A uniqueness result for a singular elliptic equation with gradient term

We prove the uniqueness of a solution for a problem whose simplest model iswith k ≥ 1, 0 f ∈ L∞(Ω) and Ω is a bounded domain of ℝN, N ≥ 2. So far, uniqueness results are known for k < 1, while existence holds for any k ≥ 1 and f positive in open sets compactly embedded in a neighbourhood of the boundary. We extend the uniqueness results to the k ≥ 1 case and show, with an example, that existence does not hold if f is zero near the boundary. We even deal with the uniqueness result when f is replaced by a nonlinear term λuq with 0 < q < 1 and λ > 0.

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Fundamental solutions and surface distributions

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500012785 ◽

1969 ◽

Vol 16 (3) ◽

pp. 255-257

Author(s):

R. A. Adams ◽

G. F. Roach

Keyword(s):

Elliptic Equation ◽

Boundary Value Problems ◽

Bounded Domain ◽

Elliptic Boundary ◽

Boundary Value ◽

Fundamental Solutions ◽

Second Order ◽

Elliptic Boundary Value Problems ◽

Elliptic Boundary Value ◽

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When studying the solutions of elliptic boundary value problems in a bounded, smoothly bounded domain D⊂Rn we often encounter the formulawhere u(x)∈C2(D)∩C′(D̄) is a solution of the second order self-adjoint elliptic equationand denotes differentiation along the inward normal to ∂D at x∈∂D.

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Solutions for a Class of Singular Elliptic Equation

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.694-697.2910 ◽

2013 ◽

Vol 694-697 ◽

pp. 2910-2913

Author(s):

Zong Hu Xiu

Keyword(s):

Elliptic Equation ◽

Variational Method ◽

Bounded Domain ◽

Existence Of Solutions ◽

Singular Elliptic Equation

In this paper, we consider a class of singular elliptic equation on bounded domain. By the variational method, we prove the existence of solutions.

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On the fractional Lazer-McKenna conjecture with critical growth

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2021.40 ◽

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 ◽

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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 $|\bar \nu | \to \infty$ .

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A uniqueness result for a singular nonlinear eigenvalue problem

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210512000418 ◽

2013 ◽

Vol 143 (4) ◽

pp. 739-744 ◽

Cited By ~ 4

Author(s):

Alfonso Castro ◽

Eunkyung Ko ◽

R. Shivaji

Keyword(s):

Eigenvalue Problem ◽

Boundary Value Problems ◽

Positive Solution ◽

Bounded Domain ◽

Smooth Boundary ◽

Nonlinear Eigenvalue Problem ◽

Uniqueness Result ◽

Singular Boundary ◽

Singular Boundary Value Problems ◽

Image Position

We consider the positive solutions to singular boundary-value problems of the form where λ > 0, β ∈ (0,1) and Ω is a bounded domain in ℝN, N ≥ 1, with smooth boundary ∂Ω. Here, we assume that f: [0, ∞) → (0, ∞) is a C1 non-decreasing function and f(s)/sβ is decreasing for s large. We establish the uniqueness of the positive solution when λ is large.

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Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210511001831 ◽

2013 ◽

Vol 143 (6) ◽

pp. 1185-1208 ◽

Cited By ~ 6

Author(s):

Rosaria Di Nardo ◽

Filomena Feo ◽

Olivier Guibé

Keyword(s):

Parabolic Equations ◽

Dirichlet Boundary ◽

Growth Conditions ◽

Dirichlet Boundary Conditions ◽

Parabolic Problems ◽

Renormalized Solutions ◽

Uniqueness Results ◽

Nonlinear Parabolic ◽

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Lower Order Terms

We consider a general class of parabolic equations of the typewith Dirichlet boundary conditions and with a right-hand side belonging to L1 + Lp′ (W−1, p′). Using the framework of renormalized solutions we prove uniqueness results under appropriate growth conditions and Lipschitz-type conditions on a(u, ∇u), K(u) and H(∇u).

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

Glasgow Mathematical Journal ◽

10.1017/s001708950900514x ◽

2009 ◽

Vol 51 (3) ◽

pp. 513-524 ◽

Cited By ~ 3

Author(s):

NGUYEN THANH CHUNG ◽

QUỐC ANH NGÔ

Keyword(s):

Bounded Domain ◽

Multiplicity Result ◽

Positive Parameter ◽

Existence And Multiplicity ◽

Carathéodory Function ◽

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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.

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An eigenvalue problem for generalized Laplacian in Orlicz—Sobolev spaces

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500027505 ◽

1999 ◽

Vol 129 (1) ◽

pp. 153-163 ◽

Cited By ~ 18

Author(s):

Vesa Mustonen ◽

Matti Tienari

Keyword(s):

Weak Solution ◽

Continuous Function ◽

Eigenvalue Problem ◽

Bounded Domain ◽

Sobolev Spaces ◽

Minimization Problem ◽

Lagrange Equation ◽

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Generalized Laplacian

Let m: [ 0, ∞) → [ 0, ∞) be an increasing continuous function with m(t) = 0 if and only if t = 0, m(t) → ∞ as t → ∞ and Ω C ℝN a bounded domain. In this note we show that for every r > 0 there exists a function ur solving the minimization problemwhere Moreover, the function ur is a weak solution to the corresponding Euler–Lagrange equationfor some λ > 0. We emphasize that no Δ2-condition is needed for M or M; so the associated functionals are not continuously differentiable, in general.

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Homogenization of a class of quasilinear elliptic equations in high-contrast fissured media

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500004911 ◽

2006 ◽

Vol 136 (6) ◽

pp. 1131-1155 ◽

Cited By ~ 8

Author(s):

B. Amaziane ◽

L. Pankratov ◽

A. Piatnitski

Keyword(s):

Elliptic Equation ◽

Asymptotic Behaviour ◽

Elliptic Equations ◽

Quasilinear Elliptic Equation ◽

Quasilinear Equation ◽

Quasilinear Elliptic Equations ◽

High Contrast ◽

Small Measure ◽

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Quasilinear Elliptic

The aim of the paper is to study the asymptotic behaviour of the solution of a quasilinear elliptic equation of the form with a high-contrast discontinuous coefficient aε(x), where ε is the parameter characterizing the scale of the microstucture. The coefficient aε(x) is assumed to degenerate everywhere in the domain Ω except in a thin connected microstructure of asymptotically small measure. It is shown that the asymptotical behaviour of the solution uε as ε → 0 is described by a homogenized quasilinear equation with the coefficients calculated by local energetic characteristics of the domain Ω.

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Morse index and symmetry breaking for an elliptic equation with negative exponent in expanding annuli

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210516000512 ◽

2017 ◽

Vol 147 (6) ◽

pp. 1215-1232

Author(s):

Zongming Guo ◽

Linfeng Mei ◽

Zhitao Zhang

Keyword(s):

Elliptic Equation ◽

Symmetry Breaking ◽

Morse Index ◽

Blow Up ◽

Semilinear Elliptic Equation ◽

Radial Solutions ◽

Entire Solutions ◽

Semilinear Elliptic ◽

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Bifurcation of non-radial solutions from radial solutions of a semilinear elliptic equation with negative exponent in expanding annuli of ℝ2 is studied. To obtain the main results, we use a blow-up argument via the Morse index of the regular entire solutions of the equationThe main results of this paper can be seen as applications of the results obtained recently for finite Morse index solutions of the equationwith N ⩾ 2 and p > 0.

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