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Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz’ya terms. (English) Zbl 1352.35047

The paper is concerned with a nonlinear elliptic differential equation that is related to a Hardy-Sobolev-Maz’ya type estimate. The independent variable is split into two components \(x=(y,z)\in{\mathbb R}^k\times{\mathbb R}^{N-k}\), and then, for suitable parameters \(s\) and \(t\) and \(2^*(r):=\frac{2(N-r)}{N-2}\), the equation reads \(-\Delta u=\mu\,\frac{|u|^{2^*(t)-2}u}{|y|^t}+\frac{|u|^{2^*(s)-2}u}{|y|^s}+a(x)u\), with Dirichlet boundary condition. Under relatively natural assumptions including \(N>6+t\) when \(\mu>0\) and \(N>6+s\) when \(\mu=0\), it is proven that the equation has infinitely many solutions. The critical exponents mean that the functional corresponding to the equation does not satisfy the Palais-Smale condition for large energy levels. Hence the mountain-pass technique can only be applied after some approximation with sub-critical exponents. The term \(a(x)u\) in the equation is needed for technical reasons.

MSC:

35J60 Nonlinear elliptic equations
35B33 Critical exponents in context of PDEs
35J20 Variational methods for second-order elliptic equations
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