By using the method of sub- and supersolutions, we study existence positive solutions for a class singular nonlinear semipositone systems involving nonlocal operators.

In this work, we study the existence of positive solutions to the singular problem −∆pu = λm(x)f(u)− u−α in Ω, u = 0 on ∂Ω, where λ is positive parameter, Ω is a bounded domain with smooth boundary, 0 < α < 1, and f : [0,∞]→ R is a continuous function which is asymptotically p-linear at ∞. The weight function is continuous satisfies m(x) > m0 > 0, ‖m‖∞ <∞. We prove the existence of a positive s...

We study semilinear elliptic equations on general bounded domains with concave semipositone nonlinearities. We prove the existence of the maximal solutions, and describe the global bifurcation diagrams. When a parameter is small, we obtain the exact global bifurcation diagram. We also discuss the related symmetry breaking bifurcation when the domains have certain symmetries.

where 1 < α < 2, 0 < βi < 1, i = 1, 2, . . . ,m – 2, 0 < η1 < η2 < · · · < ηm–2 < 1, ∑m–2 i=1 βiη α–1 i < 1, D α 0+ is the standard Riemann–Liouville derivative. Here our nonlinearity f may be singular at u = 0. As an application of Green’s function, we give some multiple positive solutions for singular positone and semipositone boundary value problems by means of the Leray–Schauder nonlinear a...

This paper studies the existence of symmetric positive solutions for a second order nonlinear semipositone boundary value problem with integral boundary conditions by applying the Krasnoselskii fixed point theorem. Emphasis is put on the fact that the nonlinear term f may take negative value. An example is presented to demonstrate the application of our main result.

By transforming the boundary value problem into the corresponding fixed-point problem of a completely continuous operator, the existence is obtained in the paper for two-point boundary value problem of fourth order superlinear singular semipositone differential system via the fixed point theorem concerning cone compression and expansion in norm type.

In this paper, we establish the existence of positive solution for the semipositone second-order three-point boundary value problem u′′(t) + λf(t, u(t)) = 0, 0 < t < 1, u(0) = αu(η), u(1) = βu(η). Our arguments are based on the well-known Guo-Krasnosel’skii fixed-point theorem in cones.

Recommended by Colin Rogers We establish the existence of multiple positive solutions for a singular nonlinear third-order periodic boundary value problem. We are mainly interested in the semipositone case. The proof relies on a nonlinear alternative principle of Leray-Schauder, together with a truncation technique.

This paper studies the boundary value problems for the fourth-order nonlinear singular difference equationsΔ4u i−2 λα i f i, u i , i ∈ 2, T 2 , u 0 u 1 0, u T 3 u T 4 0. We show the existence of positive solutions for positone and semipositone type. The nonlinear term may be singular. Two examples are also given to illustrate the main results. The arguments are based upon fixed point theorems i...

In this paper we investigate the existence of positive solutions of the following nonlinear semipositone fourth-order two-point boundary-value problem with second derivative: u(t) = f(t, u(t), u′′(t)), 0 ≤ t ≤ 1, u′(1) = u′′(1) = u′′′(1) = 0, ku(0) = u′′′(0), where −6 < k < 0, f ≥ −M , and M is a positive constant. Our approach relies on the Krasnosel’skii fixed point theorem.