scholarly journals On a logarithmic Hartree equation

2019 ◽  
Vol 9 (1) ◽  
pp. 850-865 ◽  
Author(s):  
Federico Bernini ◽  
Dimitri Mugnai
Keyword(s):  
Critical Point ◽  
Critical Point Theory ◽  
Point Theory ◽  
Hartree Equation ◽  
Poisson System ◽  
Negative Solution ◽  
Reaction Term ◽  
Nonlinear Hartree Equation ◽  
Radially Symmetric Solutions ◽  
Superlinear Reaction

Abstract We study the existence of radially symmetric solutions for a nonlinear planar Schrödinger-Poisson system in presence of a superlinear reaction term which doesn’t satisfy the Ambrosetti-Rabinowitz condition. The system is re-written as a nonlinear Hartree equation with a logarithmic convolution term, and the existence of a positive and a negative solution is established via critical point theory.

Periodic Problems with the Scalar p-Laplacian Resonant at any Eigenvalue via Critical Point Methods

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Sophia Th. Kyritsi ◽  
Donal O’ Regan ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Critical Point ◽  
Morse Theory ◽  
Critical Point Theory ◽  
Smooth Solution ◽  
Point Theory ◽  
Reaction Term ◽  
Variational Techniques ◽  
Periodic Problems

AbstractWe consider nonlinear periodic problems driven by the scalar p-Laplacian with a Carathéodory reaction term. Under conditions which permit resonance at infinity with respect to any eigenvalue, we show that the problem has a nontrivial smooth solution. Our approach combines variational techniques based on critical point theory with Morse theory.

scholarly journals Multiple positive solutions for a logarithmic Schrödinger–Poisson system with singular nonelinearity

2021 ◽  
pp. 1-15
Author(s):  
Linyan Peng ◽  
Hongmin Suo ◽  
Deke Wu ◽  
Hongxi Feng ◽  
Chunyu Lei
Keyword(s):  
Variational Method ◽  
Critical Point ◽  
Positive Solutions ◽  
Critical Point Theory ◽  
Point Theory ◽  
Multiple Positive Solutions ◽  
Poisson System ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

In this article, we devote ourselves to investigate the following logarithmic Schrödinger–Poisson systems with singular nonlinearity { − Δ u + ϕ u = | u | p−2 u log ⁡ | u | + λ u γ , i n   Ω , − Δ ϕ = u 2 , i n   Ω , u = ϕ = 0 , o n   ∂ Ω , where Ω is a smooth bounded domain with boundary 0 < γ < 1 , p ∈ ( 4 , 6 ) and λ > 0 is a real parameter. By using the critical point theory for nonsmooth functional and variational method, the existence and multiplicity of positive solutions are established.

scholarly journals On the vortical structure in a plane impinging jet

2001 ◽  
Vol 434 ◽  
pp. 273-300 ◽  
Author(s):  
J. SAKAKIBARA ◽  
K. HISHIDA ◽  
W. R. C. PHILLIPS
Keyword(s):  
Critical Point ◽  
Critical Point Theory ◽  
Vortical Structure ◽  
Strain Tensor ◽  
Symmetry Plane ◽  
Impinging Jet ◽  
Point Theory ◽  
Mixing Layers ◽  
Intensity Level ◽  
The Cross

The vortical structure of a plane impinging jet is considered. The jet was locked both in phase and laterally in space, and time series digital particle image velocimetry measurements were made both of the jet exiting the nozzle and as it impinged on a perpendicular wall. Iso-vorticity and iso-λ2 surfaces coupled with critical point theory were used to identify and clarify structure. The flow near the nozzle was much as observed in mixing layers, where the shear layer evolves into spanwise rollers, only here the rollers occurred symmetrically about the jet midplane. Accordingly the rollers were seen to depict spanwise perturbations with the wavelength of flutes at the nozzle edge and were connected, on the same side of the jet, with streamwise ‘successive ribs’ of the same wavelength. This wavelength was 0.71 of the distance between rollers and, contrary to some experiments in mixing layers, did not double when the rollers paired. Structures not reported previously but evident here with iso-vorticity, λ2 and critical point theory are ‘cross ribs’, which extend from the downstream side of each roller to its counterpart across the symmetry plane; their spanwise periodic spacing exceeds that of successive ribs by a factor of three. Cross ribs stretch because of the diverging flow as the rollers approach the wall and move apart, causing the vorticity within them to intensify. This process continues until the cross ribs reach the wall and merge with ‘wall ribs’. Wall ribs remain near the wall throughout the cycle and are composed of vorticity of the same sign as the cross ribs, but the intensity level of the vorticity within them is cyclic. Details of the expansion of fluid elements, evaluated from the rate of strain tensor, revealed that both cross and successive ribs align with the principal axis and that the vorticity comprising them is continuously amplified by stretching. It is further shown, by appeal to the production terms of the phase-averaged vorticity equation, that wall ribs are sustained by merging and stretching rather than reorientation of vorticity. Moreover production of vorticity is a maximum when cross and wall ribs merge and is greatest near the symmetry plane of the jet. The demise of successive ribs on the other hand occurs away from the symmetry plane and would appear to be less important dynamically than cross ribs merging with wall ribs.

Existence of infinitely many solutions for fractional p-Laplacian Schrödinger–Kirchhof-Type equations with general potentials

2021 ◽  
pp. 2250095
Author(s):  
Ghania Benhamida ◽  
Toufik Moussaoui
Keyword(s):  
Critical Point ◽  
Critical Point Theory ◽  
Point Theory ◽  
Infinitely Many Solutions ◽  
Kirchhoff Type ◽  
Laplacian Equations

In this paper, we use the genus properties in critical point theory to prove the existence of infinitely many solutions for fractional [Formula: see text]-Laplacian equations of Schrödinger-Kirchhoff type.

scholarly journals Perturbation results involving the 1-Laplace operator

2019 ◽  
Vol 12 (3) ◽  
pp. 277-302 ◽  
Author(s):  
Samuel Littig ◽  
Friedemann Schuricht
Keyword(s):  
Critical Point ◽  
Laplace Operator ◽  
Critical Point Theory ◽  
Eigenvalue Problems ◽  
Point Theory ◽  
Nonsmooth Critical Point Theory ◽  
Unperturbed Problem ◽  
Nonsmooth Critical Point

AbstractWe consider perturbed eigenvalue problems of the 1-Laplace operator and verify the existence of a sequence of solutions. It is shown that the eigenvalues of the perturbed problem converge to the corresponding eigenvalue of the unperturbed problem as the perturbation becomes small. The results rely on nonsmooth critical point theory based on the weak slope.

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