Positive Solutions Of Nonlinear Equations Research Articles

Existence and nonexistence of positive solutions of some elliptic problems with variable exponents

In this paper, we study the existence and the nonexistence of some positive solutions for nonlinear equations involving variable exponent Laplace operator and concave–convex second term: under Robin boundary condition in a regular open‐bounded domain of . The ‐Laplacian is given by where and . Our proofs are based on the subsupersolutions method mixed with variational arguments.

Multiplicity of positive solutions for a nonlinear equation with a Hardy potential on the Heisenberg group

In this note, we show multiplicity of positive weak solutions for nonlinear equations with a first stratum Hardy potential on the Heisenberg and stratified groups. The mountain pass theorem, Sobolev-Folland-Stein and first stratum Hardy inequalities play key roles in our proofs.

Infinitely many positive solutions for nonlinear equations with non-symmetric potentials

We consider the following nonlinear Schrodinger equation $$\begin{aligned} \left\{ \begin{array}{l} \Delta u-(1+\delta V)u+f(u)=0 \ \ \hbox { in }\mathbb {R}^N,\\ u>0 \ \hbox {in} \ \mathbb {R}^N, u\in H^1(\mathbb {R}^N) \end{array} \right. \end{aligned}$$ where $$V$$ is a continuous potential and $$ f(u)$$ is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Using localized energy method, we prove that there exists a $$\delta _0$$ such that for $$0<\delta <\delta _0$$ , the above problem has infinitely many positive solutions. This generalizes and gives a new proof of the results by Cerami et al. (Comm. Pure Appl. Math. 66, 372–413, 2013). The new techniques allow us to establish the existence of infinitely many positive bound states for elliptic systems.

Positive solutions of nonlinear equations via comparison with linear operators

We discuss positive solutions of problems that arise from nonlinear boundary value problems in the particular situation where the nonlinear term $f(t,u)$ depends explicitly on $t$ and this dependence is crucial. We give new fixed point index results using comparisons with linear operators. These prove new results on existence of positive solutions under some conditions which can be sharp.

Positive solutions of nonlocal boundary value problems involving integral conditions

We establish the existence of multiple positive solutions of nonlinear equations of the form $$-u^{\prime\prime}(t) = g(t)f(t, u(t)), t \in (0, 1),$$ where g, f are non-negative functions, subject to various nonlocal boundary conditions. The common feature is that each can be written as an integral equation, in the space C[0, 1], of the form $$u(t) = \gamma(t)\alpha [u] + \int \nolimits^1_0 k(t, s)g(s)f(s, u(s))ds$$ where α[u] is a linear functional given by a Stieltjes integral but is not assumed to be positive for all positive u. Our new results cover many non-local boundary conditions previously studied on a case by case basis for particular positive functionals only, for example, many m-point BVPs are special cases. Even for positive functionals our methods give improvements on previous work. Also we allow weaker assumptions on the nonlinear term than were previously imposed.

Positive Solutions of Nonlinear Elliptic Equations in Unbounded Domains in R 2

We study the existence of positive solutions of the nonlinear equation Δu+f(⋅,u)=0, in D with u=0 on ∂D, where D is an unbounded domain in R2 with a compact nonempty boundary ∂D consisting of finitely many Jordan curves. The aim is to prove an existence result for the above equation in a general setting by using potential theory.

Multiplicative algorithms for determining positive solutions of nonlinear equations

New schemes for developing iterative algorithms based on a special nonlinear (multiplicative) transformation are proposed. These schemes are applied to computing positive solutions of nonlinear equations. Modifications of multiplicative algorithms which use parallel and coordinatewise scheme of iterations are considered. Convergence theorems are proved. Multiplicative algorithms are used for solving some problems of urban planning, transportation research, and image reconstruction which can be described by the problem of entropy maximization. Results of computation experiments are included.