Multiple positive solutions of singular and critical elliptic problem in $${\mathbb{R}^2}$$ with discontinuous nonlinearities

Abstract

Abstract. Let Ω be a bounded domain in R 2 with smooth boundary. We consider the following singular and critical elliptic problem with discontinuous nonlinearity:where 0 ≤ q < 1, 0 < α ≤ 4π and β ∈ [0, 2) such that≤ 1 andUnder the suitable assumptions on m(x, t) we show the existence and multiplicity of solutions for maximal interval for λ.

Existence and concentration of positive solutions for a class of discontinuous quasilinear Schrödinger problems in $\mathbb{R}^{N}$

“…The critical case of (1.2) was considered in [24][25][26][27]. The supercritical results can be found in [28][29][30][31][32][33][34][35][36][37][38][39] and the references therein.…”

Elliptic Problems in with Critical and Singular Discontinuous Nonlinearities

“…The problem with jump discontinuity but without the singular term have been studied in [2], [4], [20] and [24]. In all the above mentioned works, the main methods used are variational techniques and the generalized gradient theory for locally Lipschitz functionals as developed in [11] and [12].…”

“…Now we claim that u λ is a local minimum of E a λ in H 1 0 (Ω). Here we will follow the same approach as in [15] and [24] and thus be sketchy in the proof.…”