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.

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

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 → ∞.

Bifurcation results for semilinear elliptic problems in RN

2004 ◽  
Vol 134 (1) ◽  
pp. 11-32 ◽  
Author(s):  
Marino Badiale ◽  
Alessio Pomponio
Keyword(s):  
Critical Points ◽  
Elliptic Problem ◽  
Reduction Method ◽  
Elliptic Problems ◽  
Semilinear Elliptic Problem ◽  
Finite Dimensional ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems

In this paper we obtain, for a semilinear elliptic problem in RN, families of solutions bifurcating from the bottom of the spectrum of −Δ. The problem is variational in nature and we apply a nonlinear reduction method that allows us to search for solutions as critical points of suitable functionals defined on finite-dimensional manifolds.

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 Existence and bifurcation for some elliptic problems on exterior strip domains

2006 ◽  
Vol 2006 ◽  
pp. 1-26
Author(s):  
Tsing-San Hsu
Keyword(s):  
Positive Solution ◽  
Elliptic Problem ◽  
Positive Constant ◽  
Elliptic Problems ◽  
Unique Positive Solution ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Strip Domain

We consider the semilinear elliptic problem−Δu+u=λK(x)up+f(x)inΩ,u>0inΩ,u∈H01(Ω), whereλ≥0,N≥3,1<p<(N+2)/(N−2), andΩis an exterior strip domain inℝN. Under some suitable conditions onK(x)andf(x), we show that there exists a positive constantλ∗such that the above semilinear elliptic problem has at least two solutions ifλ∈(0,λ∗), a unique positive solution ifλ=λ∗, and no solution ifλ>λ∗. We also obtain some bifurcation results of the solutions atλ=λ∗.

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.

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+.

A Priori Bounds and Existence of Solutions for Slightly Superlinear Elliptic Problems

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
J. García-Melián ◽  
L. Iturriaga ◽  
H. Ramos Quoirin
Keyword(s):  
Continuous Function ◽  
Positive Solution ◽  
Critical Points ◽  
Elliptic Problem ◽  
Existence Of Solutions ◽  
A Priori ◽  
A Priori Bounds ◽  
Abstract Theorem ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

AbstractWe consider the semilinear elliptic problemwhere a is a continuous function which may change sign and f is superlinear but does not satisfy the standard Ambrosetti-Rabinowitz condition. We show that if f is regularly varying of index one at infinity then the above problem has a positive solution, provided α satisfies some additional assumptions. Our proof uses an abstract theorem due to L. Jeanjean on critical points of functionals with mountain-pass structure, and it relies on the obtention of a priori bounds for positive solutions..

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