We present some multiplicity results concerning semilinear elliptic Dirichlet problems with jumping nonlinearities where the jumping condition involves a set of high eigenvalues not including the first one. Using a variational method we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough. Indeed, we prove that for every positive integer k there exists a positive integer n(k) such that, if the number of jumped eigenvalues is greater than n(k), then the problem has at least a solution which presents k peaks. Moreover, we describe the asymptotic behaviour of the solutions as the number of jumped eigenvalues tends to infinity; in particular, we analyse some concentration phenomena of the peaks (near points or suitable manifolds), we describe the asymptotic profile of the rescaled peaks, etc …

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

We consider a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using variational methods, we show that the number of solutions tends to infinity as the number of jumped eigenvalues tends to infinity. In order to prove this fact, for every positive integer k we prove that, when a parameter is large enough, there exists a solution which presents k interior peaks. We also describe the asymptotic behaviour and the profile of this solution as the parameter tends to infinity.

In this Note we present some results on the Fučík spectrum for the Laplace operator, that give new information on its structure. In particular, these results show that, if Ω is a bounded domain of R^N with N>1, then the Fučík spectrum has infinitely many curves asymptotic to the lines {λ_1}×R and R×{λ_1}, where λ_1 denotes the first eigenvalue of the operator -Delta in H_0^1(Ω). Notice that the situation is quite different in the case N=1; in fact, in this case, the Fučík spectrum may be obtained by direct computation and one can verify that it includes only two curves asymptotic to these lines. The method we use for the proof is completely variational.