Peter Takac
Abstract:
We look for weak solutions
of the degenerate quasilinear Dirichlet boundary value problem
It is assumed that
,
,
is the p-Laplacian,
is a bounded domain in
,
is a given function, and
stands for the (real) spectral parameter.
Such weak solutions are precisely the critical points of
the corresponding energy functional on
,
I.e., problem (P) is equivalent with
in
.
Here,
stands for the (first)
Frechet derivative
of the functional
on
and
denotes the (strong) dual space of the Sobolev space
,
.
We will describe a global minimization method for this functional
provided
, together with the (strict) convexity
of the functional for
and possible "nonconvexity"
if
.
As usual,
denotes
the first (smallest) eigenvalue of the positive p-Laplacian
.
Strict convexity will force the uniqueness of a critical point
(which is then the global minimizer for
),
whereas "nonconvexity" will be shown by constructing a saddle point
which is different from any local or global minimizer.
These methods are well-known and can be found in many textbooks
on Nonlinear Functional Analysis or Variational Calculus.
The problem becomes quite difficult if
or
, even in space dimension one (N=1).
We will restrict ourselves to the case
,
the Fredholm alternative for the p-Laplacian
at the first eigenvalue.
Even if the functional
is no longer coercive on
,
for
we will show that it is bounded from below and
does possess a global minimizer.
For
the functional
is unbounded from below and one can find a pair of
sub- and super-solutions to problem (P)
by a variational method (a simplified minimax principle)
performed in the orthogonal decomposition
induced by the inner product in
.
First, the minimum is taken in
,
and then (possibly only local) maximum in
.
The "sub-" and "super-critical" points thus obtained provide
a pair of sub- and super-solutions to problem (P).
Then a topological (Leray-Schauder) degree has to be employed
to obtain a solution to problem (P)
by a standard fixed point argument.
Finally, we will discuss the existence and multiplicity
of a solution for problem (P)
when
"nearly" satisfies the orthogonality condition
and
(with
small enough).
A crucial ingredient in our proofs are
rather precise asymptotic estimates
for possible "large" solutions to problem (P)
obtained from the linearization of problem (P) about
the eigenfunction
.
These will be briefly discussed.
Naturally, the (linear selfadjoint) Fredholm alternative for
the linearization of problem (P) about
(with
)
appears in the proofs.
Published July 10, 2010.
Math Subject Classifications: 35J20, 49J35, 35P30, 49R50.
Key Words: Nonlinear eigenvalue problem; Fredholm alternative;
degenerate or singular quasilinear Dirichlet problem;
p-Laplacian; global minimizer; minimax principle.
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Peter Takac Institut fur Mathematik, Universitat Rostock D-18055 Rostock, Germany email: peter.takac@uni-rostock.de |
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