scholarly journals ENTIRE SOLUTIONS OF SCHRÖDINGER ELLIPTIC SYSTEMS WITH DISCONTINUOUS NONLINEARITY AND SIGN‐CHANGING POTENTIAL

2006 ◽  
Vol 11 (3) ◽  
pp. 229-242 ◽  
Author(s):  
T. L. Dinu
Keyword(s):  
Blow Up ◽  
Elliptic Systems ◽  
Point Theory ◽  
Entire Solution ◽  
Mountain Pass Lemma ◽  
Discontinuous Nonlinearity ◽  
Entire Solutions ◽  
Discontinuous Nonlinearities ◽  
Locally Lipschitz ◽  
Schrödinger Systems

We establish the existence of an entire solution for a class of stationary Schrodinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow‐up at infinity. The proof is based on the critical point theory in the sense of Clarke and we apply the Mountain Pass Lemma for locally Lipschitz functionals. Our result generalizes in a nonsmooth framework the result of Rabinowitz [16] on the existence of entire solutions of the nonlinear Schrodinger equation.

scholarly journals Standing wave solutions of Schrödinger systems with discontinuous nonlinearity in anisotropic media

2006 ◽  
Vol 2006 ◽  
pp. 1-13 ◽  
Author(s):  
Teodora-Liliana Dinu
Keyword(s):  
Blow Up ◽  
Point Theory ◽  
Entire Solution ◽  
Mountain Pass Lemma ◽  
Discontinuous Nonlinearity ◽  
Entire Solutions ◽  
Discontinuous Nonlinearities ◽  
Wave Solutions ◽  
Locally Lipschitz ◽  
Schrödinger Systems

We establish the existence of an entire solution for a class of stationary Schrödinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow up at infinity. The proof is based on the critical point theory in the sense of Clarke and we apply Chang's version of the mountain pass lemma for locally Lipschitz functionals. Our result generalizes in a nonsmooth framework a result of Rabinowitz (1992) related to entire solutions of the Schrödinger equation.

scholarly journals Ground State Solutions of Nonlinear Stationary Schrödinger Systems with Discontinuous Nonlinearity and Variable Potential

2006 ◽  
Vol 13 (3) ◽  
pp. 433-445
Author(s):  
Teodora-Liliana Dinu
Keyword(s):  
Blow Up ◽  
Point Theory ◽  
Entire Solution ◽  
Mountain Pass Lemma ◽  
Entire Solutions ◽  
Discontinuous Nonlinearities ◽  
Locally Lipschitz ◽  
Schrödinger Systems ◽  
Variable Potential ◽  
Math Phys

Abstract We establish the existence of an entire solution for a class of stationary Schrödinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow-up at infinity. The proof is based on the critical point theory in the sense of Clarke and we apply the Mountain Pass Lemma for locally Lipschitz functionals. Our result generalizes in a nonsmooth framework a result of Rabinowitz [Z. Angew. Math. Phys. 43: 270–291, 1992] on the existence of entire solutions of the nonlinear Schrödinger equation.

MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH A DISCONTINUOUS NONLINEARITY

2006 ◽  
Vol 04 (01) ◽  
pp. 1-18 ◽  
Author(s):  
MICHAEL E. FILIPPAKIS ◽  
NIKOLAOS S. PAPAGEORGIOU
Keyword(s):  
Nonlinear Elliptic Equation ◽  
Point Theory ◽  
Discontinuous Nonlinearity ◽  
Locally Lipschitz Functions ◽  
Nonlinear Elliptic Problems ◽  
Nonsmooth Critical Point Theory ◽  
Nonlinear Elliptic ◽  
Nonsmooth Critical Point ◽  
Locally Lipschitz ◽  
Truncation Techniques

We consider a nonlinear elliptic equation driven by the p-Laplacian with a discontinuous nonlinearity. Such problems have a "multivalued" and a "single-valued" interpretation. We are interested in the latter and we prove the existence of at least two distinct solutions, both smooth and one strictly positive. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions, coupled with penalization and truncation techniques.

scholarly journals Multiple Solutions for Degenerate Elliptic Systems Near Resonance at Higher Eigenvalues

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Yu-Cheng An ◽  
Hong-Min Suo
Keyword(s):  
Saddle Point ◽  
Point Theorem ◽  
Blow Up ◽  
Smooth Boundary ◽  
Elliptic Systems ◽  
Point Theory ◽  
Saddle Point Theorem ◽  
Multiplicity Results ◽  
Near Resonance ◽  
Degenerate Elliptic

We study the degenerate semilinear elliptic systems of the form-div(h1(x)∇u)=λ(a(x)u+b(x)v)+Fu(x,u,v),x∈Ω,-div(h2(x)∇v)=λ(d(x)v+b(x)u)+Fv(x,u,v),x∈Ω,u|∂Ω=v|∂Ω=0, whereΩ⊂RN(N≥2)is an open bounded domain with smooth boundary∂Ω, the measurable, nonnegative diffusion coefficientsh1,h2are allowed to vanish inΩ(as well as at the boundary∂Ω) and/or to blow up inΩ¯. Some multiplicity results of solutions are obtained for the degenerate elliptic systems which are near resonance at higher eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory.

ENTIRE SOLUTIONS OF A CURVATURE FLOW IN AN UNDULATING CYLINDER

2018 ◽  
Vol 99 (1) ◽  
pp. 137-147
Author(s):  
LIXIA YUAN ◽  
BENDONG LOU
Keyword(s):  
Curvature Flow ◽  
Entire Solution ◽  
Travelling Wave ◽  
Normal Velocity ◽  
Two Dimensional ◽  
Entire Solutions ◽  
Smooth Functions ◽  
Periodic Traveling Waves ◽  
Periodic Travelling Wave ◽  
Positive Functions

We consider a curvature flow $V=\unicode[STIX]{x1D705}+A$ in a two-dimensional undulating cylinder $\unicode[STIX]{x1D6FA}$ described by $\unicode[STIX]{x1D6FA}:=\{(x,y)\in \mathbb{R}^{2}\mid -g_{1}(y)<x<g_{2}(y),y\in \mathbb{R}\}$, where $V$ is the normal velocity of a moving curve contacting the boundaries of $\unicode[STIX]{x1D6FA}$ perpendicularly, $\unicode[STIX]{x1D705}$ is its curvature, $A>0$ is a constant and $g_{1}(y),g_{2}(y)$ are positive smooth functions. If $g_{1}$ and $g_{2}$ are periodic functions and there are no stationary curves, Matano et al. [‘Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit’, Netw. Heterog. Media1 (2006), 537–568] proved the existence of a periodic travelling wave. We consider the case where $g_{1},g_{2}$ are general nonperiodic positive functions and the problem has some stationary curves. For each stationary curve $\unicode[STIX]{x1D6E4}$ unstable from above/below, we construct an entire solution growing out of it, that is, a solution curve $\unicode[STIX]{x1D6E4}_{t}$ which increases/decreases monotonically, converging to $\unicode[STIX]{x1D6E4}$ as $t\rightarrow -\infty$ and converging to another stationary curve or to $+\infty /-\infty$ as $t\rightarrow \infty$.

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

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 ◽  
Image Position

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.

scholarly journals Entire solutions of the Fisher–KPP equation on the half line

2019 ◽  
Vol 31 (3) ◽  
pp. 407-422 ◽  
Author(s):  
BENDONG LOU ◽  
JUNFAN LU ◽  
YOSHIHISA MORITA
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Traveling Wave ◽  
Wave Solution ◽  
Dirichlet Boundary ◽  
Entire Solution ◽  
Traveling Wave Solution ◽  
Half Line ◽  
Entire Solutions ◽  
Kpp Equation

In this paper, we study the entire solutions of the Fisher–KPP (Kolmogorov–Petrovsky–Piskunov) equation ut = uxx + f(u) on the half line [0, ∞) with Dirichlet boundary condition at x = 0. (1) For any $c \ge 2\sqrt {f'(0)} $, we show the existence of an entire solution ${{\cal U}^c}(x,t)$ which connects the traveling wave solution φc(x + ct) at t = −∞ and the unique positive stationary solution V(x) at t = +∞; (2) We also construct an entire solution ${{\cal U}}(x,t)$ which connects the solution of ηt = f(η) at t = −∞ and V(x) at t = +∞.

scholarly journals A Class of functional equations which have entire solutions

1988 ◽  
Vol 38 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Peter L. Walker
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Wide Class ◽  
Functional Equations ◽  
Entire Functions ◽  
Inverse Function ◽  
Entire Solution ◽  
Entire Solutions ◽  
Image Position

We consider the Abelian functional equationwhere φ is a given entire function and g is to be found. The inverse function f = g−1 (if one exists) must satisfyWe show that for a wide class of entire functions, which includes φ(z) = ez − 1, the latter equation has a non-constant entire solution.

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