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In this paper, we consider probability measures μ and ν on a d-dimensional sphere in $𝐑^{d + 1}, d \geq 1,$ and cost functions of the form $c (𝐱, 𝐲) = l (\frac{{| 𝐱 - 𝐲 |}^{2}}{2})$ that generalize those arising in geometric optics where $l (t) = - log t .$ We prove that if μ and ν vanish on $(d - 1)$ -rectifiable sets, if |l'(t)|>0, ${lim}_{t \to 0^{+}} l (t) = + \infty,$ and $g (t) : = t (2 - t) {(l^{'} (t))}^{2}$ is monotone then there exists a unique optimal map To that transports μ onto $ν,$ where optimality is measured against c. Furthermore, ${inf}_{𝐱} | T_{o} 𝐱 - 𝐱 | > 0 .$ Our approach is based on direct variational arguments. In the special case when $l (t) = - log t,$ existence of optimal maps...

Existence of positive and sign-changing solutions for $p$ -Laplace equations with potentials in $ℝ^{N}$ .

Wu, Mingzhu, Yang, Zuodong (2010)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Existence of positive bounded solutions for some nonlinear polyharmonic elliptic systems.

Gontara, Sabrine, El Abidine, Zagharide Zine (2010)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Existence of positive bounded solutions of semilinear elliptic problems.

Toumi, Faten (2010)

International Journal of Differential Equations

Existence of positive entire solutions of semilinear elliptic equations on $ℝ^{N}$ .

Hirano, N. (1996)

Abstract and Applied Analysis

Existence of positive radial solutions for a weakly coupled systems via blow up.

García-Huidobro, Marta, Guerra, Ignacio, Manásevich, Raúl (1998)

Abstract and Applied Analysis

Existence of positive radial solutions for the elliptic equations on an exterior domain

Yongxiang Li, Huanhuan Zhang (2016)

Annales Polonici Mathematici

We discuss the existence of positive radial solutions of the semilinear elliptic equation ⎧-Δu = K(|x|)f(u), x ∈ Ω ⎨αu + β ∂u/∂n = 0, x ∈ ∂Ω, ⎩ $l i m_{| x | \to \infty} u (x) = 0$ , where $Ω = x \in ℝ^{N} : | x | > r ₀$ , N ≥ 3, K: [r₀,∞) → ℝ⁺ is continuous and $0 < \int_{r ₀}^{\infty} r K (r) d r < \infty$ , f ∈ C(ℝ⁺,ℝ⁺), f(0) = 0. Under the conditions related to the asymptotic behaviour of f(u)/u at 0 and infinity, the existence of positive radial solutions is obtained. Our conditions are more precise and weaker than the superlinear or sublinear growth conditions. Our discussion is based on the fixed point...

Existence of positive solutions for boundary value problems of nonlinear functional difference equation with $p$ -Laplacian operator.

Yang, S.J., Shi, B., Zhang, D.C. (2007)

Boundary Value Problems [electronic only]

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Existence of nontrivial solution for a nonlocal elliptic equation with nonlinear boundary condition.

Existence of nontrivial solutions for semilinear problems with strictly differentiable nonlinearity.

Existence of nontrivial solutions for singular quasilinear equations with sign changing nonlinearity.

Existence of optimal maps in the reflector-type problems

Existence of positive and sign-changing solutions for $p$ -Laplace equations with potentials in $ℝ^{N}$ .

Existence of positive bounded solutions for some nonlinear polyharmonic elliptic systems.

Existence of positive bounded solutions of semilinear elliptic problems.

Existence of positive entire solutions of semilinear elliptic equations on $ℝ^{N}$ .

Existence of positive radial solutions for a weakly coupled systems via blow up.

Existence of positive radial solutions for the elliptic equations on an exterior domain

Existence of positive solutions for boundary value problems of nonlinear functional difference equation with $p$ -Laplacian operator.

Existence of positive solutions for Dirichlet problems of some singular elliptic equations.

Existence of positive solutions for higher order singular sublinear elliptic equations.

Existence of positive solutions for nonlinear boundary value problems in bounded domains of $ℝ^{n}$ .

Existence of positive solutions for nonlinear boundary-value problems in unbounded domains of $ℝ^{n}$ .

Existence of positive solutions for $p (x)$ -Laplacian problems.

Existence of positive solutions for quasilinear elliptic systems involving the $p$ -Laplacian.

Existence of positive solutions for some Dirichlet problems with an asymptotically homogeneous operator.

Existence of positive solutions for some nonlinear elliptic systems on the half space.

Existence of positive solutions for some nonlinear elliptic problems in unbounded domains of $ℝ^{n}$ .

Currently displaying 341 – 360 of 504