Pseudomonotonicity and nonlinear hyperbolic equations

summary:In this paper we consider a nonlinear hyperbolic boundary value problem. We show that this problem admits weak solutions by using a lifting result for pseudomonotone operators and a surjectivity result concerning coercive and monotone operators

ftp ejde.math.txstate.edu (login: ftp) A MULTIPLICITY RESULT FOR QUASILINEAR PROBLEMS WITH CONVEX AND CONCAVE NONLINEARITIES AND NONLINEAR BOUNDARY CONDITIONS IN UNBOUNDED DOMAINS

Abstract. We study the following quasilinear problem with nonlinear boundary conditions −∆pu = λa(x)|u | p−2 u + k(x)|u | q−2 u − h(x)|u | s−2 u, in Ω, |∇u | p−2 ∇u · η + b(x)|u | p−2 u = 0 on ∂Ω, where Ω is an unbounded domain in RN with a noncompact and smooth boundary ∂Ω, η denotes the unit outward normal vector on ∂Ω, ∆pu = div(|∇u | p−2∇u) is the p-Laplacian, a, k, h and b are nonnegative essentially bounded functions, q < p < s and p ∗ < s. The properties of the first eigenvalue λ1 and the associated eigenvectors of the related eigenvalue problem are examined. Then it is shown that if λ < λ1, the original problem admits an infinite number of solutions one of which is nonnegative, while if λ = λ1 it admits at least one nonnegative solution. Our approach is variational in character. 1

Robin boundary-value problems for quasilinear elliptic equations with subcritical and critical nonlinearities

By using variational methods we study the existence of positive solutions for a class of quasilinear elliptic problems with Robin boundary conditions

Periodic Solutions For Nonlinear Evolution Inclusions

In this paper we prove the existence of periodic solutions for a class of nonlinear evolution inclusions defined in an evolution triple of spaces (X;H;X*) and driven by a demicontinuous pseudomonotone coercive operator and an upper semicontinuous multivalued perturbation defined on T \Theta X with values in H. Our proof is based on a known result about the surjectivity of the sum of two operators of monotone type and on the fact that the property of pseudomonotonicity is lifted to the Nemitsky operator, which we prove in this paper

Periodic solutions for nonlinear Volterra integrodifferential equations in Banach spaces

summary:In this paper we examine periodic integrodifferential equations in Banach spaces. When the cone is regular, we prove two existence theorems for the extremal solutions in the order interval determined by an upper and a lower solution. Both theorems use only the order structure of the problem and no compactness condition is assumed. In the last section we ask the cone to be only normal but we impose a compactness condition using the ball measure of noncompactness. We obtain the extremal solutions for both the Cauchy and periodic problems in a constructive way, using a monotone iterative technique

Multiplicity of positive solutions for some quasilinear Dirichlet problems on bounded domains in Rn\Bbb R^n

summary:We show that, under appropriate structure conditions, the quasilinear Dirichlet problem \cases -\operatorname{div}(|\nabla u|^{p-2}\nabla u) =f(x,u), \quad & x\in\Omega, \ u=0, & x\in\partial\Omega, \endcases where $\Omega$is a bounded domain in $\Bbb R^n$, $1<p<+\infty$, admits two positive solutions $u_{0}$, $u_{1}$ in $W_{0}^{1,p}(\Omega)$ such that $0<u_{0}\leq u_{1}$ in $\Omega$, while $u_{0}$ is a local minimizer of the associated Euler-Lagrange functional