A Discourse on a Class of Schrodinger-Poisson System Involving (p,q)-Laplacian

Abstract

In this work, we consider the following singular Schrödinger–Poisson system: {(-Δ_p w-Δ_q w+(∣w∣^(p-2)+∣w∣^(q-2) )w+ϕ∣w∣^(q-2) w=f(x)∣w∣^(r-2) w+μ (h(x))/w^β ,x ∈ R^N, -Δϕ=∣w∣^q, x ∈ R^N.)┤

Here, μ > 0, 0 < β < 1 < p < q < N, 2q < r < p*, N ≥ 3. Moreover, f∈L^(q^*/(q^*-r)) (R^N), with f>0a.e. in R^N, h>0a.e. in R^N, h∈L^θ (R^N)∩L^∞ (R^N), 1/θ+(1-β)/τ=1, p<τ≤p^*, and θ>1. The operator Δ_m w=div⁡(∣∇w∣^(m-2) ∇w),m=p,q,

denotes the m-Laplace operator. The main difficulty of the problem arises from the unbalanced growth of the differential operator induced by the (p, q)-Laplacian. By applying variational methods and critical point theory for nonsmooth functionals, we establish the existence and multiplicity of positive solutions for the above system.