Existence and location of solutions to the Dirichlet problem. for a class of nonlinear elliptic equations

摘要

In a bounded open set Omega subset of R-n, we consider a Dirichlet problem of the type -Deltau = g(x, u) + h(x, u) + alpha (x) + 1/mu (f(x, u) + l(x, u) + beta (x)), in Omega, (u)partial derivative Omega = 0, where, in particular, f(x, (.)), g(x, (.)) have a subcritical growth, and h(x, (.)), l(x, (.)) are nonincreasing, with a critical growth. It is our aim to show that, for explicitly determined psi : W-0(1,2)(Ohm) --> R, and phi :]r*,+infinity[--> [0,+infinity[, with r* = inf(W0)((Ohm))(1,2) psi, for each r > r* and each mu > phi (r), the above problem has at least one weak solution that lies in psi (-1)(] -infinity, r[). A major novelty is just the precise determination of phi. (C) 2000 Elsevier Science Ltd. All rights reserved. [References: 3]