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$h$.     

Existence of periodic solutions for Hamiltonian systems with super-linear and sign-changing nonlinearities

In this paper, we consider the existence of periodic solutions for the super quadratic second order Hamiltonian system, and primitive functions of nonlinearities are allowed to be sign-changing. By using some weaker conditions, our result extends and improves some existed results in the literature.     

Symmetry results for systems involving fractional Laplacian

In this paper we investigate symmetry results for positive solutions of systems involving the fractional Laplacian (1) $\left\{ \begin{gathered} ( - \Delta )^{\alpha _1 } u_1 (x) = f_1 (u_2 (x)),x \in \mathbb{R}^\mathbb{N} , \hfill \\ ( - \Delta )^{\alpha _2 } u_2 (x) = f_2 (u_1 (x)),x \in \mathbb{R}^\mathbb{N} , \hfill \\ \lim _{|x| \to \infty } u_1 (x) = \lim _{|x| \to \infty } u_2 (x) = 0 \hfill \\ \end{gathered} \right. $ where N ≥ 2 and α ₁, α ₂ ∈ (0, 1). We prove symmetry properties by the method of moving planes.     

Existence results for asymmetric fractional ‐Laplacian problem

We investigated a class of quasi‐linear nonlocal problems with a right‐hand side nonlinearity which exhibits an asymmetric growth at and . Namely, it is linear at and superlinear at . However, it needs not satisfy the Ambrosetti–Rabinowitz condition on the positive semiaxis. Some existence results for nontrivial solution are established by using variational methods combined with the Moser–Trudinger inequality.     

Existence and location results for sign-changing solutions for three-point boundary value problems using Leray-Schauder degree

In this paper, by using the modification function technique and the Leray-Schauder degree method some existence and multiplicity results for sign- changing solutions of certain three-point boundary value problems are obtained. The main result of this paper is an improvement on a former existence result for sign-changing solutions of certain three-point boundary value problems.     

Existence and location results for sign-changing solutions for three-point boundary value problems using Leray-Schauder degree

In this paper, by using the modification function technique and the Leray-Schauder degree method some existence and multiplicity results for sign- changing solutions of certain three-point boundary value problems are obtained. The main result of this paper is an improvement on a former existence result for sign-changing solutions of certain three-point boundary value problems.     

Existence and multiplicity results for classes of elliptic resonant problems

In this paper we prove some new existence results of nontrivial solutions for classes of elliptic resonant problems. We also establish several multiplicity results. The methods used here are based on combining the minimax methods and the Morse theory especially some new observations on the critical groups of a local linking-type degenerate critical point.     

Existence and multiplicity of solutions for nonlocal systems involving fractional Laplacian with non-differentiable terms

In this paper, we devote to the study of the existence and multiplicity of solutions of nonlocal systems involving fractional Laplacian with non-differentiable terms using some extended critical point theorems for locally Lipschitz function on product spaces.     

Existence of multiple positive solutions for semilinear elliptic systems involving m critical hardy-sobolev exponents and m sign-changing weight function

In this article, we consider a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical Hardy-Sobolev exponent and one sign-changing function. The existence and multiplicity results of positive solutions are obtained by variational methods.     

Existence results for critical singular elliptic systems

In this paper, we consider semilinear elliptic systems with both singular and critical growth terms in bounded domains. The existence of a nontrivial solution is obtained by variational methods.     

Existence results for systems of vector variational-like inequalities

The purpose of this paper is to introduce and study systems of vector variational-like inequalities in Banach spaces. Under certain conditions, some existence results for systems of vector variational-like inequalities in Banach spaces are obtained by Kakutani–Fan–Glicksberg fixed point theorem.     

Existence of sign-changing solutions for semilinear singular elliptic equations with critical Sobolev exponents

We consider the semilinear elliptic problem in Ω, u=0 on ∂Ω, where 0∈Ω is a smooth bounded domain in RN, N?4, , is the critical Sobolev exponent, f(x,⋅) has subcritical growth at infinity, K(x)>0 is continuous. We prove the existence of sign-changing solutions under different assumptions when Ω is a usual domain and a symmetric domain, respectively.     

Existence and symmetry results for competing variational systems

In this paper we consider a class of gradient systems of type $$\begin{array}{ll} -c_{i} \Delta u_{i} + V_{i}(x)u_{i} = P_{u_i}(u),\qquad u_{1}, \ldots, u_{k} >\; 0\; \text{in}\; \Omega,\\ \quad u_{1} = \cdots = u_{k} = 0 \text{ on } \partial \Omega, \end{array}$$ in a bounded domain ${\Omega \subseteq \mathbb{R}^N}$ . Under suitable assumptions on V _i and P, we prove the existence of ground-state solutions for this problem. Moreover, for k = 2, assuming that the domain Ω and the potentials V _i are radially symmetric, we prove that the ground state solutions are foliated Schwarz symmetric with respect to antipodal points. We provide several examples for our abstract framework.     

Existence and multiplicity results for some elliptic systems with discontinuous nonlinearities

In this paper we complete two tasks. We first extend the critical point theorem obtained by Li and Willem [S.J. Li, M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl. 189 (1995), 6-32] to the nonsmooth case in which the energy functional is locally Lipschitz and satisfies the weaker nonsmooth Cerami condition. Then we study some semilinear elliptic systems with discontinuous nonlinearities and obtain some existence and multiplicity results.