Normalized solutions for a biharmonic Choquard equation with exponential critical growth in \(\mathbb{R}^4\). (English) Zbl 1544.35094

Chen, Wenjing; Wang, Zexi

Z. Angew. Math. Phys. 75, No. 2, Paper No. 58, 20 p. (2024).

Summary: In this paper, we study the following biharmonic Choquard-type problem \[ \begin{cases} \Delta^2 u-\beta \Delta u=\lambda u+(I_{\mu} *F(u))f(u), \quad \text{in } \mathbb{R}^4, \\ \displaystyle\int\limits_{\mathbb{R}^4}|u|^2 \mathrm{d}x=c^2 >0,\quad u\in H^2 (\mathbb{R}^4), \end{cases} \] where \(\beta \geq 0, \lambda \in \mathbb{R}, I_{\mu} =\frac{1}{|x|^{\mu}}\) with \(\mu \in (0,4), F(u)\) is the primitive function of \(f(u)\), and \(f\) is a continuous function with exponential critical growth. By using the mountain-pass argument, we prove the existence of radial ground-state solutions for the above problem.

© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

MSC:

35J30	Higher-order elliptic equations
35J61	Semilinear elliptic equations
35B33	Critical exponents in context of PDEs
35A01	Existence problems for PDEs: global existence, local existence, non-existence

Keywords:

bi-Laplacian; Choquard nonlinearity; exponential critical growth; existence of ground states

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