p-Laplacian problems with nonlinearities interacting with the spectrum

Abstract

Abstract. The aim of this paper is investigating the existence and the multiplicity of weak solutions of the quasilinear elliptic problemwith smooth boundary ∂Ω and the nonlinearity g behaves as u p−1 at infinity. The main tools of the proof are some abstract critical point theorems in Bartolo et al. (Nonlinear Anal. 7: 981-1012, 1983, but extended to Banach spaces, and two sequences of quasi-eigenvalues for the p-Laplacian operator as in Candela and Palmieri (Calc.

Periodic solutions for a fractional asymptotically linear problem

“…Under our conditions on the nonlinear term f , one can obtain not only existence critical point theorems, but also sharper multiplicity results when the functionals are symmetric. In [10,11,12,13], multiplicity results for critical points of even functionals are stated, and their proofs are based on the use of a pseudo-index theory; see, for instance, [9,14] as general references on this topics.…”

“…In addition, when the nonlinear-term is symmetric, we obtain the existence of multiple periodic solutions to (1.2). More precisely, we adapt the pseudo-index theory due to Bartolo, Benci and Fortunato [9], to prove a multiplicity result for critical points of the even energy functional J ; see also [10,11,12,13] for related topics.…”

Multiple solutions for asymptotically q‐linear (p, q)‐Laplacian problems

“…Due to the behavior of g ( x , ·) at infinity, one can expect some interaction with the spectrum of the q ‐Laplacian; compare Li and Zhou 11 and Bartolo et al 14 In particular, when ℓ ∞ is an eigenvalue of −Δ q , a stronger assumption on f is needed to get compactness; see ( f r ). Furthermore, due to the symmetry condition , the energy functional I is even, so that if u is a critical point of I at some critical value c , also − u has the same property.…”

Multiple solutions for p-Laplacian type problems with asymptotically p-linear terms via a cohomological index theory

“…under the hypotheses (h 0 ) and (h 1 ), a variational approach was first used for p = 2 and A(x, t) ≡ 1 (see the seminal papers [1,5]). On the contrary, only a few results have been obtained when p = 2, but always for A(x, t) ≡ 1 or, at worst, for A(x, t) = A(x) independent of t (see [2,4,6,16,18,21,22,23,24,25]). In fact, when p > 1 is arbitrary, the main difficulty is that, while the structure of the spectrum of −∆ in H 1 0 (Ω) is known, the full spectrum of −∆ p is still unknown, even though various authors have introduced different characterizations of eigenvalues and definitions of quasieigenvalues.…”