Multiple Solutions for Second-Order Sturm–Liouville Boundary Value Problems with Subquadratic Potentials at Zero

Dan Liu,Xuejun Zhang,Mingliang Song,Nan-Jing Huang

doi:10.1155/2021/4221459

Abstract

We deal with the following Sturm–Liouville boundary value problem: − P t x ′ t ′ + B t x t = λ ∇ x V t , x , a.e. t ∈ 0,1 x 0 cos α − P 0 x ′ 0 sin α = 0 x 1 cos β − P 1 x ′ 1 sin β = 0 Under the subquadratic condition at zero, we obtain the existence of two nontrivial solutions and infinitely many solutions by means of the linking theorem of Schechter and the symmetric mountain pass theorem of Kajikiya. Applying the results to Sturm–Liouville equations satisfying the mixed boundary value conditions or the Neumann boundary value conditions, we obtain some new theorems and give some examples to illustrate the validity of our results.

Highlights

Introduction and Main ResultsIn this paper, we consider the Sturm–Liouville boundary value problem for second-order Hamiltonian systems:⎧⎪⎪⎪⎨ − P(t)x′(t)􏼁′ + B(t)x(t) λ∇xV(t, x), ⎪⎪⎪⎩x(0)cos x(1)cos α β − −P(0)x′(0)sin α 0, P(1)x′(1)sin β 0, a.e. t ∈ [0, 1], (1)(V0) V(t, x) is measurable in t for every x ∈ Rn and continuously differentiable in x for a.e. t ∈ [0, 1]
Linear term B(t) ≥ 0 is necessary in the conditions of theorems discussing the existence of two nontrivial solutions for problem (3) in [1, 2, 9, 10]
In [24], using the linking theorem of Schechter [19, 20], Bonanno et al have discussed the existence of two nontrivial solutions for second-order Hamiltonian systems with subquadratic potentials at zero

Summary

Introduction and Main Results

We consider the Sturm–Liouville boundary value problem for second-order Hamiltonian systems:. Linear term B(t) ≥ 0 is necessary in the conditions of theorems discussing the existence of two nontrivial solutions for problem (3) in [1, 2, 9, 10]. In [24], using the linking theorem of Schechter [19, 20], Bonanno et al have discussed the existence of two nontrivial solutions for second-order Hamiltonian systems with subquadratic potentials at zero. Based on the index theory of Dong [15, 16], the linking theorem of Schechter [19, 20], and the symmetric mountain pass theorem of Kajikiya [27], we will prove the existence of two nontrivial solutions and infinitely many solutions. Some examples are given to show that our results are new even in the cases of problems (3) and (4)

Preliminaries and Variational Setting

Applications to Sturm–Liouville Equations and Examples