On the Morse index of least energy nodal solutions for quasilinear elliptic problems

2020 ◽  
Vol 59 (2) ◽  
Author(s):  
Oscar Agudelo ◽  
Daniel Restrepo ◽  
Carlos Vélez
Keyword(s):  
Morse Index ◽  
Elliptic Problems ◽  
Nodal Solutions ◽  
Quasilinear Elliptic Problems ◽  
Quasilinear Elliptic ◽  
Least Energy

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].

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