In this paper the results of some researches concerning scalar field equations are summarized. The interest is focused on the question of existence and multiplicity of stationary solutions; so the model equation $-Delta u + a(x)u = |u|^{p-1}u$ in $ eal^N$, is considered. The difficulties and the ideas introduced to face them as well as known results are discussed. Some recent advances concerning existence and multiplicity of multi-bump solutions are described in detail. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM.

In this paper the question of finding infinitely many solutions to the problem $−Delta u +a(x)u =|u|^{p−2}u$ , in $R^N$, u ∈H^1(R^N), is considered when N≥2, p∈(2, 2N/(N−2)), and the potential a(x) is a positive function which is not required to enjoy symmetry properties. Assuming that a(x)satisfies a suitable “slow decay at infinity” condition and, moreover, that its graph has some “dips”, we prove that the problem admits either infinitely many nodal solutions orinfinitely many constant sign solutions. The proof method is purely variational and allows to describe the shape of the solutions.

In this paper the equation $ -Delta u+a(x)u=|u|^{p-1}u mbox{ in }R^N$ is considered, when $N ge2$, $p>1, p<{frac{N+2}{N-2}},$ if $Nge 3.$ Assuming that the potential $a(x)$ is a positive function belonging to $L^{N/2}_ {loc}(R^N),$ such that $a(x) o a_infty > 0, mbox{as} |x| ightarrow infty$, and that satisfies slow decay assumptions, but not requiring any symmetry property, the existence of infinitely many positive solutions, by purely variational methods, is proved. The shape of the solutions is described and, furthermore, their asymptotic behavior when $|a(x) - a_infty|_ {L^ {N/2}_ {loc}(R^N)} o 0$.

In this paper we consider the equation {equation presented}. During last thirty years the question of the existence and multiplicity of solutions to (E) has been widely investigated mostly under symmetry assumptions on a. The aim of this paper is to show that, differently from those found under symmetry assumption, the solutions found in [6] admit a limit configuration and so (E) also admits a positive solution of infinite energy having infinitely many "bumps".