Nonlinear Elliptic Equations with Singular Terms and Combined Nonlinearities

We consider nonlinear elliptic Dirichlet problems with a singular term, a concave (i.e., (p − 1)-sublinear) term and a Carathéodory perturbation. We study the cases where the Carathéodory perturbation is (p − 1)-linear and (p − 1)-superlinear near +∞. Using variational techniques based on the critical point theory together with truncation arguments and the method of upper and lower solutions, we show that if the L ^∞-coefficient of the concave term is small enough, the problem has at least two nontrivial smooth solutions.     

Nonlinear Elliptic Differential Equations with Multivalued Nonlinearities

In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is de_ned on all . Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of . This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally, in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).     

Existence Results for Quasilinear Elliptic Equations with Discontinuous Nonlinearities

The nonsmooth critical point theory is applied to prove the existence of solutions and multiple solutions of a quasilinear elliptic equation with discontinuous nonlinearities.     

Existence of Nodal Solutions for Dirac Equations with Singular Nonlinearities

We prove, by a shooting method, the existence of infinitely many solutions of the form ${\psi (x^{0}, x) = {{e}}^{-i \Omega x^{0}} \chi(x)}$ of the nonlinear Dirac equation $$i \underset{\mu = 0}{\overset{3}{\sum}}\gamma^{\mu} \partial_\mu \psi-m \psi - F( \overline{\psi}\psi)\psi = 0$$ where Ω >  m >  0, χ is compactly supported and $$F(x) = \left \{\begin{array}{ll}{p|x|^{p-1}} & {\rm if} |x| > 0\\ 0 & {\rm if} x = 0 \end{array}\right.$$ with ${p \in (0,1)}$ under some restrictions on the parameters p and Ω. We study also the behavior of the solutions as p tends to zero to establish the link between these equations and the M.I.T. bag model ones.     

拟线性椭圆型方程带有混合奇异项的正整体解

在本文中,研究了方程div(｜u｜p-2u) f(x,u)=0,x∈RN,N≥3的正整体解,其中f(x,u)在u=0未假定是正则的,且f(x,u)可以同时包含超线性,亚线性项和奇异项.     

Nonlinear Elliptic Equations with Mixed Singularities

We study non-variational degenerate elliptic equations with mixed singular structures, both at the set of critical points and on the ground touching points. No boundary data are imposed and singularities occur along an a priori unknown interior region. We prove that positive solutions have a universal modulus of continuity that does not depend on their infimum value. We further obtain sharp, quantitative regularity estimates for non-negative limiting solutions.     

一种奇异临界椭圆方程的非平凡解

该文研究了一类带有Sobolev-Hardy临界指数的半线性椭圆方程,运用变分理论中的环绕定理证明了方程非平凡解的存在性     

Resonance Problems for a Class of Singular Elliptic Equations

In this article we study the resonance problems for Hardy-Sobolev operator $ L_{\mu }:= -\Delta _{p}-\fraca {(\mu }{|x|^{p})}, 0\le \mu \lt {\fraca {(N-p}{p} )}\,^{p} $ with unbounded nonlinearity. Existence of weak solutions is established using the minimax theorems in critical point theory.     

一类奇异临界椭圆方程非平凡解的存在性

研究了一类带Sobolev-Hardy临界指数的奇异椭圆方程,应用变分方法,通过能量估计和证明对应的能量泛函满足(PS)_c条件,运用山路引理得到了这类方程非平凡解的存在性.     

Boundary Value Problems with Sign Changing Nonlinearities for Second Order Singular Ordinary Differential Equations

General existence criteria, based on upper and lower solution type arguments, are presented for the Dirichlet boundary value problem y " + f ( t , y , y ') = 0, 0 < t < 1 with y (0) = y (1) = 0. Our nonlinearity may be singular in its dependent variable and is allowed to change sign.     

Multiple Positive Solutions for Semilinear Elliptic Equations Involving Subcritical Nonlinearities in RN

In this paper, we study how the shape of the graph of a（z） affects on the number of positive solutions of -△v＋μb（z）v=a（z）vp-1＋λh（z）vq-1,inRN.（0.1） We prove for large enough λ,μ〉 0, there exist at least k＋ 1 positive solutions of the this semilinear elliptic equations where 1 ≤ q 〈 2 〈 p 〈 2＊ = 2N/（N-2） forN ≥ 3.     

双曲型方程具有奇性斜导数的混合问题<英>

本文讨论了2m阶双曲型方程具有奇性斜导数的边值问题。在边界奇点(即不满足Lopatinsky边界条件的点)子流形的一定假设下,证明了所论问题在Sobolev空间H~(s,s)(Q)中解的存在性和唯一性,从而将二阶双曲方程的相应问题的已有的结果(例如[1]、[4—6])推广到了高维的情形。     

Elliptic PDEs with Fibered Nonlinearities

In ℝ ^m ×ℝ ^n−m , endowed with coordinates x=(x′,x″), we consider bounded solutions of the PDE

Weak Solutions of Nonlinear Elliptic Equations with Prescribed Singular Set

GivenΩany open and bounded subset of $\text{R}$ⁿ,n⩾4, with smooth boundary and givenΣany (n−m)-dimensional compact submanifold ofΩwithout boundary,n>m>2, we prove the existence of weak solutions to the problem[formula]which are singular onΣ, whenpis a realp>m/(m−2), close to this value.     

Positive Solutions to Boundary Value Problems for Second Order Ordinary Differential Equations with Singular Nonlinearities

This paper deals with the existence of solutions to the periodic and the Sturm-Liouville boundary value problems for second order ordinary differential equations with singular nonlinearity. The results are obtained by use of some continuation theorems in the absence of a priori bounds of Capietto-Mawhin-Zanolin.