Compactly supported solutions for a semilinear elliptic problem in ℝn with sign-changing function and non-Lipschitz nonlinearity

2011 ◽  
Vol 141 (1) ◽  
pp. 127-154 ◽  
Author(s):  
Qiuping Lu
Keyword(s):  
Elliptic Problem ◽  
Mountain Pass Theorem ◽  
Radial Symmetry ◽  
Connected Components ◽  
Compactly Supported ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Lipschitz Nonlinearity ◽  
Image Position ◽  
Compactly Supported Solutions

For a sign-changing function a(x) we consider the solutions of the following semilinear elliptic problem in ℝn with n ≥ 3:where γ > 0 and 0 < q < 1 < p < (n + 2)/(n − 2). Under an appropriate growth assumption on a− at infinity, we show that all solutions are compactly supported. When Ω+ = {x ∈ ℝn | a(x) > 0} has several connected components, we prove that there exists an interval on γ in which the solutions exist. In particular, if a(x) = a(|x|), by applying the mountain-pass theorem there are at least two solutions with radial symmetry that are positive in Ω+.

NONNEGATIVE SOLUTIONS FOR INDEFINITE SUBLINEAR ELLIPTIC PROBLEMS

2012 ◽  
Vol 14 (03) ◽  
pp. 1250021 ◽  
Author(s):  
FRANCISCO ODAIR DE PAIVA
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Solution Method ◽  
Mountain Pass Theorem ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Nonnegative Solutions ◽  
Carathéodory Function ◽  
Multiplicity Of Positive Solutions

This paper is devoted to the study of existence, nonexistence and multiplicity of positive solutions for the semilinear elliptic problem [Formula: see text] where Ω is a bounded domain of ℝN, λ ∈ ℝ and g(x, u) is a Carathéodory function. The obtained results apply to the following classes of nonlinearities: a(x)uq + b(x)up and c(x)(1 + u)p (0 ≤ q < 1 < p). The proofs rely on the sub-super solution method and the mountain pass theorem.

Decaying Solutions Of 2mth Order Elliptic Problems

1991 ◽  
Vol 43 (3) ◽  
pp. 449-460 ◽  
Author(s):  
W. Allegretto ◽  
L. S. Yu
Keyword(s):  
Positive Solution ◽  
Elliptic Problem ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Mountain Pass ◽  
Weighted Spaces ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Regularity Estimates ◽  
Open Question

AbstractWe consider a semilinear elliptic problem , (n > 2m). Under suitable conditions on f, we show the existence of a decaying positive solution. We do not employ radial arguments. Our main tools are weighted spaces, various applications of the Mountain Pass Theorem and LP regularity estimates of Agmon. We answer an open question of Kusano, Naito and Swanson [Canad. J. Math. 40(1988), 1281-1300] in the superlinear case: , and improve the results of Dalmasso [C. R. Acad. Sci. Paris 308(1989), 411-414] for the case .

scholarly journals Radial solutions of a semilinear elliptic problem

1991 ◽  
Vol 118 (3-4) ◽  
pp. 305-326
Author(s):  
M. A. Herrero ◽  
J. J. L. Velázquez
Keyword(s):  
Asymptotic Behaviour ◽  
Elliptic Problem ◽  
Nonnegative Function ◽  
Radial Solutions ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Almost Everywhere ◽  
Image Position

SynopsisWe analyse the set of nonnegative, global, and radial solutions (radial solutions, for short) of the equationwhere 0 < p < 1, and is a radial and almost everywhere nonnegative function. We show that radial solutions of (E) exist if f(r) = o(r2p/1−1−p) or if f(r) ≈ cr2p/1−p as r → ∞, whereWhen f(r) = c*r2p/1−p + h(r) with h(r) = o(r2p/1−p) as r → ∞, radial solutions continue to exist if h(r) is sufficiently small at infinity. Existence, however, breaks down if h(r) > 0,Whenever they exist, radial solutions are characterised in terms of their asymptotic behaviour as r → ∞.

Multiplicity of non-radial solutions of critical elliptic problems in an annulus

2005 ◽  
Vol 135 (1) ◽  
pp. 25-37 ◽  
Author(s):  
Djairo Guedes de Figueiredo ◽  
Olímpio Hiroshi Miyagaki
Keyword(s):  
Critical Points ◽  
Elliptic Problem ◽  
Elliptic Problems ◽  
Critical Sobolev Exponent ◽  
Radial Solutions ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Sobolev Exponent ◽  
Image Position

By looking for critical points of functionals defined in some subspaces of , invariant under some subgroups of O (N), we prove the existence of many positive non-radial solutions for the following semilinear elliptic problem involving critical Sobolev exponent on an annulus, where 2* − 1 := (N + 2)/(N − 2) (N ≥ 4), the domain is an annulus and f : R+ × R+ → R is a C1 function, which is a subcritical perturbation.

On a singular nonlinear semilinear elliptic problem

1998 ◽  
Vol 128 (6) ◽  
pp. 1389-1401 ◽  
Author(s):  
Junping Shi ◽  
Miaoxin Yao
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Boundary Value ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Image Position

We consider the singular boundary value problemWe study the existence, uniqueness, regularity and the dependency on parameters of the positive solutions under various assumptions.

scholarly journals Radial symmetry of large solutions of semilinear elliptic equations with convection

2014 ◽  
Vol 144 (1) ◽  
pp. 139-147
Author(s):  
Ehsan Kamalinejad ◽  
Amir Moradifam
Keyword(s):  
Elliptic Problem ◽  
Elliptic Equations ◽  
Radial Solution ◽  
Radial Symmetry ◽  
Semilinear Elliptic Equations ◽  
Rate Of Growth ◽  
Large Solutions ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

We study the radial symmetry of large solutions of the semilinear elliptic problem Δu + ∇h · ∇u = f (∣x∣, u), and we provide sharp conditions under which the problem has a radial solution. The result is independent of the rate of growth of the solution at infinity.

Critical semilinear biharmonic equations in RN

1992 ◽  
Vol 121 (1-2) ◽  
pp. 139-148 ◽  
Author(s):  
Ezzat S. Noussair ◽  
Charles A. Swanson ◽  
Yang Jianfu
Keyword(s):  
Fourth Order ◽  
Elliptic Problem ◽  
Critical Sobolev Exponent ◽  
Sobolev Embedding ◽  
Biharmonic Equations ◽  
Best Constant ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Sobolev Exponent ◽  
Image Position

SynopsisAn existence theorem is obtained for a fourth-order semilinear elliptic problem in RN involving the critical Sobolev exponent (N + 4)/(N − 4), N>4. A preliminary result is that the best constant in the Sobolev embedding L2N/(N–4) (RN) is attained by all translations and dilations of (1 + ∣x∣2)(4-N)/2. The best constant is found to be

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