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  1. Acta Mathematica Sinica, English Series
  2. Acta Mathematica Sinica, English Series : Volume 18
  3. Acta Mathematica Sinica, English Series : Volume 18, Issue 1, January 2002
  4. An Application of a Mountain Pass Theorem
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Acta Mathematica Sinica, English Series : Volume 32
Acta Mathematica Sinica, English Series : Volume 31
Acta Mathematica Sinica, English Series : Volume 30
Acta Mathematica Sinica, English Series : Volume 29
Acta Mathematica Sinica, English Series : Volume 28
Acta Mathematica Sinica, English Series : Volume 27
Acta Mathematica Sinica, English Series : Volume 26
Acta Mathematica Sinica, English Series : Volume 25
Acta Mathematica Sinica, English Series : Volume 24
Acta Mathematica Sinica, English Series : Volume 23
Acta Mathematica Sinica, English Series : Volume 22
Acta Mathematica Sinica, English Series : Volume 21
Acta Mathematica Sinica, English Series : Volume 20
Acta Mathematica Sinica, English Series : Volume 19
Acta Mathematica Sinica, English Series : Volume 18
Acta Mathematica Sinica, English Series : Volume 18, Issue 4, October 2002
Acta Mathematica Sinica, English Series : Volume 18, Issue 3, July 2002
Acta Mathematica Sinica, English Series : Volume 18, Issue 2, April 2002
Acta Mathematica Sinica, English Series : Volume 18, Issue 1, January 2002
L 1→L q Poincaré Inequalities for 0 < q < 1 Imply Representation Formulas
The p-Harmonic Heat Flow with Potential into Homogeneous Spaces
An Application of a Mountain Pass Theorem
Admissible Estimation for Finite Population When the Parameter Space is Restricted
Strongly Zero-Dimensional Locales
On the Reflectivity and Coreflectivity of L-Fuzzifying Topological Spaces in L-Topological Spaces
On States of Total Weighted Occupation Times for Superdiffusions
On the Hyper-Order of Solutions of Some Second-Order Linear Differential Equations
The Comparable Structure of Exchange Rings
New Families in the Stable Homotopy of Spheres Revisited
New Wavelet Bases and Isometric Between Symbolic Operators Spaces $$ {\text{OP}}S^{m}_{{1,\delta }} $$ and Kernel Distributions Spaces
The Existence of Entropy Solutions to Some Parabolic Problems with L 1 Data
A Construction of Characteristic Tilting Modules
Truncation Analysis for the Derivative Schrödinger Equation
Nash Inequalities for Markov Processes in Dimension One
Generalized Hopfian Property, a Minimal Haken Manifold, and Epimorphisms Between 3-Manifold Groups
The Star Chromatic Numbers of Some Planar Graphs Derived from Wheels
Simple C*-Algebras with Unique Tracial States and Quantized Topological Spaces
Bifurcations of Rough Heteroclinic Loops with Three Saddle Points
Acta Mathematica Sinica, English Series : Volume 17
Acta Mathematica Sinica, English Series : Volume 16
Acta Mathematica Sinica, English Series : Volume 15
Acta Mathematica Sinica, English Series : Volume 14
Acta Mathematica Sinica, English Series : Volume 13

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An Application of a Mountain Pass Theorem

Content Provider SpringerLink
Author Zhou, Huan Song
Copyright Year 2002
Abstract We are concerned with the following Dirichlet problem: (P) $$ \begin{array}{*{20}c} {{ - \Delta u{\left( x \right)} = f{\left( {x,u} \right)},}} & {{x \in \Omega ,}} & {{u \in H^{1}_{0} {\left( \Omega \right)},}} \\ \end{array} $$ where f(x, t) ∈C ( $$ {\bar\={\Omega }} $$ ×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, (AR) $$ 0 < \theta F{\left( {x,s} \right)} \leqslant f{\left( {x,s} \right)}s,\;{\text{for}}\;{\text{all}}\;{\left| s \right|} \geqslant M\;{\text{and}}\;x \in \Omega , $$ is no longer true, where $$ F{\left( {x,s} \right)} = {\int_0^s {f{\left( {x,t} \right)}dt} }. $$ As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.
Ending Page 36
Page Count 10
Starting Page 27
File Format PDF
ISSN 14398516
e-ISSN 14397617
Journal Acta Mathematica Sinica, English Series
Issue Number 1
Volume Number 18
Language English
Publisher Springer-Verlag
Publisher Date 2002-01-01
Publisher Place Berlin/Heidelberg
Access Restriction One Nation One Subscription (ONOS)
Subject Keyword Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation Mountain Pass Theorem Dirichlet problem Nonlinear boundary value problems for linear elliptic equations Mathematics Resonant problem Asymptotically linear
Content Type Text
Resource Type Article
Subject Applied Mathematics
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