Multiple solutions for a class of elliptic equations with jumping nonlinearities

Abstract

The paper deals with a semilinear elliptic Dirichlet problem with jumping\ud nonlinearity and, using variational methods, it shows that the number\ud of solutions tends to infinity as the number of jumped eigenvalues tends\ud to infinity.\ud In order to prove this fact, for every positive integer k it is proved\ud that, when a parameter is large enough, there exists a solution which\ud presents k interior peaks.\ud The asymptotic behaviour and the profile of this\ud solution, as the parameter tends to infinity, are also described

Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities

“…In this case, the methods used in [34] and [35] do not work, no solution there exists which (as e 1 α−λ 1 or e 1 λ 1 −α in the case α = λ 1 ) might be the limit of the k-peaks solutions, so more refined arguments are necessary. However, also in this case we can construct k-peaks solutions and describe their behavior.…”

“…The solutions u k,β we obtain in this case tend to −∞, as β → +∞, a.e. in Ω and the top of the peaks tends to +∞ (while in [34] and [35] the height of the peaks remains bounded).…”

“…in Ω. Notice that, if we replace e 1 by such a p in the case α < λ 1 , for example, then the method used in [34] yields solutions u k,β which tend to the solution ( + α) −1 p as β → +∞ and present k peaks localized near the minimum points of ( + α) −1 p (which are the maximum points of e 1 if p = e 1 ). On the contrary, due to the fact that α = λ 1 , the solutions u k,β we obtain in this paper for the problem u − λ 1 u − + βu + = p in Ω, u = 0 o n ∂Ω (1.7) have k peaks which are always localized near the maximum points of e 1 for any p ∈ L 2 (Ω) such that p > 0 a.e.…”

“…In fact, in [34] we fix α < λ 1 and show that, for all k ∈ N and ξ 0 ∈ L 2 (Ω), problem (1.6) has, for β and t positive and large enough, a solution u k,β,t such that, as t → +∞, 1 t u k,β,t tends to a solution u k,β of problem P + (e 1 ), which presents k peaks near the maximum points of e 1 and, as β → +∞, converges to the solution e 1 α−λ 1 (this result might be seen as a weaker version of the LazerMcKenna conjecture, which is not in contradiction with the counter-example of Dancer [14]). In [35] we consider the case α > λ 1 and, for β > 0 large enough, we construct analogous kpeaks solutions u k,β of problem P − (e 1 ), which converge to the solution e 1 λ 1 −α as β → +∞ and, for t < 0 large enough, give rise to solutions u k,β,t of problem (1.3), such that 1 |t| u k,β,t → u k,β as t → −∞.…”

“…In [34] and [35] we prove that, for any fixed domain Ω, the number of solutions may be arbitrarily large, provided the number of jumped eigenvalues is large enough. In fact, in [34] we fix α < λ 1 and show that, for all k ∈ N and ξ 0 ∈ L 2 (Ω), problem (1.6) has, for β and t positive and large enough, a solution u k,β,t such that, as t → +∞, 1 t u k,β,t tends to a solution u k,β of problem P + (e 1 ), which presents k peaks near the maximum points of e 1 and, as β → +∞, converges to the solution e 1 α−λ 1 (this result might be seen as a weaker version of the LazerMcKenna conjecture, which is not in contradiction with the counter-example of Dancer [14]).…”