Neumann Boundary Value Problems with not Coercive Potential

Non-trivial solutions for nonlinear fourth-order elastic beam equations

“…For more details on Theorem 2.1, we refer the reader to [17,18], where it has already been applied to nonlinear second-order differential problems. Throughout this paper, the following hypotheses are needed.…”

Non-trivial solutions of local and non-local Neumann boundary-value problems

“…In Section 6 we illustrate the applicability of our theory in three examples, two of which deal with solutions that change sign. The third example is taken from an interesting paper by Bonanno and Pizzimenti [3], where the authors proved the existence, with respect to the parameter λ, of positive solutions of the following BVP…”

“…Consider the BVP(6.4) − u ′′ (t) + u(t) = λte u(t) , t ∈ [0, 1], u ′ (0) = u ′ (1) = 0.In[3] the authors establish the existence of at least one positive solution such that u < 2 for λ ∈ (0, 2e −2 ).The BVP (6.4) is equivalent to the following integral problemu(t) = k(t, s)g(s)f (u(s))ds, where g(s) = s, f (u) = λe u − t) cosh s, 0 ≤ s ≤ t ≤ 1, cosh(1 − s) cosh t, 0 ≤ t ≤ s ≤ 1.The kernel k is positive, by the results Section 5 satisfies (C 1 )-(C 8 ) with [a, b] = [0, 1]. Thus we work in the cone K = {u ∈ C[0, 1] : min t∈[0,1] u(t) ≥ c u }, where c = c(0, 1) = 1/ cosh 1 = 0.648.We can compute the following constants ρ =f ρ,ρ/c = λe ρ /ρ.…”

Existence of non-trivial solutions for systems of n fourth order partial differential equations

“…We also refer the reader to the recent papers [3,12] where the local minimum theorem for differentiable functionals due to Bonanno [1: Theorem 5.1] was successfully applied to second order Neumann boundary value problems.…”