Neumann problems for nonlinear hemivariational inequalities

In this paper we study nonlinear Neumann problems driven by the p ‐Laplacian and having a nonsmooth potential. Using techniques from the nonsmooth critical point theory, we prove two existence theorems and a multiplicity result. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a class of nonlinear Neumann problems

Summary Existence theorems for nonlinear Neumann problems with inhomogeneous boundary conditions are established. It is then investigated under which conditions the solutions are uniformly bounded. Uniqueness results for positive solutions are given and the asymptotic behavior of the solutions of the corresponding parabolic equation is discussed. The main tools are fixed point theorems and the method of upper and lower solutions.     

Generalized BSDEs and nonlinear Neumann boundary value problems

Summary. We study a new class of backward stochastic differential equations, which involves the integral with respect to a continuous increasing process. This allows us to give a probabilistic formula for solutions of semilinear partial differential equations with Neumann boundary condition, where the boundary condition itself is nonlinear. We consider both parabolic and elliptic equations. Received: 27 September 1996 / In revised form: 1 December 1997     

Successively iterative method of nonlinear Neumann boundary value problems

Two successively iterative sequences are constructed for computing solutions of the nonlinear Neumann boundary value problems with time singularity. The sequences start off with some constants. Main tool is the fixed point theorem of increasing operator on the order interval. By considering convergence of the sequences, we prove the existence of nontrivial sign-changing solutions.     

Singularly perturbed nonlinear Neumann problems with a general nonlinearity

Let Ω be a bounded domain in Rn, n?3, with the boundary ∂Ω∈C³. We consider the following singularly perturbed nonlinear elliptic problem on Ω

     

Solutions to nonlinear Neumann problems with an inverse square potential

Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem

Positive solutions for nonlinear Neumann problems with concave and convex terms

We consider a nonlinear elliptic Neumann problem driven by the p-Laplacian with a reaction that involves the combined effects of a ??concave?? and of a ??convex?? terms. The convex term (p-superlinear term) need not satisfy the Ambrosetti?CRabinowitz condition. Employing variational methods based on the critical point theory together with truncation techniques, we prove a bifurcation type theorem for the equation.     

Entropy solutions for nonlinear elliptic anisotropic problems with homogeneous Neumann boundary condition

This study is about a nonlinear anisotropic problem with homogeneous Neumann boundary condition. We first prove, by using the technic of monotone operators in Banach spaces, the existence of weak solution, and by approximation methods, we achieve a result of existence and uniqueness of entropy solution     

Multiple positive solutions of nonlinear Neumann problems with time and space singularities

Allowing the nonlinear term to be singular with respect to both the time and space variables, we consider the positive solutions of a nonlinear Neumann boundary value problem. By constructing two height functions and estimating the integrations of these height functions, the existence and multiplicity of positive solutions are established.     

Exterior nonlinear Neumann problem

We consider the solvability of the Neumann problem for equation (1.1) in exterior domains in both cases: subcritical and critical. We establish the existence of least energy solutions. In the subcritical case the coefficient b(x) is allowed to have a potential well whose steepness is controlled by a parameter λ > 0. We show that least energy solutions exhibit a tendency to concentrate to a solution of a nonlinear problem with mixed boundary value conditions.     

Multiple positive radial solutions on annuli for nonlinear Neumann problems with large growth

We consider a classical semilinear elliptic equation with Neumann boundary conditions on an annulus in R ^N . The nonlinear term is the product of a radially symmetric coefficient with a pure power. We prove that if the power is sufficiently large, the problem admits at least three distinct positive and radial solutions. In case the coefficient is constant, we show that none of the three solutions is constant. The methods are variational and are based on the study of a suitable limit problem.     

On resonant Neumann problems

We consider semilinear Neumann problems at resonance and prove existence and multiplicity theorems. The existence theorems allow resonance with respect to any eigenvalue of the negative Neumann Laplacian. The multiplicity theorems concern problems resonant at 0 (the principal eigenvalue) or at the first nonzero eigenvalue. Our approach uses tools from critical point theory and from Morse theory.