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Nonlocal Lazer–McKenna-type problem perturbed by the Hardy’s potential and its parabolic equivalence
, Article Boundary Value Problems ; Volume 2021, Issue 1 , 2021 ; 16872762 (ISSN) ; Hesaaraki, M ; Karim Hamdani, M ; Thanh Chung, N ; Sharif University of Technology
Springer Science and Business Media Deutschland GmbH
2021
Abstract
In this paper, we study the effect of Hardy potential on the existence or nonexistence of solutions to the following fractional problem involving a singular nonlinearity: {(−Δ)su=λu|x|2s+μuγ+fin Ω,u>0in Ω,u=0in (RN∖Ω). Here 0 < s< 1 , λ> 0 , γ> 0 , and Ω ⊂ RN (N> 2 s) is a bounded smooth domain such that 0 ∈ Ω. Moreover, 0 ≤ μ, f∈ L1(Ω). For 0 < λ≤ Λ N,s, Λ N,s being the best constant in the fractional Hardy inequality, we find a necessary and sufficient condition for the existence of a positive weak solution to the problem with respect to the data μ and f. Also, for a regular datum of f, under suitable assumptions, we obtain some existence and uniqueness results and calculate the rate of...
Pairs of Positive Periodic Solutions of Second Order Nonlinear Equations
, M.Sc. Thesis Sharif University of Technology ; Hesaaraki, Mahmoud (Supervisor)
Abstract
In this thesis we study the problem of existence and multiplicity of positive periodic solution to the scalar ODE , , where is a positive function on , super linear at zero and sub linear at infinity, and is a -periodic and sign indefinite weight with negative mean value. We first show the nonexistence of solution for some classes of nonlinearities when is small. Then, using critical point theory, we prove the existence of at least two positive -periodic solutions for large. Then, we prove the existence of a pair of positive -periodic solutions as well as the existence of positive sub harmonic solutions of any order for the scalar second order ODE where is same as above,...
On the existence of bounded positive solutions of Schrödinger equations in two-dimensional exterior domains
, Article Applied Mathematics Letters ; Volume 20, Issue 12 , December , 2007 , Pages 1227-1231 ; 08939659 (ISSN) ; Moradifam, A ; Sharif University of Technology
2007
Abstract
We prove under quite general assumptions the existence of a bounded positive solution of the semilinear Schrödinger equation Δ u + f (x, u) = 0 in a two-dimensional exterior domain. Our results are independent of the behavior of f (x, u) when u is sufficiently small or sufficiently large and just require some knowledge about the nonlinearity f (x, u) for a ≤ u ≤ b, for some a, b > 0. We obtain solutions with a prescribed positive lower bound. © 2007 Elsevier Ltd. All rights reserved