Asymptotic Analysis

In this paper, we study the large time behaviour of the solution to parabolic problems defined on noncylindrical domains becoming unbounded in many directions when t tends to infinity. That is to say, the state variable domains are also becoming unbounded when t → ∞ . Since the steady state problem is elliptic and defined on unbounded domains we need to define in which sense the solution is understood and to deal with its existence. The convergence and its rate are also investigated with respect to the growth rate of the domain when t → ∞ . As the convergence cannot be expected on the whole domain, correctors are built to describe the asymptotic behaviour in the distant regions.

We study the following class of linearly coupled Schrödinger elliptic systems − Δ u + V 1 ( x ) u = μ | u | p − 2 u + λ ( x ) v , x ∈ R N , − Δ v + V 2 ( x ) v = | v | q − 2 v + λ ( x ) u , x ∈ R N , where N ⩾ 3 , 2 < p ⩽ q ⩽ 2 ∗ = 2 N / ( N − 2 ) and μ ⩾ 0 . We consider nonnegative potentials periodic or asymptotically periodic which are related with the coupling term λ ( x ) by the assumption | λ ( x ) | ⩽ δ V 1 ( x ) V 2 ( x ) , for some 0 < δ < 1 . We deal with three cases: Firstly, we study the subcritical case, 2 < p ⩽ q < 2 ∗ , and we prove the existence of positive ground state for all parameter μ ⩾ 0 . Secondly, we consider the critical case, 2 < p < q = 2 ∗ , and we prove that there exists μ 0 > 0 such that the coupled system possesses positive ground state solution for all μ ⩾ μ 0 . In these cases, we use a minimization method based on Nehari manifold. Finally, we consider the case p = q = 2 ∗ , and we prove that the coupled system has no positive solutions. For that matter, we use a Pohozaev identity type.

We study a nonlinear Robin problem driven by the p-Laplacian and with a reaction term depending on the gradient (convection term). Using the theory of nonlinear operators of monotone-type and the asymptotic analysis of a suitable perturbation of the original equation, we show the existence of a positive smooth solution.