Bifurcation analysis for a modified quasilinear equation with negative exponent

: In this paper, we consider the following modified quasilinear problem: where Ω ⊂ R N is a smooth bounded domain, N ≥ 3, a , b are two bounded continuous functions, α > 0, 1 < β ≤ 22 * − 1 and λ > 0 is a bifurcation parameter. We use the framework of analytic bifurcation theory to obtain an analytic global unbounded path of solutions to the problem. Moreover, we get the direction of solution curve at the asmptotic point.


Introduction
In this paper we consider the following modi ed quasilinear problem −∆u − κu∆u = λa(x)u −α + b(x)u β in Ω, u > in Ω, u = on ∂Ω, (1.1) where Ω is a smooth bounded domain in R N , N ≥ , < a(x) ∈ C(Ω), b(x) ∈ C(Ω) ∩ L ∞ (Ω) and may change sign, κ > is a real constant, α > , < β ≤ * − , * = N N− , λ > is a bifurcation parameter. Problem (1.1) is related to the standing wave solutions for the quasilinear Schrödinger equations i∂ t ψ = −∆ψ + ψ + η(|ψ| )ψ − κ∆ρ(|ψ| )ρ ′ (|ψ| )ψ, (1.2) where ψ = ψ(t, x), ψ : R× Ω → C, κ > is a real constant. Equation (1.2) has been applied extensively in many areas of physical phenomena, for the progress of this topic and the other modi ed Schrödinger equations one may refer to [3, 12, 13, 24-26, 29-32, 42, 44-46]. For κ = , problem (1.1) can be transformed into a semilinear one. In recent years, this type of equations has been studied extensively in both bounded and unbounded domains due to its wide applications in non-Newtonian uids. For instance, Lazer and McKenna [27] studied the following semilinear problem −∆u = p(x)u −γ in Ω, u = on ∂Ω. (1.3) For p(x) > with some smoothness conditions, they showed that there exists a solution which is smooth on Ω and continuous on Ω, the Lazer-McKenna obstruction then was rstly presented: the equation has a H -solution if and only if γ < . For the power of 3, Sun and Zhang [40] provided an extension of the classical Lazer-McKenna obstruction and revealed the role of , they also gave a local description of the solution set. Lair and Shaker [28] proved that (1.3) has a unique weak H -solution on a bounded domain provided ε f (s)ds < ∞ and p(x) ∈ L (Ω). For the regularity of (1.3), Gui and Lin [17] obtained the positive solutions that are Hölder-continuous up to the boundary and has even better regularity in some special cases. The problem was also studied by Ma and Wei [34] when p(x) = − , they showed that the gradient estimates, Lestimates, global upper bounds, Liouville properties, classi cation of stable and nite Morse index solutions, and symmetry properties. For the perturbed singular problem, Yang [43] considered the following problem with singular nonlinearity, −∆u = λu −γ + u p in Ω, u = on ∂Ω. (1.4) For < γ < < p ≤ (N + )/(N − ), Yang carried out a direct analysis in an H -neighborhood and proved that (1.4) has a solution which is a local minimiser with respect to the H -topology. Then the existence of the second solution was given by making use of Ekeland's variational principle. Arcoya and Moreno-Mérida [2] extended the results of [43] to all γ > in subcritical case by establishing suitable approximated problems, they showed that there exists Λ > such that (1.4) has two positive solutions for every λ ∈ ( , Λ). Apart from the existence and regularity of solutions for this type of equations, there are many researchers have obtained global bifurcation and local multiplicity results. For instance, the authors in [11] considered the following singular elliptic problem with exponential type growth in a bounded smooth domain Ω ⊂ R , −∆u = λ(u −δ + h(u)e u p ) in Ω, u = on ∂Ω, where ≤ p ≤ , < δ < and h(t) is a smooth "perturbation" of e t p as t → ∞. For the radial case, they made a detailed study of the blow-up/convergence of the solution branch as it approaches to the asymptotic bifurcation point at in nity. For the critical case p = , they also interpreted all previous works on multiplicity in terms of the corresponding bifurcation diagrams and the asymptotic pro le of large solutions along the branch at in nity. Later, Bougherara et al. [5] considered the following semilinear elliptic problem with a strong singular term in a bounded smooth domain Ω ⊂ R N (N ≥ ), They improved the results of [11] to all δ > , and obtained an analytic global unbounded path of solutions of (1.5) by using the framework of analytic bifurcation theory as developed in the work [4]. In two dimensions, for < δ < and certain classes of nonlinearities f with critical growth, it was shown that the existence of an analytic unbounded path of solutions of (1.5) whose Morse index is unbounded along the path and admits in nitely many turning points. Specially, for p-Laplacian di erential operator, such bifurcation type results were obtained by Bai et al. [6], Papageorgiou, Rădulescu and Repovš [35,36]. For more results about this type of equations, one may see [1, 8, 14, 18-20, 22, 38] and the references therein. Motivated by the above papers and by the increasing interest on problems with singular nonlinearities, our main purpose in this paper is to investigate the analytic global bifurcation in the case of κ = for (1.1). The quasilinear term u∆u makes the problem much more complicated. By using a change of variables, the authors in [30] transformed the quasilinear Schrödinger equations into a new semilinear one and showed that the existence of ground states of soliton-type solutions by variational method. Involving the quasilinear operator and singular nonlinearities in bounded domain, there are some results as well. For instance, the authors in [16] considered the following singular quasilinear problem for δ ∈ ( , ) in a ball Ω ⊂ R N , They obtained the existence of solutions of (1.6) when λ belongs to a certain neighborhood of the rst eigenvalue λ . Moameni and O n [33] obtained the same results as in [16] by considering a more general class of equations. The authors in [37] considered the following class of singular quasilinear problem, The function a(x) is nonnegative, δ > is a constant and the nonlinearity h(x, u) is continuous. They showed that the existence of a solution for the problem via sub-supersolution method when h has an arbitrary polynomial growth. For the second result, they showed that the existence of the second solution by applying the mountain pass theorem when h has subcritical growth. Recently, the authors in [39] studied problem (1.1) in the case of < α < , they showed that the existence of a minimal solution as a minimum critical point of the energy functional, and then the second solution was also given by constrained critical point theory.
As far as we know in the literature there is little research on the bifurcation analysis of solutions to this type of quasilinear equation. The present paper is mainly consider the case of κ = , and regard (1.1) as a bifurcation problem with λ being the bifurcation parameter. There seem to be some di culties to transform the quasilinear equation to a semilinear one by making a change of variables, such that the existence of solutions of (1.1) be equivalent to the existence of solutions to new transformed equation, and there holds similar properties. Inspired by [4], we use the framework of analytic bifurcation theory to obtain an analytic global unbounded path of solutions to the transformed equation and so to problem (1.1). Then the direction of the solution curve at the asmptotic point under some conditions is given by making use of local bifurcation theory.
The paper is organized as follows. In section 2, we state some preliminaries and main results including transforming quasilinear problem (1.1) into a semilinear elliptic one and give some de nitions and lemmas. In section 3, we give the proof to the main results by using bifurcation theory to the transformed problem to obtain an global unbounded path of solutions, and the properties at turning point.

Preliminaries and main results
Taking into account the ideas of [30], we can use the change of variables ω = h − (u) to transform the quasilinear equation into a semilinear one with singularity at zero and superlinear at in nity, which h is de ned by h Now, we list some properties of the function h(t) as given in [41,42].
Lemma 2.1. Assume h : R → R is given as above, then there hold: h is unique, invertible, and C ∞ (R) − function, is decreasing for t > and α > , (10) the function h β (t)h ′ (t) is increasing for t > and β ≥ .
Using Lemma 2.1, we can perform the changing of variables ω = h − (u) to infer that u ∈ H loc (Ω) is a solution of (1.1) if and only if ω ∈ H loc (Ω) is a weak solution of in the sense of the following de nition.

De nition 2.2.
We say ω ∈ H loc (Ω) ∩ C (Ω) is a weak solution of (2.1) if essinf K ω > for any compact set K ⊂ Ω, (u − ε) + ∈ H (Ω) for any ε > given, and To state our main result, we denote the following set of all classical solutions to (2.1) In the sequel, let φ be the rst positive eigenfunction for −∆ in H (Ω), φ L ∞ (Ω) = , we de ne de nes a Banach space endowed with the norm is an open convex subset of C ϕ (Ω). Now, we de ne the following solution operator associated to (2.1):
We recall some results about global analytic bifurcation theory that introduced in [7]. Let X, Y be real Banach spaces, U ⊂ R × X be an open set containing ( , ) in its closure and F : U → Y be an R-analytic function.
De ne the solution set and the non-singular solution set
At each of its points, A has a local analytic re-parameterization in the following sense: for each s * ∈ ( , ∞), there exists a continuous and injective map ρ * : (− , ) → R such that ρ * ( ) = s * and the re-parametrisation Furthermore, the map s → λ(s) is injective in a right neighborhood of s = and for each s * > there exists ε * > such that λ is injective on [s * , s * + ε * ] and [s * − ε * , s * ]. (e) One of the following holds: (i) (λ(s), u(s)) R×X → ∞ as s → ∞, (ii) the sequence {(λ(s), u(s))} approaches the boundary of U as s → ∞, (iii) A is the closed loop: In this case, chosing the smallest T > such that

then (e)(iii) occurs and |s − s | is an integer multiple of T. In particular, the map s → (λ(s), u(s)) is injective on [ , T).
From the de nition of F, we immediately have the following Lemma.
Lemma 2.6. Let F be given as above. Then

1) coincides with a path-connected portion of
A which closure containing ( , ), furthermore, the minimal solution branch is parametrised by an analytic map, (v) A has at least one asymptotic bifurcation point Λa ∈ [ , Λ], (vi) each point of A has a local analytic re-parametrization as follows: for each s * ∈ ( , ∞), there exists a continuous and injective map ρ * : (− , ) → R such that ρ * ( ) = s * and the re-parametrisation Moreover, the map s → λ(s) is injective in a right neighborhood of s = and for each s * > there exists ε * > such that λ is injective on [s * , s * + ε * ] and [s * − ε * , s * ].

Local and global bifurcation analysis
In this section, we establish local and global bifurcation to problem (2.1). Firstly, we consider the properties of the linearised operator for the corresponding function F to the problem. Proof. We split the proof in three steps.

. Analysis of the solution operator and linearised operator
Step 1. h(ω) ∈ C + ϕα for any ω ∈ C + ϕα . We just consider the case < α < , because the case α ≥ is similar. Since ϕα = φ , it follows from the properties of h(t) and φ L ∞ (Ω) = , that there exist positive constants Step 2. For any ω ∈ C ϕ −α α (Ω), by similar ideas made in the proofs of Proposition 2.3 in [5], we are able to obtain that ω → (−∆) − ω ∈ C ϕα (Ω) is a linear continuous map and hence analytic.
Step 3. Due to α > , < β ≤ * − and the properties of h(t), it is easy to see that ( ϕα . Hence, by the above three steps, we can obtain the result. Now, to show the existence of the analytic global path of solutions to F in R + × C + ϕα (Ω), we consider the following problem: where k ≥ and g(x) is a local Hölder continuous function in Ω.
To begin with, we have the following comparison principle.

Lemma 3.2.
Assume that there exist u and v satisfying the following inequalities in the weak sense, Proof. Arguing by contradiction, assume that Ω : By pointwise limit, we get where χ Ω represent the characteristic function of Ω . By taking derivatives, we get in Ω . From the above argument, we have that φϵ , ψϵ ∈ H loc (Ω) ∩ C (Ω). So, by density arguments, we are able to test the rst and second inequalities in (3.2) against φϵ and ψϵ, respectively, to obtain Now, set W := ∇ ln uϵ = ∇u uϵ and W := ∇ ln vϵ = ∇v vϵ in Ω , it follows from (3.3) and Lemma 4.2 of [23], that On the other hand, since a(x) > and h(t) −α h ′ (t) is decreasing for t > and α > , we havê (3.5) Besides, we are able to obtainˆΩ This, together with (3.4) and (3.5), implies which is a contradiction. Thus, Ω = ∅, and hence the proof is completed.
The unique positive solution ψ ε ∈ H (Ω) of is a super-solution of (3.6). Indeed, it follows from the monotonicity of By comparison principle, it is obvious that ψ ϵ < ψ ε . Then we obtain a solution wε ∈ [ψ ϵ , ψ ε ] to (3.6) by standard arguments, and uniquely by the non-increasing nature of the right hand side in (3.6). Thus, we can infer that wε is Hölder continuous on Ω through elliptic regularity. In addition, wε > in Ω by maximum principle. Now we prove that wε is monotone as ε → + by a comparison argument: let < ε ′ < ε, then we have On the other hand, assume x = arg min which is a contradiction with the last equation. Thus we have w ε ′ > wε in Ω if < ε ′ < ε. Therefore, we obtain that w = lim ε→ + wε ≥ cϕα and w ∈ C (Ω) (3.7) and satis es in the sense of distributions of (3.1). Furthermore, w ∈ W ,q (Ω) for some q > . Indeed, from g(x) ∈ L ∞ (Ω) and (3.7), it is easy to get that Then, it follows from Theorems 3 and 4 of [15], that w ∈ W ,q (Ω) for some q > . Using the comparison principle again, we can derive that w is the unique weak solution of (3.1). Now, assume that (3.1) has a super-solution ϕ ∈ C + ϕα (Ω). It is clear that ϕ is also a super-solution of (3.6). Thus, for c > small enough, we get cϕα ≤ ϕ. Hence ψ ε can be replaced by ϕ and repeat the above argument to get a solution w ∈ C + ϕα (Ω). Thus we complete the proof.
Since < a(x) ∈ C(Ω) and h(t) We have the following result.
To this end, we need the next Lemma.

Lemma 3.5. If m(x) is de ned as above, then there exists positive constant m such that m(x) ≤ m d(x, ∂Ω) − .
Proof. Since a(x) ∈ C(Ω) be bounded, it su ces to prove that there exists a positive constant C such that −(h(ω) −α h ′ (ω)) ′ ≤ Cd(x, ∂Ω) − for all ω > . In the following process, the C represents di erent positive constant. A direct calculation shows that , which turns out that I +A λ is invertible on C ϕα (Ω). On the other hand, B is compact on C ϕα (Ω). Thus, ∂ω F(λ, ω) is Fredholm with index 0.

. Local and global bifurcation analysis
In this section, we shall show the existence of minimal solution to problem (2.1) for λ ∈ ( , Λ), and then state that the full set of minimal solution can be parametrised by an analytic curve. Besides this, we shall illustrate some bifurcation results for λ = Λ, where Λ := sup{λ > : (2.1) has a weak solution}.

Remark 3.9. Suppose that there exists M > such that
Indeed, suppose ω λ is another solution and satis es ω λ C(Ω) < M with λ < λ . Let ψ λ = ω λ − ω λ , then ψ λ solves where ξ λ lies between ω λ and ω λ . It is easy to see λa( Proof. For some φ ∈ C (Ω) ∩ C ϕα (Ω), de ne the minimization problem inf By the properties of h(t), a(x) and b(x), it is easy to show that the above functional is coercive and weakly lower semicontinuous on H (Ω). Then there exists a minimiser ψ ∈ H (Ω) and is a non-trivial H -weak By standard elliptic regularity, we have ψ ∈ C (Ω). Now, in H -weak sense, let us take a comparison with the solution ξ ∈ H (Ω) of −∆ξ = M in Ω, where M = sup φ|, which infer that ψ ∈ Cφ (Ω).
Remark 3.14. It is clear that the above lemma remains true if b(x) < .
Next we consider the bifurcation analysis at λ = Λ. Then we have Λ (Λ) ≥ by the above obtained, and in fact Λ (Λ) = by the implicit function theorem and (2.1) has no solution for λ > Λ. We now verify the conditions of the local bifurcation result of Cranduall-Rabinowitz [9]. From obtained above, the map ∂ω F(λ, ω) is Fredholm with index 0 for all (λ, ω) ∈ R + × C + ϕα (Ω), then we can easily get ker(∂ω F(Λ, ω Λ )) is one dimensional and spanned by φ Λ which is the associated eigenfunction of Λ. We can also get codimRange(∂ω F(Λ, ω Λ )) = . Now we claim that

. The proof of main results
Proof of Theorem 2.7. Let U = R + × X = R + × C ϕα (Ω) and the positive cone W = C + ϕα (Ω). Clearly W is open. Conditions (H1)-(H3) of Lemma 2.5 hold because of Lemma 3.13, Proposition 3.1, Lemma 3.7 and Lemma 3.12. In fact, by Lemma 3.12, we may x A + = {(λ(s), w(s)) : < s < s } for some s > be an analytic parametrisation which is one portion of minimal solution branch which given by {(λ, ω λ ) ∈ Γ : < λ < λ }. Then Lemma 2.5 holds. Next, we apply Lemma 2.5 to prove Theorem 2.7. It is clear that assertions (iii) and (vi) of Theorem 2.7 are true. From the de nition of A and A + , assertion (i) easily obtained. Assertion (iv) can be get from Lemma 3.8 and Lemma 3.12.
It remains to prove assertion (ii), clearly, (v) is a consequence of (ii). In order to show assertion (ii), we only need verify the property (e)(i) of Lemma 2.5 occurring. If case (e)(ii) is occur, then there exists (λ(sn), ω(sn)) → ( , ω ) as sn → ∞ in U, where ( , ω ) is a boundary point. Then ∆ω(sn) → in C loc (Ω), i.e. ω = and hence ω ≡ . By Lemma 3.8, it is easy to see that ω(sn) is the minimal solution for all large sn. However, the minimal solution arc A starting from ( , ) is isolated from other solutions, and hence, the distinguished arc corresponding to all large s coincide with A, which is a contradiction with (a) of Lemma 2.5. By the same argument, we can rule out case (e)(iii). Hence case (e)(i) holds. Therefore, ω(s) C ϕα (Ω) → ∞ as s → ∞ since (2.1) has no solution for λ > Λ. This completes the proof.
Proof of Corollary 2.8. It follows from Lemma 3.15 and Lemma 2.5, that one can easily get the analytic path A turns to the left at the point (Λ, ω Λ ).