scholarly journals Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity

2020 ◽  
Vol 13 (4) ◽  
pp. 1369-1393
Author(s):  
Peter Poláčik ◽  
◽  
Darío A. Valdebenito ◽  
Keyword(s):  
Elliptic Equations ◽  
Quadratic Nonlinearity ◽  
Entire Space ◽  
Quasiperiodic Solutions

Singular solutions of semilinear elliptic equations with fractional Laplacian in entire space

2016 ◽  
Vol 18 (01) ◽  
pp. 1550024 ◽  
Author(s):  
Zhuoran Du
Keyword(s):  
Elliptic Equations ◽  
Singular Solution ◽  
Fractional Laplacian ◽  
Semilinear Elliptic Equations ◽  
Singular Solutions ◽  
Entire Space ◽  
Semilinear Elliptic ◽  
Supercritical Exponent ◽  
Extension Result

We consider the existence of positive singular solutions of the problem [Formula: see text] where [Formula: see text] and [Formula: see text]. We obtain the existence of positive singular solutions [Formula: see text] of this problem in space dimensions [Formula: see text] for the case [Formula: see text] under the condition [Formula: see text]. By the well-known Caffarelli–Silvestre extension result, the trace of [Formula: see text] is a positive singular solution for the following equation with a supercritical exponent [Formula: see text] where [Formula: see text] denotes the fractional Laplacian.

scholarly journals Elliptic problems in ℝN with discontinuous nonlinearities

2000 ◽  
Vol 43 (3) ◽  
pp. 545-558
Author(s):  
Gabriele Bonanno ◽  
Salvatore A. Marano
Keyword(s):  
Fixed Points ◽  
Lebesgue Measure ◽  
Elliptic Equations ◽  
Measure Zero ◽  
Elliptic Problems ◽  
Strong Solutions ◽  
Lebesgue Measure Zero ◽  
Zero Set ◽  
Entire Space ◽  
Discontinuous Nonlinearities

AbstractFor a class of elliptic equations in the entire space and with nonlinear terms having a possibly uncountable (but of Lebesgue measure zero) set of discontinuities, the existence of strong solutions is established. Two simple applications are then developed. The approach taken is strictly based on set-valued analysis and fixed-points arguments.

scholarly journals Multiple Positive Solutions of Nonhomogeneous Elliptic Equations in Unbounded Domains

2007 ◽  
Vol 2007 ◽  
pp. 1-19 ◽  
Author(s):  
Tsing-San Hsu
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Exterior Domain ◽  
Unbounded Domains ◽  
Multiple Positive Solutions ◽  
Entire Space ◽  
Unique Positive Solution ◽  
Strip Domain ◽  
Unbounded Cylinder

We will show that under suitable conditions onfandh, there exists a positive numberλ∗such that the nonhomogeneous elliptic equation−Δu+u=λ(f(x,u)+h(x))inΩ,u∈H01(Ω),N≥2, has at least two positive solutions ifλ∈(0,λ∗), a unique positive solution ifλ=λ∗, and no positive solution ifλ>λ∗, whereΩis the entire space or an exterior domain or an unbounded cylinder domain or the complement in a strip domain of a bounded domain. We also obtain some properties of the set of solutions.

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