Multiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin

Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros

“…A complement of this result was proved in [38]. Two positive solutions are also found in [36], in a situation similar to [5], while in [37] nonlinearities concave in the origin are again considered and the results in [8] are extended.…”

“…Equations with the bi-Laplacian or poly-Laplacian operator and several kinds of nonlinearities were studied in many works [11,23,32,35,31,30,18,24,3,28,29,14,10,36,37,38]: among them we emphasize those more related to our setting. [3] considered the problem of the existence of two positive solutions with a concaveconvex nonlinearity similar to the one in [1].…”

Nonlinearities with zeros for Laplacian, p−Laplacian and Poly-Laplacian

“…Equations with the bi-Laplacian or poly-Laplacian operator and several kinds of nonlinearities were studied in many works [23,42,53,60,56,49,30,43,6,46,47,25,22,62,64,63]. Among them we emphasize those more related to our setting: [6] considered the problem of the existence of two positive solutions with a concave-convex nonlinearity similar to the one in [1].…”

The Existence and Uniqueness of Radial Solutions for Biharmonic Elliptic Equations in an Annulus

“…Biharmonic elliptic equations arise in the study of traveling waves in suspension bridges [1,2], and the study of the static deflection of an elastic plate in a fluid [3,4]. BVP(1) is a general nonlinear biharmonic elliptic equation with the Navier boundary condition, and some of its special situations have been studied by many researchers; see [5][6][7][8][9][10][11][12][13][14][15][16] and the references therein. The authors of [5][6][7][8][9] considered the simple case where…”

“…Recently, the authors of [8,9], using the fixed point theorem on a cone, have obtained the existence and uniqueness results of positive solutions of BVP (2). Some researchers have also discussed the case where the right side of the equation with linear terms of ∆u ∆ 2 u = c ∆u + f (x, u), x ∈ Ω, u| ∂Ω = 0, ∆u| ∂Ω = 0, (3) Axioms 2024, 13, 383. https://doi.org/10.3390/axioms13060383 https://www.mdpi.com/journal/axioms see [10][11][12][13]. The authors of [10][11][12][13] mainly applied variational methods and critical theory to discuss the existence of non trivial solutions of BVP (3).…”