Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential

Abstract

We examine semilinear Neumann problems driven by the Laplacian plus an unbounded and indefinite potential. The reaction is a Carathéodory function which exhibits linear growth near ±∞. We allow for resonance to occur with respect to a nonprincipal nonnegative eigenvalue, and we prove several multiplicity results. Our approach uses critical point theory, Morse theory and the reduction method (the Lyapunov-Schmidt method).

Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential

“…We mention that the standard eigenvalue problems for the Robin Laplacian have recently been studied by D'Agui, Marano & Papageorgiou [4] and by Papageorgiou & Rȃdulescu [11]. Additional existence and multiplicity results for parametric Robin and Neumann problems can be found in the works of Papageorgiou & Rȃdulescu [12,13].…”

Quenching Time Estimates for Semilinear Parabolic Equations Controlled by Two Absorption Sources in Control System

“…We change the exponents of both two absorption sources such that the two terms have a large enough gaps in the sense of growth order, in order to reveal and compare the importance of the two factors acting on the behavior of the quenching phenomena, which are the number of the absorption source terms and the exponents of these terms. The results obtained in the preset paper suggests the dominant influence of the exponents of the absorption terms comparing the number of them, which is not only different from the classical heat equation with nonlinear power-type external force [12,13], but also different from the nonlinearities and their corresponding behaviours and affects in other models [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30].…”

Robin problems with a general potential and a superlinear reaction

“…We mention the works of Lan and Tang [14] (with ξ ≡ 0), Li and Wang [15], Miyagaki and Souto [17] (with ξ ≡ 0), Papageorgiou and Papalini [21], Qin, Tang and Zhang [29], Wu and An [34], Zhang-Liu [35]. For Neumann and Robin problems, we mention the works of D'Agui, Marano and Papageorgiou [5], Papageorgiou and Rȃdulescu [23,24,26], Papageorgiou, Rȃdulescu and Repovš [27], Papageorgiou and Smyrlis [28], Pucci et al [2,4], Shi and Li [31]. Superlinear problems were treated by Lan and Tang [14], Li and Wang [15], Miyagaki and Souto [17], who proved only existence results.…”