A highly nonlinear eigenvalue problem is studied in a Sobolev space with variable exponent. The Euler-Lagrange equation for the minimization of a Rayleigh quotient of two Luxemburg norms is derived. The asymptotic case with a "variable infinity" is treated. Local uniqueness is proved for the viscosity solutions.

This chapter concerns the state-of-the-art of the research about non-local minimal surfaces and their phase-field approximations.

We give a simple convexity-based proof of the following fact: the only eigenfunction of the p-Laplacian that does not change sign is the first one. The method of proof covers also more general nonlinear eigenvalue problems. © 2012 Springer Basel.

We consider the Dirichlet eigenvalue problem, (the eigenfunction) and ? > 0 (the eigen value), ? is an arbitrary domain in RN with finite measure, 1 < p < ?, 1 < q < p*, p* = Np/(N - p) if 1 < p < N and p* = ? if p >= N. We study several existence and uniqueness results as well as some properties of the solutions. Moreover, we indicate how to extend to the general case some proofs known in the classical case p = q. © 2010 Texas State University - San Marcos.