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Register a new userGround states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials
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In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness for a class of nonlinear functionals in $H^{2}(mathbb{R}^4)$. Using this result and the principle of symmetric criticality, we can present a relationship between the existence of the nontrivial solutions to the semilinear bi-harmonic equation of the form [ (-Delta)^{2}u+gamma u=f(u) text{in} mathbb{R}^4 ] and the range of $gammain mathbb{R}^{+}$, where $f(s)$ is the general nonlinear term having the critical exponential growth at infinity. Our next goal in this paper is to establish the existence of the ground-state solutions for the equation begin{equation}label{con} (-Delta)^{2}u+V(x)u=lambda sexp(2|s|^{2})) text{in} mathbb{R}^{4}, end{equation} when $V(x)$ is a positive constant using the Fourier rearrangement and the Pohozaev identity. Then we will explore the relationship between the Nehari manifold and the corresponding limiting Nehari manifold to derive the existence of the ground state solutions for the above equation when $V(x)$ is the Rabinowitz type trapping potential, namely it satisfies $$0<V_{0}=underset{xinmathbb{R}^{4}}{inf}V(x) <underset{ | x | rightarrowinfty}{lim}V(x) < +infty. $$ The same result and proof applies to the harmonic equation with the critical exponential growth involving the Rabinowitz type trapping potential in $mathbb{R}^2$.
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Recently, the authors of the current paper established in [9] the existence of a ground-state solution to the following bi-harmonic equation with the constant potential or Rabinowitz potential: begin{equation} (-Delta)^{2}u+V(x)u=f(u) text{in} mathbb{R}^{4}, end{equation} when the nonlinearity has the special form $f(t)=t(exp(t^2)-1)$ and $V(x)geq c>0$ is a constant or the Rabinowitz potential. One of the crucial elements used in [9] is the Fourier rearrangement argument. However, this argument is not applicable if $f(t)$ is not an odd function. Thus, it still remains open whether the above equation with the general critical exponential nonlinearity $f(u)$ admits a ground-state solution even when $V(x)$ is a positive constant. The first purpose of this paper is to develop a Fourier rearrangement-free approach to solve the above problem. More precisely, we will prove that there is a threshold $gamma^{*}$ such that for any $gammain (0,gamma^*)$, the above equation with the constant potential $V(x)=gamma>0$ admits a ground-state solution, while does not admit any ground-state solution for any $gammain (gamma^{*},+infty)$. The second purpose of this paper is to establish the existence of a ground-state solution to the above equation with any degenerate Rabinowitz potential $V$ vanishing on some bounded open set. Among other techniques, the proof also relies on a critical Adams inequality involving the degenerate potential which is of its own interest.
We consider the existence of bound and ground states for a family of nonlinear elliptic systems in $mathbb{R}^N$, which involves equations with critical power nonlinearities and Hardy-type singular potentials. The equations are coupled by what we call ``Schrodinger-Korteweg-de Vries non-symmetric terms, which arise in some phenomena of fluid mechanics. By means of variational methods, ground states are derived for several ranges of the positive coupling parameter $ u$. Moreover, by using min-max arguments, we seek bound states under some energy assumptions.
We study the existence of ground states to a nonlinear fractional Kirchhoff equation with an external potential $V$. Under suitable assumptions on $V$, using the monotonicity trick and the profile decomposition, we prove the existence of ground states. In particular, the nonlinearity does not satisfy the Ambrosetti-Rabinowitz type condition or monotonicity assumptions.
This paper is concerned with the existence of ground states for a class of Kirchhoff type equation with combined power nonlinearities begin{equation*} -left(a+bint_{mathbb{R}^{3}}| abla u(x)|^{2}right) Delta u =lambda u+|u|^{p-2}u+u^{5}quad text{for some} lambdainmathbb{R},quad xinmathbb{R}^{3}, end{equation*} with prescribed $L^{2}$-norm mass begin{equation*} int_{mathbb{R}^{3}}u^{2}=c^{2} end{equation*} in Sobolev critical case and proves that the equation has a couple of solutions $(u_{c},lambda_{c})in S(c)times mathbb{R}$ for any $c>0$, $a,b >0$ and $frac{14}{3}leq p< 6,$ where $S(c)={uin H^{1}(mathbb{R}^{3}):int_{mathbb{R}^{3}}u^{2}=c^{2}}.$ textbf{Keywords:} Kirchhoff type equation; Critical nonlinearity; Normalized ground states oindent{AMS Subject Classification:, 37L05; 35B40; 35B41.}
In this paper, a class of Schr{o}dinger-Poisson system involving multiple competing potentials and critical Sobolev exponent is considered. Such a problem cannot be studied with the same argument of the nonlinear term with only a positive potential, because the weight potentials set ${Q_i(x)|1le i le m}$ contains nonpositive, sign-changing, and nonnegative elements. By introducing the ground energy function and subtle analysis, we first prove the existence of ground state solution $v_varepsilon$ in the semiclassical limit via the Nehari manifold and concentration-compactness principle. Then we show that $v_varepsilon$ converges to the ground state solution of the associated limiting problem and concentrates at a concrete set characterized by the potentials. At the same time, some properties for the ground state solution are also studied. Moreover, a sufficient condition for the nonexistence of the ground state solution is obtained.
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