Normalized solutions to a class of $(2, q)$-Laplacian equations in the strongly sublinear regime

In this paper, we consider the existence and multiplicity of normalized solutions for the following $(2, q)$-Laplacian equation {equation}{Equation} \{{aligned} &-Δ u-Δ_q u+λ u=g(u), x ∈ R^N, &\|u\|_2^2 =c^2, {aligned}. {E_λ} {equation} where $1<q<N$, Δq=div(|∇ u|q-2 ∇ u) is the q-Laplacian operator, λ is a Lagrange multiplier and $c>0$ is a constant. The nonlinearity g:R→ R is continuous and the behaviour of g at the origin is allowed to be strongly sublinear, i.e., lim s → 0 g(s) / s=-∞, which includes the logarithmic nonlinearity g(s)= s log s². We consider a family of approximating problems that can be set in H¹(RN)∩ D1, q(RN) and the corresponding least-energy solutions. Then, we prove that such a family of solutions converges to a least-energy solution to the original problem. Additionally, under certain assumptions about g that allow us to work in a suitable subspace of H¹(RN)∩ D1, q(RN), we prove the existence of infinitely many solutions of the above $(2, q)$-Laplacian equation.