On nonlinear nonhomogeneous resonant Dirichlet equations

Abstract

We consider a (p, 2)-equation with a Carathéodory reaction f (z, x) which is resonant at ±∞ and has constant sign, z-dependent zeros. Using variational methods, together with truncation and comparison techniques and Morse theory, we establish the existence of five nontrivial smooth solutions (four of constant sign and the fifth nodal). If the reaction f (z, x) is C 1 in x ∈ ‫,ޒ‬ then we produce a second nodal solution for a total of six nontrivial smooth solutions.

On a Parametric Nonlinear Dirichlet Problem with Subdiffusive and Equidiffusive Reaction

“…Recently, there have been some existence and multiplicity results for such operators. We refer to the works of Cingolani-Degiovanni [12], Medeiros-Perera [27], Papageorgiou-Smyrlis [30], and Sun [34]. ξ with 1 < p < ∞, then this map represents the generalized p-mean curvature differential operator defined by…”

On a Class of Parametric (p, 2)-equations

“…Since ϕ λ ∈ C 2 (W 1,p 0 (Ω)), from (31) and Papageorgiou and Smyrlis [29] (see also Papageorgiou and Rȃdulescu [26]) it follows that (32) C k (ϕ λ ,ŷ) = δ k,1 Z for all k 0.…”

“…From the proof of Proposition 26 (see the claim), we know that u 0 and v 0 are local minimizers of the functionalφ λ . Therefore From the proof of Proposition 26, we have C k (φ λ , y 0 ) = 0, ⇒ C k (φ λ , y 0 ) = δ k,1 Z for all k 0 (84) (see Papageorgiou and Smyrlis [29] and Papageorgiou and Rȃdulescu [26]).…”

Double-phase problems with reaction of arbitrary growth

“…So, if we let λ 0 = λ 0 * = max{λ 0 , λ * }, then we infer from (33) that σ + λ (τû 1 (p)) < 0 for all λ > λ 0 * , ⇒σ + λ (û * λ ) < 0 =σ + λ (0) for all λ > λ 0 * (see (30)), ⇒û * λ = 0 andû * λ ∈ Kσ+ λ for all λ > λ 0 * (see (30)), ⇒û * λ = u * λ for all λ > λ 0 * (see Claim 1). By (27)…”