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.

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.

Global gradient estimates for elliptic equations of p(x)-Laplacian type with BMO nonlinearity

2016 ◽  
Vol 2016 (715) ◽  
pp. 1-38 ◽  
Author(s):  
Sun-Sig Byun ◽  
Jihoon Ok ◽  
Seungjin Ryu
Keyword(s):  
Weak Solution ◽  
Elliptic Problem ◽  
Elliptic Equations ◽  
Growth Conditions ◽  
Divergence Form ◽  
Gradient Estimates ◽  
Nonlinear Elliptic Problem ◽  
Nonstandard Growth ◽  
Nonlinear Elliptic ◽  
Nonstandard Growth Conditions

AbstractWe consider a nonlinear elliptic problem in divergence form, with nonstandard growth conditions, on a bounded domain. We obtain the global Calderón–Zygmund type gradient estimates for the weak solution of such a problem in the setting of Lebesgue and Sobolev spaces with variable

Existence and non-existence of solutions to elliptic equations with a general convection term

2014 ◽  
Vol 144 (2) ◽  
pp. 225-239 ◽  
Author(s):  
Salomón Alarcón ◽  
Jorge García-Melián ◽  
Alexander Quaas
Keyword(s):  
Elliptic Problem ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Classical Solutions ◽  
Convection Term ◽  
Nonlinear Elliptic Problem ◽  
Smooth Bounded Domain ◽  
Nonlinear Elliptic ◽  
Comparison Arguments ◽  
Increasing Function

In this paper we consider the nonlinear elliptic problem −Δu + αu = g(∣∇u∣) + λh(x) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain of ℝN, α ≥ 0, g is an arbitrary C1 increasing function and h ∈ C1() is non-negative. We completely analyse the existence and non-existence of (positive) classical solutions in terms of the parameter λ. We show that there exist solutions for every λ when α = 0 and the integral 1/g(s)ds = ∞, or when α > 0 and the integral s/g(s)ds = ∞. Conversely, when the respective integrals converge and h is non-trivial on ∂Ω, existence depends on the size of λ. Moreover, non-existence holds for large λ. Our proofs mainly rely on comparison arguments, and on the construction of suitable supersolutions in annuli. Our results include some cases where the function g is superquadratic and existence still holds without assuming any smallness condition on λ.

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.

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.

scholarly journals On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Anna Maria Micheletti ◽  
Angela Pistoia
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Elliptic Problem ◽  
Multiplicity Result ◽  
Nonlinear Elliptic Problem ◽  
Nodal Solutions ◽  
Nonlinear Elliptic ◽  
Sign Changing Solutions

Given thatis a smooth compact and symmetric Riemannian -manifold, , we prove a multiplicity result for antisymmetric sign changing solutions of the problem in . Here if and if .

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