Existence and Multiplicity Results for Nonlocal Boundary Value Problems with Strong Singularity

In this paper, we study singular φ-Laplacian nonlocal boundary value problems with a nonlinearity which does not satisfy the L1-Carathéodory condition. The existence, nonexistence and/or multiplicity results of positive solutions are established under two different asymptotic behaviors of the nonlinearity at ∞.

The condition (A1) on the odd increasing homeomorphism ϕ was first introduced by Wang in [1] where the existence, nonexistence and/or multiplicity of positive solutions to quasilinear elliptic equations were studied. Later on, the condition (A1) was weakened by some researchers. For example, Karakostas ([2,3]) introduced a sup-multiplicative-like function as an odd increasing homeomorphism ϕ satisfies the following condition.
The author investigated several sufficient conditions for the existence of positive solutions to the one dimensional ϕ-Laplacian equation with deviated arguments. Any function of the form ϕ(s) = n ∑ k=1 c k |s| p k −2 s is sup-multiplicative-like, where c k ≥ 0 and p k ∈ (1, ∞) for 1 ≤ k ≤ n and c 1 c n > 0 for some n ∈ N (see, e.g., [2,4]). Lee and Xu ( [5,6]) generalized the condition (A1) to the one with ψ 2 is a function not requiring that ψ 2 (0) = 0 and studied the existence of positive solutions to singularly weighted nonlinear systems. In [7], it was pointed out that the condition (A1) is equivalent to the one (F 1 ). Consequently, the condition (A1) is equivalent to those in [2,3,5,6]. Due to a wide range of applications in mathematics and physics (see, e.g., [8][9][10][11][12][13][14]), p-Laplacian or more generalized Laplacian problems have been extensively studied. For example, when ϕ(s) = |s| p−2 s for some p ∈ (1, ∞), w ≡ 1 and h ∈ H ϕ , Agarwal, Lü and O'Regan [15] investigated the existence and multiplicity of positive solutions to BVP (1) and (2) withα 1 =α 2 = 0 under various assumptions on the nonlinearity f = f (t, u) at u = 0 and ∞. When ϕ(s) = s, w ≡ 1 and λ = 1, Webb and Infante [16] considered problem (1) with various nonlocal boundary conditions involving a Stieltjes integral with a signed measure and gave several sufficient conditions on the nonlinearity f = f (t, u) for the existence and multiplicity of positive solutions via fixed point index theory. When ϕ(s) = |s| p−2 s for some p ∈ (1, ∞), w ≡ 1, λ = 1 and h ∈ H ϕ , Kim [17] investigated sufficient conditions on the nonlinearity f = f (t, u) for the existence and multiplicity of positive solutions to problem (1) with multi-point boundary conditions.
Xu, Qin and Li [18] studied the following three-point boundary value problem where p > 1, ϕ p (s) = |s| p−2 s, η ∈ (0, 1) and g, h ∈ C([0, ∞), [0, ∞)) are strictly increasing. Under the suitable assumptions on g and k such that g is p-sublinear at 0 and k is p-superlinear at ∞, the exact number of pseudo-symmetric positive solutions to problem (5) was studied.
Recall that we say that g : Throughout this paper, we assume h ∈ H ϕ . Since there may be a function h ∈ H ϕ \ L 1 (0, 1) (see, e.g., Remark 2 below), the nonlinearity h(t) f (t, u) in the equation (1) may not satisfy the L 1 -Carathéodory condition. Consequently, the solution space should be taken as C[0, 1], since the solutions to BVP (1) and (2) may not be in C 1 [0, 1] unlike References [20][21][22] where the nonlinearity satisfies the L 1 -Carathéodory condition. The lack of solution regularity and the boundary conditions (2) make it difficult to get the desired result. The rest of this article is organized as follows. In Section 2, we give some preliminaries which are crucial for proving the main results in this paper. In Section 3, the main results (Theorems 2-4) are proved and some examples which illustrate the main results are given. Finally, the summary of this paper is given in Section 4.
Proof. First, we show that Clearly, Since
Using Lemma 3 and (8), by the similar arguments in the proof of [17] (Lemma 2.4) and [48] (Lemma 3.3) , one can prove the complete continuity of the operator T = T(λ, u). We only state the result as follows. We recall a well-known theorem for the existence of a global continuum of solutions by Leray and Schauder [49]: Theorem 1. (see, e.g., [50] (Corollary 14.12)) Let X be a Banach space with X = {0} and let K be a cone in X. Consider where λ ∈ [0, ∞) and u ∈ K.  (1) and (2) with λ > 0.

Main Results
First, we give a list of hypotheses on f = f (t, s) which are used in this section: For convenience, let γ := α + β 2 .
Since α and β are any fixed constants in the cone K satisfying 0 < α < β < 1, When we need the assumption (F ∞ ), let α and β in the cone K be the same constants in the assumption (F ∞ ).
Proof. Let u be a positive solution to BVP (1) and (2) with λ > 0 and let σ ∈ (0, 1) be the unique point We only give the proof for the case σ ≥ γ, since the case σ < γ can be dealt similarly. Then which implies By Lemma 2 and (8), Here Proof. Suppose to the contrary that there exists a sequence {(λ n , u n )} satisfying u n is a positive solutions to BVP (1) and (2) with λ = λ n ∈ [I, ∞) and ||u n || ∞ → ∞ as n → ∞. Take By Lemma 2, Then, for sufficiently large N > 0, Let σ N ∈ (0, 1) be a unique point satisfying We only consider the case σ N ≥ γ, since the case σ N < γ can be dealt in a similar manner. By (8) and the fact that However, this contradicts the choice of C * . Thus the proof is complete.
By the compactness of T and Lemma 5, there exists a subsequence, say it again {(λ n , u n )}, satisfying from the continuity of T, it follows that Thus BVP (1) and (2) has at least one positive solution for λ = λ * .
To complete the proof of Theorem 2, it suffices to show that there are no positive solutions to BVP (1) and (2) for λ > λ * . Assume on the contrary that there exists λ 1 ∈ (λ * , ∞) such that BVP (1) and (2) has a positive solution u 1 for λ = λ 1 . We will show that there are two positive solutions to BVP (1) and (2) for all λ ∈ (0, λ 1 ), which contradicts the definition of λ * .
Since BVP (1) and (2) is equivalent to problem (20) on Ω ∩ K, by Lemmas 5 and 6 and the same argument in the proof of [51] (Theorem 1.1), one can conclude that BVP (1) and (2) has at least two positive solutions for λ * < λ < λ 1 . Thus the proof is complete. Proof. We give the proof for the case that (F 0 ) and h ∈ H ϕ , since the case (F 0 ) and h ∈ H ψ 1 can be proved in a similar manner. Set Assume to the contrary that there exists a sequence {(λ n , u n )} such that u n is a positive solution to BVP (1) and (2) with λ = λ n ∈ (0, L] and u n ∞ → ∞ as n → ∞. Set Then there exists N > 0 satisfying Consequently, by the definition of C M and (26), Let σ N be a unique point satisfying u N ∞ = u N (σ N ). Assume that σ N ≤ γ, since the case σ N > γ can be dealt in a similar manner. Then, by (8), (10) and (27), Here the choice of M is used in the last inequality. This contradiction completes the proof. Consequently, (F 0 ) does not imply (F 0 ). Since H ψ 1 ⊆ H ϕ , we give an example of h satisfying h ∈ H ϕ \ H ψ 1 . Let it follows that and ψ −1 Consequently h ∈ H ϕ \ H ψ 1 , since h ∈ C(0, 1].
Next we show that Assume to the contrary that there exists a sequence {(λ n , u n )} in C such that λ n → ∞ as n → ∞, but there exists m > 0 satisfying u n ∞ ≤ m for all n. For each n, let σ n be the unique point satisfying u n (σ n ) = u n ∞ . Suppose that σ n ≥ γ (the case σ n < γ is similar). Then, by (8), which contradicts the fact that u n ∞ ≤ m for all n. Thus, the proof is complete.