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Register a new userExistence and Non-existence of Ground states of bi-harmonic equations involving constant and degenerate Rabinowitz potentials
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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.
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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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