Existence Of Ground State Solutions Research Articles

Ground state solution for a critical fractional Kirchhoff equation with L2-constraint

In this paper, we study the existence of ground state solution to the following fractional Kirchhoff equation with Sobolev critical exponent in RN:(1+b∫RN|(−Δ)s2u|2dx)(−Δ)su+λu=μ|u|p−2u+|u|2⁎−2u, under the L2-norm constraint:∫RN|u|2dx=a2, where 0<s<1 and N>2s. b,μ are positive numbers and a>0 is a prescribed constant. The existence results are obtained when 2<p<2+4sN.

Existence of ground state solution for semilinear -Laplace equation

ABSTRACT This paper explores the existence of ground state solutions for a class of semilinear -Laplace equations where is a coercive potential, is a strongly degenerate elliptic operator, f is a function with more general superlinear conditions or asymptomatic linear conditions.

Existence of ground state solutions of elliptic system in Fractional Orlicz-Sobolev Spaces

We employing a minimization arguments on appropriate Nehari manifolds, we obtain ground state solutionsfor a non-local elliptic system driven by the fractional a(.)-Laplacian operator, with Dirichlet boundaryconditions type.

Positive solutions for nonlinear schrödinger–poisson systems with general nonlinearity

In this paper, we study a class of Schrödinger-Poisson (SP) systems with general nonlinearity where the nonlinearity does not require Ambrosetti-Rabinowitz and Nehari monotonic conditions. We establish new estimates and explore the associated energy functional which is coercive and bounded below on Sobolev space. Together with Ekeland variational principle, we prove the existence of ground state solutions. Furthermore, when the ‘charge’ function is greater than a fixed positive number, the (SP) system possesses only zero solutions. In particular, when ‘charge’ function is radially symmetric, we establish the existence of three positive solutions and the symmetry breaking of ground state solutions.

Fractional double phase Robin problem involving variable‐order exponents and logarithm‐type nonlinearity

The paper deals with the logarithmic fractional equations with variable exponents where and denote the variable ‐order ‐fractional Laplace operator and the nonlocal normal ‐derivative of ‐order, respectively, with and ( ) being continuous. Here, is a bounded smooth domain with ( ) for any and are a positive parameters, and are two continuous functions, while variable exponent can be close to the critical exponent , given with and for . Precisely, we consider two cases. In the first case, we deal with subcritical nonlinearity, that is, , for any . In the second case, we study the critical exponent, namely, for some . Then, using variational methods, we prove the existence and multiplicity of solutions and existence of ground state solutions to the above problem.

Ground state solution for a class of Choquard equation with indefinite periodic potential

This paper is concerned with a class of nonlinear Choquard equation with potential. We assume the potential satisfies general indefinite periodic condition, so the Schrödinger operator has purely continuous spectrum and the associate energy functional is strongly indefinite. Applying the generalized Nehari manifold method developed by Szulkin and Weth, we prove the existence of ground state solution.

Ground state solutions for the generalized extensible beam equations

In this paper, we study the generalized extensible beam equations with steep potential Δ2u−M‖∇u‖L22Δu+λV(x)u=f(x,u)inRN,u∈H2(RN),where N≥4, M(t)=at+b with a>0,λ>0 and b∈R are parameters, V∈C(RN,R) and f∈C(RN×R,R). Unlike most other papers on this problem, we allow the parameter b to be any negative number, which has the physical significance. Under some suitable assumptions on V(x) and f(x,u), when λ is large enough, we prove the existence of ground state solutions.

Ground states for Dirac equation with singular potential and asymptotically periodic condition

In this paper we study the existence and asymptotic analysis of ground states for nonlinear Dirac equation with singular potential. Under the asymptotically periodic condition, using variational tools from non-Nehari manifold method, we establish a global compactness result and we prove that the existence of ground state solution and the continuous dependence of ground state energy about parameter as well as the asymptotic convergence of solutions.

Existence of Ground State Solutions for Generalized Quasilinear Schrödinger Equations with Asymptotically Periodic Potential

Existence of Ground State Solutions for Generalized Quasilinear Schrödinger Equations with Asymptotically Periodic Potential

Groundstates for magnetic Choquard equations with critical exponential growth

This paper is dedicated to show the existence of ground state solution for a magnetic Choquard equation with critical exponential growth. By introducing a Moser type function involving magnetic potential and applying analytical techniques, we surmount the obstacles brought from the magnetic potential which makes it a complex-valued problem and the critical exponential growth nonlinearity which makes it difficult to show the non-vanishing of Cerami sequence. Our methods can be applied to related magnetic elliptic equations.

Planar Schrödinger-Poisson system with critical exponential growth in the zero mass case

In this paper, we develop new proof techniques and analytical methods to prove the existence of ground state solutions for the following planar Schrödinger-Poisson system with zero mass{−Δu+ϕu=f(u),x∈R2,Δϕ=2πu2,x∈R2, where f∈C(R,R) has the critical exponential growth at infinity and there is no monotonicity restriction on f(u)/u3. In particular, by using delicate estimates we obtain a desired upper bound for the Mountain Pass level just with the optimal asymptotic condition κ=liminf|t|→∞t2F(t)eα0t2>0 to restore the compactness in the presence of critical exponential growth, which significantly improves analogous assumptions on asymptotic behavior of t2F(t)eα0t2 or tf(t)eα0t2 at infinity in the previous works. Moreover, we use a different approach from the one of Du and Weth (2017) [21] dealing with the power nonlinearities to establish the Pohozaev type identity, which not only allows critical exponential growth nonlinearities, but also deals with the non-autonomous case containing a linear term V(x)u in the first equation, both of which are not covered in the existing literature. To our knowledge, there has not been any work in the literature on the subject, even for the simpler equation: −Δu=f(u) in R2.

Symmetric ground state solutions for the Choquard Logarithmic equation with exponential growth

We investigate the existence of ground state solutions for the fractional Choquard Logarithmic equation (−Δ)1/2u+V(x)u+(ln|⋅|∗|u|2)u=f(u),x∈R, where V∈C(R,[0,∞)) and the f satisfies exponential critical growth. The present paper extends and complements the result of Böer and Miyagaki (0000). In particular, our paper has two typical features. Firstly, using a weaker assumption on f, we establish the energy inequality to exclude the vanishing case of the required Cerami sequence. Secondly, with the property of radial symmetry we shall use some new variational and analytic technique to establish our final result which is different to the arguments explored in Böer and Miyagaki (0000).

Ground state solutions for a Schrödinger–Poisson–Slater‐type equation with critical growth

We study the existence of ground state solutions of a Schrödinger–Poisson–Slater‐type equation with critical growth. By using the Nehari–Pohozaev manifold, we obtain the existence of ground state solutions of this system.

Ground state solution for strongly indefinite Choquard system

In this paper, we consider the following Choquard system in RN−Δu+V(x)u=2qp+q(Iα∗|u|p)|v|q−2v,−Δv+V(x)v=2pp+q(Iα∗|v|q)|u|p−2u,u(x)→0andv(x)→0as|x|→∞, where α∈(0,N) and N+αN<p,q<2∗α, in which 2∗α denotes N+αN−2 if N≥3 and 2∗α≔∞ if N=1,2. Iα is a Riesz potential. V is continuous, 1-periodic in x and 0 belongs to a spectral gap of −Δ+V. This system is subcritical in the sense of the Hardy–Littlewood–Sobolev inequality. Based on the generalized Nehari manifold, we obtain the existence of ground state solutions. Our results can be looked on as a generalization to results by Szulkin and Weth (2009).

Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells

Abstract The purpose of this paper is to investigate the ground state solutions for the following nonlinear Schrödinger equations involving the fractional p-Laplacian ( − Δ ) p s u ( x ) + λ V ( x ) u ( x ) p − 1 = u ( x ) q − 1 , u ( x ) ≥ 0 , x ∈ R N , {\left(-\Delta )}_{p}^{s}u\left(x)+\lambda V\left(x)u{\left(x)}^{p-1}=u{\left(x)}^{q-1},\hspace{1em}u\left(x)\ge 0,\hspace{0.33em}x\in {{\mathbb{R}}}^{N}, where λ &gt; 0 \lambda \gt 0 is a parameter, 1 &lt; p &lt; q &lt; N p N − s p 1\lt p\lt q\lt \frac{Np}{N-sp} , N ≥ 2 N\ge 2 , and V ( x ) V\left(x) is a real continuous function on R N {{\mathbb{R}}}^{N} . For λ \lambda large enough, the existence of ground state solutions are obtained, and they localize near the potential well int ( V − 1 ( 0 ) ) \left({V}^{-1}\left(0)) .

The existence of ground state solutions for semi-linear degenerate Schrödinger equations with steep potential well

In this article, we study the following degenerated Schrödinger equations: { − Δ γ u + λ V ( x ) u = f ( x , u ) in R N , u ∈ E λ , where λ &gt; 0 is a parameter, Δ γ is a degenerate elliptic operator, the potential V ( x ) has a potential well with bottom and the nonlinearity f ( x , u ) is either super-linear or sub-linear at infinity in u . The existence of ground state solution be obtained by using the variational methods.

Ground states solutions for some non-autonomous Schrödinger-Bopp-Podolsky system

In this paper we study the existence of ground states solutions for non-autonomous Schrödinger–Bopp–Podolsky system { − Δ u + u + λ K ( x ) ϕ u = b ( x ) | u | p − 2 u in R 3 , − Δ ϕ + a 2 Δ 2 ϕ = 4 π K ( x ) u 2 in R 3 , where λ &gt; 0 , 2 &lt; p ≤ 4 and both K ( x ) and b ( x ) are nonnegative functions in R 3 . Assuming that lim | x | → + ∞ K ( x ) = K ∞ &gt; 0 and lim | x | → + ∞ b ( x ) = b ∞ &gt; 0 and satisfying suitable assumptions, but not requiring any symmetry property on them. We show that the existence of a positive solution depends on the parameters λ and p . We also establish the existence of ground state solutions for the case 3.18 ≈ 1 + 73 3 &lt; p ≤ 4 .

Ground state solutions for the fractional problems with dipole-type potential and critical exponent

&lt;p style='text-indent:20px;'&gt;We are concerned with ground state solutions of the fractional problems with dipole-type potential and critical exponent. Under certain conditions on the dipole-type potential and the parameter, we show that the structure of the Palais-Smale sequence goes to zero weakly, and establish the existence of ground state solution to the above problems by using a new analytical method not involving the concentration-compactness principle.&lt;/p&gt;

Existence of ground state solutions for fractional Kirchhoff Choquard problems with critical Trudinger–Moser nonlinearity

Existence of ground state solutions for fractional Kirchhoff Choquard problems with critical Trudinger–Moser nonlinearity

Global existence and non-existence analyses for a semilinear edge degenerate parabolic equation with singular potential term

This paper discusses initial boundary value problem for a semilinear edge degenerate parabolic equation and corresponding stationary problem. We first find some initial conditions with different energy levels such that the solution exists globally and blows up in finite time, respectively. We also study the asymptotic behaviors like exponential decay and exponential growth for solution and energy function. Especially, we show the solution of evolution problem will converge to the steady state solution. Additionally, we find that there are two explicit vacuum regions which are ball and annulus respectively, that is to say, there is no solution belongs to them and all solutions are isolated by them. Finally, we discuss the existence of ground state solution to the stationary problem. The instability of the ground state solution is considered and we prove that there exists initial value such that the instability occurs as a blow-up in finite time.