Existence Of Nontrivial Positive Solutions Research Articles

Existence and nonexistence of nontrivial solutions for a class of p-Kirchhoff type problems with critical Sobolev exponent

In this work, by using variational methods we study the existence of nontrivial positive solutions for a class of p-Kirchhoff type problems with critical Sobolev exponent.

Degree for weakly upper semicontinuous perturbations of quasi-m-accretive operators.

In the paper, we provide the construction of a coincidence degree being a homotopy invariant detecting the existence of solutions of equations or inclusions of the form Ax ∈ F(x), x ∈ U, where [Formula: see text] is an m-accretive operator in a Banach space E, [Formula: see text] is a weakly upper semicontinuous set-valued map constrained to an open subset U of a closed set K ⊂ E. Two different approaches are presented. The theory is applied to show the existence of non-trivial positive solutions of some nonlinear second-order partial differential equations with discontinuities. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.

Positive Solution for the Integral and Infinite Point Boundary Value Problem for Fractional-Order Differential Equation Involving a Generalized ϕ-Laplacian Operator

In this paper, we establish the existence of nontrivial positive solution to the following integral-infinite point boundary-value problem involving ϕ-Laplacian operator D0+αϕx,D0+βux+fx,ux=0,x∈0,1,D0+σu0=D0+βu0=0,u1=∫01 gtutdt+∑n=1n=+∞ αnuηn, where ϕ:0,1×R→R is a continuous function and D0+p is the Riemann-Liouville derivative for p∈α,β,σ. By using some properties of fixed point index, we obtain the existence results and give an example at last.

A Lyapunov-Type Inequality for a Laplacian System on a Rectangular Domain with Zero Dirichlet Boundary Conditions

We consider a coupled system of partial differential equations involving Laplacian operator, on a rectangular domain with zero Dirichlet boundary conditions. A Lyapunov-type inequality related to this problem is derived. This inequality provides a necessary condition for the existence of nontrivial positive solutions.

Systems of ϕ-Laplacian Three-Point Boundary-Value Problems on the Positive Half Line

We study the existence of positive solutions to boundary-value problems for two systems of two secondorder nonlinear three-point ϕ-Laplacian equations defined on the positive half line. The nonlinearities may change sign, exhibit time singularities at the origin, and depend both on the solutions and on their first derivatives. Using the fixed-point theory, we prove some results on the existence of nontrivial positive solutions on appropriate cones in some weighted Banach spaces.

Existence of positive solutions for a variational inequality of Kirchhoff type

In the present paper, we study existence of nontrivial positive solutions for a Kirchhoff type variational inequality. Our approach is based on the non-smooth critical point theory for Szulkin-type functionals.

Existence results for some nonlocal problems

We consider nonlocal elliptic boundary value problems of the form −div(A (x,u)∇u )= λ f(u) with Dirichlet boundary conditions where A is a nonlocal function. We prove existence of nontrivial positive solutions if the graph of the non linear function f is of single positive loop type. Methods of approximation and Schauder fixed point theorem are the main tools to be used here.

Existence of Nontrivial Positive Solutions to a Semilinear Elliptic System

This paper studies the nontrivial positive solutions to a semilinear elliptic system in the n dimensional Euclide space. By constructing new variational space, using the linking theorems and some embedding theorems, this paper proves the existence of positive solution to a semilinear elliptic system, and improves the results of Li and Wang.

Existence of Nontrivial Positive Solutions to a Semilinear Elliptic System with Variable Coefficients

This paper studies the nontrivial positive solutions to a semilinear elliptic system with variable coefficients in the n dimensional Euclide space. By constructing a new variational space and using some linking theorems, this paper finally proves the existence of positive solution to a semilinear elliptic system.

Entire Extremal Solutions for Elliptic Inclusions of Clarke’s Gradient Type

We consider multivalued quasilinear elliptic problems of hemivariational type in all of ℝ^N given by −∆_pu + ∂j(·, u) \ni 0 \quad \text{in } \mathcal D', and show the existence of entire extremal solutions by applying the method of sub- and supersolutions without imposing any condition at infinity. Due to the unboundedness of the domain, standard variational methods cannot be applied. The novelty of our approach is on the one hand to obtain entire solutions and on the other hand that Clarke’s generalized gradient need only satisfies a natural growth condition. In the last section conditions are provided that ensure the existence of nontrivial positive solutions.

Positive solutions and multiple solutions for periodic problems driven by scalar p ‐Laplacian

AbstractIn this paper we study a nonlinear second order periodic problem driven by a scalar p ‐Laplacian and with a nonsmooth, locally Lipschitz potential function. Using a variational approach based on the nonsmooth critical point theory for locally Lipschitz functions, we first prove the existence of nontrivial positive solutions and then establish the existence of a second distinct solution (multiplicity theorem) by strengthening further the hypotheses. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The existence of positive solution of elliptic system by a linking theorem on product space

This paper is devoted to study the existence of nontrivial positive solution of a class of elliptic system with Dirichlet Data. By using the abstract linking theorems on product space we established in Zhao et al. (Nonlinear Anal. 49 (2002) 431) we obtain the existence of three nonnegative solutions for a class of elliptic systems and the existence of a nontrivial positive solution for the problem related to the model of competing species systems involving critical Sobolev exponents.

Second order systems of ODE's realated to a ginzburg-labday model

We study the existence of nontrivial positive solutions to a boundary value problem for a second order system of two nonlinear ordinary differntial equations. This system contains as a particular case the one-dimensional Ginzburg-Landau equations for superconducting films. We also give a multiplicity result.