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
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