Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues

Morse theory and local linking for a class of boundary value problems with impulsive effects

“…On the other hand, in the last few years, Morse theory and local linking are powerful tools in modern nonlinear analysis [7,9,19], especially for the problems with resonance [11,18]. However, to the best of our knowledge, with the exception of [1], using Morse theory to study the existence of solutions of impulsive boundary value problems has received considerably less attention.…”

Existence and multiplicity of solutions for Neumann problems

“…Invoking the strict monotonicity of the eigenvalues in weighted nonlinear eigenvalue problems or directly from (37), we see that λ 1 (ξ ) > 1 and so from (37) it follows that y ≡ 0, a contradiction to the fact that y = 1.…”

“…However, his hypotheses preclude the possibility of double resonance and he imposes the conditions on the ratio ∂j (t,x) ∂x and not on 2j (t,x) x 2 , which is in general less restrictive. We should also mention that doubly resonant problems were studied exclusively in the context of semilinear smooth Dirichlet problems, starting with the work of Berestycki-de Figueiredo [3]; see also Cac [6], Hirano-Nishimura [15], Landesman-Robinson-Rumbos [26], Robinson [34], Su [37] and the references therein.…”

Nonlinear nonhomogeneous periodic problems

“…Choosing ε ∈ 0, 1 b b 0 η(t)dt and recalling that q < r, we see that for θ ∈ (0, 1) small we will have Ψ + (θ) < 0, and so, from (20) …”