In this paper we give survey of the results concerning the solvability of certain quasilinear boundary value problems which contain the p-Laplacian as a main part. Special attention is paid to the resonance problems which lead naturally to the study of spectral properties of nonlinear operators. Besides statements of principal results in the historical context we also mention some open problems...

wt t (t, x)− K (‖wx (t, ·)‖ β Lr (R))a(wx (t, x))wxx (t, x) = 0, (2) w(0, x) = 8(x), wt (0, x) = 9(x), where K is an arbitrary function, sufficiently smooth and taking only positive values; and a = a(s) behaves like |s|p−2 near s = 0. The detailed assumptions on K , r , β, and a are given in (3), (4) and Condition 1 below. For K = K (s) = c1 + c2s (c1, c2 > 0) and p = r = β = 2, we get the famo...

This is a survey on problems involving equations −divA(x,∇u) = μ, where μ is a Radon measure and A : Rn ×Rn → Rn verifies Leray-Lions type conditions. We shall discuss a potential theoretic approach when the measure is nonnegative. Existence and uniqueness, and different concepts of solutions are discussed for general signed measures. 2000 Mathematics Subject Classification: 35J60, 31C45.

Under suitable assumptions on the potential of the nonlinearity, we study the existence and multiplicity of solutions for a Steklov problem involving the p(x)-Laplacian. Our approach is based on variational methods.

We consider a number of boundary value problems involving the p-Laplacian. The model case is −∆pu = V |u|p−2u for u ∈ W 1,p 0 (D) with D a bounded domain in R. We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of V , the measure of D, and a sharp Sobolev constant. In most cases, these ineq...

In this paper we consider the continuous piecewise linear finite element approximation of the following problem: Given p € (1, oo), /, and g , find u such that -V • (\Vu\"-2Vu) = f iniîcR2, u = g on a«. The finite element approximation is defined over Í2* , a union of regular triangles, yielding a polygonal approximation to Q. For sufficiently regular solutions u , achievable for a subclass of ...

In this paper we consider the eigenvalue problem − pu = λ(m)|u|p−2u, u ∈ W 1,p 0 ( ) where p > 1, p is the p-Laplacian operator, λ > 0, is a bounded domain in R(N ≥ 1) and m is a given positive function in L( ) (r depending on p and N ). We prove that the second positive eigenvalue admits exactly two nodal domains. AMS subject classification: 35J20, 35J70, 35P05, 35P30.

In this paper, we examine some computational issues on finite element discretization of the p-Laplacian. We introduced a class of descent methods with multi-grid finite element preconditioners, and carried out convergence analysis. We showed that their convergence rate is mesh-independent. We studied the behavior of the algorithms with large p. Our numerical tests show that these algorithms are...

We consider the p-Laplacian boundary value problem −(φp(u(x)) = f(x, u(x), u′(x)), a.e. x ∈ (0, 1), (1) c00u(0) + c01u ′(0) = 0, c10u(1) + c11u ′(1) = 0, (2) where p > 1 is a fixed number, φp(s) = |s|p−2s, s ∈ R, and for each j = 0, 1, |cj0|+ |cj1| > 0. The function f : [0, 1]× R2 → R is a Carathéodory function satisfying, for (x, s, t) ∈ [0, 1]× R2, ψ±(x)φp(s)− E(x, s, t) ≤ f(x, s, t) ≤ Ψ±(x)φ...

In this paper we consider the continuous piecewise linear finite element approximation of the following problem: Given p € (1, oo), /, and g , find u such that -V • (\Vu\"-2Vu) = f iniîcR2, u = g on a«. The finite element approximation is defined over Í2* , a union of regular triangles, yielding a polygonal approximation to Q. For sufficiently regular solutions u , achievable for a subclass of ...