p < γ < p∗, λ ∈ (0, λe). Thenproblem (1) admits two distinct branches of positive solutions (u2

λ,µ), (u3λ,µ),

namely:

(10) There exists µ∗λ ∈ (µn,+λ , µe,+λ ) such that for any µ ∈ [µ∗λ, µn,−λ ), problem

(1) possesses a positive solution u2λ,µ ∈ C1,κ(Ω), κ ∈ (0, 1) such that (i)

u2λ,µ is a ground state; (ii) u2

λ,µ is linearly stable and Φ′′λ,µ(u2λ,µ) > 0; (iii)

the function µ 7→ Φλ,µ(u2λ,µ) is continuous and monotone decreasing on

(µ∗λ, µn,−λ ); (iv) Φλ,µ(u2

λ,µ) > 0 for µ ∈ (µ∗λ, µe,+λ ), Φλ,µe,+λ

(u2λ,µe,+λ

) = 0,

and Φλ,µ(u2λ,µ) < 0 for µ ∈ (µe,+λ , µn,−λ ).

(20) For any µ ∈ (−∞, µn,−λ ), problem (1) possesses a positive solutionu3λ,µ ∈ C1,κ(Ω), κ ∈ (0, 1) such that (i) u3

λ,µ is linearly unstable

and Φ′′λ,µ(u3λ,µ) < 0; (ii) the function (−∞, µn,−λ ) 3 µ 7→ Φλ,µ(u3

λ,µ)

is continuous and monotone decreasing; (iii) Φλ,µ(u3λ,µ) > 0 if µ ∈

(−∞, µe,−λ ), Φλ,µe,−λ(u3λ,µe,−λ

) = 0, and Φλ,µ(u3λ,µ) < 0 if µ ∈ (µe,−λ , µn,−λ );

(iv) Φλ,µ(u3λ,µ)→ +∞, ‖u3

λ,µ‖1 → +∞ as µ→ −∞.

Moreover, if µ ∈ (−∞, µn,+λ ), then u3λ,µ is a ground state of (1).

A weak solution uλ,µ ∈W 1,p0 \ 0 of (1) is said to be mountain pass type if

Φλ,µ(uλ,µ) = Φmλ,µ := infγ∈P

supu∈γ

Φλ,µ(u) > 0, (8)

for the paths set P := g ∈ C([0, 1];W 1,p0 ) : g(0) = 0, g(1) = w1 with some

w1 ∈W 1,p0 such that Φλ,µ(w1) < 0.

Theorem 1.3. Assume that 1 < q < α < p < γ < p∗, λ > 0, −∞ < µ < +∞.Then (1) admits a positive mountain pass type solution uλ,µ ∈ C1,κ(Ω),κ ∈ (0, 1) such that Φλ,µ(uλ,µ) > 0. Moreover, if λ ∈ (0, λe), µ ∈ (−∞, µn,+λ ),then (i) uλ,µ is a ground state of (1), i.e., uλ,µ ∈Mλ,µ; (ii) Φλ,µ(uλ,µ)→ +∞,‖uλ,µ‖1 → +∞ as µ→ −∞, (iii) Φλ,µ(uλ,µ)→ 0, ‖uλ,µ‖1 → 0 as µ→ +∞.

Theorems 1.2, 1.3 yield the following result on the existence of three distinctbranches of weak positive solutions of (1).

4

Theorem 1.4. Assume λ ∈ (0, λe) and µ ∈ [µe,−λ , µn,−λ ). Then (1) admits atleast three distinct positive solutions: u1

λ,µ, u2λ,µ, u3

λ,µ such that Φλ,µ(u1λ,µ) >

0, Φλ,µ(u2λ,µ) < 0, Φλ,µ(u3

λ,µ) ≤ 0 and Φ′′λ,µ(u2λ,µ) > 0, Φ′′λ,µ(u3

λ,µ) <

0. Furthermore, u2λ,µ is linearly stable while u1

λ,µ, u3λ,µ are linearly unstable

solutions.

In view of these result, it is natural to expect that the branches of solutionsto (1) behave as depicted on Figures 1, 2. That is there are two bifurcationvalues µ∗λ, µ∗∗λ such that: (i) 0 < µn,+λ ≤ µ∗λ ≤ µ∗λ ≤ µe,+λ < µn,−λ ≤ µ∗∗λ ;(ii) the branches u1

λ,µ, u2λ,µ continue until the bifurcation value µ∗λ, where

they coincide; (iii) u2λ,µ, u3

λ,µ continue until the bifurcation value µ∗∗λ , where

they coincide. Furthermore, we anticipate that the solution u2λ,µ is stable

for any µ ∈ (µ∗λ, µ∗∗λ ), whereas solutions u1

λ,µ for µ ∈ (−∞, µ∗λ) and u3λ,µ for

µ ∈ (µ∗∗λ ,+∞) are unstable. It is important to emphasize that such behaviormeans that the S-shaped bifurcation curve of (1) exhibits the so-called dualcusp catastrophe [17]. This type of catastrophe is characterized by the factthat the corresponding dynamic equation has an opposite behaviour, namelyit has stable states only within the cusp-shaped region in the control plane ofparameters µ, λ. Note that the cusp catastrophe, which is more common in thestudying S-shaped bifurcation curves (see, e.g., [7, 8, 24, 36]) has a stable statefor the entire range of parameters and are characterized by hysteresis behaviour[17, 31, 34]. We failed to find any reference addressing the existence of an S-shaped bifurcation curve of nonlinear PDE’s in high dimensions which exhibitsthe dual cusp catastrophe.

Remark 1.5. We anticipate that µ∗λ = µ∗λ, and u3λ,µ is a ground state for

µ ∈ (−∞, µ∗λ); u2λ,µ is a ground state for µ ∈ (µ∗λ, µ

∗∗λ ); u1

λ,µ is a ground statefor µ ∈ (µ∗∗λ ,−∞). In this regard, it is interesting to note such a phenomenonfor the ground states branch as the appearance of jump at µ∗λ of the energy levelΦλ,µ and jump at µ∗∗λ of the values of norm ‖ · ‖1 (see Figures 1, 2)

.

0

‖u‖1

‖u1µ,λ‖1

µ

‖u3µ,λ‖1

‖u2µ,λ‖1

µe,+λ

µe,−λ µn,−

λµn,+λ

µ∗λ µ∗∗

λ

Figure 1: The branches of solutions to (1) in term of the norm ‖ · ‖1

5

0

Φλ,µ

Φλ,µ(u1λ,µ

)

Φλ,µ(u2λ,µ

)

Φλ,µ(u3λ,µ

)

µµ∗λ

µ∗∗λ

µn,+λ µ

e,+λ

µe,−λ

µn,−λ

Figure 2: The branches of solutions to (1) in term of the energy levels Φλ,µ.

Remark 1.6. Evidently, if u is a solution of (1), then so is −u. Thus,Theorems 1.2-1.4 actually establish the existence at least of pairs of the branchesof solutions (u1

λ,µ,−u1λ,µ), (u2

λ,µ,−u2λ,µ), and (u3

λ,µ,−u3λ,µ), respectively.

Remark 1.7. The present article extends results obtained in our previous work[4], where (1) has been studied in the case p = 2, and the solution u2

λ,µ had

been obtained for λ ∈ (0, λe), µ ∈ (µe,+λ , µn,−λ ), whereas the solution u3λ,µ had

been obtained, under additional restrictions 1 + α < γ < 2∗, for λ ∈ (0, λn),µ ∈ (−∞, µn,−λ ). Furthermore, we succeed to develop a new approach, whichproved to be useful in obtaining solutions on wider intervals of parameters andimproving the results in general.

2. Preliminaries

Hereinafter, we use the standard notation Lr := Lr(Ω), 1 ≤ r ≤ +∞ for theLebesgue spaces endowed with the norm ‖u‖Lr := (

∫|u|r)1/r, W 1,p

0 := W 1,p0 (Ω)

for the Sobolev space endowed with the norm ‖u‖1 := (∫|∇u|p)1/p. The weak

convergence in W 1,p0 we shall denote by ””.

Lemma 2.1. Let u ∈ Nλ,µ be an extremal point of Φλ,µ(u) on the Neharimanifold. Suppose that Φ′′λ,µ(u) := DΦ′λ,µ(u)(u) 6= 0. Then DΦλ,µ(u) = 0.

Proof. Due to the assumption we may apply the Lagrange multiplier rule (seeProposition 43.19 in [38]), and thus, we have DΦλ,µ(u) + νDΦ′λ,µ(u) = 0, forsome ν ∈ R. Testing this equality by u we obtain µDΦ′λ,µ(u)(u) = 0. SinceDΦ′λ,µ(u)(u) 6= 0, ν = 0, and therefore, DΦλ,µ(u) = 0.

Observe that Φλ,µ is coercive on Nλ,µ, ∀λ > 0, ∀µ ∈ R. Indeed, by theSobolev inequality,

Φλ,µ(u) ≥ γ − ppγ‖u‖p1 − µC‖u‖α1 , ∀u ∈ Nλ,µ,

6

for some constant C > 0 independent of u ∈ Nλ,µ. Since α < p, we haveΦλ,µ(u)→ +∞ for ‖u‖1 → +∞ and u ∈ Nλ,µ.

It is easily seen that for u ∈W 1,p0 \ 0, the fibering function Φλ,µ(su), s > 0,

may have at most three nonzero critical points

0 < s1λ,µ(u) ≤ s2

λ,µ(u) ≤ s3λ,µ(u) <∞

such that Φ′′λ,µ(s1λ,µ(u)u) ≤ 0, Φ′′λ,µ(s2

λ,µ(u)u) ≥ 0, Φ′′λ,µ(s3λ,µ(u)u) ≤ 0 (see

Figure 3).To apply the Nehari manifold method, we need to find values λ, µ, where the

strong inequalities 0 < s1λ,µ(u) < s2

λ,µ(u) < s3λ,µ(u) hold. We solve this by the

recursively application of the nonlinear generalized Rayleigh quotient methodproposed in [4]. In the first step of this recursive procedure, we consider

Rnλ(u) =

∫|∇u|p + λ

∫|u|q −

∫|u|γ∫

|u|α , u ∈W 1,p0 \ 0, λ ∈ R. (9)

Notice that for u ∈W 1,p0 \ 0,

Rnλ(u) = µ ⇔ Φ′λ,µ(u) = 0

Rnλ(u) = µ, (Rnλ)′(u) > (<) 0 ⇔ Φ′′λ,µ(u) > (<) 0.(10)

In particular,Nλ,µ = u ∈W 1,p

0 \ 0 : Rnλ(u) = µ.Moreover, since for any λ > 0, Rnλ(tu) → +∞ as t ↓ 0 and Rnλ(tu) → −∞ ast→ +∞,

Nλ,µ 6= ∅, ∀µ ∈ (−∞,+∞), ∀λ ∈ (0,+∞). (11)

Using the Implicit Function Theorem we have

Proposition 2.2. If (Rnλ)′(siλ,µ(u0)u0) 6= 0 for u0 ∈ W 1,p0 \ 0, i = 1, 2, 3, then

there exists a neighbourhood Uu0⊂W 1,p

0 \ 0 of u0 such that siλ,µ(·) ∈ C1(Uu0).

Simple analysis shows that for any given u ∈W 1,p0 \ 0, the fibering function

Rnλ(tu) may have at most two non-zero critical points tn,+λ (u), tn,−λ (u), where

tn,+λ (u) is a local minimum, tn,−λ (u) is a local maximum point of Rnλ(tu), and

0 < s1λ,µ(u) ≤ tn,+λ (u) ≤ s2

λ,µ(u) ≤ tn,+λ (u) ≤ s3λ,µ(u) <∞. (12)

(see Figure 4).To split the points tn,+λ (u), tn,−λ (u), in the second step of the recursive

procedure, we apply the nonlinear generalized Rayleigh quotient method tothe functional Rnλ with respect to the parameter λ, i.e., we consider

Λn(u) :=(p− α)

∫|∇u|p − (γ − α)

∫|u|γ

(α− q)∫|u|q , u ∈W 1,p

0 \ 0. (13)

7

Notice that for any u ∈W 1,p0 \0, (Rnλ)′(tu) = 0⇔ Λn(tu) = λ. The only solution

ofd

dtΛn(tu) = 0 is a global maximum point tn(u) of the function Λn(tu) which

can be found precisely

tn(u) :=

(Cn

∫|∇u|p∫|u|γ

)1/(γ−p), ∀u ∈W 1,p

0 \ 0, (14)

where

Cn =(p− α)(p− q)(γ − α)(γ − q) .

This allow us to introduce the following NG-Rayleigh λ-quotient

λn(u) := Λn(tn(u)u) = cnq,γ(∫|∇u|p) γ−qγ−p

(∫|u|q)(

∫|u|γ)

p−qγ−p

, (15)

and the corresponding principal extremal value

λn = infu∈W 1,p

0 \0supt>0

Λn(tu),= cnq,γ(∫|∇u|p) γ−qγ−p

(∫|u|q)(

∫|u|γ)

p−qγ−p

,

where

cnq,γ =(p− α)

γ−qγ−p (p− q) p−qγ−q (γ − p)

(α− q)(γ − α)p−qγ−p (γ − q) γ−qγ−p

.

Note that this definition of λn coincides with (7).It easily follows (cf. [4])

Proposition 2.3. For any λ ∈ (0, λn) and u ∈ W 1,p0 \ 0, the function Rnλ(tu)

has precisely two distinct critical points such that 0 < tn,+λ (u) < tn,−λ (u), with

tn,+λ (·), tn,−λ (·) ∈ C1(W 1,p0 \ 0). Moreover,

• (Rnλ)′′(tn,+λ (u)u) > 0, (Rnλ)′′(tn,−λ (u)u) < 0,

• (Rnλ)′(tu) < 0 ⇔ t ∈ (0, tn,+λ (u)) ∪ (tn,−λ (u),∞),

• (Rnλ)′(tu) > 0 ⇔ t ∈ (tn,+λ (u), tn,−λ (u)).

Observe that this and (12) imply that 0 < s1λ,µ(u) < s3

λ,µ(u) < ∞ for any

λ ∈ (0, λn), u ∈ W 1,p0 \ 0. Thus, for λ ∈ (0, λn), we are able to introduce the

following NG-Rayleigh µ-quotients

µn,+λ (u) := Rnλ(tn,+λ (u)u), µn,−λ (u) := Rnλ(tn,−λ (u)u), u ∈W 1,p0 \ 0.

By Proposition 2.3 and regularity of Rnλ it follows that µn,+λ and µn,−λ are

C1(W 1,p0 \ 0), λ ∈ (0, λn). It is easily seen that the corresponding principal

8

extremal values

µn,+λ = infu∈W 1,p

0 \0µn,+λ (u), (16)

µn,−λ = infu∈W 1,p

0 \0µn,−λ (u) (17)

coincide with (4) and (5), respectively.We also need the so-called zero-energy level Rayleigh quotient

Reλ(u) =

1p

∫|∇u|p + λ

q

∫|u|q − 1

γ

∫|u|γ

1α

∫|u|α , u ∈W 1,p

0 \ 0,

which is characterized by the fact that Reλ(u) = µ ⇔ Φλ,µ(u) = 0. It is easyto see that Reλ(u) possesses similar properties to that Rnλ(u). In particular, thefibering function Reλ(tu) may have at most two non-zero fibering critical points0 < te,+λ (u) ≤ te,−λ (u) < +∞ so that te,+λ (u) is a local minimum while te,−λ (u) isa local maximum point of Reλ(tu). Moreover, the same conclusion as for Λn(u)can be drawn for the Rayleigh quotient

Λe(u) := q

(p−α)p

∫|∇u|p − (γ−α)

γ

∫|u|γ

(α− q)∫|u|q , (18)

which is characterized by the fact that (Reλ)′(tu) = 0 ⇔ Λe(tu) = 0 for any

u ∈ W 1,p0 \ 0. The unique solution of d

dtΛe(tu) = 0 is a global maximum point

of the function Λe(tu) defined by

te(u) :=

(Ce‖u‖p1‖u‖γLγ

)1/(γ−p), ∀u ∈W 1,p

0 \ 0, (19)

where

Ce =γ(p− α)(p− q)p(γ − α)(γ − q) .

Thus we have the following NG-Rayleigh quotient λe(u) := Λe(te(u)u), u ∈W 1,p

0 \ 0 with the corresponding principal extremal value

λe = infu∈W 1,p

0 \0supt>0

Λe(tu) = ceq,γ infu∈W 1,p

0 \0

‖u‖pγ−qγ−p

1

‖u‖qLq‖u‖γ p−qγ−pLγ

.

Note that this definition of λe coincides with (7). We thus have

Proposition 2.4. For any λ ∈ (0, λe) and u ∈ W 1,p0 \ 0, the function Reλ(tu)

has precisely two distinct critical points such that 0 < te,+λ (u) < te,−λ (u) with

te,+λ (·), te,−λ (·) ∈ C1(W 1,p0 \ 0). Moreover,

• (Reλ)′′(te,+λ (u)u) > 0 and (Reλ)′′(te,−λ (u)u) < 0,

9

• (Reλ)′(tu) < 0 ⇔ t ∈ (0, te,+λ (u)) ∪ (te,−λ (u),∞),

• (Reλ)′(tu) > 0 ⇔ t ∈ (te,+λ (u), te,−λ (u)).

Hence, for λ ∈ (0, λe), we are able to introduce the following zero-energylevel NG-Rayleigh µ-quotients

µe,+λ (u) := Reλ(te,+λ (u)u), µe,−λ (u) := Rnλ(te,−λ (u)u), u ∈W 1,p0 \ 0.

It is easily seen that the corresponding zero-energy level principal extremalvalues

µe,+λ = infu∈W 1,p

0 \0µe,+λ (u), µe,−λ = inf

u∈W 1,p0 \0

µe,−λ (u) (20)

coincide with (2) and (3), respectively.The relationships among the above introduced Rayleigh quotients are given

by the following lemma (see Figures 4)

Lemma 2.5. Assume that 1 < q < α < p < γ, u ∈W 1,p0 \ 0, t > 0.

(i) Λe(tu) = Λn(tu) ⇔ t = te(u),

(ii) Reλ(tu) = Rnλ(tu) ⇔ t = te,+λ (u) or t = te,−λ (u), ∀λ ∈ (0, λe),

(iii) tn,+λ (u) < te,+λ (u) < te(u) < tn,−λ (u) < te,−λ (u), ∀λ ∈ (0, λe),

(iv) Rnλ(tu) < Reλ(tu) ⇔ t ∈ (0, te,+λ (u)) or t ∈ (te,−λ (u),∞), ∀λ ∈ (0, λe).

Proof. The equality Λe(tu) = Λn(tu) is equivalent to

tp−q‖u‖p1 −(γ − α)

(p− α)tγ−q‖u‖γLγ = q

(tp−q

1

p‖u‖p1 − tγ−q

(γ − α)

γ(p− α)‖u‖γLγ

).

Hence,

0 =(p− q)(p− α)

pt1−q‖u‖p1 −

(γ − q)(γ − α)

γtγ−q−1‖u‖γLγ = (Λe(tu))′,

which implies (i).Observe, Reλ(tu) = Rnλ(tu) for t > 0 if and only if

tp−α‖u‖p1 + λtq−α‖u‖qLq − tγ−α‖u‖γLγ =αtp−α

p‖u‖p1 +

λαtq−α

q− αtγ−α

γ‖u‖γLγ ,

which is equivalent to

0 =(p− α)

ptp−α‖u‖p1 −

γ − αγ

tγ−α‖u‖γLγ −λ(α− q)

qtq−α‖u‖qLq

=(α− q)‖u‖qLq tq−α

q(Λe(tu)− λ) .

Since (Reλ)′(tu) = 0⇔ Λe(tu) = 0, we get (ii). The proof of (iii) and (iv) followfrom items (i),(ii) and simple accounts.

10

s0 s0λ,µ(u) s1λ,µ(u) s2λ,µ(u)

Φλ,µ(su)

Φ

Figure 3: Fibering function Φλ,µ(su), s ≥ 0,

u ∈W 1,p0 .

tte,+λ te,−λ

R

0

Reλ

Rnλ

tn,+λ tn,−λ

Figure 4: The functions Reλ(tu), Rnλ(tu)

We need also

Corollary 2.6. The functionals λe(u), λn(u) and µn,±λ (u) for λ < λn,

µe,±λ (u) for λ < λe are weakly lower semi-continuous on W 1,p0 . Furthermore,

tn,−λ (u), te,−λ (u) are lower semi-continuous, while tn,+λ (u), te,+λ (u) upper semi-

continuous on W 1,p0

Proof. The weakly lower semi-continuity of λe(u), λn(u) on W 1,p0 follow

straightforward from the definition of its (see (15)).Let us prove as an example that µn,+λ (u) is weakly lower semi-continuous

on W 1,p0 . Assume that λ < λn. Let (um) be a sequence in W 1,p

0 suchthat um u as m → +∞ weakly in W 1,p

0 for some u 6= 0. Thenλ < λn < λn(u) and by the weakly lower semi-continuity of λn(u) we haveλ < λn < λn(u) ≤ lim infm→+∞ λn(um). Hence by Proposition 2.3, there existtn,±λ (u), tn,±λ (um) ∈ (0,+∞), m = 1, 2, . . .. Moreover, since Λn(tu) ≤ Λn(tum),for t > 0 and for sufficiently large m = 1, 2, . . ., we have

tn,+λ (um) ≤ tn,+λ (u) < tn,−λ (u) ≤ tn,−λ (um). (21)

Beside this we have Rnλ(tu) ≤ lim infm→+∞Rnλ(tum), for all t > 0. Hence andusing the fact that (Rnλ)′(tu) < 0 for t ∈ (0, tn,+λ (u)) and (21) we infer

µn,+λ (u) = Rnλ(tn,+λ (u)u) ≤ lim infm→+∞

Rnλ(tn,+λ (um)u)

≤ lim infm→+∞

Rnλ(tn,+λ (um)um) = lim infm→+∞

µn,+λ (um).

The proof of the last part of the corollary follows from (21).

3. Proof of Lemma 1.1

First we prove

11

Lemma 3.1. Assume 1 < q < α < p < γ < p∗. Then,

(I) for any λ ∈ (0, λe),

(i) (2) has a minimizer ue,+λ ∈W 1,p0 \0 such that 0 < µe,+λ = µe,+λ (ue,+λ ),

(Reλ)′′(ue,+λ ) > 0;

(ii) (3) has a minimizer ue,−λ ∈W 1,p0 \0 such that 0 < µe,−λ = µe,−λ (ue,−λ ),

(Reλ)′′(ue,−λ ) < 0;

(II) for any λ ∈ (0, λn),

(i) (4) has a minimizer un,+λ ∈W 1,p0 \0 such that 0 < µn,+λ = µn,+λ (un,+λ ),

(Rnλ)′′(un,+λ ) > 0;

(ii) (5) has a minimizer un,−λ ∈W 1,p0 \0 such that 0 < µn,−λ = µn,−λ (un,−λ ),

(Rnλ)′′(un,−λ ) < 0.

Proof. The proofs of these assertions are similar. Let us prove as an exampleassertion (i), (I).

Let λ > 0. Define the set Zλ := u ∈W 1,p0 \ 0 : (Reλ)′(u) = 0. Notice that

Re(u) = α

[(γ−p)p ‖u‖p1 + λ (γ−q)

q ‖u‖qLq

](γ − α)‖u‖αLα

, ∀u ∈ Zλ. (22)

Hence by the Sobolev inequality we derive that Re(u) ≥ α (γ−p)p(γ−α)‖u‖

p−α1 → ∞

if u ∈ Zλ and ‖u‖1 → +∞. Thus, Reλ is coercive on Zλ, ∀λ > 0.Let (um) be a minimizing sequence of (2), i.e., µe,+λ (um) = Reλ(um)→ µe,+λ ,

where by the homogeneity of µe,+λ (u) we may assume that tm = te,+λ (um) = 1for m = 1, 2, . . .. The coerciveness of Reλ on Zλ implies that the minimizing

sequence (um) of (2) has a weak in W 1,p0 and strong in Lr, 1 < r < p∗ limit

point ueλ ∈W 1,p0 .

Let us show that ueλ 6= 0. Observe,

Reλ(um) = R0λ(um)− α‖um‖γLγ

γ‖um‖αLα, m = 1, . . . , (23)

where

R0λ(u) :=

1p‖u‖

p1 + λ

q ‖u‖qLq

1α‖u‖αLα

, u ∈W 1,p0 \ 0.

It can be shown (see e.g. [13]) that

minu∈W 1,p

0 \0mint>0R0λ(tu) = µ0 > 0. (24)

Denote am :=α‖um‖γLγγ‖um‖αLα

, m = 1, 2, . . .. Then

R0λ(um) = amγ

1p‖um‖

p1 + λ

q ‖um‖qLq

‖um‖γLγ.

12

We may assume that a0 := limm→+∞ am ≥ 0. Since µ0 > 0, a0 6= 0. Suppose,conversely to our claim, that ueλ = 0. Then

Reλ(um) ≥ am(γ

C

‖um‖γ−pLγ

− 1

)→ +∞,

which is a contradiction. Thus, ueλ 6= 0 and µe,+λ > 0. Now by the weakly lower

semi-continuity of µe,+λ (u) it follows that µe,+λ (ue,+λ ) ≤ µe,+λ , which obviously

implies µe,+λ = µe,+λ (ue,+λ ).

Corollary 3.2. (i) if λ ∈ (0, λe), then 0 < µe,+λ < µe,−λ < +∞,

(ii) if λ ∈ (0, λn), then 0 < µn,+λ < µn,−λ < +∞.

Proof. By Lemma 3.1, we have

0 < µe,+λ ≤ µe,+λ (ue,−λ ) < µe,−λ (ue,−λ ) = µe,−λ < +∞.

Thus, we get (i). The proof of (ii) is similar.

Corollary 3.3. If λ ∈ (0, λe), then (i) µe,−λ < µn,−λ ; (ii) µn,+λ < µe,+λ

Proof. By Lemma 3.1, there exists un,−λ such that µn,−λ = µn,−λ (un,−λ ). Lemma

2.5 entails that Reλ(te,−λ (un,−λ )un,−λ ) = Rnλ(te,−λ (un,−λ )un,−λ ) and the function

t 7→ Rnλ(tun,−λ ) is decreasing on the interval (tn,−λ (un,−λ ), te,−λ (un,−λ )). Hence,

µe,−λ ≤ Reλ(te,−λ (un,−λ )un,−λ ) = Rnλ(te,−λ (un,−λ )un,−λ ) < Rnλ(tn,−λ (un,−λ )un,−λ ) = µn,−λ ,

and we get (i). The proof of (ii) is similar.

The proof of Lemma 1.1: follows from Corollaries 3.2, 3.3.

Corollary 3.4. (i) The minimizer ue,+λ of (2) (perhaps, after a scaling) is

a non-negative critical point of Φλ,µe,+λ, moreover Φλ,µe,+λ

(ue,+λ ) = 0 and

Φ′′λ,µe,+λ

(ue,+λ ) > 0.

(ii) The minimizer ue,−λ of (3) (perhaps, after a scaling) is a non-negative

critical point of Φλ,µe,−λmoreover Φλ,µe,−λ

(ue,−λ ) = 0 and Φ′′λ,µe,−λ

(ue,−λ ) < 0.

Proof. (i) Let ue,+λ be a minimizer of (2). Then ue,+λ is also a minimizer of (20)

with te,+λ (ue,+λ ) = 1. Hence, 0 = Dµe,+λ (ue,+λ ) = DReλ(ue,+λ ), and consequently,

DΦλ,µe,+λ(ue,+λ ) = 0. Moreover, Reλ(ue,+λ ) = µe,+λ , (Reλ)′′(ue,+λ ) > 0 yield

Φλ,µe,+λ(ue,+λ ) = 0 and Φ′′

λ,µe,+λ(ue,+λ ) > 0, respectively. Since µe,+λ (|ue,+λ |) =

µe,+λ (ue,+λ ) one may assume that ue,+λ ≥ 0. The proof of (ii) is similar.

13

4. Proof of (1o), Theorem 1.2

We obtain the solution u2λ,µ using the following Nehari minimization problem

Φ2λ,µ = minΦλ,µ(u) : u ∈ RN 2

λ,µ, (25)

where RN 2λ,µ := u ∈ Nλ,µ : (Rnλ)′(u) ≥ 0 . Observe, RN 2

λ,µ 6= ∅, for all

λ ∈ (0, λe) and µ ∈ (µn,+λ , µn,−λ ). Indeed, if µ ∈ (µn,+λ , µn,−λ ), then µ <

µn,−λ < µn,−λ (u), ∀u ∈ W 1,p0 \ 0, and therefore, there exists u ∈ W 1,p

0 \ 0 such

that µn,+λ < µn,+λ (u) < µ < µn,−λ (u). Lemma 2.5 implies that there exists

s2λ,µ(u) ∈ (tn,+λ (u), tn,−λ (u)), and thus, s2

λ,µ(u)u ∈ RN 2λ,µ.

Furthermore, the assumption λ ∈ (0, λe), µ ∈ (µn,+λ , µn,−λ ) implies that

µn,+λ (u) ≤ µ < µn,−λ (u), ∀u ∈ RN 2λ,µ. (26)

Indeed, µ < µn,−λ ≤ µn,−λ (u) for any u ∈ W 1,p0 \ 0, whereas the conditions

Rnλ(u) = µ, (Rnλ)′(u) ≥ 0 for u ∈ RN 2λ,µ yield µn,+λ (u) ≤ µ.

Lemma 4.1. Let λ ∈ (0, λe) and µ ∈ (µn,+λ , µn,−λ ). There exists a minimizer

u2µ ∈W 1,p

0 \ 0 of (25) andΦ2λ,µ = Φλ,µ(u2

µ) > 0 if µ ∈ (µn,+λ , µe,+λ ),

Φ2λ,µ = Φλ,µ(u2

µ) = 0 if µ = µe,+λ ,

Φ2λ,µ = Φλ,µ(u2

µ) < 0 if µ ∈ (µe,+λ , µn,−λ ).

(27)

Proof. Note that by Lemma 3.1, the minimum of (25) for µ = µe,+λ attains

(perhaps, after a scaling) at the solution ue,+λ ∈W 1,p0 \0 of (2) and Φ2

λ,µ|µ=µe,+λ=

0. This easily implies the proof of the lemma in the case µ = µe,+λ .

Assume that µ ∈ (µn,+λ , µn,−λ ). Let um be a minimizing sequence of (25).

The coerciveness of Φλ,µ implies that the sequence (um) is bounded in W 1,p0

and thus, up to a subsequence,

um → u2µ strongly in Lr, and weakly in W 1,p

0 ,

for some u2µ ∈ W 1,p

0 , where r ∈ (1, p∗). Moreover, Φλ,µ(u2µ) ≤

lim infm→∞ Φλ,µ(um) = Φ2λ,µ.

If µ ∈ (µe,+λ , µn,−λ ), then there exists u ∈ RN 2λ,µ such that 1 =

s2λ,µ(u) ∈ (te,+λ (u), te,−λ (u)). Hence by (iv), Lemma 2.5, µ = Rnλ(s2

λ,µ(u)u) >

Reλ(s2λ,µ(u)u) ≡ Reλ(u), which implies 0 > Φλ,µ(u) ≥ Φ2

λ,µ. Hence, Φλ,µ(u2µ) <

0, and consequently, u2µ 6= 0 for µ ∈ (µe,+λ , µn,−λ ).

If µ ∈ (µn,+λ , µe,+λ ), then µ < µe,+λ < µe,+λ (u) = Reλ(te,+λ (u)u) for any

u ∈ RN 2λ,µ, and therefore, 1 = s2

λ,µ(u) ∈ (0, te,+λ (u)). Hence by (iv), Lemma 2.5,

Reλ(u) ≡ Reλ(s2λ,µ(u)u) > µ, and consequently, Φλ,µ(u) > 0, ∀µ ∈ (µn,+λ , µe,+λ ),

∀u ∈ RN 2λ,µ. Let us show that u2

µ 6= 0 for µ ∈ (µn,+λ , µe,+λ ). To obtain a

14

contradiction, suppose that u2µ = 0. Observe that if µ ∈ (µn,+λ , µe,+λ ), then

tn,+λ (um) < s2λ,µ(um) = 1, m = 1, 2, . . .. Consequently, tn,+λ (um)um → 0 as

m → +∞ strongly in Lr, r ∈ (1, p∗) and weakly in W 1,p0 . Analysis similar to

that in the proof of Lemma 3.1 shows that this implies Rnλ(tn,+λ (um)um)→ +∞.

However, this contradicts Rnλ(tn,+λ (um)um) ≤ Rnλ(um) = µ, m = 1, 2, . . ., and

thus, u2µ 6= 0 for µ ∈ (µn,+λ , µe,+λ ).

Let µ ∈ (µn,+λ , µe,+λ ) and suppose, contrary to our claim, that Φλ,µ(um)→ 0as m→ +∞. Then

Φλ,µ(um) = ‖um‖qLq (Reλ(um)− µ)→ 0. (28)

Since Φλ,µ(um) > 0, m = 1, 2, . . . and u2µ 6= 0, this implies Reλ(um) ↓ µ as

m → +∞. Since µ < µe,+λ ≤ µe,+λ (um) = Reλ(te,+λ (um)um), 1 = s2λ,µ(um) ∈

(0, te,+λ (um)), and therefore, Reλ(um) > Reλ(te,+λ (um)um) = µe,+λ (um) > µ, m =

1, 2, . . .. Consequently, limm→+∞ µe,+λ (um) = µ < µe,+λ , which is a contradiction

since µe,+λ (um) ≥ µe,+λ , m = 1, 2, . . .. Thus, Φ2λ,µ > 0 if µ ∈ (µn,+λ , µe,+λ ), and

we have proved (27).Let us show that u2

µ is a minimizer of (25), i.e., u2µ ∈ RN 2

λ,µ and

Φλ,µ(u2µ) = Φ2

λ,µ. By (26), µn,+λ (u2µ) ≤ µ < µn,−λ (u2

µ), and therefore ∃s2λ,µ(u2

µ) ∈(s1λ,µ(u2

µ), s3λ,µ(u2

µ)) such that

µ = Rnλ(s2λ,µ(u2

µ)u2µ) ≤ lim inf

m→∞Rnλ(s2

λ,µ(u2µ)um),

0 < (Rnλ)′(s2λ,µ(u2

µ)u2µ) ≤ lim inf

m→∞(Rnλ)′(s2

λ,µ(u2µ)um).

This means that 1 = s2λ,µ(um) ≤ s2

λ,µ(u2µ) < s3

λ,µ(um), m = 1, . . .. Hence by

Rnλ(u2µ) ≤ lim inf

m→∞Rnλ(um) = µ,

we obtain s1λ,µ(u2

µ) ≤ 1 ≤ s2λ,µ(u2

µ). Since Φ′λ,µ(su2µ) < 0, for any s ∈

(s1λ,µ(u2

µ), s2λ,µ(u2

µ)), we derive

Φλ,µ(s2λ,µ(u2

µ)u2µ) ≤ Φλ,µ(u2

µ) ≤ lim infm→∞

Φλ,µ(um) = Φ2λ,µ,

which yields that s2λ,µ(u2

µ)u2µ = u2

µ is a minimizer of (25) and thus, um → u2µ

strongly in W 1,p0 .

Consider the following subset of Mλ,µ

M2λ,µ := u ∈ RN 2

λ,µ : Φ2λ,µ = Φλ,µ(u).

Lemma 4.1 yields thatM2λ,µ 6= ∅ for any λ ∈ (0, λe) and µ ∈ (µn,+λ , µn,−λ ). Note

that the minimizer u2µ ∈ M2

λ,µ of (25) does not necessarily provide a solution

15

of (1). By Lemma 2.1 and (10), u2µ ∈ M2

λ,µ corresponds to a solution of (1), if

the strict inequality (Rnλ)′(u2µ) > 0 holds. Note that by Proposition 2.3,

(Rnλ)′(u2µ) > 0 ⇔ µn,+λ (u2

µ) < µ < µn,−λ (u2µ).

By (26), if λ ∈ (0, λe) and µ ∈ (µn,+λ , µn,−λ ), then µ < µn,−λ (u2µ) for any

u2µ ∈ M2

λ,µ. Thus, to obtain that u2µ ∈ M2

λ,µ is a weak solution of (1) it is

sufficient to show that µn,+λ (u2µ) < µ.

Corollary 4.2. Let λ ∈ (0, λe). If µ ∈ [µe,+λ , µn,−λ ), then u2µ ∈ M2

λ,µ is a weaksolution of (1).

Proof. Indeed, the inequality Φλ,µ(u2µ) = Φ2

λ,µ ≤ 0 implies µn,+λ (u2µ) <

µe,+λ (u2µ) ≤ Reλ(u2

µ) ≤ µ, and thus, µn,+λ (u2µ) < µ.

Lemma 4.3. Let λ ∈ (0, λe). There exists µ∗λ ∈ (µn,+λ , µe,+λ ) such that if µ ∈(µ∗λ, µ

e,+λ ), then u2

µ ∈M2λ,µ weakly satisfies (1), moreover Φλ,µ(u2

µ) = Φ2λ,µ > 0,

Φ′′λ,µ(u2µ) > 0.

Proof. By the above, it is sufficient to show that there exists µ∗λ ∈ (µn,+λ , µe,+λ )

such that µn,+λ (u) < µ, ∀u ∈ M2λ,µ, ∀µ ∈ (µ∗λ, µ

e,+λ ). Suppose this is false.

Then there exist sequences µm ∈ (µn,+λ , µe,+λ ) and um ∈ M2λ,µ, m = 1, 2, . . .

such that µm → µe,+λ as m → +∞ and µm = µn,+λ (um), ∀m = 1, 2, . . ..By Proposition Appendix A.3, up to a subsequence, um → u0 strongly inW 1,p

0 as m → +∞ for some u0 ∈ M2λ,µe,+λ

. Hence µn,+λ (u0) = µe,+λ , and

consequently u0 ∈ RN 2λ,µe,+λ

. Furthermore, Corollary Appendix A.2 implies

that Φλ,µe,+λ(u0) = limm→+∞ Φλ,µm(um) = Φ2

λ,µe,+λ. Hence u0 ∈ M2

λ,µe,+λand

µn,+λ (u0) = µe,+λ (u0) = µe,+λ , which contradicts (2o), Lemma 1.1.

Conclusion of the proof of (1o), Theorem 1.2.Let 0 < λ < λe. Introduce,

µ∗λ := supµ ∈ (µn,+λ , µn,−λ ) : µ = µn,+λ (u), ∃u ∈M2λ,µ. (29)

Corollary 4.2 and Lemma 4.3 imply that µn,+λ ≤ µ∗λ < µe,+λ . Hence for any

µ ∈ (µ∗λ, µn,−λ ), there holds µn,+λ (u) < µe,+λ , ∀u ∈ M2

λ,µe,+λ, and therefore, each

u2λ,µ := u2

µ ∈ M2λ,µ is a weak solution of (1). By the above, Φ′′λ,µ(u2

λ,µ) > 0,

Φλ,µ(u2λ,µ) > 0, for µ ∈ (µ∗λ, µ

e,+λ ), Φλ,µe,+λ

(u2λ,µe,+λ

) = 0, and Φλ,µ(u2λ,µ) < 0

for µ ∈ (µe,+λ , µn,−λ ). From the above, u2λ,µ is a local minimizer of Φλ,µ(u) in

the Nehari manifold Nλ,µ. This by Φ′′λ,µ(u2λ,µ) > 0 implies that u2

λ,µ is a local

minimizer of Φλ,µ(u) in W 1,p0 . Thus, u2

λ,µ is a linearly stable solution. It is

obvious that u2λ,µ is a ground state.

Note that Φλ,µ(|u2λ,µ|) = Φλ,µ(u2

λ,µ) and |u2λ,µ| ∈ RN 2

λ,µ. Hence one may

assume that u2λ,µ ≥ 0. The bootstrap argument and the Sobolev embedding

16

theorem yield that u2λ,µ ∈ L∞. Then C1,κ-regularity results of DiBenedetto [14]

and Tolksdorf [32] (interior regularity) combined with Lieberman [26] (regularityup to the boundary) yield u2

λ,µ ∈ C1,κ(Ω) for κ ∈ (0, 1). Furthermore, since

p < γ, the Harnack inequality due to Trudinger [35] implies that u2λ,µ > 0 in Ω.

From Corollary Appendix A.2 it follows that the function (µ∗λ, µn,−λ ) 3 µ 7→

Φλ,µ(u2λ,µ) is continuous and monotone decreasing.

5. Proof of (2o), Theorem 1.2

Let λ ∈ (0, λn) and µ ∈ (−∞,+∞). Consider

Φ3λ,µ = minΦλ,µ(u) : u ∈ RN 3

λ,µ, (30)

whereRN 3

λ,µ := u ∈ Nλ,µ : (Rnλ)′(u) ≤ 0, (Λn)′(u) < 0.Notice that RN 3

λ,µ = u ∈ Nλ,µ : (Rnλ)′(u) ≤ 0, tn(u) < 1, where tn(u) isdefined by (14).

Lemma 5.1. Let λ ∈ (0, λn) and µ ∈ (−∞,+∞). Then there exists aminimizer u3

λ,µ ∈ RN 3λ,µ of (30) and Φ′′λ,µ(u3

λ,µ) ≤ 0.

Proof. Since supu∈W 1,p0 \0 µ

n,−λ (u) = +∞, one can find u ∈ W 1,p

0 \ 0 for any

µ ∈ (−∞,+∞) such that µ < µn,−λ (u), and therefore, there exists s3λ,µ(u) >

tn(u). Hence s3λ,µ(u)u ∈ RN 3

λ,µ and therefore, RN 3λ,µ 6= ∅ for any λ ∈ (0, λn),

µ ∈ (−∞,+∞).Let (um) be a minimizing sequence of (30). Similar to the proof of (1o),

Theorem 1.2 one can deduce that there exists a subsequence, which we againdenote by (um), and a limit point u3

µ such that um → u3µ strongly in Lr,

r ∈ (1, p∗) and weakly in W 1,p0 . Observe that if u3

µ = 0, then by (14) we obtaina contradiction

1 > (tn(um))(γ−p) = Cn‖um‖p1‖um‖γLγ

≥ c 1

‖um‖γ−pLγ

→ +∞ as m→ +∞,

where c ∈ (0,+∞) does not depend on m. Thus u3µ 6= 0, and therefore, there

exists s3λ,µ(u3

µ) > tn(u3µ) so that s3

λ,µ(u3µ)u3

µ ∈ RN 3λ,µ.

By Corollary 2.6, we have tn,+λ (um) ≤ tn,+λ (u) < tn,−λ (u) ≤ tn,−λ (um) forsufficiently large m. From this and since Rnλ(tu3

µ) ≤ lim infm→+∞Rnλ(tum),for any t > 0, it follows that for sufficiently large m there holds s3

λ,µ(u3µ) ≤

s3λ,µ(um) = 1, and if s2

λ,µ(um) exists, s2λ,µ(um) < s2

λ,µ(u3µ) ≤ s3

λ,µ(u3µ). Hence

by the weak lower semi-continuity of Φλ,µ(u) we have

Φλ,µ(s3λ,µ(u3

µ)u3µ) ≤ lim inf

m→+∞Φλ,µ(s3

λ,µ(u3µ)um) ≤ lim inf

m→+∞Φλ,µ(um) = Φ3

λ,µ,

which implies that s3λ,µ(u3

µ)u3µ is a minimizer, and consequently, s3

λ,µ(u3µ) = 1

and um → u3λ,µ strongly in W 1,p

0 . Since λ < λn, we have (Λn)′(u3µ) < 0, and

therefore, (Rnλ)′(u3µ) ≤ 0.

17

Assume that µ ∈ (−∞, µn,−λ ). Then µ < µn,−λ < µn,−λ (u3µ), and therefore,

(Rnλ)′(u3µ) < 0. This by Lemma 2.1 and (10) implies that u3

λ,µ := u3µ is a

weak solution of (1). Moreover, since (Rnλ)′(u3λ,µ) < 0, we have Φ′′λ,µ(u3

λ,µ) < 0,

and therefore, u3λ,µ is a linearly unstable solution. Analysis similar to that in

the proof of (1o), Theorem 1.2 shows that u3λ,µ ∈ C1,κ(Ω) for κ ∈ (0, 1) and

u3λ,µ > 0. As (27) it can be shown that Φλ,µ(u3

λ,µ) > 0 if µ ∈ (−∞, µe,−λ ),

Φλ,µe,−λ(u3λ,µe,−λ

) = 0, and Φλ,µ(u3λ,µ) < 0 if µ ∈ (µe,−λ , µn,−λ ). Corollary

Appendix A.2 implies that the function (−∞, µn,−λ ) 3 µ 7→ Φλ,µ(u3λ,µ) is

continuous and monotone decreasing.Let us show (iv). From the monotonicity of Φλ,µ(u3

λ,µ) it follows

Φλ,µ(u3λ,µ) → C as µ → −∞ for some C ∈ (0,+∞]. Suppose, contrary to

our claim, that C < +∞. Since u3λ,µ ∈ Nλ,µ,

C

2<Φλ,µ(u3

λ,µ) = (31)

γ − pγp‖u3

λ,µ‖p1 + λγ − qγq‖u3

λ,µ‖qLq − µγ − αγα‖u3

λ,µ‖αLα <3C

2,

for sufficiently large |µ|. This implies that ‖u3λ,µ‖αLα → 0 as µ→ −∞, and (u3

λ,µ)

is bounded. Thus, there exists a subsequence µj → −∞ such that u3λ,µj

u

in W 1,p0 as j → +∞ for some u ∈ W 1,p

0 . Since ‖u3λ,µ‖αLα → 0 as µ → −∞,

‖u3λ,µj‖Lq → 0. Hence passing to the limit in Φ′λ,µ(u3

λ,µj) = 0 we obtain

limj→+∞ ‖u3λ,µj‖p1 = 0. This and (31) yield 0 < C/2 ≤ limµj→∞Φλ,µ(u3

λ,µj) =

0, which is a contradiction. Thus Φλ,µ(u3λ,µ) → +∞ and ‖u3

λ,µ‖1 → +∞ asµ→ −∞.

Let us show that u3λ,µ is a ground state of (1) if µ ∈ (−∞, µn,+λ ). By

Proposition 2.3 and (16), if µ ∈ (−∞, µn,+λ ), then for any u ∈ W 1,p \ 0,the fibering function Φλ,µ(su) has only critical point s3

λ,µ(u) > 0. Hence if

µ ∈ (−∞, µn,+λ ), then RN 3λ,µ = Nλ,µ and infu∈W 1,p

0 \0 maxs>0 Φλ,µ(su) = Φ3λ,µ

and we obtain the desired. This concludes the proof of (2o), Theorem 1.2.

6. Proof of Theorems 1.3, 1.4

Lemma 6.1. Let λ > 0, −∞ < µ < +∞. Then (1) has a mountain pass typesolution uλ,µ ∈ C1,κ(Ω), κ ∈ (0, 1) such that uλ,µ > 0 in Ω and Φλ,µ(uλ,µ) > 0.

Proof. The functional Φλ,µ satisfies the Palais-Smale condition. Indeed,

suppose that (un) ⊂ W 1,p0 \ 0 is a Palais-Smale sequence, i.e., Φλ,µ(un) →

c, DΦλ,µ(un)→ 0. By the Sobolev embedding theorem, we have

c+ o(1)‖un‖1 =γ − ppγ‖un‖p1 + λ

γ − qqγ‖un‖qLq − µ

γ − αα‖un‖αLα ≥

γ − ppγ‖un‖p1 − |µ|

γ − αα‖un‖α1 , as n→∞.

18

This implies that ‖un‖1 is bounded, and hence, after choosing a subsequenceif necessary, we have un u weakly in W 1,p

0 , and un → u strongly in Lr(Ω),1 ≤ r < p∗ to some u ∈ W 1,p

0 . Hence the convergence DΦλ,µ(un)(un) → 0 asn→∞ implies

lim supn→∞

〈−∆pun, un − u〉 = 0.

Thus by S+ property of the p-Laplacian operator (see [15]) it follows thatun → u strongly in W 1,p

0 , which means that Φλ,µ satisfies the Palais-Smalecondition.

The functional Φλ,µ possesses a mountain pass type geometry for λ > 0,−∞ < µ < +∞. Indeed, for each λ > 0, −∞ < µ < +∞, there existsc(λ, µ) > 0 such that (λ/q)sq − (µ/α)sα − (1/γ)sγ ≥ −c(λ, µ)sγ , ∀s > 0.Therefore, by the Sobolev embedding theorem we have

Φλ,µ(u) ≥ 1

p‖u‖p1 − c(λ, µ)‖u‖γLγ ≥ (

1

p− c(λ, µ)‖u‖γ−p1 )‖u‖p1, (32)

where c(λ, µ) > 0 does not depend on u ∈W 1,p0 . We thus can find a sufficiently

small ρ > 0 such that Φλ,µ(u) > δ for some δ > 0 provided ‖u‖1 = ρ. Evidently,

Φλ,µ(su)→ −∞ as t→ +∞ for any u ∈W 1,p0 \0, and thus, there is w1 ∈W 1,p

0 ,‖w1‖1 > ρ such that Φλ,µ(w1) < 0. Since Φλ,µ(0) = 0, Φλ,µ possesses amountain pass type geometry. It easily seen that the same conclusion holdsif we replace the function f(µ, λ, u) := |u|γ−2u + µ|u|α−2u − λ|u|q−2u by thetruncation function: f+(µ, λ, u) := f(µ, λ, u) if u ≥ 0, f+(u) := 0 if u < 0.Thus, the mountain pass theorem [2] provides us the critical point uλ,µ ofΦλ,µ(uλ,µ) such that

Φλ,µ(uλ,µ) = Φmλ,µ := infγ∈P

supu∈γ

Φλ,µ(u) > 0

and uλ,µ ≥ 0. As in the proof of (1o), Theorem 1.2, it follows that uλ,µ ∈ C1,κ(Ω)for κ ∈ (0, 1) and uλ,µ > 0 in Ω.

Proposition 6.2. If λ ∈ (0, λe), µ ∈ (−∞, µn,+λ ), then uλ,µ is a ground stateof (1). Moreover, Φλ,µ(uλ,µ)→ +∞, ‖uλ,µ‖1 → +∞ as µ→ −∞.

Proof. Let λ ∈ (0, λe), µ ∈ (−∞, µn,+λ ) and uλ,µ be a mountain pass solution.Then uλ,µ ∈ Nλ,µ, and in view of (2o), Theorem 1.2,

Φλ,µ(uλ,µ) = infg∈P

maxs∈[01]

Φλ,µ(g(s)) ≤ infu∈W 1,p

0 \0maxs>0

Φλ,µ(su) = Φ3λ,µ ≤ Φλ,µ(uλ,µ)

where P := g ∈ C([0, 1],W 1,p0 ) : g(0) = 0, g(1) = w1 with w1 ∈ W 1,p

0 such

that ‖w1‖1 > ρ and Φλ,µ(w1) < 0. Hence Φλ,µ(u3λ,µ) = Φ3

λ,µ = Φλ,µ(uλ,µ).

Thus, for λ ∈ (0, λe), µ ∈ (−∞, µn,+λ ), any mountain pass type solution uλ,µ isa ground state of (1), i.e., uλ,µ ∈ Mλ,µ. Furthermore, by (iv), (2o), Theorem

1.2, it follows that Φ3λ,µ = Φλ,µ(uλ,µ)→ +∞, ‖uλ,µ‖1 → +∞ as µ→ −∞.

19

Lemma 6.3. Let λ ∈ (0, λn). Then Φλ,µ(uλ,µ) → 0 and ‖uλ,µ‖1 → 0 asµ→ +∞.

Proof. The proof is based on the use of the following auxiliary variationalproblem

Φ1λ,µ = minΦλ,µ(u) : u ∈ RN 1

λ,µ, (33)

whereRN 1

λ,µ := u ∈ Nλ,µ : µn,−λ (u) ≤ µ,

and λ ∈ (0, λn), µ ∈ (µn,−λ ,+∞).

Lemma 6.4. Let λ ∈ (0, λn) and µ ∈ (µn,−λ ,+∞). There exists a minimizer

u1λ,µ ∈ RN 1

λ,µ of (33) such that Φ1λ,µ = Φλ,µ(u1

λ,µ) > 0.

Proof. Since λ ∈ (0, λn), the functional µn,−λ (u) is well defined on W 1,p0 \ 0.

This implies that RN 1λ,µ 6= ∅ for λ ∈ (0, λn), µ ∈ (µn,−λ ,+∞). By the proof of

Lemma 6.1, there exists ρ > 0 such that infu:‖u‖1=ρΦλ,µ(u) > 0, and therefore

Φ1λ,µ > 0 for λ ∈ (0, λn), µ ∈ (µn,−λ ,+∞).

Let (um)∞m=1 be a minimizing sequence of (33). The coerciveness of Φλ,µon Nλ,µ implies that the sequence (um) is bounded in W 1,p

0 and thus, up to asubsequence,

um → u1λ,µ strongly in Lr for r ∈ (1, p∗) and weakly in W 1,p

0 ,

for some u1λ,µ ∈W 1,p

0 . It is easily seen that if um → u1λ,µ strongly in W 1,p

0 , then

u1λ,µ is a non-zero minimizer of (33).

To obtain a contradiction, suppose that the convergence um → u1λ,µ in W 1,p

0

is not strong. Let us show that u1λ,µ 6= 0. Observe limm→+∞ ‖um‖p1 = β > 0,

since Φ1λ,µ > 0. Thus, if u1

λ,µ = 0, then 0 = limm→+∞(Φλ,µ)′(um) =

(1/p)β > 0 is a contradiction. By the weak lower-semicontinuity of µn,−λ (u)

we have µn,−λ (u1λ,µ) ≤ lim infm→∞ µn,−λ (um) ≤ µ, and therefore, there exists

s1λ,µ(u1

λ,µ) > 0 such that

µ = Rnλ(s1λ,µ(u1

λ,µ)u1λ,µ) < lim inf

m→∞Rnλ(s1

λ,µ(u1λ,µ)um).

Hence s1λ,µ(u1

λ,µ) < s1λ,µ(um), m = 1, 2, . . .. In view of that Φ′λ,µ(sum) > 0 for

s ∈ (0, s1λ,µ(um)), this implies

Φλ,µ(s1λ,µ(u1

λ,µ)u1λ,µ) <

lim infm→∞

Φλ,µ(s1λ,µ(u1

λ,µ)um) ≤ lim infm→∞

Φλ,µ(s1λ,µ(um)um) = Φ1

λ,µ,

which is a contradiction since s1λ,µ(u1

λ,µ)u1λ,µ ∈ RN 1

λ,µ.

Proposition 6.5. Let λ ∈ (0, λn). Then Φλ,µ(u1λ,µ)→ 0 as µ→ +∞.

20

Proof. Corollary Appendix A.2 implies that Φλ,µ(u1λ,µ) is monotone decreasing

on (µn,−λ ,+∞), and therefore, Φλ,µ(u1λ,µ)→ δ as µ→ +∞ for some δ ∈ (0,+∞).

Assume by contradiction that δ > 0. Then, since Φ′λ,µ(u1λ,µ) = 0, we have

δ

2< Φ(u1

λ,µ) =γ − pγp‖u1

λ,µ‖p1 +λγ − qγq‖u1

λ,µ‖qLq −µγ − αγα‖u1

λ,µ‖αLα <3δ

2, (34)

for sufficiently large µ, whence follows by the embedding W 1,p0 → Lα(Ω)

γ − pγp‖u1

λ,µ‖p1 − µCγ − αγα‖u1

λ,µ‖α1 <3δ

2, (35)

for some positive constant C. Hence (u1λ,µ) is bounded in W 1,p

0 . Consequently,

there exists a subsequence u1λ,µj

such that limj→+∞ µj = +∞ and u1λ,µj

u

weakly in W 1,p0 and u1

λ,µj→ u strongly in Lr(Ω), 1 ≤ r < p∗ as j →∞ for some

u ∈ W 1,p0 . Observe that (34) implies ‖u1

λ,µ‖αLα → 0 as µ→ +∞, which implies

u = 0. Passing to the limit in Φ′λ,µ(u1λ,µj

) = 0 we obtain limj→+∞ ‖u1λ,µj‖p1 = 0,

and consequently, (34) implies that µj‖u1λ,µj‖αLα → 0. Hence

δ

2≤ Φ(u1

λ,µ) = limµ→∞

Φ(u1λ,µ) = 0.

Thus δ = 0 and we obtain ‖u1λ,µ‖1 → 0 and Φλ,µ(u1

λ,µ)→ 0 as µ→ +∞.

Let us now conclude the proof of Lemma 6.3. Since µn,−λ (u1λ,µ) ≤ µ, the

function Φλ,µ(su1λ,µ) has a unique global maximum point s = s1

λ,µ(u1λ,µ) = 1.

Take a sufficiently large s0 > 1 such that Φλ,µ(s0u1λ,µ) < 0. Then by the above

there exists a mountain pass solution uλ,µ such that

Φλ,µ(uλ,µ) = Φmλ,µ := infγ∈P

supu∈γ

Φλ,µ(u) > 0,

where P := g ∈ C([0, 1];W 1,p0 ) : g(0) = 0, g(1) = s0u

1λ,µ. Note that g ∈ P ,

where g = su1λ,µ, s ∈ [0, s0]. Hence

Φλ,µ(u1λ,µ) = sup

s>0Φλ,µ(su1

λ,µ) ≥ infγ∈P

supu∈γ

Φλ,µ(u) = Φλ,µ(uλ,µ).

for any λ ∈ (0, λn) and µ ∈ (µn,−λ ,+∞). Then by Proposition 6.5, Φλ,µ(uλ,µ)→0, and consequently, ‖uλ,µ‖1 → 0 as µ→ +∞.

This concludes the proof of Theorem 1.3.Proof of Theorem 1.4:

The existence of three solutions uiλ,µ, i = 1, 2, 3, for λ ∈ (0, λe) and

µ ∈ [µe,−λ , µn,−λ ) follows from Theorems 1.2, 1.3, where we set u1λ,µ := uλ,µ, for

λ ∈ (0, λe), µ ∈ [µe,−λ , µn,−λ ). They are distinct since Φλ,µ(u1λ,µ) > 0 by Theorem

21

1.3, while Φλ,µ(u2λ,µ) < 0, Φλ,µ(u3

λ,µ) < 0 and Φ′′λ,µ(u2λ,µ) > 0, Φ′′λ,µ(u3

λ,µ) < 0

by Theorem 1.2. By Theorem 1.2, u2λ,µ is linearly stable while u3

λ,µ is linearlyunstable solution.

Consider

KΦmλ,µ:= u ∈W 1,p

0 : Φλ,µ(u) = Φmλ,µ, DΦλ,µ(u) = 0,

where Φλ,µ is replaced by the truncation functional as in the proof of Lemma6.1. Let us show that for λ ∈ (0, λe) and µ ∈ [µe,−λ , µn,−λ ), KΦmλ,µ

contains a

point u1λ,µ which is a linearly unstable solution. Indeed, by (32) it is easily

seen that 0 ∈ W 1,p0 is a local minimizer of Φλ,µ(u). Furthermore, by the

proof of (1o), Theorem 1.2, u2λ,µ is also a local minimizer of Φλ,µ(u) and

0 = Φλ,µ(0) > Φλ,µ(u2λ,µ). Hence, by the result of Hofer [18], Pucci, Serrin

[30] it follows that the set KΦmλ,µcontains a critical point u1

λ,µ which is not local

minimum of Φλ,µ(u), and therefore it is a linearly unstable solution.

Appendix A. Appendix

The statements below are proved using the approach introduced in [22].

Proposition Appendix A.1. Let i = 1, 2, 3, λ, µa, µb ∈ R, µb > µa. Assumethat viλ,µa , v

iλ,µb

are minimizers of (33) for i = 1, of (25) for i = 2 and (30)for i = 3. Then for sufficiently small |µb − µa| there holds

−(µb − µa)(siλ,µa(viλ,µb))

α

α‖viλ,µb‖αLα < Φλ,µb(v

iλ,µb

)− Φλ,µa(viλ,µa) <

−(µb − µa)(siλ,µb(v

iλ,µa

))α

α‖viλ,µa‖αLα .

(A.1)

Proof. Proofs for i = 1, 2, 3 are similar. As an example we prove for i = 1.Evidently, s1

λ,µb(v1λ,µa

)v1λ,µa

∈ N 1λ,µb

. Moreover, µe,−λ (v1λ,µa

) ≤ µa < µb,

and therefore, we have s1λ,µb

(v1λ,µa

)v1λ,µa

∈ RN 1λ,µb

. Hence Φλ,µb(v1λ,µb

) ≤Φλ,µb(s

1λ,µb

(v1λ,µa

)v1λ,µa

), and consequently

Φλ,µb(v1λ,µb

) ≤ Φλ,µa(s1λ,µb

(v1λ,µa)v1

λ,µa)− (µb − µa)

α‖s1λ,µb

(v1λ,µa)v1

λ,µa‖αLα .(A.2)

Observe, if µb > µa, then 0 < s1λ,µb

(v1λ,µa

) < s1λ,µa

(v1λ,µa

). Thus, since

s 7→ Φλ,µa(su1λ,µa

) is increasing in [0, s1λ,µa

(v1λ,µa

)], we obtain

Φλ,µa(s1λ,µb

(v1λ,µa)v1

λ,µa) ≤ Φλ,µa(s1λ,µa(v1

λ,µa)v1λ,µa) = Φλ,µa(v1

λ,µa),

and consequently, the second inequality in (A.1). The proof of the firstinequality in (A.1) is handled in much the same way.

22

From Lemmas 4.1, 5.1, 6.4, Proposition Appendix A.1 and using thecoerciveness of Φλ,µ on Nλ,µ it is not hard to show

Corollary Appendix A.2. The functions µ 7→ Φλ,µ(u1λ,µ) on (µn,−λ ,+∞) for

λ ∈ (0, λn); µ 7→ Φλ,µ(u2λ,µ) on (µn,+λ , µn,−λ ) λ ∈ (0, λe), µ 7→ Φλ,µ(u3

λ,µ) on(−∞,+∞) for λ ∈ (0, λe) are continuous and monotone decreasing.

Proposition Appendix A.3. Let µ0, µm ∈ (µn,+λ , µn,−λ ) (µ0, µm ∈(−∞,+∞)), m = 1, 2 . . . such that µm → µ0 as m → +∞. Then there exist asubsequence, which we again denote by (µm), and a sequence u2

λ,µm∈ M2

λ,µm

(u2λ,µm

∈M3λ,µm

) such that u2λ,µm

→ u2λ,µ0

strongly in W 1,p0 .

Proof. As an example we prove the proposition in the case i = 2. By the aboveit follows that Φλ,µn(v2

λ,µm) → Φ2

λ,µ0as m → +∞, which easily implies that

(s2λ,µ0

(v2λ,µm

)v2λ,µm

)∞m=1 is a minimizing sequence of (25) for µ = µ0. Then from

the proof of Lemma 4.1 it follows that up to a subsequence, u2λ,µm

→ u2λ,µ0

strongly in W 1,p0 , for some u2

λ,µ0∈M2

λ,µ0.

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