Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems

By Shujie Li, Zhi-Qiang Wang. 2000

89 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/f206acdef6779c945f09c9b391e7c2acce7a7426

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In Applied Mathematics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Ljusternik-Schnirelman theory in partially ordered Hilbert spaces (2002, methodology): “Another motive of our work in this paper is to continue our study in [39] and clarify the locations of critical points produced by linking type theorems in relation with the order structure.”
Sign Changing Solutions of Superlinear Schrödinger Equations (2003, methodology): “The methods and the abstract critical point theory from [2, 7] or [17, 18] require to work on a Banach space X ⊂ H(Ω) whose cone of positive functions has nonempty interior.”
Schrödinger equations with concave and convex nonlinearities (2005, background): “The main idea in [3, 17, 27 , 28, 29] is to make use of the cones of positive functions and negative functions in C 1 0 (Ω).”
Existence of five nonzero solutions with exact sign for A p-laplacian equation (2009, methodology): “Roughly speaking, from these works emerged two approaches to the problem: (a) combining the method of upper-lower solutions with variational techniques (see Dancer-Du [18, 19, 20]); (b) using invariance properties of the negative gradient flow associated to a well-chosen pseudo-gradient vector field”
Increasing radial solutions for Neumann problems without growth restrictions (2011, methodology): “In particular, this prevents us from using the techniques in [17], where the authors prove an abstract mountain pass theorem in order intervals.”
A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations (2007, background): “In fact, in previous works in the semilinear case, even when the right-hand side nonlinearity f (x) is linear near infinity, it is required that the asymptotic limit of the quotient f (x) x as |x |→∞ be bigger than λ1 (λ1 > 0 denoting the principal eigenvalue of (−∆, H 1 0 (Z ))), for reasons that w”