Superlinear Conditions Research Articles

Infinitely many solutions and least energy solutions for Klein–Gordon–Maxwell systems with general superlinear nonlinearity

This paper is concerned with the following Klein–Gordon–Maxwell system: −△u+V(x)u−(2ω+ϕ)ϕu=f(x,u),x∈R3,△ϕ=(ω+ϕ)u2,x∈R3,where ω>0 is a constant, V∈C(R3,R), f∈C(R3×R,R), and f is superlinear at infinity. Using some weaker superlinear conditions instead of the common super-cubic conditions on f, we prove that the above system has (1) infinitely many solutions when V(x) is coercive and sign-changing; (2) a least energy solution when V(x) is positive periodic. These results improve the related ones in the literature.

HOMOCLINIC SOLUTIONS FOR FOURTH ORDER DIFFERENTIAL EQUATIONS WITH SUPERLINEAR NONLINEARITIES

In this paper we investigate the existence of homoclinic solutions for a class of fourth order differential equations with superlinear nonlinearities. Under some superlinear conditions weaker than the well-known (AR) condition, by using the variant fountain theorem, we establish one new criterion to guarantee the existence of infinitely many homoclinic solutions.

Existence of multiplicity harmonic and subharmonic solutions for second‐order quasilinear equation via Poincaré‐Birkhoff twist theorem

In this paper, we investigate the existence and multiplicity of harmonic and subharmonic solutions for second‐order quasilinear equation urn:x-wiley:mma:media:mma4494:mma4494-math-0001 where , g satisfies the superlinear condition at infinity. We prove that the given equation possesses harmonic and subharmonic solutions by using the phase‐plane analysis methods and a generalized version of the Poincaré‐Birkhoff twist theorem.

Mountain pass and linking type sign-changing solutions for nonlinear problems involving the fractional Laplacian

In this paper we study the existence of sign-changing solutions for nonlinear problems involving the fractional Laplacian \t\t\t0.1{(−Δ)su−λu=f(x,u),x∈Ω,u=0,x∈Rn∖Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\left \\{ \\textstyle\\begin{array}{l@{\\quad}l} (-\\Delta)^{s}u-\\lambda u=f(x,u),&x\\in\\Omega, \\\\ u=0,&x\\in\\mathbb{R}^{n}\\setminus\\Omega, \\end{array}\\displaystyle \\right . $$\\end{document} where Omegasubsetmathbb{R}^{n} (ngeq2) is a bounded smooth domain, sin(0,1), (-Delta)^{s} denotes the fractional Laplacian, λ is a real parameter, the nonlinear term f satisfies superlinear and subcritical growth conditions at zero and at infinity. When lambdaleq0, we prove the existence of a positive solution, a negative solution and a sign-changing solution by combing minimax method with invariant sets of descending flow. When lambdageq lambda_{1}^{s} (where lambda_{1}^{s} denotes the first eigenvalue of the operator (-Delta)^{s} in Ω with homogeneous Dirichlet boundary data), we prove the existence of a sign-changing solution by using a variation of linking type theorems.

Existence results of solitons in discrete non-linear Schrödinger equations

The discrete non-linear Schrödinger equation is one of the most important inherently discrete models, having a crucial role in the modelling of a great variety of phenomena, ranging from solid-state and condensed-matter physics to biology. In this paper, a class of discrete non-linear Schrödinger equations are considered. Using critical point theory in combination with periodic approximations, we establish some new sufficient conditions on the existence results for solitons of the equation. The classical Ambrosetti–Rabinowitz superlinear condition is improved.

Existence of time homoclinic solutions for a class of discrete wave equations

In this paper, a class of nonperiodic discrete wave equations with Dirichlet boundary conditions are obtained by using the center-difference method. It is a strongly indefinite discrete Hamiltonian system. By using a variant and generalized weak linking theorem, the existence of the nontrivial time homoclinic solutions for the system will be obtained. The obtained main results here allow the classical Ambrosetti-Rabinowitz superlinear condition to be replaced by a general superquadratic condition. Such a method cannot be used for the corresponding continuous wave equations, however, it is valid for some general discrete Hamiltonian systems. Similarly, the existence of homoclinic periodic solutions can also be considered.

Existence of breathers for discrete nonlinear Schrödinger equations

In this paper, we obtain a new sufficient condition on the existence of breathers for the discrete nonlinear Schrödinger equations by using critical point theory in combination with periodic approximations. The classical Ambrosetti–Rabinowitz superlinear condition is improved.

Positive doubly periodic solutions of telegraph equations with delays

This paper deals with the existence of positive doubly periodic solutions for the nonlinear telegraph equation with delays $\mathcal{L}u=f( t, x, u(t-\tau_{1}, x),\ldots,u(t-\tau_{n}, x) )$ , $(t, x)\in\mathbb{R}^{2}$ , where $\mathcal{L}u:=u_{tt}-u_{xx}+c u_{t}+a(t, x) u $ is a linear telegraph operator acting on function $u: \mathbb{R}^{2}\to \mathbb{R}$ , $c>0$ is a constant, $a\in C(\mathbb{R}^{2}, (0, \infty))$ is 2π-periodic in t and x, $f\in C(\mathbb{R}^{2}\times[0, \infty)^{n}, [0, \infty))$ is 2π-periodic in t and x, and $\tau_{1}, \ldots, \tau_{n}\in[0, \infty)$ are constants. Some existence results of positive doubly 2π-periodic weak solutions are obtained under that $f(t, x, \eta_{1}, \ldots, \eta_{n})$ satisfies some superlinear or sublinear growth conditions on $\eta_{1}, \ldots, \eta_{n}$ . The discussion is based on the fixed point index theory in cones.

A class of stochastic differential equations with super-linear growth and non-Lipschitz coefficients

The purpose of this paper is to study some properties of solutions to one-dimensional as well as multidimensional stochastic differential equations (SDEs in short) with super-linear growth and non-Lipschitz conditions on the coefficients. Taking inspiration from [K. Bahlali, E.H. Essaky, M. Hassani, and E. Pardoux Existence, uniqueness and stability of backward stochastic differential equation with locally monotone coefficient, C.R.A.S. Paris. 335(9) (2002), pp. 757–762; K. Bahlali, E. H. Essaky, and H. Hassani, Multidimensional BSDEs with super-linear growth coefficients: Application to degenerate systems of semilinear PDEs, C. R. Acad. Sci. Paris, Ser. I. 348 (2010), pp. 677-682; K. Bahlali, E. H. Essaky, and H. Hassani, p-Integrable solutions to multidimensional BSDEs and degenerate systems of PDEs with logarithmic nonlinearities, (2010). Available at arXiv:1007.2388v1 [math.PR]], we introduce a new local condition which ensures the pathwise uniqueness, as well as the non-contact property. We moreover show that the solution produces a stochastic flow of continuous maps and satisfies a large deviations principle of Freidlin–Wentzell type. Our conditions on the coefficients go beyond the existing ones in the literature. For instance, the coefficients are not assumed uniformly continuous and therefore cannot satisfy the classical Osgood condition. The drift coefficient could not be locally monotone and the diffusion is neither locally Lipschitz nor uniformly elliptic. Our conditions on the coefficients are, in some sense, near the best possible. Our results are sharp and mainly based on Gronwall lemma and the localization of the time parameter in concatenated intervals.

Degenerate Kirchhoff problems involving the fractional p-Laplacian without the (AR) condition

In this paper, we use a variant of the mountain pass theorem to investigate the existence of nontrivial solutions for a degenerate Kirchhoff type problem driven by a nonlocal fractional -Laplace operator with homogeneous Dirichlet boundary conditions:where , is a smooth bounded domain of , is the fractional -Laplace operator with and , and is a Carathéodory function. The main feature of this paper is the fact that the coefficient of fractional Laplace operator can be zero at zero, that is, the problem is degenerate. The novelty here is that we may consider nonlinearities that satisfy a local -superlinear condition and may change sign.

Existence of Ground State Solutions for Hamiltonian Elliptic Systems with Gradient Terms

Abstract In this paper we consider the following Hamiltonian elliptic terns in RN: [XXX] Where V(x) > 0 is a periodic continuous real Function, b̅(x) = (b1....bN) € C1(RN,RN) satisfies the gauge condition div b̅(x) =0, g(x,v), f(x,u) are super- linear at infinity. We establish the existence of ground state solutions without the classical Ambrosetti-Rabinowitz superlinear condition.

On quasilinear elliptic problems without the Ambrosetti–Rabinowitz condition

Using variational arguments, we establish the existence of multiple solutions for quasilinear elliptic problems driven by the Φ-Laplacian operator. A major point is that we ensure compactness without the well-known Ambrosetti–Rabinowitz superlinearity condition.

Existence of Solutions for a Class of Quasilinear Parabolic Equations with Superlinear Nonlinearities

Working in a weighted Sobolev space, this paper is devoted to the study of the boundary value problem for the quasilinear parabolic equations with superlinear growth conditions in a domain of RN. Some conditions which guarantee the solvability of the problem are given.

Infinitely many standing wave solutions for the nonlinear Klein–Gordon–Maxwell system with sign-changing potential

In this paper, we study the infinitely many solutions for the nonlinear Klein–Gordon–Maxwell system {−Δu+V(x)u−(2ω+ϕ)ϕu=f(x,u),x∈R3,Δϕ=(ω+ϕ)u2,x∈R3, where ω>0 is a constant, u, ϕ:R3→R, the potential V(x) is allowed to be sign-changing, and the primitive of the nonlinearity f is of super-linear growth near infinity. Our super-linear conditions are weaker than the usual super-linear conditions.

On the Palais–Smale condition

We present a new sufficient assumption weaker than the classical Ambrosetti–Rabinowitz condition which guarantees the boundedness of (PS) sequences. Moreover, we relax the standard subcritical polynomial growth condition ensuring the compactness of a bounded (PS) sequences. We also revise the Costa–Magalhaes condition [8] to obtain Cerami condition. As a consequence, some existence results derived by minimax methods were proved. Finally, we establish the existence of positive solution under the subcritical polynomial growth condition, while the strong superlinear condition is only required along an unbounded sequence. In other words, a certain degraded oscillation is allowed.

Infinitely many homoclinic solutions for a nonperiodic fourth-order differential equation without (AR)-condition

In this paper, we study the existence of infinitely homoclinic solutions for a nonperiodic fourth-order differential equation under the superlinear condition, but not assuming the well-known (AR)-condition. The proof is based on variational methods.

Infinitely Many Nontrivial Solutions of Resonant Cooperative Elliptic Systems with Superlinear Terms

We study a class of resonant cooperative elliptic systems and replace the Ambrosetti-Rabinowitz superlinear condition with general superlinear conditions. We obtain ground state solutions and infinitely many nontrivial solutions of this system by a generalized Nehari manifold method developed recently by Szulkin and Weth.

Homoclinic Solutions of a Class of Nonperiodic Discrete Nonlinear Systems in Infinite Higher Dimensional Lattices

By using critical point theory, we obtain a new sufficient condition on the existence of homoclinic solutions of a class of nonperiodic discrete nonlinear systems in infinite lattices. The classical Ambrosetti-Rabinowitz superlinear condition is improved by a general superlinear one. Some results in the literature are improved.

Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity

In this paper, we consider the existence of homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity. The classical Ambrosetti-Rabinowitz superlinear condition is improved by a general superlinear one. The proof is based on the critical point theory in combination with periodic approximations of solutions.

On a class of elliptic problems with indefinite nonlinearities

In this paper we apply variational and sub-supersolution methods to study the existence and multiplicity of nonnegative solutions for a class of indefinite semilinear elliptic problems that depend on a parameter. The results on the existence of solutions do not impose any growth condition at infinity on the term which depends on the parameter. To derive such results, first we find a positive supersolution by solving an auxiliary problem. Then we use a truncation argument and a global minimization method. The main hypothesis for the existence of two nonzero solutions is that the indefinite term is the product of a weight function, having a thick zero set, and a nonlinear function which satisfies the Ambrosetti–Rabinowitz superlinear condition. Results for some corresponding indefinite problems are also established.