Multiple solutions of a sublinear Schrödinger equation

Abstract

Abstract. In this paper we study the Schrödinger equation of the formwhere λ > 0 is a parameter, a and b are positive potentials, while the nonlinear term f : R → R is sublinear at infinity. Two cases will be considered:(i) f is superlinear at the origin; (ii) f does not satisfy any asymptotical property at the origin. In both situations, the existence of certain multiple weak solutions of (P λ ) are established for some λ > 0.2000 Mathematics Subject Classification: 35J60, 35J65.

Remarks on infinitely many solutions for a class of Schrödinger equations with sublinear nonlinearity

“…11 For the case that nonlinearity contains a parameter , the existence of certain multiple weak solutions of (2) with (x, u) = b(x)̃(u) was established for some > 0 in Kristály. 12 Subsequently, Moschetto 13 improved the result in Kristály. 12 A very recent paper 14 studied energy sign of solutions and the case of lacking compactness.…”

A multiplicity result for hemivariational inequalities

“…We point out that the same type of equations, in the case of continuous nonlinearities, was studied by Kristály in [10].…”

“…This abstract result has a natural application in the field of differential equations: already in [18], a class of boundary value problems for semilinear equations with smooth nonlinearities on bounded domains was studied, achieving the existence of at least three solutions. Further developments of Ricceri's result deal with more general classes of problems: in [10], Kristály examines a Schrödinger equation on R N ; in [6], Faraci and Iannizzotto study, through an extension of the abstract result from Hilbert spaces to a wider class of Banach spaces, boundary value problems involving the p-Laplacian on bounded domains. This paper represents a new step in the progress resumed above: the multiplicity result (first in the alternative form) is proved for a class of locally Lipschitz functionals, defined on Banach spaces, and applied to hemivariational inequalities on unbounded domains.…”

Multiplicity Results for a Perturbed Nonlinear Schrödinger Equation

“…We refer the reader to [1], [2], [6]. Very recently, in [4], Kristaly obtained two results concerning three weak solutions for the Schrödinger equation of the form …”

“…The proofs of our theorems are all based on a recent two local minima result of Ricceri (see [8]), while in [4] the aim is achieved using a three critical points theorem of Bonanno (see [3]). …”