Solvability of the -Laplacian with nonlocal boundary conditions
Introduction
In this paper we study the existence of positive solutions of the nonlocal boundary value problem of the formHere are linear operators, c is a measurable function, is a specific (increasing) homeomorphism of the real line onto itself (called sup-multiplicative-like function as it was defined in [23], [24]) and F is an operator having the property defined below.
First let us make a convention: C is the Banach space of all continuous functions endowed with the sup-norm ; is the set of all nonnegative functions in C; for any the symbol represents the set of all points belonging to the ball . Property 1.1 There are (nonempty) sets with and J Borel with nonempty interior, satisfying the following condition:
The infimum of all such numbers m corresponding to M is a characteristic quantity which expresses the level of growth of the operator F with respect to the sets I and J. It is clear that the correspondence is a lower semicontinuous function and, as we shall see later, it plays an important role in our approach.
An operator F satisfying Property 1.1 is allowed to be singular1 and its prototype is a function of the form given in the following example: Example 1.2 Consider an operator of the formwhere are nonnegative real valued functions defined on , is bounded and are mappings of into itself. Assume that and are continuous functions and for some it holds and . If the exponents are such that , then the function F satisfies Property 1.1.
Indeed, to show this fact, take a with and for some defineandNow for each we let be a positive real number satisfying the inequalitywhere (here and in the sequel) is the sup-norm of the function . Also let us take any with and . Then we see that for all it holds andThis argument shows that Property 1.1 is satisfied. Obviously, in this example, the quantity is a positive real number satisfying , where is the least (positive) root of the equation(Notice that .) Another example of a general operator which satisfies Property 1.1 will be given later in Section 2.
During the last two decades, multi-point boundary value problems have been extensively studied and many results have been established. Boundary value problems having closed connection or are specific cases of the problem (1.1), (1.2) are investigated elsewhere in the literature, see, for instance, [1], [2], [3], [4], [5], [8], [9], [10], [11], [12], [13], [14], [15], [17], [16], [18], [19], [20], [21], [28], [30], [31], [33], [34], [35], [36], [37], [38] and the references therein. Among others a short, but informative, survey on this subject was presented by Liu [26], but our interest is focused mainly on the papers [4], [22], [38], which are more closely related to our subject under investigation.
In [4] Bai and Fang, deal with the problemwhere , , , for and . They, in order to apply a fixed point theorem in cones based on index theory to get the existence of multiple positive solutions, among others, assume that and there exists such that .
Also a paper, which was our main motivation to investigate equation (1.1), is the paper [38] due to Zhang and Wang, where the multi-point boundary value problemis investigated. Here , and moreover withandUnder certain conditions on f existence results for positive solutions of (1.5) are established. A little more general version including the p-Laplacian case of (1.5) was discussed in [22] as well as in its closely related [9].
It is clear that in case is a function of the form , Eq. (1.1) may be produced by a nonautonomous m-Laplacian elliptic equation in the n-dimensional space which has radially symmetric solutions. Also (1.1) includes the form of the equationwhere , for all , by settingWang [32], motivated by Erbe and Wang [7], has considered the function , and he studied existence of solutions of an equation of the form satisfying some specific boundary conditions.
In [6], where an equation of the form (1.1) is discussed (but without deviating arguments and with simple Dirichlet conditions), the leading factor depends on an odd homeomorphism , which, in order to guarantee the nonexistence of solutions, actually, satisfies the conditionfor all .
It is well known that in order to seek for positive solutions of operator equations Krasnoselskii presented in [25] a fixed point theorem, which is stated below and which has been proved as a powerful tool in investigating the existence of positive solutions of boundary value problems, see, e. g., most of the papers cited above. Theorem 1.3 Krasnoselskii [25] Let be a Banach space and let be a cone in . Assume that , are open, bounded subsets of E, with , and letbe a completely continuous operator such that eitherorThen A has a fixed point in .
The paper is organized as follows: Section 2 is devoted to the conditions of our situation as well as to the formulation of the main theorem concerning the problem. Also we present two examples to illustrate the results. Finally, in Section 3, we give the proof of the main results.
Section snippets
The conditions, the main results and some examples
In this paper we are discussing the problem (1.1), (1.2), where the items and are as follows: the function is a sup-multiplicative-like function as it was defined in [23] (see, also [24]), in the sense that is an increasing homeomorphism of the real line onto itself, vanished at zero, such that there exists a (increasing) homeomorphism of onto , which supports , namely for all it holdsAny function of the formis a
Proof of Theorem 2.1
The whole body of this section is devoted to the proof of the main theorem. But in order to simplify things, some facts will be given in the form of lemmas. We start with the following obvious result: Lemma 3.1 If is a concave function, then for all it holds thatNext, for any fixed and define the quantitywhere is the inverse function of . The quantity has some
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