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On a nonlocal p(x)-Laplacian Dirichlet problem involving several critical Sobolev-Hardy exponents

Costa Augusto César dos Reis, Silva Ronaldo Lopes da

Opuscula Mathematica

2024

Vol. 44, no. 6

789--814

The aim of this work is to present a result of multiplicity of solutions, in generalized Sobolev spaces, for a non-local elliptic problem with p(x)-Laplace operator containing k distinct critical Sobolev–Hardy exponents combined with singularity points [formula] where Ω ⊂ RN is a bounded domain with 0 ∈ Ω and 1 < p− ≤ p(x) ≤ p+ < N. The real function M is bounded in [0,+∞) and the functions hi (i = 1, . . . , k) and f are functions whose properties will be given later. To obtain the result we use the Lions’ concentration-compactness principle for critical Sobolev–Hardy exponent in the space W 0 1,p(x) (Ω) due to Yu, Fu and Li, and the Fountain Theorem.