Sign-changing tower of bubbles to an elliptic subcritical equation

2019 ◽  
Vol 21 (07) ◽  
pp. 1850052 ◽  
Author(s):  
Yessine Dammak ◽  
Rabeh Ghoudi
Keyword(s):  
Critical Exponent ◽  
Elliptic Problem ◽  
Smooth Domain ◽  
Positive Parameter ◽  
Nonlinear Elliptic Problem ◽  
Bounded Smooth Domain ◽  
Nonlinear Elliptic ◽  
Small Positive Parameter ◽  
Bounded Smooth

This paper is concerned with the following nonlinear elliptic problem involving nearly critical exponent [Formula: see text]: [Formula: see text] in [Formula: see text], [Formula: see text] on [Formula: see text], where [Formula: see text] is a bounded smooth domain in [Formula: see text], [Formula: see text], [Formula: see text] is a small positive parameter. As [Formula: see text] goes to zero, we construct a solution with the shape of a tower of sign changing bubbles.

A NONEXISTENCE RESULT OF SINGLE PEAKED SOLUTIONS TO A SUPERCRITICAL NONLINEAR PROBLEM

2003 ◽  
Vol 05 (02) ◽  
pp. 179-195 ◽  
Author(s):  
M. BEN AYED ◽  
K. EL MEHDI ◽  
O. REY ◽  
M. GROSSI
Keyword(s):  
Bounded Domain ◽  
Elliptic Problem ◽  
Critical Sobolev Exponent ◽  
Positive Parameter ◽  
Nonlinear Elliptic Problem ◽  
Smooth Bounded Domain ◽  
Nonlinear Elliptic ◽  
Nonexistence Result ◽  
Small Positive Parameter ◽  
Sobolev Exponent

This paper is concerned with the nonlinear elliptic problem (Pε): -Δu = up+ε, u > 0 in Ω; u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝn, n ≥ 3, p + 1 = 2n/(n - 2) is the critical Sobolev exponent and ε is a small positive parameter. In contrast with the subcritical problem (P- ε) studied by Han [11] and Rey [17], we show that (Pε) has no single peaked solution for small ε.

Vanational elliptic problems involving noncoercive functionals

1989 ◽  
Vol 112 (1-2) ◽  
pp. 177-185 ◽  
Author(s):  
Miguel Ramos ◽  
Luis Sanchez
Keyword(s):  
Elliptic Problem ◽  
First Eigenvalue ◽  
Nonlinear Elliptic Problem ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Nonlinear Elliptic ◽  
At Resonance ◽  
Problem At Resonance ◽  
Bounded Smooth ◽  
First Eigenfunction

SynopsisWe consider the nonlinear elliptic problem at resonance, Δu + λ1u + f(x, u) = h(x) in Ω, u = 0 on ∂Ω, where Ω is a bounded smooth domain in ℝN, λl is the first eigenvalue of –Δ in Ω and h(x) is orthogonal to the first eigenfunction. We give some conditions of solvability in terms of the primitive of f with respect to u.

Multiple boundary blow-up solutions for nonlinear elliptic equations

2003 ◽  
Vol 133 (2) ◽  
pp. 225-235 ◽  
Author(s):  
Amandine Aftalion ◽  
Manuel del Pino ◽  
René Letelier
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
A Priori ◽  
Nonlinear Elliptic Equations ◽  
Positive Parameter ◽  
Number Of Solutions ◽  
Bounded Smooth Domain ◽  
Nonlinear Elliptic ◽  
Bounded Smooth

We consider the problem Δu = λf(u) in Ω, u(x) tends to +∞ as x approaches ∂Ω. Here, Ω is a bounded smooth domain in RN, N ≥ 1 and λ is a positive parameter. In this paper, we are interested in analysing the role of the sign changes of the function f in the number of solutions of this problem. As a consequence of our main result, we find that if Ω is star-shaped and f behaves like f(u) = u(u−a)(u−1) with ½ < a < 1, then there is a solution bigger than 1 for all λ and there exists λ0 > 0 such that, for λ < λ0, there is no positive solution that crosses 1 and, for λ > λ0, at least two solutions that cross 1. The proof is based on a priori estimates, the construction of barriers and topological-degree arguments.

Nonexistence Results of Sign-changing solutions for a Supercritical Problem of the Scalar Curvature Type

2012 ◽  
Vol 12 (1) ◽  
Author(s):  
Kamal Ould Bouh
Keyword(s):  
Critical Exponent ◽  
Scalar Curvature ◽  
Elliptic Problem ◽  
Nonlinear Elliptic Problem ◽  
Nonlinear Elliptic ◽  
Sign Changing Solutions ◽  
Supercritical Problem ◽  
Curvature Type

AbstractThis paper is devoted to the study of the nonlinear elliptic problem with supercritical critical exponent (P

scholarly journals Liouville Theorem for Some Elliptic Equations with Weights and Finite Morse Indices

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Qiongli Wu ◽  
Liangcai Gan ◽  
Qingfeng Fan
Keyword(s):  
Elliptic Problem ◽  
Elliptic Equations ◽  
Morse Index ◽  
Liouville Theorem ◽  
Bounded Solution ◽  
Positive Parameter ◽  
Nonlinear Elliptic Problem ◽  
Nonlinear Elliptic ◽  
Morse Indices

We establish the nonexistence of solution for the following nonlinear elliptic problem with weights:-Δu=(1+|x|α)|u|p-1uinRN, whereαis a positive parameter. Suppose that1<p<N+2/N-2,α>(N-2)(p+1)/2-NforN≥3orp>1,α>-2forN=2; we will show that this equation does not possess nontrivial bounded solution with finite Morse index.

scholarly journals Sign-changing solutions for a Schrödinger–Kirchhoff–Poisson system with 4-sublinear growth nonlinearity

2021 ◽  
pp. 1-21
Author(s):  
Shubin Yu ◽  
Ziheng Zhang ◽  
Rong Yuan
Keyword(s):  
Type System ◽  
Invariant Sets ◽  
Smooth Domain ◽  
Positive Parameter ◽  
Poisson System ◽  
Bounded Smooth Domain ◽  
Existence And Multiplicity ◽  
Bounded Smooth ◽  
Sign Changing Solutions ◽  
Poisson Type

In this paper we consider the following Schrödinger–Kirchhoff–Poisson-type system { − ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u + λ ϕ u = Q ( x ) | u | p − 2 u in   Ω , − Δ ϕ = u 2 in   Ω , u = ϕ = 0 on   ∂ Ω , where Ω is a bounded smooth domain of R 3 , a > 0 , b ≥ 0 are constants and λ is a positive parameter. Under suitable conditions on Q ( x ) and combining the method of invariant sets of descending flow, we establish the existence and multiplicity of sign-changing solutions to this problem for the case that 2 < p < 4 as λ sufficient small. Furthermore, for λ = 1 and the above assumptions on Q ( x ) , we obtain the same conclusions with 2 < p < 12 5 .

PROFILE AND EXISTENCE OF SIGN-CHANGING SOLUTIONS TO AN ELLIPTIC SUBCRITICAL EQUATION

2008 ◽  
Vol 10 (06) ◽  
pp. 1183-1216 ◽  
Author(s):  
MOHAMED BEN AYED ◽  
RABEH GHOUDI
Keyword(s):  
Critical Exponent ◽  
Elliptic Problem ◽  
Blow Up ◽  
Low Energy ◽  
Nonlinear Elliptic Problem ◽  
Smooth Bounded Domain ◽  
Nonlinear Elliptic ◽  
Robin Function ◽  
Sign Changing Solutions ◽  
Two Bubbles

In this paper, we study the nonlinear elliptic problem involving nearly critical exponent (Pε) : Δ2 u = |u|(8/(n-4))-εu, in Ω, Δu = u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝn, n ≥ 5. We characterize the low energy sign-changing solutions (uε) of (Pε). We prove that (uε) are close to two bubbles with different signs and they have to blow up either at two different points with the same speed or at a critical point of the Robin function. Furthermore, we construct families of each kind of these solutions and we prove that the bubble-tower solutions exist in our case.

A variational approach to multiplicity in elliptic problems near resonance

1997 ◽  
Vol 127 (2) ◽  
pp. 385-394 ◽  
Author(s):  
M. Ramos ◽  
L. Sanchez
Keyword(s):  
Variational Approach ◽  
Multiple Solutions ◽  
First Eigenvalue ◽  
Nonlinear Elliptic Problem ◽  
Bounded Smooth Domain ◽  
Nonlinear Elliptic ◽  
Near Resonance ◽  
Existence Of Positive Solutions ◽  
Bounded Smooth ◽  
First Eigenfunction

We consider the nonlinear elliptic problem ± (Δu + λu) + f(x, u) = h(x) in Ω, u = 0 on ∂Ω, where Ω is a bounded smooth domain in ℝN, λ is near the first eigenvalue and h(x) is orthogonal to the first eigenfunction. We give some conditions of existence of positive solutions and of multiple solutions in terms of the primitive of f with respect to u.

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