Generalized abstract economy and systems of generalized vector quasi-equilibrium problems

The present paper is in two-fold. The first fold is devoted to the existence theory of equilibria for generalized abstract economy with a lower semicontinuous constraint correspondence and a fuzzy constraint correspondence defined on a noncompact/nonparacompact strategy set. In the second fold, we consider systems of generalized vector quasi-equilibrium problems for multivalued maps (for short, SGVQEPs) which contain systems of vector quasi-equilibrium problems, systems of generalized mixed vector quasi-variational inequalities and Debreu-type equilibrium problems for vector valued functions as special cases. By using the results of first fold, we establish some existence results for solutions of SGVQEPs.     

Existence results for semilinear elliptical hemivariational inequalities

We are interested in studying the existence of solutions to an elliptical hemivariational inequality, depending on a real parameter λ$\lambda$. The main tool in the proof of our results is a critical point theorem recently established. We obtain the existence of solution through a direct method, both with a changing sign nonlinearity of the kind p(x)f(ξ)$p\left(x\right)f\left(\xi \right)$ and in the classical one P(x,ξ)$P(x,\xi )$ too.     

Nontrivial solutions for elliptic resonant problems

Nontrivial solutions for elliptic resonant problems are obtained via Morse theory. To compute the critical groups at infinity of the relevant functional, we propose a new approach by combining the homotopy and reduction methods, and the Alexander Duality Theorem.     

Existence results for semilinear elliptic variational inequalities with changing sign nonlinearities

In this paper we present some existence results for a class of semilinear elliptic variational inequalities, depending on a real parameter λ, with changing sign nonlinearities. The fundamental tool to prove the existence result is a penalization method combined with the Mountain Pass Theorem and the Linking Theorem, respectively in the case λ < λ ₁ and λ ≥ λ ₁, where λ₁ is the first eigenvalue of the uniformly elliptic operator A involved in the variational inequality.     

On resonant Neumann problems: Multiplicity of solutions

We consider a semilinear Neumann problem with an asymptotically linear reaction term. We assume that resonance occurs at infinity. Using variational methods based on the critical point theory, together with the reduction technique and Morse theory, we show that the problem has at least four nontrivial smooth solutions.     

Nontrivial solutions of some quasilinear problems via a cohomological local splitting

This paper deals with a generalization of the p-Laplacian type boundary value problem

     

Existence of radial solutions with prescribed number of zeros for elliptic equations and their Morse index

In this paper we consider radially symmetric solutions of the nonlinear Dirichlet problem Δu+f(|x|,u)=0 in Ω, where Ω is a ball in RN, N?3 and f satisfies some appropriate assumptions. We prove existence of radially symmetric solutions with k prescribed number of zeros. Moreover, when f(|x|,u)=K(|x|)|u|^p−1u, using the uniqueness result due to Tanaka (2008) [21], we verify that these solutions are non-degenerate and we prove that their radial Morse index is exactly k.     

Existence and multiplicity results for quasilinear elliptic exterior problems with nonlinear boundary conditions

In this paper, we study a class of quasilinear elliptic exterior problems with nonlinear boundary conditions. Existence of ground states and multiplicity results are obtained via variational methods.     

Multiplicity results for quasi-linear problems

Applying the minimax arguments and Morse theory, we establish some results on the existence of multiple nontrivial solutions for a class of p$p$-Laplacian elliptic equations.     

Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities

Multiple critical points theorems for non-differentiable functionals are established. Applications both to elliptic variational-hemivariational inequalities and eigenvalue problems with discontinuous nonlinearities are then presented.     

Asymptotically linear elliptic boundary value problems

We study semilinear problems in which the nonlinear term has different asymptotic behavior at ± with the limits (1.2) spanning a finite number of eigenvalues of the linear operator.Research supported in part by an NSF grant.     

Sign-changing and constant-sign solutions for p-Laplacian problems with jumping nonlinearities

By variational methods, we prove the existence of a sign-changing solution for the p-Laplacian equation under Dirichlet boundary condition with jumping nonlinearity having relation to the Fu?ík spectrum of p-Laplacian. We also provide the multiple existence results for the p-Laplacian problems.     

Existence and multiplicity of solutions for Neumann problems

In this paper we examine semilinear and nonlinear Neumann problems with a nonsmooth locally Lipschitz potential function. Using variational methods based on the nonsmooth critical point theory, for the semilinear problem we prove a multiplicity result under conditions of double resonance at higher eigenvalues. Our proof involves a nonsmooth extension of the reduction method due to Castro-Lazer-Thews. The nonlinear problem is driven by the p-Laplacian. So first we make some observations about the beginning of the spectrum of (−Δp,W1,p(Z)). Then we prove an existence and multiplicity result. The existence result permits complete double resonance. The multiplicity result specialized in the semilinear case (i.e. p=2) corresponds to the super-sub quadratic situation.     

Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed points

In this paper, we present a new multi-step iterative method. We prove the strong convergence of the method to a common fixed point of a finite number of nonexpansive mappings that also solves a suitable equilibrium problem.     

Existence results for variational-hemivariational problems with lack of convexity

We establish existence results of Hartmann-Stampacchia type for a class of variational-hemivariational inequalities on closed and convex sets (either bounded or unbounded) in a Hilbert space.     

Critical point theorems in cones and multiple positive solutions of elliptic problems

We obtain critical point variants of the compression fixed point theorem in cones of Krasnoselskii. Critical points are localized in a set defined by means of two norms. In applications to semilinear elliptic boundary value problems this makes possible the use of local Moser-Harnack inequalities for the estimations from below. Multiple solutions are found for problems with oscillating nonlinearity.