Positive Solutions of One-Dimensional p-Laplacian Problems with Superlinearity

We study one-dimensional p-Laplacian problems and answer the unsolved problem. Our method is to study the property of the operator, the concavity of the solutions and the continuity of the first eigenvalues. By the above study, the main difficulty is overcome and the fixed point theorem can be applied for the corresponding compact maps. An affirmative answer is given to the unsolved problem with superlinearity. A global growth condition is not imposed on the nonlinear term f. The assumptions of this paper are more general than the usual, thus the existing results cannot be utilized. Some recent results are improved from weak solutions to classical solutions and from p ≥ 2 to p ∈ ( 1 , ∞ ) .


Introduction
It is well-known that one-dimensional p-Laplacian problems −∆ p z(x) = f (x, z(x)) for almost every (a.e.) x ∈ (0, 1), z(0) = z(1) = 0. ( are of great importance in the fields of Newtonian fluids (p = 2) and non-Newtonian fluids (p = 2); Dilatant fluids and pseudoplastic fluids may be characterized by p > 2 and 1 < p < 2, respectively (e.g., see [1]), where ∆ p z(x) = (φ p (z (x))) , z (x) = dz dx denotes the usual derivative, φ p (s) = |s| p−2 s, s ∈ R The existence of positive solutions of Equation ( 1) has been widely investigated via various methods and a lot of results have been proved under various assumptions.Let us mention just a few.Using the fixed point index, Wang [2] and Webb and Lan [3] studied Equation (1).In [2], f (x, u) = g(x) f 0 (u) and f 0 was assumed to satisfy lim u→∞ f 0 (u) u p−1 = 0 and lim and in [3], p = 2 and was imposed on f .Rynne [4] and Dai and Ma [5] investigated Equation (1) with suitable boundary conditions using bifurcation theory.When p ≥ 2, the existence of positive weak solutions was studied by Ćwiszewski and Maciejewski [6] under the sublinear conditions: or under the superlinear conditions: where µ p is the first eigenvalue of the corresponding homogeneous Dirichlet boundary value problem and µ 2 = π 2 .Actually, Ćwiszewski et al. [6] covered PDE cases, where f was not required to be nonnegative, but Ćwiszewski et al. [6] only studied weak solutions and requires both a global growth condition on f and p ≥ 2. Hence, they [6] obtained less restrictive solution under stronger assumptions.
In 2015, Lan et al. [7] proved the existence of positive (classical) solutions for Equation (1)under the general conditions (see (H 1 ) and (H 2 ) in Theorem 2.11 [7], which cover Equation ( 3)) involving the first eigenvalues of the corresponding problems.However, the problem in Equation (1) under the superlinear case is left unsolved [7], that is, whether Equation (1) has positive (classical) solutions under the superlinear conditions in Equation (4).
The core of this paper is to give an affirmative answer to the unsolved problem with superlinearity [7].Our method is to study the property of the operator (see Lemma 4), the concavity of the solutions (see Lemma 5) and the continuity of the first eigenvalues (see Lemma 12).By the study in the above aspects, the difficulty such as of lacking linearity of the operator is overcome, the fixed point theorem can be applied for the corresponding compact maps and new results are obtained.Since we do not assume that f satisfies a global growth condition (see, for example, [6,8,9]) and the assumptions of this paper are more general than the usual that (see, for example, [6]), the existing results cannot be utilized in this paper.In addition, some recent results are improved from weak solutions to classical solutions and from p ≥ 2 to p ∈ (1, ∞).

Preliminaries
Let AC[0, 1] denote the space of all the absolutely continuous functions defined on [0 [11], then we call z being a positive (classical) solution of Equation (1).
Let W 1,p 0 (0, 1) denote the standard Sobolev space with norm and P denote the positive cone in W 1,p 0 (0, 1), that is, We recall some facts (see, for example, [7]) and establish several Lemmas.The first fact (Lemma 2.2 in [7]) is used to prove the limit property of the first eigenvalue µ g (Lemma 8) and the main result, where c 0 > 0 is a constant.
The following two Lemmas are the maximum principle and the weak comparison principle.
It is easy to verify that T satisfies Lemma 4. The map T : Proof.We may assume that, by Lemma 3, By Lemma 2, we have that u 1 ≤ u 2 and Tw 1 ≤ Tw 2 .
Define a map A from P to D(∆ p ) by where and T is in Equation (7).
By Theorem 2.8 in [7] and Lemma 1, we have Lemma 6.Under the assumption (C 1 ) and (C 2 ), the following conclusions hold.
(i) A(P) ⊆ P and A is compact, where A defined in Equation (9) and A(P) = {Ax : x ∈ P}.
(ii) z ∈ P \ {0} satisfying z = Az is equivalent to z being a positive solution of Equation (1).
For any > 0, by Equation ( 11), there exists v ∈ W By Equation ( 5), v ∈ C[0, 1] and lim n→∞ This, together with 1 0 g(x) ds > 0, shows that there exists n 0 > 0 such that Lemma 9. Let z n , e ∈ P \ {0} with Te ∈ P \ {0} and t n > 0 such that z n = T(Fz n + t n e).If Proof.In fact, if it is false, then we have a constant r 1 > 0 and a subset {n i } ⊆ N (N is the natural number set) satisfying 0 ≤ z n i (x) ≤ r 1 for all i and x ∈ [0, 1].Obviously, we may assume {n i } = N.
By Lemma 4, z n ≥ T(t n e) = t 1 p−1 n Te, we see that {t n } is bounded.Let r 2 > 0 be a constant such that 0 ≤ t n ≤ r 2 for all n.Let ξ n ∈ (0, 1) such that z n (ξ n ) = 0 and Let r > 0 and let P r = {z ∈ P : z < r}, ∂P r = {z ∈ P : z = r} and P r = {z ∈ P : z ≤ r}.Lemma 10. (i) If A : P r → P is compact and satisfies z = tAz for z ∈ ∂P r and t ∈ (0, 1], then i P (A, P r ) = 1 [7,13].
(ii) If A : P r → P is compact and z = Az for z ∈ P r , then i P (A, P r ) = 0.
(iv) If i P (A, P r ) = 1 and i P (A, P ρ ) = 0 for some ρ ∈ (r, ∞), then A has a fixed point in P ρ \ P r .

Main Result and Proof
Now, we state and prove our main result.
Next, some results are improved and the existing results cannot be used in this paper.
In [6], Ćwiszewski and Maciejewski studied positive weak solutions under the superlinear conditions in Equation ( 4) or (21), where a global growth condition on f and p ≥ 2 were required.Corollary 2 improves Ćwiszewski and Maciejewski's results (Theorem 1.1 with n = 1, [6]) from p ≥ 2 to p ∈ (1, ∞) and from weak solutions to classical solutions under the superlinear conditions.
The following example shows that the assumptions (H 1 ) and (H 2 ) of this paper are more general than the usual superlinear conditions in Equation (21).does not hold and the global growth condition (see, see for example, [6,8,9]) 0 ≤ | f (x, s)| ≤ C 0 (1 + s q−1 ) for all x ∈ Ω and s ∈ [0, ∞) is not imposed on f .Hence the existing results such as [5,6,[8][9][10] can not be used to treat this case.
Finally, in the study of boundary value problems, the linearity of the corresponding operators was applied in an essential way in [3,12].However, when p = 2, the corresponding operators of Equation ( 1) is nonlinear, which is the main difficulty we encounter in this paper.We expect the results obtained in this paper to be applied to other areas and, under (H 1 ) and (H 2 ) (p = 2, see [12]), Equation (1) to be studied further for the case of f taking negative values.