Concave Term Research Articles

Nontrivial solutions of nonlocal fourth order elliptic equation of Kirchhoff type in [formula omitted

This paper is devoted to a class of important and general nonlocal fourth order elliptic problemΔ2u−(1+λ∫R3|∇u|2dx)Δu+V(x)u=f(x,u) in R3, where Δ2=Δ(Δ) is the bi-harmonic operator, λ≥0 is a constant. We focus on the case that f(x,u) involves a combination of convex and concave terms and the potential V(x) is allowed to be sign-changing. By new techniques, multiplicity results of two different type of solutions are established. Our results improves and generalizes that obtained in the literature.

Multiplicity of solutions for nonlinear nonhomogeneous Robin problems

We consider a nonlinear nonhomogeneous Robin problem, with an indefinite concave term near the origin and a perturbation of arbitrary growth. By modifying the perturbation and using a variant of the symmetric mountain pass theorem due to Heinz (J. Diff. Equ. 66 (1987)), we show that the problem has a whole sequence of distinct nontrivial smooth solutions converging to the trivial solution.

Infinitely many solutions for a class of quasilinear equation with a combination of convex and concave terms

We consider the following quasilinear elliptic equation with convex and concave nonlinearities: \begin{equation*} -\Delta_p u-(\Delta_pu^2)u+V(x)|u|^{p-2}u=\lambda K(x) |u|^{q-2}u+\mu g(x,u),\quad \text{in }\mathbb{R}^N, \end{equation*} where $2\leq p< N$, $1< q< p$, $\lambda,\mu\in\mathbb{R}$, $V$ and $K$ are potential functions, and $g\in C(\mathbb{R}^N\times\mathbb{R},\mathbb{R})$ is a continuous function. Under some suitable conditions on $V,K$ and $g$, the existence of infinitely many solutions is established.

Multiple nodal solutions for semilinear Robin problems with indefinite linear part and concave terms

We consider a semilinear Robin problem driven by Laplacian plus an indefinite and unbounded potential. The reaction function contains a concave term and a perturbation of arbitrary growth. Using a variant of the symmetric mountain pass theorem, we show the existence of smooth nodal solutions which converge to zero in $C^1(\overline{\Omega})$. If the coefficient of the concave term is sign changing, then again we produce a sequence of smooth solutions converging to zero in $C^1(\overline{\Omega})$, but we cannot claim that they are nodal.

Positive solutions for a Schrödinger–Poisson system involving concave–convex nonlinearities

In this paper, we study the critical growth Schrödinger–Poisson system with a concave term, and establish the existence of multiple positive solutions via using the variational method.

Stationary Schrödinger equations in ℝ2 with unbounded or vanishing potentials and involving concave nonlinearities

ABSTRACTWe study the existence and multiplicity of solutions for the following class of stationary nonlinear Schrödinger equations:where is a nonnegative parameter, V and Q are unbounded or decaying radial potentials, the nonlinearity f(s) may exhibit exponential growth and g(x, s) is a concave term. The approach used here is based on a version of the Trudinger–Moser inequality, mountain-pass theorem and the Ekeland’s variational principle in a suitable weighted Sobolev space.

Multiple Solutions to the Problem of Kirchhoff Type Involving the Critical Caffareli-Kohn-Niremberg Exponent, Concave Term and Sign-Changing Weights

In this paper, we establish the existence of at least four distinct solutions to an Kirchhoff type problems involving the critical Caffareli-Kohn-Niremberg exponent, concave term and sign-changing weights, by using the Nehari manifold and mountain pass theorem.

Nonlinear elliptic problems with superlinear reaction and parametric concave boundary condition

We study parametric nonlinear elliptic boundary value problems driven by the p-Laplacian with convex and concave terms. The convex term appears in the reaction and the concave in the boundary condition (source). We study the existence and nonexistence of positive solutions as the parameter λ > 0 varies. For the semilinear problem (p = 2), we prove a bifurcation type result. Finally, we show the existence of nodal (sign changing) solutions.

Multiple and Nodal Solutions for a Class of Nonlinear, Nonhomogeneous Periodic Eigenvalue Problems

We consider a nonlinear parametric periodic problem driven by a nonhomogeneous differential operator. We assume that the reaction is (p \(-\) 1)-superlinear near \({\pm }\infty \) (without satisfying the AR-condition) and exhibits concave terms near the origin. We show the existence of five nontrivial solutions providing sign information for all of them.

Robust Secure Beamforming in MISO Full-Duplex Two-Way Secure Communications

Considering worst-case channel uncertainties, we investigate the robust secure beamforming design problem in multiple-input-single-output (MISO) full-duplex, two-way secure communications. Our objective is to maximize worst-case sum secrecy rate under weak secrecy conditions and individual transmit power constraints. Since the objective function of the optimization problem includes both convex and concave terms, we propose to transform convex terms into linear terms. We decouple the problem into four optimization problems and employ the alternating optimization algorithm to obtain the locally optimal solution. Simulation results demonstrate that our proposed robust secure beamforming scheme outperforms the nonrobust one. It is also found that when the regions of channel uncertainties and the individual transmit power constraints are sufficiently large, because of self-interference, the proposed two-way robust secure communication is proactively degraded to one-way communication.

A Path-following Algorithm for Robust Point Matching

We propose a new point matching algorithm in this letter by minimizing a concave geometric matching cost function coming from the objective function of the robust point matching algorithm. Due to concavity of this function, naive optimization strategies such as gradient descent will fail. To address this problem, we use a path following strategy for optimization which works by adding a convex quadratic term to the objective function and then gradually transitioning from the state that there is only weight of the convex term to the state that there is only weight of the concave geometric matching cost term. Extensive experimental results demonstrate strong robustness of the method over several state-of-the-art methods and it also has good computational efficiency.

Path following algorithm for matching partially overlapping feature sets

To address feature matching problem where there is only partial overlap between the two feature sets, minimisation of an energy function consisting of feature matching costs and a concave regularisation term capable of handling partial overlaps is proposed. To improve robustness to disturbances, a path following strategy for minimisation which dynamically combines weights of the two terms is adopted. Experimental results verify better performance of the method over several state-of-the-art methods.

Nonlinear Neumann problems with indefinite potential and concave terms

In this paper we conduct a detailed study of Neumann problems driven by a nonhomogeneous differential operator plus an indefinite potential and with concave contribution in the reaction. We deal with both superlinear and sublinear (possibly resonant) problems and we produce constant sign and nodal solutions. We also examine semilinear equations resonant at higher parts of the spectrum and equations with a negative concavity.

Nonlinear Parametric Robin Problems with Combined Nonlinearities

Abstract We consider a nonlinear parametric Robin problem driven by the p-Laplacian. We assume that the reaction exhibits a concave term near the origin. First we prove a multiplicity theorem producing three solutions with sign information (positive, negative and nodal) without imposing any growth condition near ±∞ on the reaction. Then, for problems with subcritical reaction, we produce two more solutions of constant sign, for a total of five solutions. For the semilinear problem (that is, for p = 2), we generate a sixth solution but without any sign information. Our approach is variational, coupled with truncation, perturbation and comparison techniques and with Morse theory.

Combined effects of concave–convex nonlinearities and indefinite potential in some elliptic problems

We consider a nonlinear Dirichlet problem driven by the p-Laplacian and a reaction which exhibits the combined effects of concave (that is, sublinear) terms and of convex (that is, superlinear) terms. The concave term is indefinite and the convex term need not satisfy the usual in such cases Ambrosetti-Rabinowitz condition. We prove a bifurcation-type result describing the set of positive solutions as the positive parameter λ varies.

Neumann problems with indefinite and unbounded potential and concave terms

We consider a semilinear parametric Neumann problem driven by the negative Laplacian plus an indefinite and unbounded potential. The reaction is asymptotically linear and exhibits a negative concave term near the origin. Using variational methods together with truncation and perturbation techniques and critical groups, we show that for all small values of the parameter the problem has at least five nontrivial solutions, four of which have constant sign.

Four Nontrivial Solutions for Kirchhoff Problems with Critical Potential, Critical Exponent and a Concave Term

In this paper, we consider the existence of multiple solutions to the Kirchhoff problems with critical potential, critical exponent and a concave term. Our main tools are the Nehari manifold and mountain pass theorem.

Existence and Nonexistence for Elliptic Equation with Cylindrical Potentials, Subcritical Exponent and Concave Term

In this paper, we consider the existence and nonexistence of non-trivial solutions to elliptic equations with cylindrical potentials, concave term and subcritical exponent. First, we shall obtain a local minimizer by using the Ekeland’s variational principle. Secondly, we deduce a Pohozaev-type identity and obtain a nonexistence result.

On Elliptic Problem with Singular Cylindrical Potential, a Concave Term, and Critical Caffarelli-Kohn-Nirenberg Exponent

In this paper, we establish the existence of at least four distinct solutions to an elliptic problem with singular cylindrical potential, a concave term, and critical Caffarelli-Kohn-Nirenberg exponent, by using the Nehari manifold and mountain pass theorem.

Nonlinear nonhomogeneous Neumann eigenvalue problems

We consider a nonlinear parametric Neumann problem driven by a nonho- mogeneous differential operator with a reaction which is (p 1)-superlinear near ¥ and exhibits concave terms near zero. We show that for all small values of the parame- ter, the problem has at least five solutions, four of constant sign and the fifth nodal. We also show the existence of extremal constant sign solutions.