Generalized anisotropic Neumann problems of Ambrosetti–Prodi type with nonstandard growth conditions
Section snippets
Introduction and main results
Let () be a bounded -extension domain whose boundary is an upper d-set with respect to an appropriate and a measure μ supported on Γ. (see Definition 3, Definition 4). As the domain above may include domains with non-smooth boundaries and non-Lipschitz domains, in order to pose our boundary value problem, an appropriate interpretation of the normal outward vector needs to be addressed. The following definition is presented in advance (e.g. [20]).
Definition 1 Given , let μ be
Preliminaries and intermediate results
In this section we present some important (well-known) definitions, fix the notations that will be carried out in the subsequent sections, and state some known results that will be used in the later sections.
A sub-supersolution method of anisotropic type
In this part we will derive an alternative sub-supersolution method for the problem (1.4), which will be very useful in the establishment of the main results of the paper. Enlightened with the approach employed by Le [49], we derive an appropriate sub-supersolution method for anisotropic equations of type (1.4) (see also [23], [22]). In our knowledge, such method has not been developed for an anisotropic model equation, up to the present time. Furthermore, the subsolution method established by
Global regularity for weak solutions
The following section is devoted to establish global regularity results for weak solutions of problem (1.4), under the supposition that such solutions exist. Existence results will be established in a subsequent section (as part of the main result of the paper). However, in order to obtain multiplicity results, we need regularity results that may enable us to use techniques, such as degree theory. For the achievement of the results of this section, we will assume all the conditions of
Key auxiliary results
In this section we focus our attention in deriving several key results which will be useful in establishing the main result of this paper. All the arguments will be carried out assuming all the conditions in Assumption 1. For the first two results, we follow arguments similar as in [73]. However, as our problem under consideration is much more general and all results require substantial generalizations and modifications, complete details will be given, when needed. In particular, we will give
Proof of Theorem 1
The proof will be based in the approach applied in [73]. Nevertheless, since there are several modifications and generalizations, we will provide a complete proof of the main result of the paper.
To begin, assume all the conditions of the theorem. Given , we notice that the zero function 0 is a weak supersolution of (1.4) if and only if for a.e. . By assumption one has a.e. in Ω, so letting , one can define the parameter
Examples
The general structure of the differential equation in problem (1.4) allows us to establish the Ambrosetti–Prodi problem for several boundary value problems which are more classical and have been investigated more (for other structures, but not necessarily for Ambrosetti–Prodi-type problems). The following example gives further details on this assertion.
Example 1 We consider the following types of situations: Consider the realization of problem (1.4) for the coefficient Then one has
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