This paper investigates the existence and multiplicity of positive solutions for the Kirchhoff-type elliptic equation

where λ ≥ 0 is a parameter, Ω ⊂ ℝ N ( N ≥ 1 ) is a bounded smooth domain, M : [ 0 , + ∞ ) → ℝ and f : Ω × ℝ → ℝ are smooth functions. The Kirchhoff-type equation is a class of non-classical elliptic partial differential equations. It was proposed by Kirchhoff [1] to describing the equilibrium state of strings or membrane vibrations, and it has a wide range of applications in theoretical physics, biochemistry, engineering mathematics, and other disciplines. When the Kirchhoff-type equation does not contain a nonlocal term, that is, when M ≡ 1 in Equation (1), there have been some results on the existence of positive solutions for this class of equations; see references [2]-[5]. For example, Carmona and Martnez [2] studied the semilinear elliptic problem

where Ω ⊂ ℝ N ( N ≥ 1 ) is a bounded smooth open set, and f is a nonnegative continuous real-valued function with multiple zeros. Based on topological degree theory and the Rabinowitz global bifurcation theorem, they obtained.

Theorem A ([2] Theorem 1.1) Let f be a nonnegative continuous function, and let 0 < α < β be two adjacent positive zeros of f , that is, f ( α ) = f ( β ) = 0 , and f ( s ) > 0 for any s ∈ ( α , β ) . Assume that for σ = α and σ = β , the following condition holds: there exist M > 0 , ϵ > 0 such that the function f ( s ) + M s is increasing on the interval ( σ − ϵ , σ ) . Define

Then there exists an unbounded connected component Σ ⊂ S α , β , and the following conclusions hold:

1) λ α β = inf { λ : ( λ , u ) ∈ Σ } ;

2) λ α β > 0 , and for any λ ≥ λ α β , problem (2) has at least one solution u , satisfying α < ‖ u ‖ L ∞ ( Ω ) < β ;

3) for any λ > λ α β , there exist at least two distinct solutions u 1 ≠ u 2 in Σ , such that ( λ , u 1 ) , ( λ , u 2 ) ∈ Σ , and α < ‖ u i ‖ L ∞ ( Ω ) < β ( i = 1 , 2 ) .

Unlike semilinear elliptic equations, a prominent feature of Equation (1) is that it contains a nonlocal term ∫ Ω | ∇ u | 2 d x , which makes Equation (1) not a pointwise

identity, and its solutions cannot be verified pointwise, thus bringing essential difficulties to the study of the problem. In recent years, many scholars have adopted different methods to study this class of equations and have obtained a series of results; see references [6]-[12]. Recently, Ambrosetti and Arcoya [6] used the Rabinowitz global bifurcation theorem to study the existence of positive solutions for problem (1), and assumed the following:

(A1) f ( x , 0 ) = 0 , ∀ x ∈ Ω , a ( x ) : = D u f ( x , 0 ) > 0 ;

(A2) f ( x , u ) = b ( x ) u + g ( x , u ) , where b ( x ) > 0 , lim u → ∞ g ( x , u ) u = 0 ;

(A3) there exists α > 0 such that f ( x , u ) ≥ α u , ∀ ( x , u ) ∈ Ω × ℝ ;

(A4) there exists m 0 > 0 such that M ( t ) ≥ m 0 , ∀ t ≥ 0 ;

(A5) there exists M ∞ > 0 such that lim t → + ∞ M ( t ) = M ∞ .

Denote

where λ 1 [ a ] is the principal eigenvalue of

and a ( x ) ∈ C ( Ω ¯ ) with a ( x ) > 0 . The main results obtained in this paper are as follows:

Theorem B ([6] Theorem 2.6) Assume that (A1)-(A5) hold. Then there exists a connected branch Γ of positive solutions of problem (1) bifurcating from ( Λ 0 , 0 ) , satisfying the following properties:

1) Γ connects to the point ( Λ ∞ , ∞ ) ;

2) if lim t → ∞ M ( t ) = ∞ , then Proj ℝ Γ = [ l , + ∞ ) , where l ≤ Λ 1 0 .

It should be pointed out that the results in reference [6] depend on condition (A1), and their branch of positive solutions emanates from ( Λ 0 , 0 ) . However, this paper allows f ( x , 0 ) > 0 , in which case u ≡ 0 is no longer a solution of the original equation, and the original bifurcation framework is no longer applicable; furthermore, when the nonlinear term f has two zeros on [ 0 , + ∞ ) , this paper will continue to explore the multiplicity of positive solutions of problem (1). This makes the model possess a broader physical background. Therefore, this paper investigates the existence and multiplicity of positive solutions of problem (1) under the condition f ( x , 0 ) > 0 .

The general assumptions of this paper are as follows:

(A6) f ( x , 0 ) > 0 , ∀ x ∈ Ω ;

(A7) there exist 0 < s ′ < s ″ such that when u ∈ [ s ′ , s ″ ] , f ( x , u ) ≤ 0 , and when u ∈ ( 0 , s ′ ) ∪ ( s ″ , + ∞ ) , f ( x , u ) > 0 , ∀ x ∈ Ω .

Let u , v belong to the Sobolev space X : = H 0 1 ( Ω ) , then the inner product and norm on it are respectively

Theorem 1 Assume that (A2)-(A4) and (A6) hold. Then problem (1) has two unbounded connected continua Σ 0 ⊂ R + × X and Σ ∞ ⊂ R + × X , satisfying:

1) Assume that (A5) holds. Then Σ 0 connects to the point ( Λ ∞ , ∞ ) , where Λ ∞ : = M ∞ λ 1 [ b ] . That is, Σ 0 = Σ ∞ ;

2) if lim t → ∞ M ( t ) = ∞ , then Proj ℝ Σ 0 = Proj ℝ Σ ∞ = ( 0 , + ∞ ) , where Proj ℝ Σ 0 and Proj ℝ Σ ∞ denote the projections of Σ 0 and Σ ∞ in the R direction, respectively.

Theorem 2 Assume that (A2), (A4)-(A7) hold. Then problem (1) has no positive solution u satisfying max x ∈ Ω ¯ | u ( x ) | ∈ [ s ′ , s ″ ] . In this case, the connected continua Σ 0 and Σ ∞ are disjoint.

Corollary 1 Assume that (A2)-(A4) and (A6) hold. Then

1) Assume that (A5) holds. If λ ∈ ( 0 , Λ ∞ ) , then problem (1) has one positive solution.

2) If lim t → ∞ M ( t ) = ∞ , then when λ > 0 , problem (1) has one positive solution.

Corollary 2 Assume that (A2), (A4)-(A7) hold. If λ > Λ ∞ , then problem (1) has at least two positive solutions u 1 , u 2 , where u 1 satisfies max x ∈ Ω ¯ | u 1 ( x ) | < s ′ , and u 2 satisfies max x ∈ Ω ¯ | u 2 ( x ) | > s ″ .

The paper is organized as follows. Section 2 is devoted to the preliminary knowledge. More precisely, we introduce the solution operator T associated with the Kirchhoff-type elliptic equation, prove that T is compact, and then establish the existence of an unbounded connected continuum Σ 0 connecting ( 0 , 0 ) to infinity as well as an unbounded connected continuum Σ ∞ emanating from ( Λ ∞ , ∞ ) . The proofs of our main results are provided in Section 3. We first obtain the existence of positive solutions by using the global bifurcation method, and then, under the assumption that f possesses two zeros on [ 0 , + ∞ ) , we prove the multiplicity of positive solutions and show that the two branches Σ 0 and Σ ∞ are disjoint.

Let F denote the Nemitski operator associated with f , and define the linear operator L : L 2 ( Ω ) → X by

Since M ( ‖ u ‖ 2 ) > 0 , we define the operator T : ℝ × X → X by

Clearly, a solution of u = T ( λ , u ) is a solution of (1).

Lemma 1Assume that (A2), (A4) and (A5) hold, and that f : Ω × ℝ → ℝ is a Carathodory function satisfying the growth condition

Then T : ℝ × X → X is a compact operator.

Proof. According Theorem 1.2.1 in [13] and (A2), the Nemitski operator F is continuous from X to L 2 ( Ω ) . To see this, note that the convergence u n → u in X implies strong convergence in L p ( Ω ) and pointwise convergence almost everywhere. Combining this with the growth condition and Lebesgue’s dominated convergence theorem yields F ( u n ) → F ( u ) in L 2 ( Ω ) . Since the inverse Laplacian L = ( − Δ ) − 1 : L 2 ( Ω ) → X is a compact mapping from L 2 ( Ω ) to X , the L F : X → X is consequently compact.

Now choose an arbitrary bounded sequence { u k } ⊂ X , In view of (A4) and (A5), after passing to a subsequence, we may assume ‖ u k ‖ → r , or at least that { ‖ u k ‖ } is bounded. The continuity of M then gives

where C 0 is a constant. Hence

admits a convergent subsequence in X , this proves that T is compact. □

Lemma 2Assume that (A2), (A4) and (A5) hold. Then problem (1) possesses an un-bounded connected continuum Σ 0 ⊂ ℝ + × X connecting ( 0 , 0 ) to infinity.

Proof. Since the operator T : ℝ × X → X is continuous and compact, and T ( 0 , u ) ≡ 0 , By Theorem 3.2 in [14] that the equation u = T ( λ , u ) has an unbounded connected continuum Σ 0 contained in ℝ + × X and containing ( 0 , 0 ) , that is,

and Σ 0 is unbounded in ℝ + × X , i.e., it connects ( 0 , 0 ) to infinity.

On the other hand, from f ( x , 0 ) > 0 we have

Hence ( λ , 0 ) is a solution of problem (1) only when λ = 0 , while ( 0 , u ) is a solution of problem (1) only when u = 0 . By (A6) and the continuity of the positive solution branch, it follows that those sufficiently small solutions bifurcating from ( 0 , 0 ) satisfy u > 0 . Hence all solutions in Σ 0 \ { ( 0 , 0 ) } are positive. Therefore, problem (1) possesses an unbounded connected continuum Σ 0 ⊂ ℝ + × X , which connects ( 0 , 0 ) to infinity. □

Lemma3 Assume that (A2), (A4) and (A5) hold. Then problem (1) possesses an unbounded connected continuum Σ ∞ emanating from ( Λ ∞ , ∞ ) , and ( Λ ∞ , ∞ ) is the unique bifurcation point of problem (1) at infinity.

Proof. Let

Then problem (1) is transformed into

The linearization of problem (3) at z = 0 is

Since the bifurcation of (1) at infinity corresponds to the bifurcation of (3) at the trivial solution, by the Rabinowitz global bifurcation theorem ([14], Theorem 2.3) there exists an unbounded connected continuum Σ ∞ of positive solutions of problem (3) bifurcating from ( Λ ∞ , ∞ ) .

Since the principal eigenvalue of Equation (4) is M ∞ λ 1 [ b ] , and the corresponding eigenfunction does not change sign, ( Λ ∞ , ∞ ) is the unique bifurcation point of problem (1) at infinity. □

Lemma 4Assume that (A3)-(A5) hold. Then there exists λ * > 0 such that u = T ( λ , u ) has no positive solution ( λ , u ) whenever λ > λ * .

Proof. Suppose, on the contrary, that there exists a positive solution ( λ , u ) of u = T ( λ , u ) . By integration by parts, we obtain

where λ 1 = λ 1 [ 1 ] is the principal eigenvalue of

and ϕ is the corresponding eigenfunction. By (A4)-(A5), the function M is bounded above. Set

Then

From (A3) it follows that

that is,

Taking λ * = λ 1 M ¯ α , we conclude that u = T ( λ , u ) has no positive solution ( λ , u ) when λ > λ * . □

Proof of Theorem 1

1) By Lemma 2, problem (1) has an unbounded connected continuum Σ 0 ⊂ R + × X connecting ( 0 , 0 ) to infinity. Then by Lemma 3, problem (1) has an unbounded connected continuum Σ ∞ emanating from ( Λ ∞ , ∞ ) . Furthermore, ( Λ ∞ , ∞ ) is proven to be the sole bifurcation point of positive solutions of problem (1) at infinity. On the other hand, by Lemma 4, both Σ 0 and Σ ∞ are bounded in the λ direction, so the unbounded connected continuum Σ 0 must intersect ( Λ ∞ , ∞ ) , that is, Σ 0 = Σ ∞ .

2) By Lemma 2, problem (1) has an unbounded connected branch Σ 0 . However, if lim t → + ∞ M ( t ) = ∞ , then no bifurcation occurs at infinity. For this reason, we will prove that the solutions u of problem (1) have a priori bounds.

Let λ ≥ 0 , and suppose u ∈ X is a positive solution of problem (1) corresponding to λ . Taking u as a test function in problem (1), we obtain

From condition (A2), it follows that there exists a constant K > 0 such that for any u ≥ 0 , f ( x , u ) ≤ K u . Hence

that is,

Moreover, since lim t → + ∞ M ( t ) = ∞ , there exists a constant C λ > 0 such that ‖ u ‖ 2 ≤ C λ . Therefore ‖ u ‖ 2 has a priori bounds, and since the connected branch Σ 0 is unbounded, we have Proj ℝ Σ 0 = ( 0 , + ∞ ) . □

Proof of Theorem 2

Assume that u ∈ X and u ( x 0 ) = max x ∈ Ω ¯ | u ( x ) | = s 0 ∈ [ s ′ , s ″ ] . Then

On the one hand, since u ( x 0 ) = max x ∈ Ω ¯ | u ( x ) | = s 0 ∈ [ s ′ , s ″ ] , we have

On the other hand, by condition (A4), we have M ( ‖ u ‖ 2 ) ≥ m 0 > 0 , so

Combining the assumption s 0 ∈ [ s ′ , s ″ ] with condition (A7), we get f ( x 0 , s 0 ) ≤ 0 . Therefore

Hence Δ u ( x 0 ) = 0 . Applying maximum principle ([15], Theorem 2.2), if the function u is not identically constant, it cannot achieve a nonnegative maximum within the interior of Ω + . This forces u ≡ 0 , which stands in direct contradiction to the initial condition, which contradicts u ( x 0 ) = s 0 > 0 . Therefore, problem (1) has no positive solution satisfying max x ∈ Ω ¯ | u ( x ) | ∈ [ s ′ , s ″ ] .

Next, we prove that the connected branches Σ 0 and Σ ∞ are disjoint. By Lemma 2, problem (1) has a connected branch Σ 0 ⊂ R + × X emanating from ( 0 , 0 ) and going to infinity, and it has been proved that the solutions on this

branch do not satisfy max x ∈ Ω ¯ | u ( x ) | ∈ [ s ′ , s ″ ] . By the connectedness of solutions, all positive solutions on Σ 0 satisfy

On the other hand, by Lemma 3, problem (1) has an unbounded connected branch Σ ∞ emanating from ( Λ ∞ , ∞ ) , and ( Λ ∞ , ∞ ) is the unique bifurcation point of positive solutions of problem (1) at infinity. From ‖ u ‖ → + ∞ , together with the connectedness of solutions and the fact that solutions cannot lie in the interval [ s ′ , s ″ ] , all positive solutions on Σ ∞ satisfy

Since 0 < s ′ < s ″ , in summary, we finally obtain that Σ 0 and Σ ∞ are disjoint.

□

Supported by the National Natural Science Foundation of China (No.12461036).