scholarly journals Multiplicity of positive solutions to semi-linear elliptic problems on metric graphs

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Masataka Shibata
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Existence Result ◽  
Elliptic Problems ◽  
Positive Parameter ◽  
Metric Graphs ◽  
Multiplicity Of Positive Solutions

<p style='text-indent:20px;'>We consider positive solutions of semi-linear elliptic equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ - \epsilon^2 u'' +u = u^p $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>on compact metric graphs, where <inline-formula><tex-math id="M1">\begin{document}$ p \in (1,\infty) $\end{document}</tex-math></inline-formula> is a given constant and <inline-formula><tex-math id="M2">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is a positive parameter. We focus on the multiplicity of positive solutions for sufficiently small <inline-formula><tex-math id="M3">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula>. For each edge of the graph, we construct a positive solution which concentrates some point on the edge if <inline-formula><tex-math id="M4">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is sufficiently small. Moreover, we give the existence result of solutions which concentrate inner vertices of the graph.</p>

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.

An existence result on positive solutions for a class of semilinear elliptic systems

2004 ◽  
Vol 134 (1) ◽  
pp. 137-141 ◽  
Author(s):  
D. D. Hai ◽  
R. Shivaji
Keyword(s):  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Existence Result ◽  
Elliptic Systems ◽  
Positive Parameter ◽  
Semilinear Elliptic Systems ◽  
Semilinear Elliptic ◽  
Sign Conditions ◽  
Image Position

Consider the system where λ is a positive parameter and Ω is a bounded domain in RN. We prove the existence of a large positive solution for λ large when limx → ∞ (f(Mg(x))/x) = 0 for every M > 0. In particular, we do not need any monotonicity assumptions on f, g, nor any sign conditions on f(0), g(0).

scholarly journals Multiplicity of Positive Solutions for Weighted Quasilinear Elliptic Equations Involving Critical Hardy-Sobolev Exponents and Concave-Convex Nonlinearities

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equations ◽  
Analysis Techniques ◽  
Multiplicity Of Positive Solutions ◽  
Quasilinear Elliptic

By variational methods and some analysis techniques, the multiplicity of positive solutions is obtained for a class of weighted quasilinear elliptic equations with critical Hardy-Sobolev exponents and concave-convex nonlinearities.

Symmetry properties of positive solutions of elliptic equations in an infinite sectorial cone

1992 ◽  
Vol 122 (1-2) ◽  
pp. 137-160
Author(s):  
Chie-Ping Chu ◽  
Hwai-Chiuan Wang
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Semilinear Elliptic Equations ◽  
Spherically Symmetric ◽  
Semilinear Elliptic ◽  
Symmetry Properties

SynopsisWe prove symmetry properties of positive solutions of semilinear elliptic equations Δu + f(u) = 0 with Neumann boundary conditions in an infinite sectorial cone. We establish that any positive solution u of such equations in an infinite sectorial cone ∑α in ℝ3 is spherically symmetric if the amplitude α of ∑α is not greater than π.

scholarly journals Existence and multiplicity results for quasilinear elliptic equations

2005 ◽  
Vol 71 (3) ◽  
pp. 377-386 ◽  
Author(s):  
Wei Dong
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equations ◽  
Laplacian Operator ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Bounded Open Subset ◽  
Multiplicity Of Positive Solutions ◽  
Quasilinear Elliptic ◽  
Laplacian Operators

The goal of this paper is to study the multiplicity of positive solutions of a class of quasilinear elliptic equations. Based on the mountain pass theorems and sub-and supersolutions argument for p-Laplacian operators, under suitable conditions on nonlinearity f(x, s), we show the follwing problem: , where Ω is a bounded open subset of RN, N ≥ 2, with smooth boundary, λ is a positive parameter and ∆p is the p-Laplacian operator with p > 1, possesses at least two positive solutions for large λ.

scholarly journals Multiple Positive Solutions for Semilinear Elliptic Equations with Sign-Changing Weight Functions in

2011 ◽  
Vol 2011 ◽  
pp. 1-16
Author(s):  
Tsing-San Hsu
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equation ◽  
Nehari Manifold ◽  
Weight Functions ◽  
Multiple Positive Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Multiplicity Of Positive Solutions ◽  
The Nehari Manifold

Existence and multiplicity of positive solutions for the following semilinear elliptic equation: in , , are established, where if if , , satisfy suitable conditions, and maybe changes sign in . The study is based on the extraction of the Palais-Smale sequences in the Nehari manifold.

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