| Abstract: |
| Though the existence of the nontrivial solutions for the bi-harmonic equation with the critical exponential growth has been studied in the literature, nothing is known about the existence of the ground-state solutions for this class of equations involving the constant and Rabinowitz type potential. This paper shows the first attempt in this direction. Since the Rabinowitz type potential is not necessarily symmetric, classical radial method cannot apply to solving this problem. In order to overcome this difficulty, we first establish the existence of the ground-state solutions for the equation involving the constant potential 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 equation involving Rabinowitz type potential,
The same result and proof applies to the harmonic equation with the critical exponential growth involving the Rabinowitz type potential in $\mathbb{R}^2$. |
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