Article
Mathematics, Applied
Yutong Chen, Jiabao Su, Mingzheng Sun, Rushun Tian
Summary: This paper studies the existence and properties of nontrivial solutions for a specific form of elliptic system equation on a bounded domain with smooth boundary.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics
Yuhua Long, Huan Zhang
Summary: This paper investigates the existence and multiplicity of nontrivial solutions for discrete elliptic Dirichlet problems using a variational technique and Morse theory. Two examples and numerical simulations are provided to illustrate the theoretical results.
ELECTRONIC RESEARCH ARCHIVE
(2022)
Article
Mathematics, Applied
Yuhua Long
Summary: This article studies discrete Kirchhoff-type problems with nonlinearity resonant at both zero and infinity. By combining variational method with Morse theory, a series of results on the existence of nontrivial solutions are established. Several examples are provided to illustrate the applications of the results.
ADVANCES IN NONLINEAR ANALYSIS
(2022)
Article
Mathematics, Applied
Mingzheng Sun, Jiabao Su, Binlin Zhang
Summary: In this paper, the Kirchhoff type equation with an additional critical nonlinear term is studied using Morse theory. The main results include computation of critical groups, including cases where zero is a mountain pass solution and the nonlinearity is resonant at zero. The estimates of the critical groups are found to be new even for Kirchhoff type equations with subcritical nonlinearities.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2021)
Article
Mathematics, Applied
Huan Zhang, Yin Zhou, Yuhua Long
Summary: In this paper, the existence and multiplicity of nontrivial solutions to a second order partial difference equation with Dirichlet boundary conditions are considered using Morse theory. Multiple results are established under suitable conditions, showing that the problem has at least two nontrivial solutions. Furthermore, five examples are provided to illustrate the applications of the theorems.
Article
Mathematics, Applied
Yutong Chen, Jiabao Su
Summary: In this paper, we prove the existence of nontrivial solutions for fractional Laplacian equations, utilizing the homotopy invariance of critical groups and the topological version of linking methods.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
(2021)
Article
Mathematics
Duong Trong Luyen, Le Thi Hong Hanh
Summary: In this paper, we study the multiplicity of weak solutions to a boundary value problem with subcritical growth and without satisfying the AR condition. We establish the existence of three nontrivial solutions using Morse theory.
BOLETIM SOCIEDADE PARANAENSE DE MATEMATICA
(2022)
Article
Mathematics, Applied
Wenguo Shen
Summary: This work investigates the existence of one-sign solutions without signum condition for a specific problem. The study considers the properties of the functions a(x) and f(u) and proves the main results using bifurcation techniques and the approximation of connected components.
Article
Mathematics, Applied
Shulin Zhang
Summary: This paper concerns the existence of nontrivial positive solutions for generalized quasilinear elliptic equations with critical growth. By applying variational methods and under certain conditions, it is proven that the equation has a nontrivial positive solution. Our results improve and supplement some existing relevant studies.
Article
Mathematics, Applied
Wan-Tong Li, Julian Lopez-Gomez, Jian-Wen Sun
Summary: This paper examines the existence of positive solutions for a semilinear elliptic problem under certain conditions, such as the special case when epsilon is equal to zero. It also investigates the sharp patterns of positive solutions as epsilon approaches zero and infinity. The study shows that the existence of sharp profiles is determined by the behavior of b(x).
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Isabella Ianni, Alberto Saldana
Summary: This study examines the asymptotic behavior of radial solutions to equations with different boundary conditions and demonstrates the lack of uniform a priori bound for nodal solutions under Neumann or Dirichlet boundary conditions. This contrasts with the existence of uniform bounds for positive solutions under certain conditions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
David Ruiz, Pieralberto Sicbaldi, Jing Wu
Summary: This paper proves the existence of nontrivial contractible domains in S-d, where a positive solution exists for an overdetermined elliptic problem. Additionally, by perturbing a small geodesic ball in S-d, these contractible domains can be obtained.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2023)
Article
Multidisciplinary Sciences
Huan Zhang, Yuhua Long
Summary: In this paper, the existence and multiplicity of nontrivial solutions for discrete elliptic Dirichlet problems with a symmetric structure are considered. By constructing a suitable variational functional and seeking nontrivial critical points, the problem is transformed into finding nontrivial solutions. A series of results are established based on the Morse theory and local linking under reasonable assumptions.
Article
Mathematics
Yutong Chen, Jiabao Su, Mingzheng Sun, Rushun Tian
Summary: This paper studies the existence of nontrivial solutions to the elliptic system by Morse theory and the Gromoll-Meyer pair, obtaining multiple nontrivial vector solutions to this system.
ACTA MATHEMATICA SCIENTIA
(2021)
Article
Mathematics, Applied
Yuhua Long
Summary: In this paper, the variational technique together with the local linking theory or the fountain theorem is applied to study a class of discrete Kirchhoff type problems with Dirichlet boundary conditions. Examples and numerical simulations are provided to illustrate the applications of the results.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2023)
