Solutions and positive solutions for superlinear Robin problems

Abstract

We consider nonlinear, nonhomogeneous Robin problems with a (p − 1)-superlinear reaction term, which need not satisfy the Ambrosetti-Rabinowitz condition. We look for positive solutions and prove existence and multiplicity theorems. For the particular case of the p-Laplacian, we prove existence results under a different geometry near the origin.

Constant sign and nodal solutions for parametric anisotropic (p, 2) -equations

“…then from hypothesis H 1 (ii) we have (23) ϕ + λ (tu) → −∞ as t → +∞. Moreover, (18) and Proposition 4.1 of Gasiński-Papageorgiou [8], implies that (24) ϕ + λ (•) satisfies the C-condition (see hypothesis H 1 (iii)). From ( 22), ( 23) and (24), we see that we can use the mountain pass theorem and obtain u ∈ W 1,p(z) 0…”

“…They produce at most three nontrivial solutions, but no nodal solutions. We also mention the recent isotropic works of Li-Rong-Liang [11], Papageorgiou-Vetro-Vetro [18] producing two positive solutions for (p, 2)and (p, q)-equations respectively, and the recent work of Papageorgiou-Scapellato [15] who consider a different class of parametric equations (superlinear perturbations of the standard eigenvalue problem) and produce seven solutions, all with sign information.…”

Multiple solutions for superlinear double phase Neumann problems

“…In problems with balanced growth (( p, q)equations), for which a powerful global regularity theory exists (see Lieberman [10]), the main tools are truncation and comparison techniques, critical point theory and Morse theory (critical groups). We refer to the works of Liu and Papageorgiou [12], Papageorgiou and Rȃdulescu [17,18], Papageorgiou et al [21], Papageorgiou and Zhang [23]. Here instead, we use the Nehari manifold method as this was developed by Brown and Wu [2], Brown and Zhang [3], Szulkin and Weth [25] and Willem [26].…”