On a New Notion of Linking and Application to Elliptic Problems at Resonance

Abstract

In this paper, we introduce a new notion of linking which includes in particular the notions of homotopical linking and local linking. Critical point theorems for continuous functionals on metric spaces are presented. Finally, an application to nonlinear elliptic problems at resonance is given. Academic Press

Degenerate Semilinear Elliptic Problems Near Resonance With a Nonprincipal Eigenvalue

“…In the present paper, we extend the main results of [7] to the variational degenerate elliptic problem (1) by Local Saddle Point Theorem [12,8] and Mountain Pass Lemma. Our main results are the following theorems.…”

“…Theorem B (Local Saddle Point Theorem [12,8]). Let H = X 1 ⊕ X 2 be a Hilbert space where X 1 has finite dimension, J ∈ C 1 (H, R) satisfying the (P S) condition and such that for given ρ 1 , ρ 2 > 0, …”

The existence of infinitely many solutions for p-Laplacian type equations on R^N with linking geometry

“…In [18], Frigon introduced a new notion of linking, which includes many notions of linking, such as homotopically linking, homologically linking, etc. In considering continuous functionals, Frigon stated a deformation property in an abstract setting.…”

“…In considering continuous functionals, Frigon stated a deformation property in an abstract setting. Then, with the new notion of linking, minimax critical point theorems for continuous functionals on metric spaces were presented (Theorem 3.1 in [18]). After the publication of [18], there are many papers on problem (1.4) for the sign-changing potential case.…”

“…Then, with the new notion of linking, minimax critical point theorems for continuous functionals on metric spaces were presented (Theorem 3.1 in [18]). After the publication of [18], there are many papers on problem (1.4) for the sign-changing potential case. In [11], Degiovanni and Lancelotti considered problem (1.4) under the case a(x) = −λ and g(x, u) = |u| p * −2 u, where…”