  We discuss the existence and non-existence of periodic in one variable and
compactly supported in the other variables least energy solutions for equations
with non-Lipschitz nonlinearity of the form: $-\Delta u=\lambda u^p - u^q$ in
$\mathbb{R}^{N+1}$, where $ 0< q < p \leq 1$, $\lambda \in \mathbb{R}$. The
approach is based on the Nehari manifold method supplemented by a one-sided
constraint given through the functional of the suitable Pohozaev identity. The
limit value of the parameter $\lambda$, where the approach is applicable,
corresponds to the existence of periodic in one variable and compactly
supported in the other variables least energy solutions. This value is found
through the extrem values of nonlinear generalized Rayleigh quotients and the
so-called curve of the critical exponents of $p,q$. Important properties of the
solutions are derived, such as that they are not trivial with respect to the
periodic variable and do not coincide with compactly supported solutions on the
entire space $\mathbb{R}^{N+1}$.
