Existence of three weak solutions for fourth-order Leray–Lions problem with indefinite weights

Three Weak Solutions for a Neumann p(x)-Laplacian-like Problem with Two Control Parameters

“…therefore Ψ is indeed well defined. Moreover, Ψ(0) = Φ(0) = 0 and Ψ is compact according to KEFI-IRZI-SHOMRANI (Lemma 3.1 in [18]).…”

Existence and multiplicity of solutions for α(x)-Kirchhoff Equation with indefinite weights

“…According to the critical theorem of Bonanno and Marano [6] and variational methods, the authors in [21] provided the existence of at least three weak solutions for the following problem…”

“…We recall that such problems are proposed by Kirchhoff in [22] as an existence of the classical D'Alembert wave equations for free vibration of elastic strings. Note that elliptic and singular elliptic problems with α(x)-Laplace operator can be found in [1,2,3,4,5,8,12,13,15,17,18,19,20,21,24,25,26].…”