Superquadratic Condition Research Articles

Multiplicity of nontrivial solutions for Kirchhoff type equations with zero mass and a critical term

In this paper, a class of Kirchhoff type equations in ℝN (N ⩾ 3) with zero mass and a critical term is studied. Under some appropriate conditions, the existence of multiple solutions is obtained by using variational methods and a variant of Symmetric Mountain Pass theorem. The Second Concentration Compactness lemma is used to overcome the lack of compactness in critical problem. Compared to the usual Kirchhoff-type problems, we only require the nonlinearity to satisfy the classical superquadratic condition (Ambrosetti-Rabinowitz condition).

Ground states for a system of nonlinear Schrödinger equations with singular potentials

<p style='text-indent:20px;'>In this paper, we consider the existence and asymptotic behavior of ground state solutions for a class of Hamiltonian elliptic system with Hardy potential. The resulting problem engages three major difficulties: one is that the associated functional is strongly indefinite, the second difficulty we must overcome lies in verifying the link geometry and showing the boundedness of Cerami sequences when the nonlinearity is different from the usual global super-quadratic condition. The third difficulty is singular potential, which does not belong to the Kato's class. These enable us to develop a direct approach and new tricks to overcome the difficulties caused by singularity of potential and the dropping of classical super-quadratic assumption on the nonlinearity. Our approach is based on non-Nehari method which developed recently, we establish some new existence results of ground state solutions of Nehari-Pankov type under some mild conditions, and analyze asymptotical behavior of ground state solutions.</p>

Homoclinic orbits for first-order Hamiltonian system with local super-quadratic growth condition

In this paper we study the first-order Hamiltonian system Moreover precisely, assuming that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition, we obtain new existence results of ground state homoclinic orbits and infinitely many geometrically distinct homoclinic orbits by using a variational method. An interesting problem is that the nonlinearity may be super-quadratic on some domains and asymptotically quadratic on other domains under local super-quadratic condition. Since we are without more global information on the nonlinearity, we apply a perturbation approach and some special techniques in the proofs.

Multiple entire solutions of fractional Laplacian Schrödinger equations

<abstract> We consider the semi-linear fractional Schrödinger equation <p class="disp_formula">$ \begin{align*} \begin{cases} (-\Delta)^s u+V(x)u = f(x,u), \; \; \; x\in \mathbb{R}^N ,\\ u\in H^{s}(\mathbb {R}^N) , \\ \end{cases} \end{align*} $ where both $ V(x) $ and $ f(x, u) $ are periodic in $ x $, $ 0 $ belongs to a spectral gap of the operator $ (-\Delta)^s+V $ and $ f(x, u) $ is subcritical in $ u $. We obtain the existence of nontrivial solutions by using a generalized linking theorem, and based on this existence we further establish infinitely many geometrically distinct solutions. We weaken the super-quadratic condition of $ f $, which is usually assumed even in the standard Laplacian case so as to obtain the existence of solutions. </abstract>

Ground state solution of semilinear Schrödinger system with local super-quadratic conditions

In this paper, we dedicate to studying the following semilinear Schrödinger systemequation*-Δu+V1(x)u=Fu(x,u,v)amp;mboxin~RN,r-Δv+V2(x)v=Fv(x,u,v)amp;mboxin~RN,ru,v∈H1(RN),endequation*where the potentialViare periodic inx,i=1,2, the nonlinearityFis allowed super-quadratic at somex∈RNand asymptotically quadratic at the otherx∈RN. Under a local super-quadratic condition ofF, an approximation argument and variational method are used to prove the existence of Nehari–Pankov type ground state solutions and the least energy solutions.

On locally superquadratic Hamiltonian systems with periodic potential

In this paper, we study the second-order Hamiltonian systems u¨−L(t)u+∇W(t,u)=0,\\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}$$ \\ddot{u}-L(t)u+\\nabla W(t,u)=0, $$\\end{document} where tin mathbb{R}, uin mathbb{R}^{N}, L and W depend periodically on t, 0 lies in a spectral gap of the operator -d^{2}/dt^{2}+L(t) and W(t,x) is locally superquadratic. Replacing the common superquadratic condition that lim_{|x|rightarrow infty }frac{W(t,x)}{|x|^{2}}=+infty uniformly in tin mathbb{R} by the local condition that lim_{|x|rightarrow infty }frac{W(t,x)}{|x|^{2}}=+infty a.e. tin J for some open interval Jsubset mathbb{R}, we prove the existence of one nontrivial homoclinic soluiton for the above problem.

Ground states and multiple solutions for Hamiltonian elliptic system with gradient term

AbstractThis paper is concerned with the following nonlinear Hamiltonian elliptic system with gradient term−Δu+b→(x)⋅∇u+V(x)u=Hv(x,u,v)inRN,−Δv−b→(x)⋅∇v+V(x)v=Hu(x,u,v)inRN.$$\begin{array}{} \displaystyle \left\{\,\, \begin{array}{ll} -{\it\Delta} u +\vec{b}(x)\cdot \nabla u+V(x)u = H_{v}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N},\\[-0.3em] -{\it\Delta} v -\vec{b}(x)\cdot \nabla v +V(x)v = H_{u}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N}.\\ \end{array} \right. \end{array}$$Compared with some existing issues, the most interesting feature of this paper is that we assume that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition. This case allows the nonlinearity to be super-quadratic on some domains and asymptotically quadratic on other domains. Furthermore, by using variational method, we obtain new existence results of ground state solutions and infinitely many geometrically distinct solutions under local super-quadratic condition. Since we are without more global information on the nonlinearity, in the proofs we apply a perturbation approach and some special techniques.

Improved results of nontrivial solutions for a nonlinear nonhomogeneous Klein\u2013Gordon\u2013Maxwell system involving sign-changing potential

This paper is concerned with the following system: {−Δu+λA(x)u−K(x)(2ω+ϕ)ϕu=f(x,u)+h(x),x∈R3,Δϕ=K(x)(ω+ϕ)u2,x∈R3,\\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}$$ \\textstyle\\begin{cases} {-\\Delta u+\\lambda A(x) u-K(x)(2 \\omega +\\phi ) \\phi u=f(x, u)+h(x), \\quad x \\in \\mathbb{R}^{3}}, \\\\ {\\Delta \\phi =K(x)(\\omega +\\phi ) u^{2}, \\quad x \\in \\mathbb{R}^{3}}, \\end{cases} $$\\end{document} where lambda geq 1 is a parameter, omega >0 is a constant and the potential A is sign-changing. Under the classic Ambrosetti–Rabinowitz condition and other suitable conditions, nontrivial solutions are obtained via the linking theorem and Ekeland’s variational principle. Especially speaking, we use a super-quadratic condition to replace the 4-superlinear condition which is usually used to show the existence of nontrivial solutions in many references. Our results improve the previous results in the literature.

Remarks on infinitely many solutions for a class of Schr\xf6dinger equations with sign-changing potential

In this paper, we study the existence of infinitely many nontrivial solutions for the following semilinear Schrödinger equation: {−Δu+V(x)u=f(x,u),x∈RN,u∈H1(RN),\\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}$$ \\textstyle\\begin{cases} -\\Delta u+V(x)u=f(x,u), \\quad x\\in\\mathbb{R}^{N},\\\\ u\\in H^{1}(\\mathbb{R}^{N}), \\end{cases} $$\\end{document} where the potential V is continuous and is allowed to be sign-changing. By using a variant fountain theorem, we obtain the existence of infinitely many high energy solutions under the condition that the nonlinearity f(x,u) is of super-linear growth at infinity. The super-quadratic growth condition imposed on F(x,u)=int _{0}^{u}f(x,t),dt is weaker than the Ambrosetti–Rabinowitz type condition and the similar conditions employed in the references.

Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions

Consider the semilinear Schrödinger equation{−△u+V(x)u=f(x,u),x∈RN,u∈H1(RN), where both V(x) and f(x,u) are periodic in x, 0 belongs to a spectral gap of −△+V, and f(x,u) is subcritical and allowed to be super-linear at some x∈RN and asymptotically linear at the other x∈RN. In the existing works in the literature, it is commonly assumed that lim|u|→∞⁡∫0uf(x,s)dsu2=∞ uniformly in x∈RN, to obtain the existence of ground state solutions or infinitely many geometrically distinct solutions. In this paper, for the first time, we prove the existence of ground state solutions and infinitely many geometrically distinct solutions under the weaker super-quadratic condition lim|u|→∞⁡∫0uf(x,s)dsu2=∞, a.e. x∈G just for some domain G⊂RN.

Multiple solutions for Grushin operator without odd nonlinearity

We deal with existence and multiplicity results for the following nonhomogeneous and homogeneous equations, respectively: [Formula: see text] and [Formula: see text] where [Formula: see text] is the strongly degenerate operator, [Formula: see text] is allowed to be sign-changing, [Formula: see text], [Formula: see text] is a perturbation and the nonlinearity [Formula: see text] is a continuous function does not satisfy the Ambrosetti–Rabinowitz superquadratic condition ((AR) for short). First, via the mountain pass theorem and the Ekeland’s variational principle, existence of two different solutions for [Formula: see text] are obtained when [Formula: see text] satisfies superlinear growth condition. Moreover, we prove the existence of infinitely many solutions for [Formula: see text] if [Formula: see text] is odd in [Formula: see text] thanks an extension of Clark’s theorem near the origin. So, our main results considerably improve results appearing in the literature.

Nontrivial solutions for a class of Hamiltonian elliptic system with gradient term

In this paper we study the following nonlinear Hamiltonian elliptic system with gradient term −Δu+b→(x)⋅∇u+V(x)u=Hv(x,u,v)inRN,−Δv−b→(x)⋅∇v+V(x)v=Hu(x,u,v)inRN.Under a local super-quadratic condition on the nonlinearity, we obtain a new existence result of nontrivial solutions by using variational method. Here we do not need the global super-quadratic condition, and in this case our result allows the nonlinearity to be super-quadratic at some domains and asymptotically quadratic at other domains. In the proofs we apply some special techniques to demonstrate the link geometry of the energy functional.

On a nonlocal asymmetric Kirchhoff problem

This work is devoted to study the existence of nontrivial solutions to nonlocal asymmetric problems involving the [Formula: see text]-Laplacian. [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary, [Formula: see text] is a Kirchhoff function, [Formula: see text] and [Formula: see text] is of subcritical polynomial or subcritical exponential growth. Moreover, the existence of nontrivial solutions for the above problem is obtained by using variational methods combined with the Moser–Trudinger inequality. Our interest then is to study [Formula: see text] without the analogue of Ambrosetti–Rabinowitz superquadratic condition ([Formula: see text] condition for short) in the positive semi-axis. To the best of our best knowledge, our results are new even in the asymmetric Kirchhoff Laplacian and [Formula: see text]-Laplacian cases.

Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials

In this paper we study a class of weakly coupled Schrödinger system arising in several branches of sciences, such as nonlinear optics and Bose-Einstein condensates. Instead of the well known super-quadratic condition that $\lim_{|z|\to∞}\frac{F(x,z)}{|z|^2} = ∞$ uniformly in $x$, we introduce a new local super-quadratic condition that allows the nonlinearity $F$ to be super-quadratic at some $x∈ \mathbb{R}^N$ and asymptotically quadratic at other $x∈ \mathbb{R}^N$. Employing some analytical skills and using the variational method, we prove some results about the existence of ground states for the system with periodic or non-periodic potentials. In particular, any nontrivial solutions are continuous and decay to zero exponentially as $|x| \to ∞$. Our main results extend and improve some recent ones in the literature.

Multiple homoclinic solutions for a class of nonhomogeneous Hamiltonian systems

By introducing a new superquadratic condition, we obtain the existence of two nontrivial homoclinic solutions for a class of perturbed second order Hamiltonian systems which are obtained by the mountain pass theorem and Ekeland’s variational principle.

Existence of ground state solutions for quasilinear Schrödinger equations with super-quadratic condition

In this paper, we study the following quasilinear Schrödinger equation − Δ u + u − Δ ( u 2 ) u = h ( u ) , x ∈ R N , where N ≥ 3 , 2 ∗ = 2 N N − 2 , h is a continuous function. By using a change of variable, we obtain the existence of ground state solutions. Unlike the condition lim | u | → ∞ ∫ 0 u h ( s ) d s | u | 4 = ∞ , we only need to assume that lim | u | → ∞ ∫ 0 u h ( s ) d s | u | 2 = ∞ .

Local super-quadratic conditions on homoclinic solutions for a second-order Hamiltonian system

In this paper, we study the homoclinic solutions of the following second-order Hamiltonian system u ̈ − L ( t ) u + ∇ W ( t , u ) = 0 , where t ∈ R , u ∈ R N , L : R → R N × N and W : R × R N → R . Applying the Mountain Pass Theorem, we prove the existence of nontrivial homoclinic solutions under new weaker conditions. In particular, we use a local super-quadratic condition lim | x | → ∞ W ( t , x ) | x | 2 = ∞ , uniformly in t ∈ ( a , b ) for some − ∞ < a < b < + ∞ , instead of the common one lim | x | → ∞ W ( t , x ) | x | 2 = ∞ , uniformly in t ∈ R , which is essential to show the existence of nontrivial homoclinic solutions for the above system in all existing literature.

New Super-quadratic Conditions for Asymptotically Periodic Schrödinger Equations

AbstractWe study the semilinear Schrödinger equationwherefis a superlinear, subcritical nonlinearity. It focuses on the casewhereV(x) =V0(x)+V1(x),V0∊C(RN),V0(x) is 1-periodic in each of x1 , x2 , . . . , xN, supinf, and. A new super-quadratic condition is obtained that is weaker than some well-known results.

Homoclinic orbits for the second-order Hamiltonian systems

We deal with the following second-order Hamiltonian systems ü−L(t)u+∇W(t,u)=0, where L∈C(R,RN2) is a symmetric and positive define matrix for all t∈R, W∈C1(R×RN,R) and ∇W(t,u) is the gradient of W with respect to u. Under the superquadratic condition, we obtain the existence of ground state homoclinic orbits by means of the generalized Nehari manifold developed by Szulkin and Weth. Under the subquadratic condition, we employ variational techniques and the concentration-compactness principle to establish new criteria guaranteeing the multiplicity of classical homoclinic orbits. Recent results in literature are generalized and significantly 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.