scholarly journals Critical elliptic systems involving multiple strongly–coupled Hardy–type terms

2019 ◽  
Vol 9 (1) ◽  
pp. 866-881
Author(s):  
Dongsheng Kang ◽  
Mengru Liu ◽  
Liangshun Xu
Keyword(s):  
Explicit Form ◽  
Elliptic Equations ◽  
Elliptic Systems ◽  
Asymptotic Behaviors ◽  
Strongly Coupled ◽  
Critical Nonlinearities ◽  
Hardy Type ◽  
Analysis Methods ◽  
Least Energy Solutions ◽  
Least Energy

Abstract In this paper, we study the radially–symmetric and strictly–decreasing solutions to a system of critical elliptic equations in RN, which involves multiple critical nonlinearities and strongly–coupled Hardy– type terms. By the ODEs analysis methods, the asymptotic behaviors at the origin and infinity of solutions are proved. It is found that the singularities of u and v in the solution (u, v) are at the same level. Finally, an explicit form of least energy solutions is found under certain assumptions, which has all of the mentioned properties for the radial decreasing solutions.

Least energy solutions for elliptic equations in unbounded domains

1996 ◽  
Vol 126 (1) ◽  
pp. 195-208 ◽  
Author(s):  
Manuel A. del Pino ◽  
Patricio L. Felmer
Keyword(s):  
Critical Point ◽  
Unbounded Domain ◽  
Elliptic Equations ◽  
Unbounded Domains ◽  
Asymptotic Results ◽  
Semilinear Elliptic ◽  
Superlinear Growth ◽  
Image Position ◽  
Least Energy Solutions ◽  
Least Energy

In this paper we study the existence of least energy solutions to subcritical semilinear elliptic equations of the formwhere Ω is an unbounded domain in RN and f is a C1 function, with appropriate superlinear growth. We state general conditions on the domain Ω so that the associated functional has a nontrivial critical point, thus yielding a solution to the equation. Asymptotic results for domains stretched in one direction are also provided.

scholarly journals A monotonicity result under symmetry and Morse index constraints in the plane

2020 ◽  
pp. 1-31
Author(s):  
Francesca Gladiali
Keyword(s):  
Elliptic Equations ◽  
Morse Index ◽  
Symmetric Domain ◽  
Angular Variable ◽  
Multiplicity Results ◽  
Semilinear Elliptic ◽  
Monotonicity Result ◽  
Image Position ◽  
Least Energy Solutions ◽  
Least Energy

This paper deals with solutions of semilinear elliptic equations of the type \[ \left\{\begin{array}{@{}ll} -\Delta u = f(|x|, u) \qquad & \text{ in } \Omega, \\ u= 0 & \text{ on } \partial \Omega, \end{array} \right. \] where Ω is a radially symmetric domain of the plane that can be bounded or unbounded. We consider solutions u that are invariant by rotations of a certain angle θ and which have a bound on their Morse index in spaces of functions invariant by these rotations. We can prove that or u is radial, or, else, there exists a direction $e\in \mathcal {S}$ such that u is symmetric with respect to e and it is strictly monotone in the angular variable in a sector of angle θ/2. The result applies to least-energy and nodal least-energy solutions in spaces of functions invariant by rotations and produces multiplicity results.

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