Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains

1987 ◽  
Vol 105 (1) ◽  
pp. 205-213 ◽  
Author(s):  
Donato Fortunato ◽  
Enrico Jannelli
Keyword(s):  
Boundary Value Problem ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Boundary Value ◽  
Sobolev Embedding ◽  
Infinitely Many Solutions ◽  
Positive Parameter ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic ◽  
Image Position

SynopsisWe consider the boundary value problemwhere Ω ⊂ ℝn is a bounded domain, n≧3, 2* = 2n/(n − 2) is the critical exponent for the Sobolev embedding and λ is a real positive parameter. We prove the existence of infinitely many solutions of (*) when Ω exhibits suitable symmetries.

scholarly journals EXISTENCE OF INFINITELY MANY SOLUTIONS FOR SUBLINEAR ELLIPTIC PROBLEMS

2012 ◽  
Vol 54 (3) ◽  
pp. 535-545
Author(s):  
X. ZHONG ◽  
W. ZOU
Keyword(s):  
Boundary Value Problem ◽  
Bounded Domain ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Elliptic Problems ◽  
Infinitely Many Solutions ◽  
Dirichlet Boundary Value Problem ◽  
Image Position

AbstractWe study the following nonlinear Dirichlet boundary value problem: where Ω is a bounded domain in ℝN(N ≥ 2) with a smooth boundary ∂Ω and g ∈ C(Ω × ℝ) is a function satisfying $\displaystyle \underset{|t|\rightarrow 0}{\lim}\frac{g(x, t)}{t}= \infty$ for all x ∈ Ω. Under appropriate assumptions, we prove the existence of infinitely many solutions when g(x, t) is not odd in t.

UNIQUENESS FOR SINGULAR SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS II

2015 ◽  
Vol 58 (2) ◽  
pp. 461-469 ◽  
Author(s):  
D. D. HAI ◽  
R. C. SMITH
Keyword(s):  
Boundary Value Problem ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Positive Parameter ◽  
Elliptic Boundary Value ◽  
Semilinear Elliptic ◽  
Formula Group ◽  
Image Position

AbstractWe prove uniqueness of positive solutions for the boundary value problem \begin{equation*} \left\{ \begin{array}{l} -\Delta u=\lambda f(u)\text{ in }\Omega , \\ \ \ \ \ \ \ \ u=0\text{ on }\partial \Omega , \end{array} \right. \end{equation*} where Ω is a bounded domain in $\mathbb{R}$n with smooth boundary ∂ Ω, λ is a large positive parameter, f:(0,∞) → [0,∞) is nonincreasing for large t and is allowed to be singular at 0.

On the fractional Lazer-McKenna conjecture with critical growth

2021 ◽  
pp. 1-33
Author(s):  
Qi Li ◽  
Shuangjie Peng
Keyword(s):  
Elliptic Equation ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Reduction Method ◽  
Smooth Boundary ◽  
Infinitely Many Solutions ◽  
Positive Eigenfunction ◽  
Image Position ◽  
Fractional Laplace Operator

This paper deals with the following fractional elliptic equation with critical exponent \[ \begin{cases} \displaystyle (-\Delta )^{s}u=u_{+}^{2_{s}^{*}-1}+\lambda u-\bar{\nu}\varphi_{1}, & \text{in}\ \Omega,\\ \displaystyle u=0, & \text{in}\ {{\mathfrak R}}^{N}\backslash \Omega, \end{cases}\] where $\lambda$ , $\bar {\nu }\in {{\mathfrak R}}$ , $s\in (0,1)$ , $2^{*}_{s}=({2N}/{N-2s})\,(N>2s)$ , $(-\Delta )^{s}$ is the fractional Laplace operator, $\Omega \subset {{\mathfrak R}}^{N}$ is a bounded domain with smooth boundary and $\varphi _{1}$ is the first positive eigenfunction of the fractional Laplace under the condition $u=0$ in ${{\mathfrak R}}^{N}\setminus \Omega$ . Under suitable conditions on $\lambda$ and $\bar {\nu }$ and using a Lyapunov-Schmidt reduction method, we prove the fractional version of the Lazer-McKenna conjecture which says that the equation above has infinitely many solutions as $|\bar \nu | \to \infty$ .

scholarly journals UNIQUENESS FOR SINGULAR SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS

2013 ◽  
Vol 55 (2) ◽  
pp. 399-409 ◽  
Author(s):  
D. D. HAI ◽  
R. C. SMITH
Keyword(s):  
Boundary Value Problems ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Positive Parameter ◽  
Elliptic Boundary Value ◽  
Semilinear Elliptic ◽  
Formula Group ◽  
Image Position

AbstractWe prove uniqueness of positive solutions for the boundary value problems \[ \{\begin{array}{ll} -\Delta u=\lambda f(u)\ \ &\text{in}\Omega, \ \ \ \ \ u=0 &\text{on \partial \Omega, \] where Ω is a bounded domain in ℝn with smooth boundary ∂Ω, λ is a positive parameter and f:(0,∞) → (0,∞) is sublinear at ∞ and is allowed to be singular at 0.

scholarly journals Existence and multiplicity of solutions for Kirchhof-type problems with Sobolev–Hardy critical exponent

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hongsen Fan ◽  
Zhiying Deng
Keyword(s):  
Variational Method ◽  
Boundary Value Problem ◽  
Critical Exponent ◽  
Positive Solution ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problem ◽  
Nontrivial Solutions ◽  
Elliptic Boundary Value ◽  
Existence And Multiplicity

AbstractIn this paper, we discuss a class of Kirchhof-type elliptic boundary value problem with Sobolev–Hardy critical exponent and apply the variational method to obtain one positive solution and two nontrivial solutions to the problem under certain conditions.

scholarly journals Radially symmetric solutions of a class of singular elliptic equations

1990 ◽  
Vol 33 (2) ◽  
pp. 169-180 ◽  
Author(s):  
Juan A. Gatica ◽  
Gaston E. Hernandez ◽  
P. Waltman
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Open Unit ◽  
Open Unit Ball ◽  
Singular Elliptic Equations ◽  
Existence Of Positive Solutions ◽  
Radially Symmetric Solutions ◽  
Image Position ◽  
Special Case

The boundary value problemis studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.This boundary value problem arises in the search of positive radially symmetric solutions towhere Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.

scholarly journals A MULTIPLICITY RESULT FOR A CLASS OF EQUATIONS OFp-LAPLACIAN TYPE WITH SIGN-CHANGING NONLINEARITIES

2009 ◽  
Vol 51 (3) ◽  
pp. 513-524 ◽  
Author(s):  
NGUYEN THANH CHUNG ◽  
QUỐC ANH NGÔ
Keyword(s):  
Bounded Domain ◽  
Multiplicity Result ◽  
Positive Parameter ◽  
Existence And Multiplicity ◽  
Carathéodory Function ◽  
Image Position

AbstractUsing variational arguments we study the non-existence and multiplicity of non-negative solutions for a class equations of the formwhere Ω is a bounded domain inN,N≧ 3,fis a sign-changing Carathéodory function on Ω × [0, +∞) and λ is a positive parameter.

scholarly journals POSITIVE SOLUTIONS OF A SECOND-ORDER NEUMANN BOUNDARY VALUE PROBLEM WITH A PARAMETER

2012 ◽  
Vol 86 (2) ◽  
pp. 244-253 ◽  
Author(s):  
YANG-WEN ZHANG ◽  
HONG-XU LI
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Existence And Uniqueness Theorem ◽  
Neumann Boundary Value Problem ◽  
Image Position ◽  
Continuous Dependence Of Solutions

AbstractIn this paper, we consider the Neumann boundary value problem with a parameter λ∈(0,∞): By using fixed point theorems in a cone, we obtain some existence, multiplicity and nonexistence results for positive solutions in terms of different values of λ. We also prove an existence and uniqueness theorem and show the continuous dependence of solutions on the parameter λ.

scholarly journals Positive solutions and eigenvalues of conjugate boundary value problems

1999 ◽  
Vol 42 (2) ◽  
pp. 349-374 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Martin Bohner ◽  
Patricia J. Y. Wong
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Lower Bounds ◽  
Boundary Value ◽  
Upper And Lower Bounds ◽  
Special Cases ◽  
Image Position

We consider the following boundary value problemwhere λ > 0 and 1 ≤ p ≤ n – 1 is fixed. The values of λ are characterized so that the boundary value problem has a positive solution. Further, for the case λ = 1 we offer criteria for the existence of two positive solutions of the boundary value problem. Upper and lower bounds for these positive solutions are also established for special cases. Several examples are included to dwell upon the importance of the results obtained.

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