Mountain pass theorem in ordered Banach spaces and its applications to semilinear elliptic equations

Abstract

Abstract. In this paper, we establish a mountain pass theorem in an ordered Banach space, which is a Riesz-Banach space such that the absolute value is continuous. Applying our theorem to semilinear elliptic equations with sign-changing nonlinear terms, we prove the existence of a positive solution. Mathematics Subject Classification (2000). 35J20, 35J25, 35J60.

Positive solutions of semilinear elliptic equations with small perturbations

“…The purpose of this paper is to prove the existence of a positive solution for |λ| small enough under the mountain pass assumption on f (x, u) only without any conditions on g(x, u). The nonlinear term f (x, u) = a(x)u p was studied by Afrouzi and Brown [1], Alama and Tarantello [2], Brown and Zhang, [4], Li and Wang [6] and the author [5]. However the assumptions in this paper are more general than those of the papers above.…”

Existence and Multiplicity Results for a Class of Nonlinear Schrödinger Equations with Magnetic Potential Involving Sign-Changing Nonlinearity

“…Other works that also dealt with the convex case in the bounded domain with the usual Laplacian can be seen in [1,8,9,16,25].…”

“…The conditions (D 4 ) and (D 5 ) appear first in [16,Example 4.3] in which the existence of positive solutions to a problem with the usual Laplacian and bounded domain is studied. Also Jalilian and Szulkin in [15] use the above hypotheses to treat an elliptic problem in R N .…”

Existence and multiplicity results for a class of non-linear Schrödinger equations with magnetic potential involving sign-changing non linearity

“…Other works that also dealt with the convex case in the bounded domain with the usual Laplacian can be seen in [1,8,9,16,24].…”