Weak perturbations of the p-Laplacian

Weak Perturbations of the P–laplacian

We consider the p-Laplacian in R perturbed by a weakly coupled potential. We calculate the asymptotic expansions of the lowest eigenvalue of such an operator in the weak coupling limit separately for p > d and p = d and discuss the connection with Sobolev interpolation inequalities. AMS Mathematics Subject Classification: 49R05, 35P30

Existence of weak solutions for a class of (p,q)$(p,q)$-Laplacian systems

EQUIVALENCE OF VISCOSITY AND WEAK SOLUTIONS FOR THE p(x)-LAPLACIAN

We consider different notions of solutions to the p(x)-Laplace equation − div(|Du(x)| Du(x)) = 0 with 1 < p(x) < ∞. We show by proving a comparison principle that viscosity supersolutions and p(x)-superharmonic functions of nonlinear potential theory coincide. This implies that weak and viscosity solutions are the same class of functions, and that viscosity solutions to Dirichlet problems are u...

L∞-BOUNDS FOR WEAK SOLUTIONS OF AN EVOLUTIONARY EQUATION WITH THE p-LAPLACIAN

We investigate the smoothing effect of the parabolic part of a quasilinear evolutionary equation on its solution as time evolves. More precisely, the following initial-boundary value problem with Dirichlet boundary conditions is considered: (P) 8 >< >: ∂u ∂t = ∆pu + f(x, t, u(x, t)) for (x, t) ∈ Ω × (0, T ); u(x, t) = 0 for (x, t) ∈ ∂Ω × (0, T ); u(x, 0) = u0(x) for x ∈ Ω. Here, ∆p stands for t...

Infinitely Many Weak Solutions of the p-Laplacian Equation with Nonlinear Boundary Conditions

We study the following p-Laplacian equation with nonlinear boundary conditions: -Δ(p)u + μ(x)|u|(p-2)u = f(x,u) + g(x,u),x ∈ Ω, | ∇u|(p-2)∂u/∂n = η|u|(p-2)u and x ∈ ∂Ω, where Ω is a bounded domain in ℝ(N) with smooth boundary ∂Ω. We prove that the equation has infinitely many weak solutions by using the variant fountain theorem due to Zou (2001) and f, g do not need to satisfy the (P.S) or (P.S...