Behavior of multiple solutions for systems of semilinear elliptic equations

Abstract

In this work we present some partial results that will appear in a completed from in a forthcoming paper, [7]. We discuss the existence and particularly the multiplicity of solutions for the nonlinear system of elliptic equations.

Δu_i + Î»f_i (x, u₁, . . . u_m) = 0    in    â„¦      (1.1)

u_iÇ€_{ðœƒâ„¦} = 0,    i = 1 , . . . , m                     (1.2)

Where f_i (x, 0, . . . 0) > 0 for all x â‹² â„¦,  i = 1, 2, . . . , m. The functions f_i,  i = 1, . . . , m, satisfy the quasimonotone condition and a certain blow up rate us to be made precise in the assumptions (H1) and (H2) below. Then results similar to those of the scalar equation case (see [6]) can be established.