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

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.

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 → ∞.

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+.

Multiplicity results for nonlinear elliptic equations involving critical Sobolev exponent

1986 ◽  
Vol 103 (3-4) ◽  
pp. 275-285 ◽  
Author(s):  
A. Capozzi ◽  
G. Palmieri
Keyword(s):  
Elliptic Equations ◽  
Multiple Solutions ◽  
Nonlinear Elliptic Equations ◽  
Critical Sobolev Exponent ◽  
Sobolev Embedding ◽  
Multiplicity Results ◽  
Variational Techniques ◽  
Nonlinear Elliptic ◽  
Sobolev Exponent ◽  
Image Position

SynopsisIn this paper we study the following boundary value problemwhere Ω is a bounded domain in Rn, n≧3, x ∈Rn, p* = 2n/(n – 2) is the critical exponent for the Sobolev embedding is a real parameter and f(x, t) increases, at infinity, more slowly than .By using variational techniques, we prove the existence of multiple solutions to the equations (0.1), in the case when λ belongs to a suitable left neighbourhood of an arbitrary eigenvalue of −Δ, and the existence of at least one solution for any λ sufficiently large.

On inhomogeneous biharmonic equations involving critical exponents

1999 ◽  
Vol 129 (5) ◽  
pp. 925-946 ◽  
Author(s):  
Yinbin Deng ◽  
Gengsheng Wang
Keyword(s):  
Variational Principle ◽  
Boundary Value Problem ◽  
Multiple Solutions ◽  
Critical Sobolev Exponent ◽  
Mountain Pass Lemma ◽  
Biharmonic Equations ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Image Position

In this paper, we consider the existence of multiple solutions of biharmonic equations boundary value problemwhere Ω is a bounded smooth domain in ℝN, N ≥ 5; λ ∈ ℝ1 is a given constant; p = 2N/(N − 4) is the critical Sobolev exponent for the embedding ; Δ2 = ΔΔ denotes iterated N-dimensional Laplacian; f(x) is a given function. Some results on the existence and non-existence of multiple solutions for the above problem have been obtained by Ekeland's variational principle and the mountain-pass lemma under some assumptions on f(x) and N.

Positive Finite Energy Solutions of Critical Semilinear Elliptic Problems

1992 ◽  
Vol 44 (5) ◽  
pp. 1014-1029 ◽  
Author(s):  
Ezzat S. Noussair ◽  
Charles A. Swanson ◽  
Yang Jianfu
Keyword(s):  
Unbounded Domain ◽  
Smooth Boundary ◽  
Existence Theorems ◽  
Asymptotic Properties ◽  
Finite Energy ◽  
Critical Sobolev Exponent ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Sobolev Exponent ◽  
Image Position

Existence theorems and asymptotic properties will be obtained for boundary value problems of the form in an unbounded domain Ω⊆ RN(N ≥3) with smooth boundary, where Δ denotes the TV-dimensional Laplacian, τ — (N+ 2)/ (N — 2) is the critical Sobolev exponent, and is the completion of in the L2(Ω) norm of .

scholarly journals On the existence of nontrivial solutions for a fourth-order semilinear elliptic problem

2005 ◽  
Vol 2005 (6) ◽  
pp. 673-683
Author(s):  
Aixia Qian ◽  
Shujie Li
Keyword(s):  
Boundary Conditions ◽  
Nontrivial Solution ◽  
Fourth Order ◽  
Elliptic Problem ◽  
Nontrivial Solutions ◽  
Navier Boundary Conditions ◽  
Minimax Theory ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

By means of Minimax theory, we study the existence of one nontrivial solution and multiple nontrivial solutions for a fourth-order semilinear elliptic problem with Navier boundary conditions.

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