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

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

Peter Takac

We look for weak solutions $u\in W_0^{1,p}(\Omega)$ of the degenerate quasilinear Dirichlet boundary value problem
  - \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 $W_0^{1,p}(\Omega)$,
  \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.

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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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