scholarly journals A Morse index formula for radial solutions of Lane–Emden problems

2017 ◽  
Vol 322 ◽  
pp. 682-737 ◽  
Author(s):  
Francesca De Marchis ◽  
Isabella Ianni ◽  
Filomena Pacella
Keyword(s):  
Morse Index ◽  
Index Formula ◽  
Radial Solutions

Morse index and symmetry breaking for an elliptic equation with negative exponent in expanding annuli

2017 ◽  
Vol 147 (6) ◽  
pp. 1215-1232
Author(s):  
Zongming Guo ◽  
Linfeng Mei ◽  
Zhitao Zhang
Keyword(s):  
Elliptic Equation ◽  
Symmetry Breaking ◽  
Morse Index ◽  
Blow Up ◽  
Semilinear Elliptic Equation ◽  
Radial Solutions ◽  
Entire Solutions ◽  
Semilinear Elliptic ◽  
Image Position

Bifurcation of non-radial solutions from radial solutions of a semilinear elliptic equation with negative exponent in expanding annuli of ℝ2 is studied. To obtain the main results, we use a blow-up argument via the Morse index of the regular entire solutions of the equationThe main results of this paper can be seen as applications of the results obtained recently for finite Morse index solutions of the equationwith N ⩾ 2 and p > 0.

scholarly journals Morse index of radial nodal solutions of Hénon type equations in dimension two

2017 ◽  
Vol 19 (03) ◽  
pp. 1650042 ◽  
Author(s):  
Ederson Moreira dos Santos ◽  
Filomena Pacella
Keyword(s):  
Elliptic Equations ◽  
Morse Index ◽  
Radial Solution ◽  
Index Formula ◽  
Connected Components ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Semilinear Elliptic ◽  
Bounded Sets ◽  
Least Energy

We consider non-autonomous semilinear elliptic equations of the type [Formula: see text] where [Formula: see text] is either a ball or an annulus centered at the origin, [Formula: see text] and [Formula: see text] is [Formula: see text] on bounded sets of [Formula: see text]. We address the question of estimating the Morse index [Formula: see text] of a sign changing radial solution [Formula: see text]. We prove that [Formula: see text] for every [Formula: see text] and that [Formula: see text] if [Formula: see text] is even. If [Formula: see text] is superlinear the previous estimates become [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] denotes the number of nodal sets of [Formula: see text], i.e. of connected components of [Formula: see text]. Consequently, every least energy nodal solution [Formula: see text] is not radially symmetric and [Formula: see text] as [Formula: see text] along the sequence of even exponents [Formula: see text].

scholarly journals The asymptotically linear Hénon problem

2020 ◽  
pp. 2050042
Author(s):  
Anna Lisa Amadori
Keyword(s):  
Boundary Conditions ◽  
Morse Index ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Radial Solutions ◽  
Asymptotically Linear ◽  
Planar Case ◽  
Asymptotic Profile ◽  
Least Energy Solutions ◽  
Least Energy

In this paper, we consider the Hénon problem in the ball with Dirichlet boundary conditions. We study the asymptotic profile of radial solutions and then deduce the exact computation of their Morse index when the exponent [Formula: see text] is close to [Formula: see text]. Next we focus on the planar case and describe the asymptotic profile of some solutions which minimize the energy among functions which are invariant for reflection and rotations of a given angle [Formula: see text]. By considerations based on the Morse index we see that, depending on the values of [Formula: see text] and [Formula: see text], such least energy solutions can be radial, or nonradial and different one from another.

scholarly journals Symmetry breaking and Morse index of solutions of nonlinear elliptic problems in the plane

2016 ◽  
Vol 18 (05) ◽  
pp. 1550087 ◽  
Author(s):  
Francesca Gladiali ◽  
Massimo Grossi ◽  
Sérgio L. N. Neves
Keyword(s):  
Symmetry Breaking ◽  
Morse Index ◽  
Bifurcation Theory ◽  
Careful Study ◽  
Positive Parameter ◽  
Radial Solutions ◽  
Real Numbers ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic ◽  
Linearized Operator

In this paper, we study the problem [Formula: see text] where [Formula: see text] is the unit ball of [Formula: see text], [Formula: see text] is a smooth nonlinearity and [Formula: see text], [Formula: see text] are real numbers with [Formula: see text]. From a careful study of the linearized operator, we compute the Morse index of some radial solutions to ([Formula: see text]). Moreover, using the bifurcation theory, we prove the existence of branches of nonradial solutions for suitable values of the positive parameter [Formula: see text]. The case [Formula: see text] provides more detailed informations.

scholarly journals Morse index computation for radial solutions of the Hénon problem in the disk

2022 ◽  
Vol 215 ◽  
pp. 112645
Author(s):  
Anna Lisa Amadori ◽  
Francesca De Marchis ◽  
Isabella Ianni
Keyword(s):  
Morse Index ◽  
Radial Solutions

Asymptotic profile and Morse index of the radial solutions of the Hénon equation

2021 ◽  
Vol 287 ◽  
pp. 212-235
Author(s):  
Wendel Leite da Silva ◽  
Ederson Moreira dos Santos
Keyword(s):  
Morse Index ◽  
Radial Solutions ◽  
Asymptotic Profile ◽  
Hénon Equation
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