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.

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 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].

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 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.

Multiple positive solutions of nonhomogeneous semilinear elliptic equations in ℝN

1996 ◽  
Vol 126 (2) ◽  
pp. 443-463 ◽  
Author(s):  
Cao Dao-Min ◽  
Zhou Huan-Song
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Multiple Positive Solutions ◽  
Semilinear Elliptic ◽  
Sobolev Constant ◽  
Image Position

We consider the following problemwhere for all ≦f(x,u)≦c1up-1 + c2u for all x ∈ℝN,u≧0 with c1>0,c2∈(0, 1), 2<p<(2N/(N – 2)) if N ≧ 3, 2 ≧ + ∝ if N = 2. We prove that (*) has at least two positive solutions ifand h≩0 in ℝN, where S is the best Sobolev constant and

Some results for semilinear elliptic equations with critical potential

2002 ◽  
Vol 132 (1) ◽  
pp. 1-24 ◽  
Author(s):  
B. Abdellaoui ◽  
I. Peral
Keyword(s):  
Elliptic Equations ◽  
Elliptic Problems ◽  
Semilinear Elliptic Equations ◽  
Critical Potential ◽  
Semilinear Elliptic ◽  
Image Position

This paper is devoted to the study of the elliptic problems with a critical potential, where N ≥ 3, λ ≥ 0 and 0 < q < 1 < p ≤ (N + 2)/(N − 2). Existence, multiplicity, behaviour in x = 0 and bifurcation are considered under some hypotheses in h and g.

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