Classification of positive D1,p(RN)-solutions to the critical p-Laplace equation in RN

MULTIPLE POSITIVE SOLUTIONS OF A SCALAR FIELD EQUATION IN Rn

(see for example [3], [8], [9], [14], [17]–[19], [24] and the references therein). Here, λ ∈ R is a positive parameter, k is a given smooth function on R, n ≥ 3, and 2∗ = 2n/(n− 2) is the critical Sobolev exponent. Problem (0) has a geometrical relevance, since for λ = 0 every solution to (0) gives rise, up to a stereographic projection, to a metric g on the sphere whose scalar curvature is pro...

Blow-up of Solutions to a p-Laplace Equation

Consider two perfectly conducting spheres in a homogeneous medium where the current-electric field relation is the power law. Electric field E blows up in the L∞-norm as δ, the distance between the conductors, tends to zero. We give here a concise rigorous justification of the rate of this blow-up in terms of δ. If the current-electric field relation is linear, see similar results obtained earl...

STABILITY OF POSITIVE SOLUTIONS TO p&2–LAPLACE TYPE EQUATIONS

In this article, we first show the existence of a positive solution to { −Δpu−αΔu = λ(u− f (u)) in Ω, u = 0 on ∂Ω, by the method of lower and upper solutions and then under certain conditions on f , we show the stability of positive solution. Mathematics subject classification (2010): 35J92, 35B35, 35B05.

Solutions for singular p-Laplacian equation in Rn

BIFURCATION OF POSITIVE SOLUTIONS FOR THE ONE-DIMENSIONAL (p, q)-LAPLACE EQUATION

In this article, we study the bifurcation of positive solutions for the one-dimensional (p, q)-Laplace equation under Dirichlet boundary conditions. We investigate the shape of the bifurcation diagram and prove that there exist five different types of bifurcation diagrams. As a consequence, we prove the existence of multiple positive solutions and show the uniqueness of positive solutions for a...