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articleOpen Access
Multiple normalized solutions for Choquard equation involving the biharmonic operator and competing potentials in $ℝ^{N}$
- Shuaishuai Liang,
- Jiaying Ma,
- Shaoyun Shi, and
- Yueqiang Song
Bulletin of Mathematical Sciences18 Jan 2025
Preview Abstract
This paper is concerned with the existence of multiple normalized solutions for a class of Choquard equation involving the biharmonic operator and competing potentials in $ℝ^{N}$ :
$\{\begin{matrix} Δ^{2} u + V (𝜀 x) u = λ u + G (𝜀 x) (I_{μ} * F (u)) f (u) & in ℝ^{N}, \\ \int_{ℝ^{N}} | u |^{2} d x = c^{2}, \end{matrix}$
where $Δ^{2}$ is the biharmonic operator, $c, 𝜀 > 0$ , $μ \in (0, N)$ , $λ \in ℝ$ is an unknown parameter that appears as the Lagrange multiplier, the absorption potential $V$ , reaction potential $G$ and nonlinear term $f$ are continuous functions and satisfy some assumptions. With the help of the minimization techniques and Lusternik–Schnirelmann category, we show that the number of normalized solutions is not less than the number of global minimum points of $V$ when the parameter $𝜀$ is sufficiently small. Furthermore, and more interestingly, we can prove the existence of multiple solutions for this problem by using the Morse theory. As far as we know, this study seems to be the first contribution regarding the concentration behavior for Choquard equation involving the biharmonic operator.
articleNo Access
GENERALIZED HARDY–RELLICH INEQUALITIES IN CRITICAL DIMENSION AND ITS APPLICATIONS
- ADIMURTHI and
- SANJIBAN SANTRA
Communications in Contemporary Mathematics01 Jun 2009
Preview Abstract
In this paper, we study the Hardy–Rellich inequalities for polyharmonic operators in the critical dimension and an analogue in the p-biharmonic case. We also develop some optimal weighted Hardy–Sobolev inequalities in the general case and discuss the related eigenvalue problem. We also prove W^2,q(Ω) estimates in the biharmonic case.
articleNo Access
Symmetry in the composite plate problem
- Francesca Colasuonno and
- Eugenio Vecchi
Communications in Contemporary Mathematics27 Feb 2019
Preview Abstract
In this paper, we deal with the composite plate problem, namely the following optimization eigenvalue problem:
$inf_{ρ \in P} inf_{u \in 𝒲 \ {0}} \frac{\int_{Ω} {(Δ u)}^{2}}{\int_{Ω} ρ u^{2}},$
where $P$ is a class of admissible densities, $𝒲 = H_{0}^{2} (Ω)$ for Dirichlet boundary conditions and $𝒲 = H^{2} (Ω) \cap H_{0}^{1} (Ω)$ for Navier boundary conditions. The associated Euler–Lagrange equation is a fourth-order elliptic PDE governed by the biharmonic operator $Δ^{2}$ . In the spirit of [S. Chanillo, D. Grieser, M. Imai, K. Kurata and I. Ohnishi, Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes, Comm. Math. Phys.214 (2000) 315–337], we study qualitative properties of the optimal pairs $(u, ρ)$ . In particular, we prove existence and regularity and we find the explicit expression of $ρ$ . When $Ω$ is a ball, we can also prove uniqueness of the optimal pair, as well as positivity of $u$ and radial symmetry of both $u$ and $ρ$ .
chapterNo Access
Remarks on a priori estimates for superlinear elliptic problems
- Miguel Ramos
Topological Methods, Variational Methods and Their Applications01 Mar 2003
Preview Abstract
We make some remarks on the method of obtaining a priori estimates for the solutions of superlinear elliptic equations having prescribed Morse index. In particular, we show that sometimes these estimates can be obtained without performing blow-up procedures.