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Multiplicity and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent. (English) Zbl 1430.35173

Summary: In present paper, we study the fractional Choquard equation \[ \varepsilon^{2s}(-\Delta)^s u+V(x)u=\varepsilon^{\mu -N}\left( \frac{1}{|x|^\mu }*F(u)\right) f(u)+|u|^{2^*_s-2}u \] where \(\varepsilon >0\) is a parameter, \(s\in (0,1)\), \([N>2s\), \(2^*_s=\frac{2N}{N-2s}\) and \(0<\mu <\min \{2s,N-2s\} \). Under suitable assumption on \(V\) and \(f\), we prove this problem has a nontrivial nonnegative ground state solution. Moreover, we relate the number of nontrivial nonnegative solutions with the topology of the set where the potential attains its minimum values and their’s concentration behavior.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35R11 Fractional partial differential equations
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