ﻻ يوجد ملخص باللغة العربية
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 $mathbb{R}^2$.
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
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 cal
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 state
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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,