Multiplicity of solutions for a fourth-order quasilinear nonhomogeneous equation

Abstract

We consider a fourth-order quasilinear nonhomogeneous equation which is equivalent to a nonhomogeneous Hamiltonian system. The purpose of this work is to prove the existence of at least two solutions for such equation when a certain parameter is small enough. Furthermore, under an additional hypothesis on positiveness of the nonhomogeneous part we prove that our solutions are positive.

Ground state and non-ground state solutions of some strongly coupled elliptic systems

“…The argument follows the lines of [19,Theorem 1.1] in the bounded domain case. We split the proof into two cases, according to whether…”

“…Here we follow a different (more direct) approach which works fine under (H1). Indeed, see [19,Theorem 1.1] and Proposition 2.1 hereafter, it is known that (1.1) is equivalent to (1.2) Δ |Δu| 1 p −1 Δu = |u| q−1 u in Ω, u = Δu = 0 on ∂Ω, in the sense that weak solutions of (1.2) correspond to classical solutions of (1.1).…”

On a fourth-order quasilinear elliptic equation of concave–convex type

“…Our approach is variational and due to the form of the corresponding functional, it is based on a detailed study of the Nehari's manifold together with Ekeland's Variational Principle [15], a Riesz's representation theorem for some Sobolev spaces [12], the Mountain Pass Theorem [2], the Fountain and Dual Fountain Theorems [4,6,23], some arguments based in the Krasnoselskii genus theory found in [17] and compactness arguments, either derived from the subcriticality nature of the problem, or from the Concentration-Compactness Principle [20] employed in the case when the equation has a critical exponential growth.…”

“…This work is motivated by the earlier works [5][6][7][8][9][10]12,14,17,22] and the purpose of this paper is to extend some of the results in those papers to problem (P λ ) presented below.…”

“…Following the same ideas of the proofs of Theorem 1.1 in [12] and Theorem 1.1 in [14], we present the equivalence between (P λ ) and (S λ ). Theorem 1.1.…”