### Abstract:

Let
i and
o be two bounded open subsets of Rn containing 0. Let
Gi be a (nonlinear) map from @
i × Rn to Rn. Let ao be a map from @
o to the
set Mn(R) of n × n matrices with real entries. Let g be a function from @
o to Rn.
Let
be a positive valued function defined on a right neighborhood of 0 in the real
line. Let T be a map from ]1−(2/n),+1[×Mn(R) to Mn(R). Then we consider the
problem
8<
:
div (T(!,Du)) = 0 in
o \ cl
i ,
−T(!,Du(x))
i(x) = 1
( )Gi(x/ ,
( ) −1(log )− 2,nu(x)) 8x 2 @
i ,
T(!,Du(x)) o(x) = ao(x)u(x) + g(x) 8x 2 @
o ,
where
i and o denote the outward unit normal to @
i and @
o, respectively,
and where > 0 is a small parameter. Here (! − 1) plays the role of ratio between
the first and second Lam´e constants, and T(!, ·) denotes (a constant multiple of)
the linearized Piola Kirchhoff stress tensor, and 2,n denotes the Kronecker symbol.
Under the condition that
generates a very strong singularity, i.e., the case in which
lim !0+
( )
n−1 exists in [0,+1[, we prove that under suitable assumptions the above
problem has a family of solutions {u( , ·)} 2]0, 0[ for 0 sufficiently small and we analyze
the behavior of such a family as is close to 0 by an approach which is alternative
to those of asymptotic analysis.