We study a nonlinear elliptic system of Lane--Emden type \[\left\{ \begin{array}{ll} -\Delta u\ =\ { m sgn}(v) |v|^{p-1} & ext{in $\R^{N}$,} \\ -\Delta v\ =\ - { m sgn}(u) |u|^{\frac1{p-1}} + f(u) & ext{in $\R^{N}$,} \\ u,\ v\ ightarrow 0 \quad ext{as}\quad \left\vert x ight\vert ightarrow +\infty,& \end{array} ight. \] which is equivalent to a fourth order elliptic equation. By using variational methods the existence of radial solutions of the given problem is proved. To this aim, new compact imbeddings are stated.

We study the following nonlinear elliptic system of Lane–Emden type $−Δu = sgn(v)|v|^{p−1}$ in Ω, $−Δv = −λ sgn(u)|u|^{1/{p−1}} + f(x, u)$ in Ω, u = v = 0 on ∂Ω, where λ ∈ R. If λ ≥ 0 and Ω is an unbounded cylinder, i.e., Ω = Ω ×$R^{N−m} ⊂ R^N$ , N − m ≥ 2, m ≥ 1, existence and multiplicity results are proved by means of the Principle of Symmetric Criticality and some compact imbeddings in partially spherically symmetric spaces. We are able to state existence and multiplicity results also if λ ∈ R and Ω is a bounded domain in $R^N$ , N ≥ 3. In particular, a good finite dimensional decomposition of the Banach space in which we work is given.

We find multiple solutions for a nonlinear perturbed Schr\"odinger equation by means of the so--called Bolle's method.

We study an elliptic system equivalent to a fourth order elliptic equation. By using variational and perturbative methods, we prove the existence of infinitely many solutions both in the symmetric and in the non-symmetric case.

We study the following nonlinear elliptic problem $u + m u = g(x,u) + f(x)$ in $R^N \setminus \bar{B}_R$, $u = h$ on $\partial B_R$, $u o 0$ as $jxj o +\infty$, with $N \ge 3$, $m > 0$, under suitable conditions on the radial functions $g, f$ and $h$. Multiplicity results are proved in the perturbed case $f \not\equiv 0$ and $h \not\equiv 0$ by using Bolle's Perturbation Method and suitable growth estimates on min-max critical levels.

The aim of this paper is investigating the multiplicity of weak solutions of the quasilinear elliptic equation \[ -\Delta_p u + V(x)|u|^{p-2}u\ =\ g(x, u), \quad x \in\R^N, \] where $1<p<+\infty$, $\Delta_p u= { m div}(|\nabla u|^{p-2}\nabla u)$, the nonlinearity $g$ behaves as $|u|^{p-2}u$ at infinity and $V$ is a potential satisfying the assumptions in \cite{bf}, so that a suitable embedding theorem for weighted Sobolev spaces holds. Both the non--resonant and the resonant case are analyzed.

Starting from a new sum decomposition of $ W^{1,p}(\R^N)\cap W^{1,q}(\R^N)$ and using a variational approach, we investigate the existence of multiple weak solutions of a (p,q)-Laplacian equation on $\R^N$, for 1<q<p<N, with a sign-changing potential and a Carath\'eodory reaction term satisfying the celebrated Ambrosetti-Rabinowitz condition. Our assumptions are mild and different from those used in related papers and moreover our results improve or complement previous ones for the single p-Laplacian.

The aim of this paper is investigating the existence of solutions of the semilinear elliptic problem \[\left\{\begin{array}{ll} \displaystyle{-\Delta u\ =\ p(x, u) + \varepsilon g(x, u)} & \mbox{ in } \Omega,\\ \displaystyle{u=0} & \mbox{ on } \partial\Omega,\\ \end{array} ight.\] where $\Omega$ is an open bounded domain of $\R^N$, $\varepsilon\in\R$, $p$ is subcritical and asymptotically linear at infinity and $g$ is just a continuous function. Even when this problem has not a variational structure on $H^1_0(\Omega)$, suitable procedures and estimates allow us to prove that the number of distinct crtitical levels of the functional associated to the unperturbed problem is ``stable'' under small perturbations, in particular obtaining multiplicity results if $p$ is odd, both in the non-resonant and in the resonant case.

The aim of this paper is investigating the existence and the multiplicity of solutions of a quasilinear elliptic problem of p-Laplacian type on an open bounded domain of $R^N$ with smooth boundary and the nonlinearity g behaves as $u^{p-1}$ at infinity. The main tools of the proof are some abstract critical point theorems, but extended to Banach spaces, and two sequences of quasi-eigenvalues for the p-Laplacian operator. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM

We look for homoclinic solutions q : R ! RN to the class of second order Hamiltonian systems - q'' + L(t)q = a(t)rG1(q) - b(t)rG2(q) + f(t), t 2 R, where L : R ! R^NXN and a; b : R ! R are positive bounded functions, G1;G2 : RN ! R are positive homogeneous functions and f : R ! RN. Using variational techniques and the Pohozaev fibering method, we prove the existence of infinitely many solutions if f = 0 and the existence of at least three solutions if f is not trivial but small enough.

We study the existence of weak solutions for a nonlinear elliptic system of Lane–Emden type $−Δu = sgn(v)|v|^{p−1}$ in $R^N$ , $−Δv = −ρ(x)sgn(u)|u|^{1/{p−1}} + f(x, u)$ in $R^N$ , u, v → 0 as |x| → +∞, by means of the Mountain Pass Theorem and some compact imbeddings in weighted Sobolev spaces.