In considering a class of quasilinear elliptic equations on a Riemannian manifold with nonnegative Ricci curvature, we give a new proof of Tolksdorf’s result on the construction of separable p-harmonic functions in a cone. 1991 Mathematics Subject Classification. 35K60 .

Using the method of explosive sub and supper solution, the existence and boundary behavior of positive boundary blow up solutions for some quasilinear elliptic systems with singular weight function are obtained under more extensive conditions.

In this article, we consider three types of solutions in Orlicz spaces for the quasilinear elliptic problem − div(a(|∇u|)∇u) = 0. By applying a comparison principle, we establish the relationships between viscosity supersolutions, weak supersolutions, and superharmonic functions.

A strictly convex hypersurface in Rn can be endowed with a Riemannian metric in a way that is invariant under the group of (equi)affine motions. We study the corresponding isometric embedding problem for surfaces in R3. This problem is formulated in terms of a quasilinear elliptic system of PDE for the Pick form. A negative result is obtained by attempting to invert about the standard embedding...

This paper is concerned with the following Dirichlet problem for a quasilinear elliptic system with variable growth: –divσ (x,u(x),Du(x)) = f in , u(x) = 0 on ∂ , where ⊂Rn is a bounded domain. By means of the Young measure and the theory of variable exponent Sobolev spaces, we obtain the existence of solutions in W 0 ( ,R m) for each f ∈ (W 0 ( ,Rm))∗.

We consider the multiplicity of nontrivial solutions of the following quasilinear elliptic system -div(|x|(-ap)|∇u|(p-2)∇u) + f₁(x)|u|(p-2) u = (α/(α + β))g(x)|u| (α-2) u|v| (β) + λh₁(x)|u| (γ-2) u + l₁(x), -div(|x|(-ap) |∇v| (p-2)∇v) + f₂(x)|v| (p-2) v = (β/(α + β))g(x)|v|(β-2) v|u|(α) + μh 2(x)|v|(γ-2)v + l 2(x), u(x) > 0, v(x) > 0, x ∈ ℝ(N), where λ, μ > 0, 1 < p < N, 1 < γ < p < α + β < p* ...

We prove the existence of solutions to Dirichlet problems associated with the p(x)-quasilinear elliptic equation Au = − div a(x, u,∇u) = f(x, u,∇u). These solutions are obtained in Sobolev spaces with variable exponents.

The aim of this study is to prove Lyapunov-type inequalities for a quasilinear elliptic equation in [Formula: see text]. Also the lower bound for the first positive eigenvalue of the boundary value problem is obtained.