Multiple solutions for the nonlinear scalar field equation

We investigate the multiplicity of solutions to the following nonlinear scalar field equation$ \begin{equation*} - \Delta u+u = h(u)\quad\text{in}\ {{\mathbb{R}}^{N}}, \quad u\in H^1({\mathbb{R}}^{N}), \end{equation*} $where $ N\geq 3 $ and the nonlinearity $ h $ satisfying a nonlipschitzian property that $ \lim\limits_{s\rightarrow0}{h(s)}/{s} = -\infty $. Inspired by [T. Cazenave, Nonlinear Anal, 7 (1983), 1127-1140], we first show that the corresponding variational functional is well-defined in a subspace of $ H^1({\mathbb{R}}^{N}) $. Then, by applying the arguments of mountain pass Lemma and symmetric mountain pass Lemma, we establish the multiplicity of radial and nonradial solutions. The keys to our approach are the monotonicity trick and some new decomposition results for bounded (PS) sequences.