Variant Fountain Theorem Research Articles

A class of double phase problem without Ambrosetti-Rabinowitz-type growth condition: infinitely many solutions

This paper concerns with a class of double phase problem without Ambrosetti-Rabinowitz-type growth condition. Under reasonable hypotheses, we establish the existence of infinitely many solutions by using the variant fountain theorems due to Zou \cite{71}.

On the High Energy Solitary Waves Solutions for a Generalized KP Equation in Bounded Domain

In this paper, we are mainly concerned with the existence of infinitely many high energy solitary waves solutions for a class of generalized Kadomtsev Petviashvili equation (KP equation) in bounded domain. The aim of this paper is to fill the gap in the relevant literature stated in a previous paper ( Xu, J., Wei, Z., Ding, Y.: Stationary solutions for a generalized Kadomtsev-Petviashvili equation in bounded domain. Electronic Journal of Qualitative Theory of Differential Equations. (2012)(68), 1-18 (2012)). Under more relaxed assumption on the nonlinearity involved in KP equation, we obtain a new result on the existence of infinitely many high energy solitary waves solutions via a variant fountain theorems.

SOLVABILITY OF STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS FOR A CLASS OF FRACTIONAL ADVECTION-DISPERSION EQUATIONS THROUGH VARIATIONAL APPROACH

In this paper, we probe into the solvability of Sturm-Liouville problem for fractional advection-dispersion equations without traditional Ambrosetti-Rabinowitz conditions. Some existence results of infinitely many small negative energy and large energy solutions are obtained by employing variant fountain theorems. The nonlinearity $f$ and $l_i$ ($i$=1,2,..., $m$) are considered under certain appropriate assumptions which are distinct from those assumed in previous articles. In addition, the main result is confirmed by an example which is provided.

Multiplicity of solutions for a fractional Schrödinger-Poisson system without (PS) condition

<abstract> In this paper, we study the following fractional Schrödinger-Poisson system <p class="disp_formula">$ \begin{equation*} \begin{cases} (-\Delta)^su+V(x)u+\phi u = f(x, u)&amp; x\in\mathbb{R}^3, \\ (-\Delta)^s\phi = u^2&amp; x\in\mathbb{R}^3. \end{cases} \end{equation*} $ Using the variant fountain theorem introduced by Zou ^{[<xref ref-type="bibr" rid="b32">32</xref>]}, we get the existence of infinitely many large energy solutions without the Ambrosetti-Rabinowitz's 4-superlinearity condition. Recent results from the literature are extended and improved. </abstract>

Existence of infinitely many high energy solutions for a class of fractional Schr\xf6dinger systems

In this paper, we investigate a class of nonlinear fractional Schrödinger systems {(−△)su+V(x)u=Fu(x,u,v),x∈RN,(−△)sv+V(x)v=Fv(x,u,v),x∈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}$$ \\left \\{ \\textstyle\\begin{array}{l@{\\quad}l}(-\\triangle)^{s} u +V(x)u=F_{u}(x,u,v),& x\\in \\mathbb{R}^{N}, \\\\(-\\triangle)^{s} v +V(x)v=F_{v}(x,u,v),& x\\in\\mathbb{R}^{N}, \\end{array}\\displaystyle \\right . $$\\end{document} where sin(0, 1), N>2. Under relaxed assumptions on V(x) and F(x, u, v), we show the existence of infinitely many high energy solutions to the above fractional Schrödinger systems by a variant fountain theorem.

Remarks on infinitely many solutions for a class of Schrödinger equations with sublinear nonlinearity

In this paper, we study the existence of multiple nontrivial solutions for the following semilinear Schrödinger equation: where the potential V is continuous and allowed to be sign‐changing and f(x,u) satisfies more general sublinear growth conditions than those in previous studies. The first result of this paper obtains the existence of a nontrivial solution of (1). The second establishes an existent result on infinitely many nontrivial solutions of (1) by using a variant fountain theorems.

Multiple solutions for fractional p-Laplace equation with concave-convex nonlinearities

In this paper, we investigate the existence of solutions for the fractional p-Laplace equation (−Δ)psu+V(x)|u|p−2u=h1(x)|u|q−2u+h2(x)|u|r−2uin 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}$$ (-\\Delta)_{p}^{s}u+V(x) \\vert u \\vert ^{p-2}u=h_{1}(x) \\vert u \\vert ^{q-2}u+h_{2}(x) \\vert u \\vert ^{r-2}u \\quad \\mbox{in } \\mathbb{R}^{N}, $$\\end{document} where N>sp, 0< s<1<p, 1< q< p< r< p_{s}^{*}:=frac{Np}{N-sp}, and the potential function V(x)>0 and h_{1}(x), h_{2}(x) are allowed to change sign in mathbb {R}^{N}. By using variant fountain theorem, we prove that the above equation admits infinitely many small and high energy solutions.

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.

New Homoclinic Orbits for Hamiltonian Systems with Asymptotically Quadratic Growth at Infinity

In this paper, we study the existence and multiplicity of homoclinic solutions for following Hamiltonian systems with asymptotically quadratic nonlinearities at infinity \begin{eqnarray*} \ddot{u}(t)-L(t)u+\nabla W(t,u)=0. {eqnarray*} We introduce a new coercive condition and obtain a new embedding theorem. With this theorem, we show that above systems possess at least one nontrivial homoclinic orbits by Generalized Mountain Pass Theorem. By Variant Fountain Theorem, infinitely many homoclinic orbits are obtained for above problem with symmetric condition. Our asymptotically quadratic conditions are different from previous ones in the references.

Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems

<p style='text-indent:20px;'>In the present paper, we deal with a new continuous and compact embedding theorems for the fractional Orlicz-Sobolev spaces, also, we study the existence of infinitely many nontrivial solutions for a class of non-local fractional Orlicz-Sobolev Schrödinger equations whose simplest prototype is <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (-\triangle)^{s}_{m}u+V(x)m(u) = f(x,u),\ x\in\mathbb{R}^{d}, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;s&lt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ d\geq2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ (-\triangle)^{s}_{m} $\end{document}</tex-math></inline-formula> is the fractional <inline-formula><tex-math id="M4">\begin{document}$ M $\end{document}</tex-math></inline-formula>-Laplace operator. The proof is based on the variant Fountain theorem established by Zou.

Infinitely many high energy solutions for fractional Schr\xf6dinger equations with magnetic field

In this paper we investigate the existence of infinitely many solutions for nonlocal Schrödinger equation involving a magnetic potential (−△)Asu+V(x)u=f(x,|u|)u,in 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}$$ (-\\triangle )_{A}^{s}u+V(x)u=f\\bigl(x, \\vert u \\vert \\bigr)u, \\quad\\text{in } {\\mathbb {R}}^{N}, $$\\end{document} where sin (0,1) is fixed, N>2s, V:{mathbb {R}}^{N}rightarrow {mathbb {R}}^{+} is an electric potential, the magnetic potential A:{mathbb {R}}^{N}rightarrow {mathbb {R}}^{N} is a continuous function, and (-triangle )_{A}^{s} is the fractional magnetic operator. Under suitable assumptions for the potential function V and nonlinearity f, we obtain the existence of infinitely many nontrivial high energy solutions by using the variant fountain theorem.

Existence and Multiplicity of Solutions for Sublinear Schrödinger Equations with Coercive Potentials

In this study, we consider the following sublinear Schrödinger equations−Δu+Vxu=fx,u,for x∈ℝN,ux⟶0,asu⟶∞,wherefx,usatisfies some sublinear growth conditions with respect touand is not required to be integrable with respect tox. Moreover,Vis assumed to be coercive to guarantee the compactness of the embedding from working space toLpℝNfor allp∈1,2∗. We show that the abovementioned problem admits at least one solution by using the linking theorem, and there are infinitely many solutions whenfx,uis odd inuby using the variant fountain theorem.

Infinitely many solutions for fractional Schr\xf6dinger equation with potential vanishing at infinity

The paper investigates the following fractional Schrödinger equation: \t\t\t(−Δ)su+V(x)u=K(x)f(u),x∈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} $$\\begin{aligned} (-\\Delta )^{s}u+V(x)u=K(x)f(u), \\quad x\\in \\mathbb{R}^{N}, \\end{aligned}$$ \\end{document} where 0< s<1, 2s< N, (-Delta )^{s} is the fractional Laplacian operator of order s. V(x), K(x) are nonnegative continuous functions and f(x) is a continuous function satisfying some conditions. The existence of infinitely many solutions for the above equation is presented by using a variant fountain theorem, which improves the related conclusions on this topic. The interesting result of this paper is the potential V(x) vanishing at infinity, i.e., lim_{|x|rightarrow +infty }V(x)=0.

Infinitely many solutions for impulsive fractional boundary value problem with p-Laplacian

This paper deals with the existence of infinitely many solutions for a class of impulsive fractional boundary value problems with p-Laplacian. Based on a variant fountain theorem, the existence of infinitely many nontrivial high or small energy solutions is obtained. In addition, two examples are worked out to illustrate the effectiveness of the main results.

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.

Homoclinic solutions for fractional Hamiltonian systems with indefinite conditions

In this paper we are concerned with the existence of infinitely many homoclinic solutions for the following fractional Hamiltonian systems {-tD??(-?D?tu(t))-L(t)u(t) + ?W(t,u(t)) = 0, u ? H?(R,Rn), (FHS) where ? ? (1/2,1), t ? R, u ? Rn, L ? C(R,Rn2) is a symmetric matrix for all t ? R, W ? C1(R x Rn,R) and ?W(t,u) is the gradient of W(t,u) at u. The novelty of this paper is that, when L(t) is allowed to be indefinite and W(t,u) satisfies some new superquadratic conditions, we show that (FHS) possesses infinitely many homoclinic solutions via a variant fountain theorem. Recent results are generalized and significantly improved.

Multiplicity of solutions to fractional Hamiltonian systems with impulsive effects

In this paper, we study the existence of infinitely many solutions to a class of boundary value problems for impulsive fractional Hamiltonian systems. The main tool is the use of variant Fountain theorems, which allow to give some sufficient conditions to guarantee that the impulsive problems object of our study have infinitely many solutions.

Existence of weak solutions for a class of fractional Schrödinger equations with periodic potential

We consider a time-independent fractional Schrödinger equation (−△)αu+V(x)u=f(x,u) in RN, u∈Hα(RN), where α∈(0,1), N>2α, V(x) is a periodic potential, f is superlinear and has a general subcritical growth. Based on a generalized linking theorem and a variant fountain theorem for strongly indefinite functional, we obtain a ground state solution and infinitely many solutions.

Existence and Multiplicity of Nontrivial Solutions for a Class of Semilinear Fractional Schrödinger Equations

This paper is concerned with the existence of solutions to the following fractional Schrödinger type equations: -∆su+Vxu=fx,u, x∈RN, where the primitive of the nonlinearity f is of superquadratic growth near infinity in u and the potential V is allowed to be sign-changing. By using variant Fountain theorems, a sufficient condition is obtained for the existence of infinitely many nontrivial high energy solutions.

Multiple Solutions for a Nonlinear Fractional Boundary Value Problem via Critical Point Theory

This paper is concerned with the existence of multiple solutions for the following nonlinear fractional boundary value problem: DT-αaxD0+αux=fx,ux, x∈0,T, u0=uT=0, where α∈1/2,1, ax∈L∞0,T with a0=ess infx∈0,Tax&gt;0, DT-α and D0+α stand for the left and right Riemann-Liouville fractional derivatives of order α, respectively, and f:0,T×R→R is continuous. The existence of infinitely many nontrivial high or small energy solutions is obtained by using variant fountain theorems.