## Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains.(English)Zbl 1430.35100

Summary: The paper is concerned with the existence and multiplicity of positive solutions of the nonhomogeneous Choquard equation over an annular type bounded domain. Precisely, we consider the following equation $-\varDelta u = \left(\int\limits_{\varOmega} \frac{|u(y)|^{2^\ast_{\mu}}}{|x-y|^{\mu}} dy\right) |u|^{2^\ast_{\mu}-2} u+f \text{ in } \varOmega, \quad u = 0 \text{ on } \partial\varOmega,$ where $$\varOmega$$ is a smooth bounded annular domain in $$\mathbb{R}^N$$ ($$N \geq 3$$), $$2^\ast_{\mu}=\frac{2N-\mu}{N-2}$$, $$f \in L^\infty(\varOmega)$$ and $$f \geq 0$$. We prove the existence of four positive solutions of the above problem using the Lusternik-Schnirelmann theory and variational methods, when the inner hole of the annulus is sufficiently small.

### MSC:

 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35B09 Positive solutions to PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35J20 Variational methods for second-order elliptic equations
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