Multiple positive solutions for a quasilinear elliptic equation with critical exponential nonlinearity

Let Ω be a bounded domain in R^N,N≥2, with C² boundary. In this work, we study the existence of multiple positive solutions of the following problem:

     

Multiple positive solutions of superlinear elliptic problems with sign-changing weight

We study the existence of multiple positive solutions for a superlinear elliptic PDE with a sign-changing weight. Our approach is variational and relies on classical critical point theory on smooth manifolds. A special care is paid to the localization of minimax critical points.     

Multiple positive solutions for semi-linear elliptic systems with sign-changing weight

In this paper, we study the multiplicity results of positive solutions for a semi-linear elliptic system involving both concave–convex and critical growth terms. With the help of the Nehari manifold and the Lusternik–Schnirelmann category, we investigate how the coefficient h(x)$h\left(x\right)$ of the critical nonlinearity affects the number of positive solutions of that problem and get a relationship between the number of positive solutions and the topology of the global maximum set of h$h$.     

Multiple solutions of an even-order nonlinear problem with convex-concave nonlinearity

We study multiple solutions of an even-order nonlinear partial differential equation with a Dirichlet boundary condition. A related class of nonlinear systems is investigated.     

Radial solutions of an elliptic equation with singular nonlinearity

For the equation

     

Homoclinic solutions for a semilinear elliptic equation with an asymptotically linear nonlinearity

We study the following semilinear elliptic equation where b is periodic and f is assumed to be asymptotically linear. The purpose of this paper is to establish the existence of infinitely many homoclinic type solutions for this class of nonlinearities.Received: 30 December 2002, Accepted: 26 August 2003, Published online: 15 October 2003Mathematics Subject Classification (2000): 35J60,35B05, 58E05     

Multiple positive solutions for a semilinear elliptic equation with critical Sobolev exponent

In this paper, we study a class of semilinear elliptic equations with Hardy potential and critical Sobolev exponent. By means of the Ekeland variational principle and Mountain Pass theorem, multiple positive solutions are obtained.     

Multiple positive and sign-changing solutions for a singular Schrödinger equation with critical growth

Variational methods are used to prove the existence of multiple positive and sign-changing solutions for a Schrödinger equation with singular potential having prescribed finitely many singular points. Some exact local behavior for positive solutions obtained here are also given. The interesting aspects are two. One is that one singular point of the potential V(x)$V(x)$ and one positive solution can produce one sign-changing solution of the problem. The other is that each sign-changing solution changes its sign exactly once.     

Multiple positive and 2-nodal symmetric solutions of elliptic problems with critical nonlinearity

We consider the problem −Δu+a(x)u=f(x)|u|^2*−2u in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN, N?4, is the critical Sobolev exponent, and a,f are continuous functions. We assume that Ω, a and f are invariant under the action of a group of orthogonal transformations. We obtain multiplicity results which contain information about the symmetry and symmetry-breaking properties of the solutions, and about their nodal domains. Our results include new multiplicity results for the Brezis-Nirenberg problem −Δu+λu=|u|^2*−2u in Ω, u=0 on ∂Ω.     

Positive solutions of a semilinear elliptic equation with singular nonlinearity

This paper is devoted to the study of positive solutions of the semilinear elliptic equation Δu+K(|x|)u−p=0, x∈Rn with n?3 and p>0. Asymptotic behaviours of sky states and uniqueness of singular sky states are obtained via invariant manifold theory of dynamical systems. The Dirichlet problem in exterior domains is also studied. It is proved that this problem has infinitely many positive solutions with fast growth.     

零点有次线性项的椭圆问题的变号解

Abstract. It is proved that the semilinear elliptic problem with zero boundary value     

Positive solutions for an elliptic equation in an annulus with a superlinear nonlinearity with zeros

We study existence, multiplicity, and the behavior, with respect to λ, of positive radially symmetric solutions of in annular domains in . The nonlinear term has a superlinear local growth at infinity, is nonnegative, and satisfies for a suitable positive and concave function a. For this, we combine several methods such as the sub and supersolutions method, a priori estimates and degree theory.     

Exact local behavior of positive solutions for a semilinear elliptic equation with Hardy term

We characterize an exact growth order near zero for positive solutions of a semilinear elliptic equation with Hardy term. This result strengthens an existence result due to E. Jannelli [The role played by space dimension in elliptic critical problems, JDE 156 (1999), 407-426].

     

Multiplicity of weak solutions to a fourth-order elliptic equation with sign-changing weight function

ABSTRACT

In this paper, a fourth-order elliptic equation with a sign-changing weight function is investigated. By the modified logarithmic Sobolev inequality and Nehari manifold it is shown that the problem admits at least two nontrivial weak solutions, which shows how the sign-changing weight function affect the existence and multiplicity of weak solutions to the problem.     

On the structure of positive solutions to an elliptic problem with a singular nonlinearity

We consider the following boundary value problem

     

Positive mountain pass solutions for a semilinear elliptic equation with a sign-changing weight function

We use the mountain pass theorem to study the existence and multiplicity of positive solutions of the generalisation of the well-known logistic equation -Δu=λg(x)u(x)(1-u(x))$-\mathrm{\Delta }u=\lambda g(x)u(x)(1-u(x))$ with Dirichlet boundary conditions to the case where g changes sign.     

Existence of positive solutions to a boundary value problem for a delayed singular high order fractional differential equation with sign-changing nonlinearity

In this paper, we discuss the existence of positive solutions to the boundary value problem for a high order fractional differential equation with delay and singularities including changing sign nonlinearity. By using the properties of the Green function, Guo-krasnosel"skii fixed point theorem, Leray-Schauder"s nonlinear alternative theorem, some existence results of positive solutions are obtained, respectively.