Nontrivial solutions of second-order singular Dirichlet systems
Zhao and Wang Boundary Value Problems
Nontrivial solutions of second-order singular Dirichlet systems
Jin Zhao
Yanchao Wang
We study the existence of nontrivial solutions for second-order singular Dirichlet systems. The proof is based on a well-known fixed point theorem in cones and the Leray-Schauder nonlinear alternative principle. We consider a very general singularity and generalize some recent results. MSC: 34B15
nontrivial solutions; singular Dirichlet systems; Leray-Schauder alternative principle; fixed point theorem in cones
-
< t < ,
lim
u→,u∈R+N v, f (t, u) = +∞
uniformly in t.
However, the word ‘singularity’ has a more general meaning in our case because we do
not need all components of the nonlinear term f (t, u) to be singular at the origin as those
in [, ]. A nontrivial solution of (.) is a function u = (u, . . . , uN )T ∈ C([, ], RN ) ∩
C((, ), RN ) that satisfies (.) and v, u(t) = for all t ∈ (, ).
Singular differential equations arise from different applied sciences. For example, the
singular problem (.) occurs in chemical reactor theory [, ], boundary layer theory [],
and the transport of coal slurries down conveyor belts []. Because of these wide
applications, during the last few decades, different types of singular differential equations have
been considered. Among those, the problem of looking for nontrivial solutions becomes
one of the central topics, and so it has drawn the attention of many researchers. See, for
example, [–] for one-dimensional Dirichlet problems, [, ] for one-dimensional
p-Laplacian problems, [–] for problems of partial differential equations, and [, ,
] for periodic problems. For instance, Agarwal and O’Regan [] showed that the scalar
singular system
u¨ + q(t)f (t, u) = , < t < ,
u() = , u() = ,
has at least two nontrivial solutions in some reasonable cases by a well-known fixed point
theorem in cones and the Leray-Schauder alternative principle. The result of [] was
extended in [] to systems.
In this work, we establish existence results for system (.). Our aim is to generalize and
improve the results in [] in the following direction: we do not need each component of the
nonlinear term f (t, u) to be singular at the origin, so that we can work out some systems
that cannot be dealt with in []. To illustrate our new results, we consider two systems
and
⎧⎪ u¨ + (u + w)–α + μ (u + w)β + e(t) = ,
⎨ w¨ + (u + w)–α + μ (u + w)β + e(t) = ,
⎪⎩ u() = u() = , w() = w() = ,
⎧⎪ u¨ + (u + w)–α + e(t) = ,
⎨ w¨ + μ (u + w)β + e(t) = ,
⎪⎩ u() = u() = , w() = w() = ,
(.)
(.)
in which α, β > and μ ∈ R is a parameter. Note that (.) cannot be dealt with the results
used in the literature.
Finally, we give some notation used in this paper. Given u, w ∈ RN , their inner product
is denoted by
u, w =
uiwi.
|u|v =
vi|ui|,
N
i=
N
i=
N
i=
Let |u|v denote the usual v-norm, that is,
where v ∈ R+N is a fixed vector. We will denote by · the supremum norm of C([, ], R)
and take X = C([, ], R) × · · · × C([, ], R) (N times). For any u = (u, . . . , uN ) ∈ X, the
v-norm becomes
|u|v =
vi ui =
N
i=
vi · mtax ui(t) .
Obviously, X is a Banach space.
2 Preliminaries
Let us first recall the following inequality, which can be found in [].
Lemma . Let
Then for all u ∈ A,
A = u ∈ C [, ], R : u(t) ≥ , t ∈ [, ], and u(t) is concave on [, ] .
u(t) ≥ t( – t) u ,
To prove our main results, we shall apply the following two well-known results.
Let K be a cone in X, and let D be a subset of X. We set DK = D ∩ K and ∂K D = (∂D) ∩ K .
Lemma . ([]) Let X be a Banach space, and let K be a cone in X. Assume that ,
are open bounded subsets of X with K = ∅, K ⊂ K . Let
S : K → K
be a continuous and completely continuous operator such that
(i) u = λSu for λ ∈ [, ) and u ∈ ∂K , and
(ii) there exists w ∈ K \ {} such that u = Su + λw for all u ∈ ∂K
Then S has a fixed point in K \ K .
and all λ > .
The following three restricted conditions need to be required throughout this paper. For
a given vector v ∈ R+N ,
(D) v, f (t, u) : [, ] × RN \ {} → R+ is continuous;
(D) q(t) ∈ C(, ), q(t) > on (,), and t( – t)q(t) dt < ∞;
(D) v, e(t) : [, ] → R is continuous, and t( – t)| v, e(t) | dt < ∞.
By condition (D) we get that the linear system
u¨ + e(t) = ,
u() = ,
< t < ,
u() = ,
γ (t) =
G(t, s)e(s) ds,
has a unique solution γ (t), which can be given as
where
and
G(t, s) =
( – t)s, ≤ s ≤ t ≤ ,
( – s)t, ≤ t < s ≤ ,
is the Green’s function. To simplify the notation, let
(t) = v, γ (t) ,
(t) = γ (t) v =
vi γi(t) ,
N
i=
∗ = min (t),
t
∗ = max (t).
t
It is obvious that ∗ ≤ .
v, f (t, u) ≥ φr+ ∗ (t)
≤ v, f (t, u) ≤ g |u|v + h |u|v
3 Main results
In this section, we always assume that (D)-(D) are satisfied and ∗ = .
Theorem . Given a vector v ∈ R+N , suppose that there exists a constant r > such that
(H) there exists a (...truncated)