Let Ω⊂RN (N≥3) be a C2 bounded domain and Σ⊂Ω be a compact, C2 submanifold without boundary, of dimension k with 0≤k1, we show that problem (P) admits several critical exponents in the sense that singular solutions exist in the subcritical cases (i.e. p is smaller than a critical exponent) and singularities are removable in the supercritical cases (i.e. p is greater than a critical exponent). Finally, we establish various necessary and sufficient conditions expressed in terms of appropriate capacities for the solvability of (P).