Abstract. In this paper, we prove a range and existence theorem for multivalued pseudomonotone perturbations of maximal monotone operators. We assume a general coercivity condition on the sum of a maximal monotone and a pseudomonotone operator instead of a condition on the pseudomonotone operator only. An illustrative example of a variational inequality in a Sobolev space with variable exponent is given.
Let us consider the boundary value problem
{‐div(A(|∇u|2)∇u)+F(x,u)=0,in Ωu=0,on ∂Ω,(1)
where Ω ⊂ RN is a bounded domain with smooth boundary (for example, such that certain Sobolev imbedding theorems hold). Let
ϕ:R→R, ϕ(s)=A(s2)s
Then, if ϕ(s) = ∣s∣p−1s, p > 1, problem (1) is fairly well understood and a great variety of existence results are available. These results are usually obtained using variational methods, monotone operator methods or fixed point and degree theory arguments in the Sobolev space W01,p(Ω). If, on the other hand, we assume that ϕ is an odd nondecreasing function such that
ϕ(0)=0, ϕ(t)>0, t>0,
limt→∞ϕ(t)=∞,
and
ϕ is right continuous,
then a Sobolev space setting for the problem is not appropriate and very general results are rather sparse. The first general existence results using the theory of monotone operators in Orlicz–Sobolev spaces were obtained in [5] and in [9, 10]. Other recent work that puts the problem into this framework is contained in [2] and [8].
It is in the spirit of these latter papers that we pursue the study of problem (1) and we assume that F:Ω×R→R is a Carathéodory function that satisfies certain growth conditions to be specified later.
We note here that the problems to be studied, when formulated as operator equations, lead to the use of the topological degree for multivalued maps (cf. [4, 16]).
We shall see that a natural way of formulating the boundary value problem will be a variational inequality formulation of the problem in a suitable Orlicz–Sobolev space. In order to do this we shall have need of some facts about Orlicz–Sobolev spaces which we shall give now.
The paper is concerned with an eigenvalue problem for the prescribed mean curvature equation. We formulate the problem as a variational inequality and show that under some growth conditions on the lower order term, the relaxed problem has at least two nontrivial solutions in a space of functions of bounded variation when the parameter is small.
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