Existence of multiple semiclassical solutions for a critical Choquard equation with indefinite potential. (English) Zbl 1437.35295
Summary: We study the existence and multiplicity of semiclassical states for the Choquard equation with critical growth
\[-\varepsilon^2\varDelta u+V(x)u=\left(\int_{\mathbb{R}^N} \frac{G(y,u(y))}{|x-y|^\mu}dy\right)g(x,u)\quad\text{in }\mathbb{R}^N,\]
where \(N\geq 3\), \(0 < \mu < \min\{4,N\}\), \(V(x)\) is sign changing and \(G\) is the primitive of \(g\) which is of critical growth due to the well known Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on \(V\) and \(g\), we prove the existence and multiplicity of semiclassical states by critical point theory.
\[-\varepsilon^2\varDelta u+V(x)u=\left(\int_{\mathbb{R}^N} \frac{G(y,u(y))}{|x-y|^\mu}dy\right)g(x,u)\quad\text{in }\mathbb{R}^N,\]
where \(N\geq 3\), \(0 < \mu < \min\{4,N\}\), \(V(x)\) is sign changing and \(G\) is the primitive of \(g\) which is of critical growth due to the well known Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on \(V\) and \(g\), we prove the existence and multiplicity of semiclassical states by critical point theory.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
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