Nonlinear elliptic systems

Lorentz Spaces and Nonlinear Elliptic Systems

In this paper we study the following system of semilinear elliptic equations:  −∆u = g(v) , in Ω, −∆v = f(u) , in Ω, u = 0 and v = 0 , on ∂Ω, where Ω is a bounded domain in R , and f, g ∈ C(R) are superlinear nonlinearities. The natural framework for such systems are Sobolev spaces, which give in most cases an adequate answer concerning the ”maximal growth” on f and g such that the proble...

Nonlinear Elliptic Systems and Mean Field Games ∗

We consider a class of quasilinear elliptic systems of PDEs consisting of N HamiltonJacobi-Bellman equations coupled with N divergence form equations, generalising to N > 1 populations the PDEs for stationary Mean-Field Games first proposed by Lasry and Lions. We provide a wide range of sufficient conditions for the existence of solutions to these systems: either the Hamiltonians are required t...

Quasilinear Elliptic Systems of Resonant Type and Nonlinear Eigenvalue Problems

Solvability of monotone systems of fully nonlinear elliptic PDE's

Multiple Positive Solutions for Some Nonlinear Elliptic Systems

where k1, k2 > 0 are positive constants, Ω ⊂ R is a bounded domain with a smooth boundary ∂Ω and V (u, v) ∈ C(R,R). We refer to [CdFM], [CM], [dFF], [dFM] and [HvV] for variational study of such elliptic systems. However, it seems that the multiplicity of positive solutions for such elliptic systems is not well studied. Here, we study a case related to some models (with diffusion) in mathematic...