Z. Angew. Math. Phys. (2018) 69:75 c  2018 Springer International Publishing AG, part of Springer Nature https://doi.org/10.1007/s00033-018-0966-1 Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP Solutions for a Kirchhoff equation with critical Caffarelli–Kohn–Nirenberg growth and discontinuous nonlinearity Gelson G. dos Santos and Giovany M. Figueiredo Abstract. In this paper, we study the existence of nonegative solutions to a class of nonlinear boundary value problems of the Kirchhoff type. We prove existence results when the problem has discontinuous nonlinearity and critical Caffarelli– Kohn–Nirenberg growth. Mathematics Subject Classification. 35A15, 35B33, 35B25, 35J60. Keywords. Variational methods, Critical exponents, Singular perturbations, Kirchhoff equation, Nonlinear elliptic equations, Discontinuous nonlinearity. 1. Introduction In the last years, the attention toward problems of the type ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ −M ⎛ ⎝  Ω |∇u| 2 dx ⎞ ⎠ Δu = g(x, u) in Ω, u =0 on ∂ Ω (1) has grown more and more thanks, in particular, to their intriguing analytical structure due to the presence of the term M   Ω |∇u| 2 dx  , which makes the equation no longer a pointwise identity. This type of problem is called Kirchhoff problem. Indeed, this operator appears in the Kirchhoff equation [23], which arises in nonlinear vibrations, namely ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u tt − M ⎛ ⎝  Ω |∇u| 2 dx ⎞ ⎠ Δu = g(x, u) in Ω × (0,T ) u = 0 on ∂ Ω × (0,T ) u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x). The reader may consult [1, 2, 14, 16, 27, 28] and the references therein, for more physical motivation on Kirchhoff problem. As already mencioned in [15], the big difficulty when we study Kirchhoff problem is the competition between the Kirchhoff operator and the growth of the nonlinearity. For instance, when m(t)=1+ t and g(x, t)= |t| p−2 t, the functional associated with Problem (1) is given by I (u)= 1 2 ‖u‖ 2 + 1 4 ‖u‖ 4 − 1 p |u| p p . Then, this functional has the geometry of Mountain Pass Theorem if 4 <p< 2 ∗ = 2N N−2 . But in the case N = 4, we obtain 2 ∗ = 4. In the general case, 2 ∗ → 2 when N → +∞, which shows the difficulty in studying the Kirchhoff problem. In order to overcome this difficulty, in last years, the authors have 0123456789().: V,-vol