Nonlinear Elliptic Equations with Dirichlet Boundary Conditions

This chapter studies nonlinear Dirichlet boundary value problems through various methods such as degree theory, variational methods, lower and upper solutions, Morse theory, and nonlinear operators techniques. The combined application of these methods enables us to handle, under suitable hypotheses, a large variety of cases: sublinear, asymptotically linear, superlinear, coercive, noncoercive, parametric, resonant, and near resonant. In many situations we are able to provide multiple solutions with additional information about their properties, for instance, constant-sign (i.e., positive or negative) solutions and nodal (sign-changing) solutions. The first section of the chapter is devoted to the study of nonlinear elliptic problems through degree theory. The second section focuses on the variational approach, specifically for investigating coercive problems and (p − 1)-superlinear parametric problems. The third section makes use of Morse theory in studying (p − 1)-linear noncoercive equations and p-Laplace equations with concave terms. The fourth section deals with general elliptic inclusion problems treated via nonlinear, possibly multivalued, operators. The last section highlights related remarks and bibliographical comments.