Existence of classical solutions of quasi-linear non-cooperative elliptic systems (original) (raw)
Uniqueness of positive solutions to some coupled cooperative variational elliptic systems
Transactions of the American Mathematical Society
The uniqueness of positive solutions to some semilinear elliptic systems with variational structure arising from mathematical physics is proved. The key ingredient of the proof is the oscillatory behavior of solutions to linearized equations for cooperative semilinear elliptic systems of two equations on one-dimensional domains, and it is shown that the stability of the positive solutions for such a semilinear system is closely related to the oscillatory behavior.
Zeitschrift für angewandte Mathematik und Physik, 2020
We seek solutions for non-cooperative elliptic systems of two Schrödinger equations which can model phenomena in Physics and Biology. General conditions are assumed under the potentials, which produce convenient spectral properties on the elliptic operator concerned; hence, the non-cooperation characterizes it as a strongly indefinite problem. Spectral theory is employed along with an abstract linking theorem developed previously by the first two authors, which is the core of our argument. Furthermore, super-and asymptotically quadratic nonlinearities are considered.
On a class of nonvariational elliptic systems with nonhomogenous boundary conditions
Differential and Integral Equations
Using a fixed-point theorem of cone expansion/compression type, we show the existence of at least three positive radial solutions for the class of quasi-linear elliptic systems -Δ p u=λk 1 (|x|)f(u,v), -Δ q v=λk 2 (|x|)g(u,v) in Ω, (u,v)=(a,b) on ∂Ω, where the nonlinearities f,g are superlinear at zero and sublinear at ∞. The parameters λ,a and b are positive, Ω is the ball in ℝ N , with N≥3 of radius R 0 which is centered at the origin, 1<p,q≤2 and k 1 ,k 2 ∈C([0,R 0 ];[0,∞)).
Symmetry breaking of solutions of non-cooperative elliptic systems
Journal of Mathematical Analysis and Applications, 2013
In this article we study the symmetry breaking phenomenon of solutions of noncooperative elliptic systems. We apply the degree for G-invariant strongly indefinite functionals to obtain simultaneously a symmetry breaking and a global bifurcation phenomenon.
Regularity and coexistence problems for strongly coupled elliptic systems
2007
Boundedness and Hölder regularity of solutions to a class of strongly coupled elliptic systems are investigated. The Hölder estimates for the gradients of solutions are also established. Finally, the fixed point theory is applied to prove existence of positive solution(s) for general cross diffusion elliptic systems.
Solutions for a Resonant Elliptic System with Coupling in
Communications in Partial Differential Equations, 2002
Existence and multiplicity of solutions are established, via the Variational Method, for a class of resonant semilinear elliptic system in R N under a local nonquadraticity condition at infinity. The main goal is to consider systems with coupling where one of the potentials does not satisfy any coercivity condition. The existence of solution is proved under a critical growth condition on the nonlinearity.
Some qualitative properties of solutions of quasilinear elliptic systems
Nonlinear Analysis-theory Methods & Applications, 2002
We study quasilinear elliptic equations of Leray Lions type in W 1, p (0), maximum principles, nonexistence and existence of solutions, the control of lower (upper) bound for essential supremum (essential infimum) of solutions, sign-changing solutions, local and global oscillation of solutions, geometry of domain, generating singularities of solutions, and lower bounds on constants appearing in Schauder, Agmon, Douglis, and Nirenberg estimates.