We study the existence and nonexistence of positive (super) solutions to the nonlinear p-Laplace equation −∆pu− μ |x|p u = C |x|σ u in exterior domains of R (N ≥ 2). Here p ∈ (1,+∞) and μ ≤ CH , where CH is the critical Hardy constant. We provide a sharp characterization of the set of (q, σ) ∈ R such that the equation has no positive (super) solutions. The proofs are based on the explicit const...

Geometric interpretation of the Hirota equation is presented as equation describing the Laplace sequence of two-dimensional quadrilateral lattices. Different forms of the equation are given together with their geometric interpretation: (i) the discrete coupled Volterra system for the coefficients of the Laplace equation, (ii) the gauge invariant form of the Hirota equation for projective invari...

this paper presents an analytical model characterizing unsteady groundwater flow in an unconfined aquifer resting on a sloping impervious bed. the aquifer is in contact with a constant water level at one end. the other end is connected to a stream whose level is increasing form an initial level to a final level at a known exponentially decaying function of time. moreover, the aquifer is repleni...

We study the p-Laplace equation with Potentials − div(|∇u|p−2∇u) + λV (x)|u|p−2u = |u|q−2u, u ∈ W 1,p(RN ), x ∈ RN where 2 ≤ p, p < q < p∗. Using a concentrationcompactness principle from critical point theory, we obtain existence, multiplicity solutions, and concentration of solutions.