Ground State Solutions Research Articles

Linear non-degeneracy and uniqueness of the bubble solution for the critical fractional Hénon equation in $ \mathbb{R}^N $

We study the equation$\begin{equation*} (-\Delta)^s u = |x|^{\alpha} u^{\frac{N+2s+2\alpha}{N-2s}}\mbox{ in }\mathbb{R}^N, \end{equation*} \quad\quad\quad\quad\quad\quad\text{(P)}$where $ (-\Delta)^s $ is the fractional Laplacian operator with $ 0 < s < 1 $, $ \alpha>-2s $ and $ N>2s $. We prove the linear non-degeneracy of positive radially symmetric solutions of the equation (P) and, as a consequence, a uniqueness result of those solutions with Morse index equal to one. In particular, the ground state solution is unique. Our non-degeneracy result extends in the radial setting some known theorems done by Dávila, Del Pino and Sire (see [15, Theorem 1.1]), and Gladiali, Grossi and Neves (see [28, Theorem 1.3]).

Ground state solution for fractional problem with critical combined nonlinearities

This paper is concerned with the following nonlocal problem with combined critical nonlinearities ( − Δ ) s u = − α | u | q − 2 u + β u + γ | u | 2 s ∗ − 2 u in Ω , u = 0 in R N ∖ Ω , where s ∈ ( 0 , 1 ) , N > 2 s , Ω ⊂ R N is a bounded C 1 , 1 domain with Lipschitz boundary, α is a positive parameter, q ∈ ( 1 , 2 ) , β and γ are positive constants, and 2 s ∗ = 2 N / ( N − 2 s ) is the fractional critical exponent. For γ > 0 , if N ⩾ 4 s and 0 < β < λ 1 , s , or N > 2 s and β ⩾ λ 1 , s , we show that the problem possesses a ground state solution when α is sufficiently small.

Ground state and nodal solutions for a class of fractional Schrödinger equations involving exponential growth

Ground state and nodal solutions for a class of fractional Schrödinger equations involving exponential growth

A sharp Gagliardo-Nirenberg inequality and its application to fractional problems with inhomogeneous nonlinearity

<p style='text-indent:20px;'>The purpose of this paper is threefold. Firstly, we establish a Gagliardo-Nirenberg inequality with optimal constant, which involves a fractional norm and an inhomogeneous nonlinearity. Secondly, as an application of this inequality, we study ground state standing waves to a nonlinear Schrödinger equation (NLS) with a mixed fractional Laplacians and a inhomogeneous nonlinearity, and consider a minimization problem which gives the existence of ground state solutions with prescribed mass. In particular, by making use of this Gagliardo-Nirenberg inequality and its optimal constant, we give a sufficient and necessary condition for the existence results. Finally, we develop local wellposedness theory for NLS with a mixed fractional Laplacians and a inhomogeneous nonlinearity. In the process, we prove Strichartz estimates in Lorentz spaces which may be of independent interest.</p>

A ground state solution for a nonhomogeneous elliptic Kirchhoff type problem involving critical growth and Hardy term

A ground state solution for a nonhomogeneous elliptic Kirchhoff type problem involving critical growth and Hardy term

A new deduction of the strict sub-additive inequality and its application: Ground state normalized solution to Schrödinger equations with potential

In the present paper, we prove the existence of solutions $(\lambda, u)\in \mathbb R\times H^1(\mathbb R^N)$ to the following elliptic equations with potential $$\displaystyle -\Delta u+(V(x)+\lambda)u=g(u)\;\hbox{in}\;\mathbb R^N, $$ satisfying the normalization constraint $$ \int_{\mathbb R^N}u^2=a>0, $$ which is deduced by searching for solitary wave solution to the time-dependent nonlinear Schrödinger equations. Besides the importance in the applications, not negligible reasons of our interest for such problems with potential $V(x)$ are their stimulating and challenging mathematical difficulties. We develop an interesting way based on iteration and give a new proof of the so-called ``sub-additive inequality", which can simplify the standard process in the traditional sense. Under some mild assumption on the potential $V(x)$ and some other suitable assumptions on $g$, we can obtain the existence of ground state solution for prescribed $a > 0$.

Homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign nonlinearities

In this paper, we obtain the multiplicity of homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign potentials. The concentration-compactness principle is applied to show the compactness. As a byproduct, we obtain the uniqueness of the positive ground state solution for a class of autonomous Hamiltonian systems and the best constant for Sobolev inequality which are of independent interests.

The Existence of Ground State Solutions for Schrödinger-Kirchhoff Equations Involving the Potential without a Positive Lower Bound

In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R2 and f has critical exponential growth. By using a version of Trudinger-Moser inequality and variational methods, we obtain the existence of ground state solutions for this problem.

Nonexistence and existence of positive ground state solutions for generalized quasilinear Schrödinger equations

Nonexistence and existence of positive ground state solutions for generalized quasilinear Schrödinger equations

Convergence from power-law to logarithm-law in nonlinear fractional Schrödinger equations

In this paper, we show a connection between fractional Schrödinger equations with power-law nonlinearity and fractional Schrödinger equations with logarithm-law nonlinearity. We prove that ground state solutions of power-law fractional equations, as p → 2+, converge to a ground state solution of logarithm-law fractional equations. In particular, we provide a new proof to the existence of a ground state of logarithm-law fractional Schrödinger equations.

Well-posedness and dynamic properties of solutions to a class of fourth order parabolic equation with mean curvature nonlinearity

The well-posedness and dynamic properties of solutions to a class of fourth order parabolic equations with mean curvature nonlinearity are studied. More specifically, we first prove the existence and uniqueness of strong solutions by semigroup method. Then, we establish conditions for the global existence and finite time blow-up of solutions with subcritical, critical and supercritical initial energies. At the same time, we study the exponential decay and $ \omega $-limit set of the global solutions, and the upper bound of the blow-up time of the blow-up solutions. In addition, we also study the existence and regularity of ground state solutions for the steady state problems.

Existence and asymptotical behavior of the ground state solution for the Choquard equation on lattice graphs

<abstract><p>In this paper, we study the nonlinear Choquard equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} - \Delta u + V(x)u = \left( {\sum\limits_{y \ne x \atop y \in { \mathbb {Z} ^{N}} } {\frac{|u(y)|^p}{|x-y|^{N-\alpha}}} }\right )|u|^{p-2}u \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>on lattice graph $ \mathbb {Z}^{N} $. Under some suitable assumptions, we prove the existence of a ground state solution of the equation on the graph when the function $ V $ is periodic or confining. Moreover, when the potential function $ V(x) = \lambda a(x)+1 $ is confining, we obtain the asymptotic properties of the solution $ u_\lambda $ which converges to a solution of a corresponding Dirichlet problem as $ \lambda\rightarrow \infty $.</p></abstract>

Multiplicity results for ground state solutions of a semilinear equation via abrupt changes in magnitude of the nonlinearity

Given $ k\in\mathbb N $, we define a class of continuous piecewise functions $ f $ having abrupt but controlled magnitude changes so that the problem$ \Delta u +f(u) = 0, \quad x\in \mathbb R^N, N> 2, $has at least $ k $ radially symmetric positive solutions.

Positive ground state solutions for a class of fractional coupled Choquard systems

<abstract><p>In this paper, we combine the critical point theory and variational method to investigate the following a class of coupled fractional systems of Choquard type</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{l} (-\Delta)^{s}u+\lambda_{1}u& = (I_{\alpha}*|u|^{p})|u|^{p-2}u+\beta v \quad &&\text{in}\\ \mathbb{R}^{N}, \ (-\Delta)^{s}v+\lambda_{2}v& = (I_{\alpha}*|v|^{p})|v|^{p-2}v+\beta u \quad &&\text{in}\ \mathbb{R}^{N}, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>with $ s\in(0, 1), \ N\geq 3, \ \alpha\in(0, N), \ p > 1 $, $ \lambda_{i} > 0 $ are constants for $ i = 1, \ 2 $, $ \beta > 0 $ is a parameter, and $ I_{\alpha}(x) $ is the Riesz Potential. We prove the existence and asymptotic behaviour of positive ground state solutions of the systems by using constrained minimization method and Hardy-Littlewood-Sobolev inequality. Moreover, nonexistence of nontrivial solutions is also obtained.</p></abstract>

Groundstates and infinitely many solutions for the Schrödinger-Poisson equation with magnetic field

<p style='text-indent:20px;'>In this paper, we investigate the nonlinear Schrödinger-Poisson equation with magnetic field. By combining non-Nehari manifold method and some new energy estimate inequalities, we obtain the existence of a ground state solution, where the strict monotonicity condition and the Ambrosetti-Rabinowitz growth condition are not needed. Moreover, when both the potential and the nonlinearity are sign-changing, by applying the Fountain Theorem and some analytical techniques, we prove the existence of infinitely many solutions. Our results extend and improve the present ones in the literature.</p>

一类p-Laplace方程基态解的存在性

一类p-Laplace方程基态解的存在性

Existence result for the critical Klein-Gordon-Maxwell system involving steep potential well

<abstract><p>The Klein-Gordon-Maxwell system has received great attention in the community of mathematical physics. Under a special superlinear condition on the nonlinear term, the existence of solution for the critical Klein-Gordon-Maxwell system with a steep potential well has been solved. In this paper, under two general superlinear conditions, we obtain the existence of ground state solution for the critical Klein-Gordon-Maxwell system with a steep potential well. The general superlinear conditions bring challenge in proving the boundedness of Cerami sequence, which is a key step in the proof of the existence. To solve this, we construct a Pohožaev identity and adopt some analytical techniques. Our results extend the previous results in the literature.</p></abstract>

A nonexistence result for a class of quasilinear Schrödinger equations with Berestycki-Lions conditions

In this paper, we study the following quasilinear Schr?dinger equation ??u + V(x)u ? [?(1 + u2)1/2] u/2(1 + u2)1/2 = h(u), x ? RN, where N ? 3, 2+ = 2N/ N?2, V(x) is a potential function. Unlike V ? C2(RN), we only need to assume that V ? C1(RN). By using a change of variable, we prove the non-existence of ground state solutions with Berestycki-Lions conditions, which contain the superliner case: lim s?+? h(s) s = +? and asymptotically linear case: lim s?+? h(s) s = ?. Our results extend and complement the results in related literature.

Existence and concentration of ground state solutions for a class of subcritical, critical or supercritical problems with steep potential well

Existence and concentration of ground state solutions for a class of subcritical, critical or supercritical problems with steep potential well

Ground State Solutions for Schrödinger Equations with Periodic Potentials

Ground State Solutions for Schrödinger Equations with Periodic Potentials