2007 Conference on Variational and Topological Methods: Theory, Applications,
Numerical Simulations, and Open Problems. 
Electron. J. Diff. Eqns., Conference 18 (2010), pp. 67-105

Title: Variational methods and linearization tools
       towards the spectral analysis  of the p-Laplacian,
       especially for the Fredholm alternative

Author: Peter Takac (Univ. Rostock, Germany)
 
Abstract: 
 We look for weak solutions $u\in W_0^{1,p}(\Omega)$
 of the degenerate quasilinear Dirichlet boundary value problem
 $$
 \eqno{(P)}
  - \Delta_p u = \lambda |u|^{p-2} u + f(x)
    \quad \hbox{in } \Omega \,;\quad
  u = 0 \quad \hbox{on } \partial\Omega \,.
 $$
 It is assumed that $1<p<\infty$, $p\neq 2$,
 $\Delta_p u\equiv \hbox{div} ( |\nabla u|^{p-2} \nabla u )$
 is the p-Laplacian,
 $\Omega$ is a bounded domain in ${\mathbb{R}}^N$,
 $f\in L^\infty(\Omega)$ is a given function, and
 $\lambda$ stands for the (real) spectral parameter.
 Such weak solutions are precisely the critical points of
 the corresponding energy functional on $W_0^{1,p}(\Omega)$,
 $$
 \eqno{(J)}
  \mathcal{J}_{\lambda}(u) := 
  \frac{1}{p}  \int_\Omega |\nabla u|^p \,dx
  - \frac{\lambda}{p} \int_\Omega |u|^p \,dx
  - \int_\Omega f(x)\, u\,dx \,,\quad
    u\in W_0^{1,p}(\Omega) \,.
$$
I.e., problem  (P) is equivalent with
$\mathcal{J}_{\lambda}'(u) = 0$ in $W^{-1,p'}(\Omega)$.
Here,
$\mathcal{J}_{\lambda}'(u)$ stands for the (first) Frechet derivative
of the functional $\mathcal{J}_{\lambda}$ on $W_0^{1,p}(\Omega)$ and
$W^{-1,p'}(\Omega)$ denotes the (strong) dual space of the Sobolev space
$W_0^{1,p}(\Omega)$, $p'= p/(p-1)$.

We will describe a global minimization method for this functional
provided $\lambda < \lambda_1$, together with the (strict) convexity
of the functional for $\lambda\leq 0$ and possible "nonconvexity"
if $0 < \lambda < \lambda_1$.
As usual, $\lambda_1$ denotes
the first (smallest) eigenvalue of the positive p-Laplacian $-\Delta_p$.
Strict convexity will force the uniqueness of a critical point
(which is then the global minimizer for $\mathcal{J}_{\lambda}$),
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 $\lambda = \lambda_1$ or
$\lambda > \lambda_1$, even in space dimension one (N=1).
We will restrict ourselves to the case $\lambda = \lambda_1$,
the  Fredholm alternative for the p-Laplacian
at the first eigenvalue.
Even if the functional $\mathcal{J}_{\lambda_1}$
is no longer coercive on $W_0^{1,p}(\Omega)$,
for $p>2$ we will show that it is bounded from below and
does possess a global minimizer.
For $1<p<2$ the functional $\mathcal{J}_{\lambda_1}$
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
$$
  W_0^{1,p}(\Omega) =
  \hbox{lin} \{ \varphi_1\} \oplus W_0^{1,p}(\Omega)^\top
$$
induced by the inner product in $L^2(\Omega)$.
First, the minimum is taken in $W_0^{1,p}(\Omega)^\top$,
and then (possibly only local) maximum in
$\hbox{lin} \{ \varphi_1\}$.
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 $f$ "nearly" satisfies the orthogonality condition
$\int_\Omega f\varphi_1 \,dx = 0$
and $\lambda < \lambda_1 + \delta$
(with $\delta > 0$ 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 $\varphi_1$.
These will be briefly discussed.
Naturally, the (linear selfadjoint) Fredholm alternative for
the linearization of problem  (P) about $\varphi_1$
(with $\lambda = \lambda_1$)
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.