Singular φ-Laplacian Research Articles

Multiplicity of periodic solutions for dynamic Liénard equations with delay and singular φ-Laplacian of relativistic type

Multiplicity of periodic solutions for dynamic Liénard equations with delay and singular φ-Laplacian of relativistic type

Multiple positive solutions of second-order nonlinear difference equations with discrete singular φ-Laplacian

ABSTRACTWe consider the discrete boundary value problems with mean curvature operator in the Minkowski space where is a parameter, n>4 and q>1. Using upper and lower solutions, topological methods and Szulkin's critical point theory for convex, lower semicontinuous perturbations of -functionals, we show that there exists such that the above problem has zero, at least one or two positive solutions according to , or . Moreover, Λ is strictly decreasing with respect to n.

Positive radial solutions for multiparameter Dirichlet systems with mean curvature operator in Minkowski space and Lane–Emden type nonlinearities

We deal with Dirichlet systems involving the mean curvature operator in Minkowski spaceM(w)=div(∇w1−|∇w|2) in a ball in RN. Using the fixed point index and the lower and upper solutions method, we first obtain the existence of positive solutions for a class of differential systems with a singular φ-Laplacian, subjected to homogeneous mixed boundary conditions. The main application of the general results concerns Dirichlet systems with Lane–Emden type nonlinearities in a superlinear case, depending on two parameters λ1,λ2. So, we prove that for such a system there exists a continuous curve Γ which separates the first quadrant in two disjoint unbounded open sets O1 and O2 such that the system has zero, at least one or at least two radial positive solutions, according to (λ1,λ2)∈O1, (λ1,λ2)∈Γ or (λ1,λ2)∈O2.

Multiple positive solutions for a class of semipositone Neumann problems with singular φ-laplacian

We study the existence of multiple positive solutions for a Neumann problem with singular φ-Laplacian {-(φ(u′))′=λf(u), x∈(0,1),u′(0)=0=u′(1),where λ is a positive parameter, φ(s)=s1-s2,f∈C1([0,∞),ℝ),f′(u)>0 for u>0, and for some 0<β<θ such that f(u)<0 for u∈[0,β) (semipositone) and f(u)>0 for u > β. Under some suitable assumptions, we obtain the existence of multiple positive solutions of the above problem by using the quadrature technique. Further, if f ∈ C2([0,β)∪(β,∞),ℝ), f″(u)≥0 for u ∈[0,β) and f″(u) ≤0 for u ∈(β,∞), then there exist exactly 2n+1 positive solutions for some interval of λ, which is dependent on n and θ. Moreover, We also give some examples to apply our results.

Existence and multiplicity results for some nonlinear problems with singular ϕ-Laplacian via a geometric approach

New results on the existence and multiplicity of the solutions for some nonlinear boundary value problems with singular ϕ-Laplacian are obtained via a bend-twist fixed point theorem. These results improve related theorems in the previous literature. Moreover, the geometric approach in this paper provides a new method to investigate the existence and multiplicity of periodic motions of charged particles in a three-dimensional electromagnetic field.

Developing an [formula omitted] Lie-group shooting method for a singular ϕ-Laplacian in a nonlinear ODE

In the present paper, we develop an SL(2,R) Lie-group shooting method for a regular or singular ϕ-Laplacian in a nonlinear ordinary differential equation (ODE). By using the closure property of the Lie-group, one-step Lie-group transformation between the boundary values at two ends is established. Hence, we can derive a closed-form formula in terms of r∈[0,1] to determine the missing left-boundary condition by matching the right-boundary condition through a few iterations. The present method is easy to numerical implementation with a cheap computational cost. Numerical examples are examined to show that the present SL(2,R) Lie-group shooting method is effective to treat the regular and singular ϕ-Laplacian with a high accuracy.

Multiple solutions for Neumann and periodic problems with singular ϕ-Laplacian

We use the critical point theory for convex, lower semicontinuous perturbations of C 1 -functionals to establish existence of multiple radial solutions for some one parameter Neumann problems involving the operator v ↦ div ( ∇ v 1 − | ∇ v | 2 ) . Similar results for periodic problems are also provided.

Heteroclinic solutions of boundary value problems on the real line involving singular Φ-Laplacian operators

We discuss the solvability of the following strongly nonlinear BVP: { ( a ( x ( t ) ) Φ ( x ′ ( t ) ) ) ′ = f ( t , x ( t ) , x ′ ( t ) ) , t ∈ R , x ( − ∞ ) = α , x ( + ∞ ) = β where α < β , Φ : ( − r , r ) → R is a general increasing homeomorphism with bounded domain ( singular Φ-Laplacian), a is a positive, continuous function and f is a Carathéodory nonlinear function. We give conditions for the existence and non-existence of heteroclinic solutions in terms of the behavior of y ↦ f ( t , x , y ) and y ↦ Φ ( y ) as y → 0 , and of t ↦ f ( t , x , y ) as | t | → + ∞ . Our approach is based on fixed point techniques suitably combined to the method of upper and lower solutions.

Nonhomogeneous boundary value problems for some nonlinear equations with singular ϕ-Laplacian

Using Leray–Schauder degree theory we obtain various existence results for the quasilinear equation problems ( ϕ ( u ′ ) ) ′ = f ( t , u , u ′ ) submitted to nonhomogeneous Dirichlet or nonlinear Neumann–Steklov boundary conditions on [ 0 , T ] , when ϕ : ] − a , a [ → R is an increasing homeomorphism, ϕ ( 0 ) = 0 . We compare the results with the ones proved earlier in the homogeneous case.

Existence and multiplicity results for some nonlinear problems with singular ϕ-Laplacian

Using Leray–Schauder degree theory we obtain various existence and multiplicity results for nonlinear boundary value problems ( ϕ ( u ′ ) ) ′ = f ( t , u , u ′ ) , l ( u , u ′ ) = 0 where l ( u , u ′ ) = 0 denotes the Dirichlet, periodic or Neumann boundary conditions on [ 0 , T ] , ϕ : ] − a , a [ → R is an increasing homeomorphism, ϕ ( 0 ) = 0 . The Dirichlet problem is always solvable. For Neumann or periodic boundary conditions, we obtain in particular existence conditions for nonlinearities which satisfy some sign conditions, upper and lower solutions theorems, Ambrosetti–Prodi type results. We prove Lazer–Solimini type results for singular nonlinearities and periodic boundary conditions.