scholarly journals Multiplicity of solutions for singular semilinear elliptic equations with critical Hardy-Sobolev exponents

2008 ◽  
Vol 2 (2) ◽  
pp. 158-174 ◽  
Author(s):  
Qianqiao Guo ◽  
Pengcheng Niu ◽  
Jingbo Dou
Keyword(s):  
Boundary Condition ◽  
Variational Methods ◽  
Elliptic Problem ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

We consider the semilinear elliptic problem with critical Hardy-Sobolev exponents and Dirichlet boundary condition. By using variational methods we obtain the existence and multiplicity of nontrivial solutions and improve the former results.

Semilinear Robin problems resonant at both zero and infinity

2018 ◽  
Vol 30 (1) ◽  
pp. 237-251
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Boundary Condition ◽  
Variational Methods ◽  
Elliptic Problem ◽  
Robin Boundary Condition ◽  
Critical Groups ◽  
Smooth Solutions ◽  
Reaction Term ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Robin Boundary

Abstract We consider a semilinear elliptic problem, driven by the Laplacian with Robin boundary condition. We consider a reaction term which is resonant at {\pm\infty} and at 0. Using variational methods and critical groups, we show that under resonance conditions at {\pm\infty} and at zero the problem has at least two nontrivial smooth solutions.

SINGULAR LIMITS FOR 2-DIMENSIONAL ELLIPTIC PROBLEM WITH EXPONENTIALLY DOMINATED NONLINEARITY AND SINGULAR DATA

2011 ◽  
Vol 13 (04) ◽  
pp. 697-725 ◽  
Author(s):  
SAMI BARAKET ◽  
INES BEN OMRANE ◽  
TAIEB OUNI ◽  
NIHED TRABELSI
Keyword(s):  
Boundary Condition ◽  
Elliptic Problem ◽  
Source Term ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Singular Limits ◽  
Singular Source ◽  
Singular Data

We study existence of solutions with singular limits for a 2-dimensional semilinear elliptic problem with exponential dominated nonlinearity and a singular source term given by Dirac masses, imposing Dirichlet boundary condition. This paper extends previous results obtained in [3, 8]. We mainly use the method of domain decomposition.

scholarly journals On some classes of generalized Schrödinger equations

2020 ◽  
Vol 10 (1) ◽  
pp. 522-533
Author(s):  
Amanda S. S. Correa Leão ◽  
Joelma Morbach ◽  
Andrelino V. Santos ◽  
João R. Santos Júnior
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Stationary Problems

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim Vλi pairs of nontrivial solutions if a parameter involved in the equation is large enough, where Vλi denotes the eigenspace associated to the i-th eigenvalue λi of laplacian operator with homogeneous Dirichlet boundary condition.

scholarly journals On a class of semilinear elliptic equations with boundary conditions and potentials which change sign

2005 ◽  
Vol 2005 (2) ◽  
pp. 95-104
Author(s):  
M. Ouanan ◽  
A. Touzani
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Elliptic Equations ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Semilinear Elliptic ◽  
Superlinear Growth ◽  
Bounded Smooth

We study the existence of nontrivial solutions for the problemΔu=u, in a bounded smooth domainΩ⊂ℝℕ, with a semilinear boundary condition given by∂u/∂ν=λu−W(x)g(u), on the boundary of the domain, whereWis a potential changing sign,ghas a superlinear growth condition, and the parameterλ∈]0,λ1];λ1is the first eigenvalue of the Steklov problem. The proofs are based on the variational and min-max methods.

scholarly journals Positive Solutions for Elliptic Problems with the Nonlinearity Containing Singularity and Hardy-Sobolev Exponents

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yong-Yi Lan ◽  
Xian Hu ◽  
Bi-Yun Tang
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Elliptic Problems ◽  
Semilinear Elliptic Equations ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Multiplicity Of Positive Solutions

In this paper, we study multiplicity of positive solutions for a class of semilinear elliptic equations with the nonlinearity containing singularity and Hardy-Sobolev exponents. Using variational methods, we establish the existence and multiplicity of positive solutions for the problem.

Pointwise Lower Bounds for Solutions of Semilinear Elliptic Equations and Applications

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Asadollah Aghajani ◽  
Alireza Mosleh Tehrani ◽  
Nassif Ghoussoub
Keyword(s):  
Bounded Domain ◽  
Lower Bounds ◽  
Elliptic Problem ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Smooth Bounded Domain ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

AbstractWe consider the semilinear elliptic problem −Δu = f (x, u), posed in a smooth bounded domain Ω of ℝ

Existence and Regularizing Effect of Degenerate Lower Order Terms in Elliptic Equations Beyond the Hardy Constant

2018 ◽  
Vol 18 (4) ◽  
pp. 775-783 ◽  
Author(s):  
David Arcoya ◽  
Alexis Molino ◽  
Lourdes Moreno-Mérida
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Lower Order ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Model Problem ◽  
Elliptic Problems ◽  
Hardy Potential ◽  
Lower Order Terms

AbstractIn this paper, we study the regularizing effect of lower order terms in elliptic problems involving a Hardy potential. Concretely, our model problem is the differential equation-\Delta u+h(x)|u|^{p-1}u=\lambda\frac{u}{|x|^{2}}+f(x)\quad\text{in }\Omega,with Dirichlet boundary condition on {\partial\Omega}, where {p>1} and {f\in L^{m}_{h}(\Omega)} (i.e. {|f|^{m}h\in L^{1}(\Omega)}) with {m\geq\frac{p+1}{p}}. We prove that there is a solution of the above problem even for λ greater than the Hardy constant; i.e., {\lambda\geq\mathcal{H}=\frac{(N-2)^{2}}{4}} and nonnegative functions {h\in L^{1}(\Omega)} which could vanish in a subset of Ω. Moreover, we show that all the solutions are in {L^{pm}_{h}(\Omega)}. These results improve and generalize the case {h(x)\equiv h_{0}} treated in [2, 10].

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