Elliptic equations with jumping nonlinearities involving high eigenvalues

Abstract

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, et

Infinitely many positive solutions of nonlinear Schrödinger equations

“…In Remark 5.5 we present some examples of potentials a(x) which satisfy all the assumptions required in Theorem 1.1. The proof method is fully variational and it is a variant of the arguments introduced in [20,21] and already applied in [10][11][12]22,23]. Of course it is well known that solutions of (1.1) correspond to free critical points of E or, equivalently, to critical points of E on the Nehari natural constraint.…”

The Lazer-McKenna conjecture for an anisotropic planar elliptic problem with exponential Neumann data

“…In the early 1980s Lazer and McKenna conjectured that if α < λ 1 < β = +∞ and g(t) does not grow too fast at infinity, problem (1.11) has an unbounded number of solutions as s → +∞ (see [13]). In fact, it is well known that the Lazer-McKenna conjecture holds true for problem (1.11) with many different types of nonlinearities, which can be found in [10] for the exponential nonlinear case, in [17,18,19] for the asymptotically linear case, in [3,8,11] for the superlinear homogeneous case, in [4] for the superlinear nonhomogeneous case, in [6,7] for the subcritical case, and in [5,12,15,16,22,23] for the critical case. In particular when N = 2 and g(t) = e t , del Pino and Muñoz in [10] proved the Lazer-McKenna conjecture by constructing solutions of problem (1.11) with the accumulation of arbitrarily many bubbles around maximum points of φ 1 in the domain.…”

Variational properties of the first curve of the Fučík spectrum for elliptic operators

“…In fact (see [36,37]) using this method we have proved that if N ≥ 2 there exist infinitely many curves in Σ, asymptotic to the lines {λ 1 } × R and R × {λ 1 } (while, if N = 1, Σ has only two curves asymptotic to these lines). More precisely (see also Theorem 3.1) we have proved that, if N ≥ 2 and k ∈ N, for β > 0 large enough there exists α k,β > λ 1 such that (α k,β , β) ∈ Σ; moreover, for all k ∈ N, lim β→+∞ α k,β = λ 1 , α k,β depends continuously on β and α k,β < α k+1,β (notice that the method developed in [33][34][35] has been also used in the study of some nonlinear scalar field equations (see [11][12][13])). The following natural question remains still open: where do these curves come from?…”

“…In particular, they show that this curve is asymptotic to the lines {λ 1 } × R and R × {λ 1 }, give a new proof of the fact that these lines are isolated in Σ and deduce that all the eigenfunctions corresponding to points of the first curve have exactly two nodal domains (extending the well known Courant nodal domains theorem). Recently (see [33][34][35]) we have obtained new existence and multiplicity results for a class of Dirichlet problems of type (1.1) (in particular for semilinear problems with jumping nonlinearities) using a variational method that does not require to know whether or not the pair (α, β) belongs to Σ and, in addition, may be used to give new information on the structure of Σ. In fact (see [36,37]) using this method we have proved that if N ≥ 2 there exist infinitely many curves in Σ, asymptotic to the lines {λ 1 } × R and R × {λ 1 } (while, if N = 1, Σ has only two curves asymptotic to these lines).…”