Existence and Multiplicity of Solutions for Neumann p-Laplacian-Type Equations

Abstract

We consider nonlinear Neumann problems driven by p-Laplacian-type operators which are not homogeneous in general. We prove an existence and a multiplicity result for such problems. In the existence theorem, we assume that the right hand side nonlinearity is p-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. In the multiplicity result, when specialized to the case of the p-Laplacian, we allow strong resonance at infinity and resonance at 0.

Superlinear Neumann problems with the p-Laplacian plus an indefinite potential

“…The next proposition summarizes the main properties of this map (see, for example, GasinskiPapageorgiou [14]). …”

“…So, problem (P λ ) exhibits the competing effects of concave and convex nonlinearities. Such problems were investigated in the context of Dirichlet problems with β ≡ 0 by AmbrosettiBrezis-Cerami [2] (semilinear equations) and by Garcia Azorero-Manfredi-Peral Alonso [12], Gasinski-Papageorgiou [14] (nonlinear equations driven by the p-Laplacian). The first two works focus on positive solutions, and the authors prove bifurcation-type results (see Sect.…”

“…6 of this paper). In [14], the authors produce nodal solutions. We mention that in all the above works the reaction is more restrictive than ours.…”

Pairs of nontrivial solutions for resonant Neumann problems

“…Consequently their conclusions are different. Some other types of Neumann boundary value problems can be found in recent papers [13][14][15][16].…”

Nonlinear Neumann Problems with Constraints

“…Recently Motreanu-Papageorgiou [23] extended the result to more general nonlinear nonhomogeneous di¤erential operators and presented an alternative simpler proof, which avoids the complicated estimations of the previous ones. Finally, we mention the related works of Gasiń ski-Papageorgiou [8,9,11,12] (for Neumann problems with p-Laplacian), Fan [5] and Gasiń ski-Papageorgiou [10] (for anisotropic Sobolev spaces), Khan-Motreanu [18] (for general Banach spaces), Winkert [26] (for unconstrained Neumann problems) and Winkert [27] (for nonsmooth functionals with nonlinear Neumann boundary condition).…”