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Existence and multiplicity of normalized solutions to lower critical Choquard equation with kinds of bounded potentials. (English) Zbl 07959963

Summary: This paper studies the existence and multiplicity of normalized solutions to the lower critical Choquard equation with a \(L^2\)-subcritical local perturbation and kinds of bounded potentials \[ \begin{cases} -\Delta u+V(x)u \\ \qquad =\lambda u+\big(I_{\alpha}\ast |u|^{(N+\alpha)/N} \big) |u|^{(N+\alpha)/N-2}u+\mu |u|^{q-2}u & \text{in } \mathbb{R}^N, \\ \displaystyle\int_{\mathbb{R}^N}|u|^2 dx=a^2, \end{cases} \] where \(N\geq 1\), \(\mu, a> 0\), \(2< q< 2+4/N\), \(\alpha\in (0,N)\), \(I_{\alpha}\) is the Riesz potential, \(V(x)\) is a bounded potential and \(\lambda\in \mathbb{R}\) is an unknown parameter that appears as a Lagrange multiplier.

MSC:

35J61 Semilinear elliptic equations
35J15 Second-order elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
Full Text: DOI

References:

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