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a b s t r a c tIn this paper, by using the Lusternik-Schnirelmann category, we obtain a multiplicity result for a quasilinear elliptic system with both concave and convex nonlinearities and critical growth terms in bounded domains.

In this paper, we extend the result of Yan and Yang [16] on equations to an elliptic system involving critical Sobolev and Hardy–Sobolev exponents in bounded domains satisfying some geometric condition. In addition, we weaken the conditions on the dimension N and on the potential {a(x)} set in [16]. Our main result asserts, by a variational global-compactness argument, that the condition on the dimension N can be refined from {N\geq 7} to {N>\max(4,\lfloor 2s\rfloor+2)}, where {0<s<2} and still end up with infinitely many solutions.

In this paper, we study a class of Brezis–Nirenberg problems for nonlocal systems, involving the fractional Laplacian $(-\unicode[STIX]{x1D6E5})^{s}$ operator, for $0<s<1$ , posed on settings in which Sobolev trace embedding is noncompact. We prove the existence of infinitely many solutions in large dimension, namely when $N>6s$ , by employing critical point theory and concentration estimates.

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