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A class of semilinear elliptic equations with dependence on the gradient is con- sidered. The existence of a positive and a negative solution is stated through an iterative method based on Mountain–pass technique.

We construct entire solutions of an elliptic equation whose nonlinearity depends on the gradient of u. Such solutions are superpositions of odd, periodic functions and linear ones, with prescribed integer or rational slope.

In this paper we state some existence results for the semilinear elliptic equation $-\Delta u(x)-\lambda u(x) = W(x)f(u)$ where $W(x)$ is a function possibly changing sign, $f$ has a superlinear growth and $\lambda$ is a positive real parameter. We discuss both the cases of subcritical and critical growth for $f$, and prove the existence of Linking type solutions.

A formulation in terms of degenerate parabolic variational inequalities for the price of an American put option is stated. The usual uniform ellipticity condition is weakened by the presence of a weight function. Then the appropriate framework is that of some suitable "weighted" Sobolev spaces. One applies a result previously stated by the authors in Ref. 8.

IN A PREVIOUS paper [3] the authors stated a result about the existence of T-periodic solutions (for any T > 0) of autonomous Hamiltonian systems having minimal period T or T/2, in the case where the Hamiltonian function H has a superquadratic behaviour of the type considered by Ekeland and Hofer in [2], but without a global convexity assumption, which is replaced by a local convexity condition. In this paper we use the notion of essentiality for critical points of linking type (see [I, 4, 71 for an exhaustive exposition) in order to construct a T-periodic solution having T as its minimal period. We remark that this result really generalizes the main theorem by Ekeland and Hofer in [2]. The solution is found through a Riesz-Galerkin method as exhibited in [3] (following the basic scheme introduced in [6] by Rabinowitz) and the use of the Marino-Prodi perturbation method. More precisely, as a first step, one finds a critical point of linking type 2, for the restriction F, of F to a suitable finite-dimensional space, for any n E N. Then some suitable estimates for .?,, , derived by the linking construction, enable the use of the Marino-Prodi theorem in order to “approximate” each 2, by a sequence (Zig,“‘], in such a way that .z$” is a nondegenerate critical point of a functional Fim’, with (Ff”] + F, in the C2-norm, as m + +oo. Indeed 2:“‘) can be chosen as a critical point of linking type satisfying the same estimates as 2, and verifying the further property of essentiality. Roughly speaking, one says that a linking critical point of a functional is not essential or avoidable if, by looking at its linking definition, one can find a local deformation (i.e. in the neighbourhood of w), which enables the construction, for any minimizing sequence of “surfaces” having infinitesimal distances from w, of another minimizing sequence of “surfaces” which keep away from w. The solution z of the Hamiltonian system is found as the limit of a suitable “diagonal” subsequence 1~:: = Zi)j. The fact that the minimal period of z can be either T or T/2 derives from some arguments which were already exploited in [3], based on some estimates for the Morse indexes of finite-dimensional linking points (see [7]). As a final step, one shows that T/2 cannot be a period for z (hence the conclusion that T is the minimal period of z), by showing that the property of T/2-periodicity for z should yield the avoidability of the points of a subsequence of 1~~). First of all, one looks at the maximum negative eigenvalue ,uj of D2F~~‘(zj).

Existence results are presented for classical solutions to some nonvariational problems through a suitable approximation method of Mountain Pass critical points.

The existence of a positive and a negative solution of a problem of the type - Δ u + f ( x , u , ∇ u ) = 0 in Ω , u = 0 on ∂ Ω is proved, when f has a growth at infinity depending both on u and on ∇ u . The techniques are based on an iterative scheme of Mountain Pass “approximated” solutions and the use of a suitable truncature method.

The main result stated in the present paper is the existence of a lion-negative solution for a semilinear variational inequality through the use of some estimates for the Mountain-Pass critical points obtained for the penalized equations associated with the variational inequality. The positivity of the solution is achieved through a regularity result and the strong maximum principle.

Then, it is well known that Rabinowitz’s theorem (see [8]) guarantees the existence, for every T > 0, of at least one nontrivial solution to (l), but he left open the question to find a solution having minimal period T. When H is convex, some results have been recently obtained about such a problem via duality methods. The problem has been completely solved by Ambrosetti & Mancini (see [2,7]), with a suitable hypothesis on the second derivative of H, and by Clarke & Ekeland (see [4]), replacing (Hj), (HA) and (H5) with a subquadratic hypothesis. In another framework, Rabinowitz determines in [9] a family of solutions to (l), having minimal periods which diverge, in the superquadratic case (see theorem 2.34) under the

Dato un sistema Hamiltoniano, nel quale la funzione Hamiltoniana e la somma di un termine quadratico definito positivo e di un termine superquadratico convesso, si dimostra l'esistenza di soluzioni periodiche di periodo minimo T fissato, per ogni T e (0, 2π/ΩN), dove ΩN e il massimo autovalore della forma quadratica. Si utilizzano alcune tecniche relative alla teoria dell'indice di Morse, introdutte in [13], [15], [16] ed una opportuna formulazione del principio di dualita di Clarke ed Ekeland (vedi [10], [11]).