Normalised solutions for p-Laplacian equations with Lᵖ-supercritical growth

For N≥ 3 and $2<p<N$, we find normalised solutions to the equation {align*} -Δ_p u+(1+V(x))|u|ᵖ⁻²u+λu&=|u|q-2u RN\\ \|u\|_2&=ρ{align*} in the mass supercritical and Sobolev subcritical case, that is q∈(pN+2/N,Np/N-p), at least if $ρ>0$ is small enough. The function V∈ LN/p(RN), which plays the role of potential, is assumed to be non-positive and vanishing at infinity. Moreover, we will prove the compactness of the embedding of the space of radial functions W1,prad(RN)⊂ Lq(RN) for p∈(1,N) and q∈(pN+2/N,Np/N-p).