Global Hölder continuity of weak solutions to quasilinear divergence form elliptic equations

Quasilinear Elliptic Systems in Divergence Form with Weak Monotonicity

We consider the Dirichlet problem for the quasilinear elliptic system − div σ(x, u(x), Du(x)) = f on Ω u(x) = 0 on ∂Ω for a function u : Ω → Rm, where Ω is a bounded open domain in Rn. For arbitrary right hand side f ∈W−1,p (Ω) we prove existence of a weak solution under classical regularity, growth and coercivity conditions, but with only very mild monotonicity assumptions.

Positive Solutions of Quasilinear Elliptic Equations

(1.2) { −∆pu = λa(x)|u|p−2u, u ∈ D 0 (Ω), has the least eigenvalue λ1 > 0 with a positive eigenfunction e1 and λ1 is the only eigenvalue having this property (cf. Proposition 3.1). This gives us a possibility to study the existence of an unbounded branch of positive solutions bifurcating from (λ1, 0). When Ω is bounded, the result is well-known, we refer to the survey article of Amann [2] and t...

Large Solutions for Quasilinear Elliptic Equations

Existence of at least three weak solutions for a quasilinear elliptic system

In this paper, applying two theorems of Ricceri and Bonanno, we will establish the existence of three weak solutions for a quasilinear elliptic system. Indeed, we will assign a differentiable nonlinear operator to a differential equation system such that the critical points of this operator are weak solutions of the system. In this paper, applying two theorems of R...

Global solutions of quasilinear wave equations

has a global solution for all t ≥ 0 if initial data are sufficiently small. Here the curved wave operator is ̃g = g ∂α∂β, where we used the convention that repeated upper and lower indices are summed over α, β = 0, 1, 2, 3, and ∂0 = ∂/∂t, ∂i = ∂/∂x i, i = 1, 2, 3. We assume that gαβ(φ) are smooth functions of φ such that gαβ(0)= mαβ , where m00=−1, m11= m22= m33=1 and mαβ= 0, if α 6=β. The resul...