A singular nonlinear elliptic equation with natural growth in the gradient

Abstract

This paper is devoted to the existence and regularity of the homogenous Dirichlet boundary value problem for a singular nonlinear elliptic equation with natural growth in the gradient. By certain transformations, the problem can be transformed formally into either a Dirichlet problem or boundary blowup problems without gradient term, for which the corresponding existence results are also derived, which is a partial extension and supplement to the previous works. A SINGULAR NONLINEAR ELLIPTIC EQUATION 1705When the lower-order term B (x, u, ) has no singularity at u = 0, the following nonlinear elliptic problems have been studied extensively (see [2,[5][6][7][8][9][10][11][12][13][14] and references therein):among which the p-Laplacian is regarded as a typical example and B is a nonlinear lower-order term with 'natural' growth with respect to ∇u. Under some additional restrictions on A and B, some existence results were established by many authors, for example Bensoussan et al. [5], Boccardo et al. [2], Cho and Choe [7], Drábek and Nicolosi [8], Ferone et al. [9] and Grenon and Trombetti [10].When the lower-order term B is singular at u = 0, however, it received a little attention. In this direction, we would like to mention Reference [4] by Michaux et al., in which they studied a class of quasilinear mixed equations of Leray-Lions type; however, they considered only the case where the boundary function has a positive lower bound and studied the existence and local regularity of solutions with the same lower bound, whereas the positive lower bound means disappearance of singularity of the lower-order term and plays a dominant role in their proof. It is very necessary and interesting to consider the singularity of the lower-order term and to study the existence of solutions in the class of functions: W 1, p 0 ( )∩ L ∞ ( ). To indicate some ideas dealing with similar problems, in this paper we consider a typical equation (1) in which the p-Laplacian is a special case. The existence and regularity of solutions will be studied in Section 2.We note that under the hypotheses >0 and (H 5) (see Section 2), Equation (1) can be transformed formally into other equations without gradient term after the change of variables:

Quasilinear equations with degenerate coerciveness and having multiple singularities

“…We could, for instance, quote [1][2][3][4][5][6][7][8]. In these last cited references, the divergence part of the equation, which is in general of the form −div A(x, u) |∇u| q−2 ∇u , q ≥ 2, is assumed to be uniformly coercived, i.e.…”

Multiplicity of solutions for a singular p-laplacian elliptic equation

“…It is of interest to consider the singularity of the lower order term and to study the existence of solutions in W 1,p 0 (Ω) ∩ L ∞ (Ω). In the present paper, we investigate the existence and multiplicity of solutions of problem (1.1) with m > 1 and λ > 0; we extend the existence results of [AMA,PV,Z1] and the multiplicity result of [Z2].…”