Existence and multiplicity of solutions to a class of nonlinear elliptic equation with gradient terms
[Objective]The main purpose of this article is to investigate the existence and multiplicity of nontrivial solutions for nonlinear elliptic equations with gradient terms,Lu:=-Δu-1/2(x·▽u)=f(x,u),Where f ∈ C(RN× R,R),N≥3.This type of elliptic equation has wide applications in fields such as mathematics,physics,engineering,and economics.By analyzing the existence and multiplicity of solutions to such equations,we can further deepen our understanding of the essence of such equations.[Methods]To achieve our research objectives,we use the variational methods to prove our conclusions.The basic idea of the variational methods is to reduce the problem of solving nonlinear partial differential equations and operator equations to the problem of finding critical points(especially extreme points)of a generalized function in a certain function space.Variational methods have wide applications in partial differential equations,integral equations,and many mathematical and physical problems.The variational methods began with Johann Bernoulli's public proposal of the brachistochrone curve problem in June 1696.The 18th century was the pioneering period of the variational methods,which established the Euler equation that extreme values should satisfy and solved a large number of specific problems based on it.In the 19th century,the variational methods was widely applied to mathematics and physics,and sufficient conditions for extreme value functions were established.At the beginning of the 20th century,three of the 23 famous mathematical problems mentioned by Hilbert in his speech at the International Congress of Mathematicians in Paris were related to variational methods.The idea of variational methods runs through Courant and Hilbert's book"Mathematical and Physical Methods".The minimax method,represented by the Mountain Pass Theorem established by Ambresetti and Rabinowitz in the 1970s,is an important part of modern variational theory,and Morse's extensive variational methods(Morse Theory)is a hallmark of the development of variational methods in the 20th century.Specifically,we mainly applied mathematical theories such as the Linking Theorem and Fountain Theorem.The Linking Theorem is an important result in modern variational methods,which provides a sufficient condition for the existence of continuous differentiable functional(PS)sequences,and a critical point can be obtained by assuming the(PS)compactness condition.The Fountain Theorem is a special form of the Linking Theorem.The Fountain Theorem is used to prove the existence of infinite solutions for nonlinear elliptic equations.[Results]We obtain three results.Result 1:By assuming that the nonlinear term is subcritical,sublinear near the origin,and satisfies the(AR)condition,we use the Linking Theorem to prove the existence of a nontrivial solution to the equation.Result 2:Based on the assumption of Result 1,we also assume that the nonlinear term satisfies the symmetry condition,and then prove the existence of infinite solutions to the equation using the Fountain Theorem.Result 3:By assuming that the nonlinear term is subcritical and satisfies the local wrapping condition near the origin,we prove the existence of a trivial solution to the equation.[Conclusions]Nonlinear elliptic partial differential equations are not only a discipline that combines theoretical and practical applications,but also an important tool for solving mathematical and real-life problems.The boundary value problem of nonlinear elliptic partial differential equations has long been widely studied by many mathematicians.In recent years,nonlinear elliptic partial differential equations with sublinear(or singular)and superlinear terms have received extensive research.Many scholars have conducted extensive research on nonlinear elliptic partial differential equations without gradient terms.However,there is currently relatively little research on nonlinear elliptic partial differential equations with gradient terms.We demonstrate for the first time the existence of nontrivial solutions when the nonlinear term satisfies the local wrapping condition using the linking theorem,and for the first time,we obtain the existence of infinite solutions to the equation using the Fountain Theorem.
gradient termexistence of solutionlinking theoremfountain theorem