Pseudomonotone Operators and Quasi-Linear Elliptic Differential Equations

Abstract

In this chapter our goal is to treat, in contrast to Section 26.5, more general quasi-linear elliptic equations having terms of lower order which satisfy no monotonicity condition. We introduce pseudomonotone operators for this purpose, i.e., we study the solvability of operator equations of the form

$$$$

(E)

, where A ₁ + A ₂: X → X* is a. pseudomonotone and coercive operator on the real reflexive B-space X. The prototype of a pseudomonotone operator is the sum operator A ₁ + A ₂, where

• (i) the operator A ₁: X → X* is monotone and hemicontinuous; and

• (ii) the operator A ₂: X → X* is strongly continuous.

The classical development of nonlinear functional analysis arose contemporaneously with the beginnings of linear functional analysis at about the beginning of the twentieth century in the work of such men as Picard, S. Bernstein, Ljapunov, E. Schmidt, and Lichtenstein, and was motivated by the desire to study the existence and properties of boundary value problems for nonlinear partial differential equations. Its most classical tool was the Picard contraction principle (put in its sharpest form by Banach in his thesis of 1920—the Banach fixed-point theorem).

Beyond the early development of bifurcation theory by Ljapunov and E. Schmidt around 1905, the second, and even more fruitful, branch of the classical methods in nonlinear functional analysis was developed in the theory of compact nonlinear map**s in Banach spaces in the late 1920’s and early 1930’s. These included Schauder’s well-known fixed-point theorem and the extension of the Brouwer topological degree by Leray and Schauder in 1934 to map**s in Banach spaces of the form I + C with C compact (as well as interesting related results of Caccioppoli on nonlinear Fredholm map**s).

The central role of compact map**s in this phase of the development of nonlinear functional analysis was due in part to the nature of the technical apparatus being developed, but also in part to a not always fruitful tendency to see the theory of integral equations as the predestined domain of application of the theory to be developed. Since, however, the more significant analytical problems lie in the somewhat different domain of boundary value problems for partial differential equations, and since the efforts to apply the theory of compact operators (and in particular the Leray-Schauder theory) to the latter problems have given rise to demands for ever more inaccessible (and sometimes, invalid) a priori estimates in these problems; the hope of applying nonlinear functional analysis to problems of this type centers on a general program of creating new theories for significant classes of noncompact nonlinear operators. The focus of this study is then to find such classes of operators which have the opposed characteristics of being narrow enough to have a significant structure of results while also being wide enough to have a significant variety of applications....

From the point of view of applications to partial differential equations, the most important class is that of monotone-like operators.

Felix E. Browder (1968)

The first substantial results concerning monotone operators were obtained by G. Minty (1962), (1963) and F. E. Browder (1963). Then the properties of monotone operators were studied systematically by F. E. Browder in order to obtain existence theorems for quasi-linear elliptic and parabolic partial differential equations. The existence theorems of F. E. Browder were generalized to more general classes of quasi-linear elliptic differential equations by J. Leray and J. L. Lions (1965), and P. Hartman and G. Stampacchia (1966).

In this paper we introduce two vast classes of operators, namely, operators of type (M) and pseudomonotone operators. These classes of operators contain many of those monotone-like operators which were used by the authors mentioned above.

Haïm Brézis (1968)

Many problems in analysis reduce to solving an equation of the form $$, where A is an operator on a space into another space. In this paper we assume that A is a map on a subset D(A) of a Banach space X into another Banach space Y ⁺, where {Y, Y ⁺} is a dual pair of Banach spaces. In an ideal situation, our equation will have a solution u for every b ∈ Y ⁺. There is a large literature on the “surjectivity” of this kind, including those related to monotone operators and their generalizations.

In the present paper we want to generalize the problem and seek sufficient conditions for our equation to have a solution u for all “sufficiently small b” Our theorem, together with its companion for evolution equations,¹ have been found useful in applications to many nonlinear partial differential equations.

Tosio Kato (1984)