Consider the nonlinear coupled elliptic system  -Δ_pu-λV(x)|u|^α|v|^βv=μ|u|^α|v|^βv, in Ω,  -Δ_qv-λV(x)|u|^α|v|^βu=μ|u|^α|v|^βu, in Ω,  u=v=0, in Ω,  where Δ_{ρ}ζ=∇⋅(|∇ζ|^{ρ-2}∇ζ),ρ > 1, Ω is a bounded domain and V(x) is a potential weight function. We prove that for any real parameter λ, there is at least a sequence of eigencurves (μ_k(λ))_k by using an energy variational method. We prove also via an homogeneity type condition that the eigenvector corresponding to the principal frequency μ_1(λ) is unique modulo scaling, bounded, regular and positive, without any condition on regularity of the domain. We end this work by giving a new proof technique to prove the simplicity of μ_1(λ) via a new version of Picones' identity.

Asymptotic large- and short-time behavior of solutions of the linear dispersion equation u_t = u_{xxx} in R×R_+, and its (2k+1)th-order extensions are studied. Such a refined scattering is based on a "Hermitian" spectral theory for a pair {B,B^∗} of non self-adjoint rescaled operators  B=D^3_y+\frac13yD_y+\frac13I, and the adjoint one B^∗=D^3_y-\frac13yD_y, with the discrete spectrum σ(B)=σ(B^∗)={λ_l=-l/3, l=0,1,2,...} and eigenfunctions for B, {ψ_l(y)=[(-1)^l/\sqrt{l!}]D^l_yAi(y), l ≥ 0}, where Ai(y) isAiry's classic function. Eigenfunctions of B^∗ are then generalized Hermite polynomials. Applications to very singular similarity solutions (VSSs) of the semilinear dispersion equation with absorption, u_S(x,t)=t^{-\frac{1}{p-1}}f(\frac{x}{t^{\frac13}}): u_t=u_{xxx}-|u|^{p-1}u in R×R_+, p > 1, and to its higher-order counterparts are presented. The goal is, by using various techniques, to show that there exists a countable sequence of critical exponents {p_l=1+3/(l+1), l=0,1,2,...} such that, at each p= p_l , a p-branch of VSSs bifurcates from the corresponding eigenfunction ψ_l of the linear operator B above.

We investigate in this work necessary and sufficient conditions for having a Maximum Principle for a cooperative elliptic system on the whole R^N. Moreover, we prove the existence of solutions by an approximation method for the considered system.

We investigate the question whether certain parabolic systems in the sense of Petrovskii fulfill the resolvent estimate required for the generation of an analytic semigroup and apply the result to a problem concerning the diffusion of gases.