Existence results for a Kirchhoff-type second-order differential equation on the half-line with impulses

Abstract

We shall discuss the existence of at least one weak solution and infinitely many weak solutions for a Kirchhoff-type second-order impulsive differential equation on the half-line. Our technical approach is based on variational methods. Some recent results are extended and improved. Some examples are presented to demonstrate the application of our main results.

Keywords

Half-line impulsive differential equation Kirchhoff-type problem one weak solution infinitely many weak solutions variational methods

1. Introduction

In this paper, we consider the following Kirchhoff-type second-order impulsive differential problem $\begin{matrix} (P_{λ}^{f, g}) & \{\begin{matrix} K (\int_{0}^{+ \infty} (| u^{'} (t) |^{2} + q (t) | u (t) |^{2}) d t) (- u^{″} (t) + q (t) u (t)) \\ = λ f (t, u (t)), t \in [0, + \infty), t \neq t_{j}, \\ Δ (u^{'} (t_{j})) = λ I_{j} (u (t_{j})), \\ u^{'} (0^{+}) = g (u (0)), u^{'} (+ \infty) = 0 \end{matrix} \end{matrix}$ where $K : [0, + \infty [\to R$ is a continuous function such that there are two positive constants $m_{0}$ and $m_{1}$ with $m_{0} ⩽ K (x) ⩽ m_{1}$ for all $x ⩾ 0$ , $q \in L^{\infty} [0, + \infty)$ with $0 < \underline{q} : = ess {inf}_{[0, + \infty)} q < \overline{q} : = ess {sup}_{[0, + \infty)} q$ , λ referred to as a control parameter, $I_{j} \in C (R, R)$ for $1 ⩽ j ⩽ p$ , $0 = t_{0} < t_{1} < t_{2} < \dots < t_{p} < + \infty$ , $Δ (u^{'} (t_{j})) = u^{'} (t_{j}^{+}) - u^{'} (t_{j}^{-}) = {lim}_{t \to t_{j}^{+}} u^{'} (t) - {lim}_{t \to t_{j}^{-}} u^{'} (t)$ , $f : [0, + \infty) \times R \to R$ is an $L^{2}$ -Carathéodory function and $g : R \to R$ is a Lipschitz continuous function with the Lipschitz constant $L > 0$ ; i.e. $\begin{array}{l} | g (s_{1}) - g (s_{2}) | ⩽ L | s_{1} - s_{2} | \end{array}$ for all $s_{1}, s_{2} \in R$ , satisfying $g (0) = 0$ .

Problems like ( P λ f , g ) are usually called nonlocal problems because of the presence of the integral over the entire domain, and this implies that the first equation in ( P λ f , g ) is no longer a pointwise identity. In fact, such kind of problem can be traced back to the work of Kirchhoff. In [29] Kirchhoff proposed the equation $\begin{matrix} (1.1) & ρ \frac{\partial^{2} u}{\partial t^{2}} - (\frac{ρ_{0}}{h} + \frac{E}{2 L} \int_{0}^{L} {| \frac{\partial u}{\partial x} |}^{2} d x) \frac{\partial^{2} u}{\partial x^{2}} = 0, \end{matrix}$ as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. The problem ( P λ f , g ) is related to the stationary analogue of the problem (1.1). Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Similar nonlocal problems also model several physical and biological systems where u describes a process which depends on the average of itself, for example the population density. Lions [32] has proposed an abstract framework for the Kirchhoff-type equations. After the work by Lions, various equations of Kirchhoff-type have been studied extensively, for instance see [15,20,36].

Impulsive differential equations serve as basic models to study the dynamics of processes in which sudden changes occur. In the last few years, variational methods have been used to determine the existence of solutions for impulsive differential equations possessing variational structures under certain boundary conditions, see for instance [39] and the references therein for detailed discussions.

Boundary value problems (BVPs) on the half-line occur naturally in the study of radially symmetric solutions of nonlinear elliptic equations, see [28], and various physical phenomena [2,26], such as an unsteady flow of gas through semi-infinite porous media, the theory of drain flows, plasma physics, in determining the electrical potential in an isolated neutral atom. In all these applications, it is frequent that only solutions that are positive are useful. Recently there have been many papers investigated the positive solutions of boundary value problem on the half-line, see [9,17,19,31,33,37,38,40–42,44] and the references therein. For example, Gomes and Sanchez in [19] by using a variational method and critical point theory, studied the existence of solutions for the following equations $\begin{matrix} (1.2) & \{\begin{matrix} u^{″} (t) + p (t) u (t) = q (t) h (u (t)), a.e. t \in [0, + \infty) \\ u (0) = α, u (+ \infty) = 0 \end{matrix} \end{matrix}$ where $α > 0$ and $p, q \in C (] 0, + \infty [)$ and $h \in C ([0, + \infty [)$ and $\begin{matrix} (1.3) & \{\begin{matrix} {(t^{k} u^{'})}^{'} = t^{k} u - t^{α} u^{β}, \\ u (0) = u (M) = 0 \end{matrix} \end{matrix}$ where M belongs to $(0, + \infty]$ , $α ⩾ k > 1$ and $β > 1$ . By establishing special Banach spaces, the authors proved the existence of solutions for problems (1.2) and (1.3), respectively. Tian and Ge in [38] by employing fixed-point theorems on cones, obtained the existence of positive solutions for a nonlinear multi-point boundary value problem on the half-line. Ma and Zhu in [33] by using a fixed point theorem in cones, obtained the existence of positive solutions for the following boundary value problem on the half-line $\begin{matrix} \{\begin{matrix} x^{″} (t) - k^{2} (t) x (t) + λ m (t) f (t, x (t)) = 0, t \in (0, + \infty) \\ x (0) = 0, {lim}_{t \to + \infty} x (t) = 0 \end{matrix} \end{matrix}$ where $k : [0, + \infty) \to (0, + \infty)$ and $f : [0, + \infty) \times (0, + \infty) \to R$ are continuous.

We also refer to [4,10,11] in which multiplicity of solutions for a parametric equation driven by the p-Laplacian operator on unbounded intervals via variational methods and critical point theory was studied.

Recently, impulsive boundary value problems (IBVP) on the half-line, have been studied by some researchers, for instance see [13,14,16,27,30,43] and the references therein. For example, in [30], Li and Nieto considered the existence of multiple positive solutions of the following IBVP on the half-line $\begin{matrix} \{\begin{matrix} u^{″} (t) + q (t) f (t, u (t)) = 0, t \in [0, + \infty), t \neq t_{j}, \\ Δ (u (t_{j})) = I_{j} (u (t_{j})), j = 1, \dots, p, \\ α u (0) = \sum_{i = 1}^{m - 2} α_{i} u (ξ_{i}), {lim}_{t \to + \infty} u^{'} (t) = 0 \end{matrix} \end{matrix}$ where $q \in C ([0, + \infty), [0, + \infty))$ and $f : [0, + \infty) \times [0, + \infty) \to [0, + \infty)$ is continuous. By using a fixed point theorem due to Avery and Peterson, the existence of at least three positive solutions was obtained. Dai and Zhang in [16] by using a variational method and a variant fountain theorem, obtained the existence and multiplicity of solutions for the following nonlinear impulsive problem on the half-line $\begin{matrix} \{\begin{matrix} - u^{″} (t) + u (t) = λ f (t, u (t)), a.e. t \in [0, + \infty), \\ Δ u^{'} (t_{j}) = I_{j} (u (t_{j})), j = 1, \dots, p, \\ u^{'} (0^{+}) = g (u (0)), u^{'} (+ \infty) = 0 \end{matrix} \end{matrix}$ where λ is a positive parameter, $g, I_{j} \in C (R, R)$ for $j = 1, \dots, p$ and $f \in C ([0, + \infty) \times R, R)$ . Zhao et al. in [43] by using a variational method and some critical points theorems, established some results on the multiplicity of solutions for a second-order impulsive differential equation depending on two real parameters on the half-line of the following form $\begin{matrix} \{\begin{matrix} - u^{″} (t) + m^{2} u (t) = λ f (t, u (t)), t \in [0, + \infty), t \neq t_{j}, \\ Δ (u^{'} (t_{j})) = μ I_{j} (u (t_{j})), \\ u^{'} (0^{+}) = g (u (0)), u^{'} (+ \infty) = 0 . \end{matrix} \end{matrix}$

Motivated by the above facts, in the present paper, using a smooth version of Theorem 2.1 of [8] which is a more precise version of Ricceri’s Variational Principle [35] we investigate the existence of at least one weak solution and infinitely many weak solutions for the problem ( P λ f , g ). In fact, we shall study the existence of at least one non-trivial weak solution for the problem ( P λ f , g ) under an asymptotical behaviour of the nonlinear datum at zero, see Theorem 3.1. We present Example 3.2 in which the hypotheses of Theorem 3.1 are fulfilled. We give some remarks on our results. Furthermore, under suitable oscillating behaviour at infinity of the nonlinear datum we shall discuss the existence of infinitely many weak solutions for the problem ( P λ f , g ). We ensure the existence of a definite interval about λ in which the problem ( P λ f , g ) admits a sequence of solutions which is unbounded in the space X which will be introduced later (Theorem 4.1). Furthermore, some consequences of Theorem 4.1 are listed. Replacing the conditions at infinity on the nonlinear terms, by a similar one at zero, we obtain a sequence of pairwise distinct solutions strongly converging at zero; see Theorem 4.8. We present two examples to illustrate Theorems 4.3 and 4.9, respectively (see Examples 4.4 and 4.10). It is worth noticing that in our results neither symmetric not monotonic condition on the nonlinear term is assumed.

The remainder of the paper is organized as follows. In Section 2, we shall recall the definitions and some properties of variable exponent Sobolev spaces. In Sections 3 and 4 we shall state and prove the main results of the paper.

For a through on the subject, we also refer the reader to [3].

2. Preliminaries

Our main tool is a smooth version of Theorem 2.1 of [8] which is a more precise version of Ricceri’s Variational Principle [35] that we recall here.

Theorem 2.1.
Let X be a reflexive real Banach space, let $Φ, Ψ : X \to R$ be two Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, strongly continuous, and coercive and Ψ is sequentially weakly upper semicontinuous. For every $r > {inf}_{X} Φ$ , let us put $\begin{array}{l} φ (r) : = inf_{u \in Φ^{- 1} (- \infty, r)} \frac{{sup}_{u \in Φ^{- 1} (- \infty, r)} Ψ (u) - Ψ (u)}{r - Φ (u)}, \\ γ : = \underset{r \to + \infty}{lim inf} φ (r), δ : = \underset{r \to {({inf}_{X} Φ)}^{+}}{lim inf} φ (r) . \end{array}$ Then, one has
for every $r > {inf}_{X} Φ$ and every $λ \in] 0, \frac{1}{φ (r)} [$ , the restriction of the functional $I_{λ} = Φ - λ Ψ$ to $Φ^{- 1} (] - \infty, r [)$ admits a global minimum, which is a critical point (local minimum) of $I_{λ}$ in X.

If $γ < + \infty$ then, for each $λ \in] 0, \frac{1}{γ} [$ , the following alternative holds:
either
$I_{λ}$ possesses a global minimum,
or
there is a sequence ${u_{n}}$ of critical points (local minima) of $I_{λ}$ such that $\begin{matrix} lim_{n \to + \infty} Φ (u_{n}) = + \infty . \end{matrix}$

If $δ < + \infty$ then, for each $λ \in] 0, \frac{1}{δ} [$ , the following alternative holds:
either
there is a global minimum of Φ which is a local minimum of $I_{λ}$ ,
or
there is a sequence of pairwise distinct critical points (local minima) of $I_{λ}$ which weakly converges to a global minimum of Φ.

We refer the interested reader to the papers [1,18,22,23,25,34] in which Theorem 2.1 has been successfully employed to the existence of at least one non-trivial solution for boundary value problems and to the papers [5–7,21,23,24] in which Theorem 2.1 has been successfully employed to the existence infinitely many solutions for boundary value problem.

In this section, we first introduce some notations and some necessary definitions. Set $\begin{matrix} W = {u : [0, + \infty) \to R is absolutely continuous, u^{'} \in L^{2} [0, + \infty)} . \end{matrix}$ Denote the Sobolev space X by $\begin{matrix} X = {u \in W : \int_{0}^{+ \infty} ({| u^{'} (t) |}^{2} + q (t) {| u (t) |}^{2}) d t < + \infty}, \end{matrix}$ with the inner product $\begin{matrix} (u, v) = \int_{0}^{+ \infty} (u^{'} (t) v^{'} (t) + q (t) u (t) v (t)) d t, \end{matrix}$ which induces the norm $\begin{matrix} ‖ u ‖ = {(\int_{0}^{+ \infty} ({| u^{'} (t) |}^{2} + q (t) {| u (t) |}^{2}) d t)}^{\frac{1}{2}} . \end{matrix}$ Obviously, X is a reflexive Banach space. We define the norm in $L^{2} ([0, + \infty))$ as $\begin{matrix} ‖ u ‖_{2} = {(\int_{0}^{+ \infty} {| u (t) |}^{2} d t)}^{\frac{1}{2}}, \end{matrix}$ and let $C = {u \in C ([0, + \infty)) : {sup}_{t \in [0, + \infty)} | u (t) | < + \infty}$ , with the norm $‖ u ‖_{\infty} = {sup}_{t \in [0, + \infty)} | u (t) |$ . Then $C$ is a Banach space.

We have the following proposition.
Proposition 2.2.
One has $\begin{array}{l} (2.1) & ‖ u ‖_{\infty} ⩽ M_{1} ‖ u ‖ \end{array}$ for every $u \in X$ where $\begin{matrix} M_{1} = \frac{1}{{\overline{q}}^{\frac{1}{4}}} . \end{matrix}$
Proof.
We follow the arguments in [12, Theorem 4, p. 138, Formula (4.61)] and in [4, Proposition 2.2], taking the equivalence of the norms into account. Let $v \in X$ . Since $\begin{matrix} v (z) - v (w) = \int_{w}^{z} v^{'} (x) d x, \end{matrix}$ bearing in mind that ${lim}_{| y | \to + \infty} v (y) = 0$ , one has $\begin{matrix} - v (t) = \int_{t}^{+ \infty} v^{'} (x) d x . \end{matrix}$ Therefore, $\begin{matrix} | v (t) | ⩽ \int_{t}^{+ \infty} | v^{'} (x) | d x ⩽ \int_{0}^{+ \infty} | v^{'} (x) | d x \end{matrix}$ for all $v \in X$ . Now, let $u \in X$ . By choosing $v (t) = {\overline{q}}^{\frac{1}{2}} | u (t) |^{2}$ for every $t \in R$ , one has $\begin{matrix} {\overline{q}}^{\frac{1}{2}} {| u (t) |}^{2} ⩽ \int_{0}^{+ \infty} 2 {\overline{q}}^{\frac{1}{2}} | u (x) | | u^{'} (x) | d x . \end{matrix}$ From the Hölder inequality one has $\begin{matrix} {\overline{q}}^{\frac{1}{2}} {| u (t) |}^{2} ⩽ 2 {\overline{q}}^{\frac{1}{2}} ‖ u ‖_{2} {‖ u^{'} ‖}_{2}, \end{matrix}$ that is $\begin{matrix} | u (t) | ⩽ 2^{\frac{1}{2}} \frac{1}{{\overline{q}}^{\frac{1}{4}}} {({\overline{q}}^{\frac{1}{2}} ‖ u ‖_{2})}^{\frac{1}{2}} {‖ u^{'} ‖}_{2}^{\frac{1}{2}} . \end{matrix}$ Since $x^{α} y^{1 - α} ⩽ α^{α} {(1 - α)}^{(1 - α)} (x + y)$ , $x, y ⩾ 0$ , $0 < α < 1$ (see [12, p. 130, Formula (4.47)]), one has $\begin{matrix} | u (t) | ⩽ 2^{\frac{1}{2}} \frac{1}{{\overline{q}}^{\frac{1}{4}}} {(\frac{1}{2})}^{\frac{1}{2}} {(\frac{1}{2})}^{\frac{1}{2}} \times [{(\int_{0}^{+ \infty} {| u^{'} (t) |}^{2} d t)}^{\frac{1}{2}} + {(\overline{q} \int_{0}^{+ \infty} {| u (t) |}^{2} d t)}^{\frac{1}{2}}] . \end{matrix}$ Hence, using the classical inequality $\begin{matrix} a^{\frac{1}{2}} + b^{\frac{1}{2}} ⩽ 2^{\frac{1}{2}} {(a + b)}^{\frac{1}{2}}, \end{matrix}$ one has $\begin{matrix} | u (t) | ⩽ \frac{1}{{\overline{q}}^{\frac{1}{4}}} \times {[(\int_{0}^{+ \infty} {| u^{'} (t) |}^{2} d t) + (\overline{q} \int_{0}^{+ \infty} {| u (t) |}^{2} d t)]}^{\frac{1}{2}} \end{matrix}$ that is, $\begin{matrix} ‖ u ‖_{\infty} ⩽ M_{1} ‖ u ‖, \end{matrix}$ and the proof is complete. □
Remark 2.3.
Let $f : [0, + \infty) \times R \to R$ be an $L^{2}$ -Carathéodory function, that means:
$t \mapsto f (t, x)$ is measurable for every $x \in R$ ,

$x \mapsto f (t, x)$ is continuous for a.e. $t \in [0, + \infty)$ ,

for every $ρ > 0$ there exists a function $l_{ρ} \in L^{2} ([0, + \infty))$ such that $\begin{matrix} sup_{| x | ⩽ ρ} | f (t, x) | ⩽ l_{ρ} (x) \end{matrix}$ for a.e. $t \in [0, + \infty)$ .

We say that a function $u \in X$ is a weak solution of the problem ( P λ f , g ) if $\begin{array}{l} K (\int_{0}^{+ \infty} ({| u^{'} (t) |}^{2} + q (t) {| u (t) |}^{2}) d t) \int_{0}^{+ \infty} (u^{'} (t) v^{'} (t) + q (t) u (t) v (t)) d t \\ - λ \int_{0}^{+ \infty} f (t, u (t)) v (t) d t + λ \sum_{j = 1}^{p} I_{j} (u (t_{j})) v (t_{j}) + g (u (0)) v (0) = 0 \end{array}$ holds for all $v \in X$ .

Corresponding to the functions f and K, we introduce the functions $F : [0, + \infty) \times R \to R$ and $\tilde{K} : [0, + \infty [\to R$ , respectively, as follows $\begin{matrix} F (t, x) = \int_{0}^{x} f (t, ξ) d ξ for all (t, x) \in [0, + \infty) \times R \end{matrix}$ and $\begin{matrix} \tilde{K} (x) = \int_{0}^{x} K (ξ) d ξ for all x ⩾ 0 . \end{matrix}$ We assume throughout and without further mention, that $I_{j}$ is a Lipschitz continuous function with the Lipschitz constant $\frac{2 k}{p} > 0$ ; i.e. $\begin{array}{l} (2.2) & | I_{j} (s_{3}) - I_{j} (s_{4}) | ⩽ \frac{2 k}{p} | s_{3} - s_{4} |, j = 1, \dots, p \end{array}$ for all $s_{3}, s_{4} \in R$ , satisfying $I_{j} (0) = 0$ .

We suppose that the Lipschitz constant $L > 0$ of the function g satisfies the condition $m_{0} > M_{1}^{2} L$ .
3. One solution

In this section, we formulate our main results on the existence of at least one weak solution for the problem ( P λ f , g ).

Theorem 3.1.
Assume that $\begin{array}{l} (D_{F}) & sup_{θ > 0} \frac{θ^{2}}{\int_{0}^{+ \infty} {sup}_{| x | ⩽ θ} F (t, x) d t + k θ^{2}} > \frac{2 M_{1}^{2}}{m_{0} - M_{1}^{2} L} \end{array}$ and there are a non-empty open set $D \subseteq [0, + \infty)$ and $B \subset D$ of positive Lebesgue measure such that $\begin{matrix} (3.1) & \underset{ξ \to 0^{+}}{lim sup} \frac{ess {inf}_{t \in B} F (t, ξ)}{| ξ |^{2}} = + \infty \end{matrix}$ and $\begin{matrix} (3.2) & \underset{ξ \to 0^{+}}{lim inf} \frac{ess {inf}_{t \in D} F (t, ξ)}{| ξ |^{2}} > - \infty . \end{matrix}$ Then, for each $\begin{matrix} λ \in Λ = (0, \frac{m_{0} - M_{1}^{2} L}{2 M_{1}^{2}} sup_{θ > 0} \frac{θ^{2}}{\int_{0}^{+ \infty} {sup}_{| x | ⩽ θ} F (t, x) d t + k θ^{2}}) \end{matrix}$ the problem ( P λ f , g ) admits at least one non-trivial weak solution $u_{λ} \in X$ .
Proof.
Our aim is to apply Theorem 2.1 to the problem ( P λ f , g ). Consider the functionals Φ, Ψ for every $u \in X$ , defined by $\begin{array}{l} Φ (u) : = \frac{1}{2} \tilde{K} (‖ u ‖^{2}) + \int_{0}^{u (0)} g (s) d s \end{array}$ and $\begin{matrix} Ψ (u) : = \int_{0}^{+ \infty} F (t, u (t)) d t - \sum_{j = 1}^{p} \int_{0}^{u (t_{j})} I_{j} (s) d s, \end{matrix}$ and put $I_{λ} (u) = Φ (u) - λ Ψ (u)$ for every $u \in X$ . The functionals Φ and Ψ satisfy the regularity assumptions of Theorem 2.1. It is well known that Ψ is a differentiable functional whose differential at the point $u \in X$ is $\begin{matrix} Ψ^{'} (u) (v) = \int_{0}^{+ \infty} f (t, u (t)) v (t) d t - \sum_{j = 1}^{p} I_{j} (u (t_{j})) v (t_{j}) \end{matrix}$ for every $v \in X$ , as well as is sequentially weakly upper semicontinuous. Since $| g (s) | ⩽ L | s |$ for every $s \in R$ , for any $u \in X$ , one has $\begin{array}{l} \frac{m_{0} - M_{1}^{2} L}{2} ‖ u ‖^{2} & ⩽ \frac{m_{0}}{2} ‖ u ‖^{2} - \int_{0}^{u (0)} L | s | d s ⩽ Φ (u) \\ (3.3) & ⩽ \frac{m_{1}}{2} ‖ u ‖^{2} + \int_{0}^{u (0)} L | s | d s ⩽ \frac{m_{1} + M_{1}^{2} L}{2} ‖ u ‖^{2} . \end{array}$ Bearing the condition $m_{0} > M_{1}^{2} L$ in mind, the first inequality of (3.3) follows ${lim}_{‖ u ‖ \to + \infty} Φ (u) = + \infty$ , namely Φ is coercive. By standard arguments, we find that Φ is a $C^{1}$ function and sequentially weakly lower semicontinuous and its Gâteaux derivative is the functional $Φ^{'} (u) \in X^{}$ , given by $\begin{array}{l} Φ^{'} (u) (v) = & K (\int_{0}^{+ \infty} ({| u^{'} (t) |}^{2} + q (t) {| u (t) |}^{2}) d t) \int_{0}^{+ \infty} (u^{'} (t) v^{'} (t) + q (t) u (t) v (t)) d t \\ + g (u (0)) v (0) \end{array}$ for every $v \in X$ . Therefore, we observe that the regularity assumptions on Φ and Ψ, as requested in Theorem 2.1, are verified. Arguing in a standard way, it is easy to prove that the critical points of the functional $I_{λ}$ are the weak solutions of the problem ( P λ f , g ). We now search for the existence of a critical point of the functional $I_{λ}$ in X. By using the condition ( D F ), there exists $\bar{θ} > 0$ such that $\begin{array}{l} \frac{{\bar{θ}}^{2}}{\int_{0}^{+ \infty} {sup}_{| x | ⩽ \bar{θ}} F (t, x) d t + k {\bar{θ}}^{2}} > \frac{2 M_{1}^{2}}{m_{0} - M_{1}^{2} L} . \end{array}$ Choose $\begin{matrix} r = \frac{m_{0} - M_{1}^{2} L}{2 M_{1}^{2}} {\bar{θ}}^{2} . \end{matrix}$ From (2.1) and the first inequality of (3.3), for every $u \in X$ such that $Φ (u) < r$ , we have $\begin{matrix} sup_{t \in [0, + \infty)} | u (t) | ⩽ \bar{θ} . \end{matrix}$ Furthermore, since $| I_{j} (u) | ⩽ \frac{2 k}{p} | u |$ for each $u \in X$ , we have $\begin{array}{l} (3.4) & - k ‖ u ‖_{\infty}^{2} ⩽ \sum_{j = 1}^{p} \int_{0}^{u (t_{j})} I_{j} (s) d s ⩽ k ‖ u ‖_{\infty}^{2} \end{array}$ and this in conjunction with the second inequality in (3.4) ensures $\begin{array}{l} Ψ (u) & ⩽ sup_{u \in Φ^{- 1} (- \infty, r)} \int_{0}^{+ \infty} F (t, u (t)) d t - \sum_{j = 1}^{p} \int_{0}^{u (t_{j})} I_{j} (s) d s \\ ⩽ \int_{0}^{+ \infty} sup_{| x | ⩽ \bar{θ}} F (t, x) d t + k ‖ u ‖_{\infty}^{2} \\ ⩽ \int_{0}^{+ \infty} sup_{| x | ⩽ \bar{θ}} F (t, x) d t + k {\bar{θ}}^{2} \end{array}$ for every $u \in X$ such that $Φ (u) < r$ . Then $\begin{matrix} sup_{Φ (u) < r} Ψ (u) ⩽ \int_{0}^{+ \infty} sup_{| x | ⩽ \bar{θ}} F (t, x) d t + k {\bar{θ}}^{2} . \end{matrix}$ By simple calculations and from the definition of $φ (r)$ , since $0 \in Φ^{- 1} (- \infty, r)$ and $Φ (0) = Ψ (0) = 0$ , one has $\begin{array}{rcl} φ (r) & = & inf_{u \in Φ^{- 1} (- \infty, r)} \frac{({sup}_{u \in Φ^{- 1} (- \infty, r)} Ψ (u)) - Ψ (u)}{r - Φ (u)} ⩽ \frac{{sup}_{u \in Φ^{- 1} (- \infty, r)} Ψ (u)}{r} \\ ⩽ & \frac{2 M_{1}^{2}}{m_{0} - M_{1}^{2} L} \frac{\int_{0}^{+ \infty} {sup}_{| x | ⩽ \bar{θ}} F (t, x) d t + k {\bar{θ}}^{2}}{{\bar{θ}}^{2}} . \end{array}$ Hence, putting $\begin{matrix} λ^{} = \frac{m_{0} - M_{1}^{2} L}{2 M_{1}^{2}} sup_{θ > 0} \frac{θ^{2}}{\int_{0}^{+ \infty} {sup}_{| x | ⩽ θ} F (t, x) d t + k θ^{2}} . \end{matrix}$ At this point, thanks to Theorem 2.1, for every $λ \in (0, λ^{}) \subseteq (0, \frac{1}{φ (r)})$ , the functional $I_{λ}$ admits at least one critical point (local minima) $u_{λ} \in Φ^{- 1} (- \infty, r)$ . We shall show that the function $u_{λ}$ cannot be trivial. Let us prove that $\begin{matrix} (3.5) & \underset{‖ u ‖ \to 0^{+}}{lim sup} \frac{Ψ (u)}{Φ (u)} = + \infty . \end{matrix}$ Owing to the assumptions (3.1) and (3.2), we can consider a sequence ${ξ_{n}} \subset R^{+}$ converging to zero and two constants σ, κ (with $σ > 0$ ) such that $\begin{matrix} lim_{n \to + \infty} \frac{ess {inf}_{t \in B} F (t, ξ_{n})}{| ξ_{n} |^{2}} = + \infty \end{matrix}$ and $\begin{matrix} ess inf_{t \in D} F (t, ξ) ⩾ κ | ξ |^{2} \end{matrix}$ for every $ξ \in [0, σ]$ . We consider a set $G \subset B$ of positive measure and a function $v \in X$ such that
$v (t) \in [0, 1]$ for every $t \in [0, + \infty)$ ,

$v (t) = 1$ for every $t \in G$ ,

$v (t) = 0$ for every $t \in [0, + \infty) ∖ D$ .
Hence, fix $M > 0$ and consider a real positive number η with $\begin{matrix} M < \frac{η meas (G) + κ \int_{D ∖ G} | v (t) |^{2} d t - k M_{1}^{2} ‖ v ‖^{2}}{\frac{m_{1} + M_{1}^{2} L}{2} ‖ v ‖^{2}} . \end{matrix}$ Then, there is $n_{0} \in N$ such that $ξ_{n} < σ$ and $\begin{matrix} ess inf_{t \in B} F (t, ξ_{n}) ⩾ η | ξ_{n} |^{2} \end{matrix}$ for every $n > n_{0}$ . Now, taking the second inequality of (3.3) into account, for every $n > n_{0}$ , by considering the properties of the function v (that is $0 ⩽ ξ_{n} v (t) < σ$ for n large enough), we have $\begin{array}{l} \frac{Ψ (ξ_{n} v)}{Φ (ξ_{n} v)} = & \frac{\int_{G} F (t, ξ_{n}) d t + \int_{D ∖ G} F (t, ξ_{n} v (t)) d t - \sum_{j = 1}^{p} \int_{0}^{ξ_{n} v (t_{j})} I_{j} (s) d s}{Φ (ξ_{n} v)} \\ > & \frac{η meas (G) + κ \int_{D ∖ G} | v (t) |^{2} d t - k M_{1}^{2} ‖ v ‖^{2}}{\frac{m_{1} + M_{1}^{2} L}{2} ‖ v ‖^{2}} > M . \end{array}$ Since M could be arbitrarily large, it is concluded that $\begin{matrix} lim_{n \to \infty} \frac{Ψ (ξ_{n} v)}{Φ (ξ_{n} v)} = + \infty, \end{matrix}$ from which (3.5) clearly follows. Hence, there exists a sequence ${w_{n}} \subset X$ strongly converging to zero such that, for n large enough, $w_{n} \in Φ^{- 1} (- \infty, r)$ and $\begin{matrix} I_{λ} (w_{n}) = Φ (w_{n}) - λ Ψ (w_{n}) < 0 . \end{matrix}$ Since $u_{λ}$ is a global minimum of the restriction of $I_{λ}$ to $Φ^{- 1} (- \infty, r)$ , we obtain $\begin{matrix} I_{λ} (u_{λ}) < 0, \end{matrix}$ hence that $u_{λ}$ is not trivial. The proof is complete. □

Here we present an example in which the hypotheses of Theorem 3.1 are satisfied.
Example 3.2.
Consider the problem $\begin{matrix} (3.6) & \{\begin{matrix} K (\int_{0}^{+ \infty} (| u^{'} (t) |^{2} + q (t) | u (t) |^{2}) d t) (- u^{″} (t) + q (t) u (t)) \\ = λ f (t, u (t)), t \in [0, + \infty), t \neq t_{1}, \\ Δ (u^{'} (t_{1})) = λ I_{1} (u (t_{1})), \\ u^{'} (0^{+}) = g (u (0)), u^{'} (+ \infty) = 0 \end{matrix} \end{matrix}$ where $K (x) = \frac{1 + e^{cos (x) + 1}}{2}$ for every $x \in [0, + \infty)$ , $q (t) = e^{- t} + 15$ for every $t \in [0, + \infty)$ , $f (t, x) = \frac{e^{- t}}{10^{2}} (2 x (1 - 10^{2}) + e^{x})$ for every $(t, x) \in [0, + \infty) \times R$ and $I_{1} (ξ) = sin (ξ)$ for every $ξ \in R$ . Choose $k = 1$ , then (2.2) is satisfied. By the expression of f, we have $\begin{matrix} F (t, x) = \frac{e^{- t}}{10^{2}} (x^{2} + e^{x} - 1) - e^{- t} x^{2} \end{matrix}$ for every $(t, x) \in [0, + \infty) \times R$ and $g (x) = 2 sin (x)$ for every $x \in R$ . By simple calculations, we obtain $m_{0} = 1$ , $M_{1} = \frac{1}{2}$ and $L = 2$ . Since $\begin{matrix} sup_{θ > 0} \frac{θ^{2}}{\int_{0}^{+ \infty} {sup}_{| x | ⩽ θ} F (t, x) d t + θ^{2}} = 10^{2} > 1, \end{matrix}$ we observe that all assumptions of Theorem 3.1 are fulfilled. Hence, Theorem 3.1 implies that for each $λ \in (0, 10^{2})$ the problem (3.6) admits at least one non-trivial weak solution $u_{λ} \in X$ .

Now, we give some remarks of our results. Remark 3.3.
If f is non-negative and $I_{j}$ is negative for $j = 1, \dots, p$ , then the weak solution ensured in Theorem 3.1 is non-negative. Indeed, let $u_{0}$ be a (non-trivial) weak solution of the problem ( P λ f , g ). Arguing by a contradiction, assume that the set $A = {t \in [0, + \infty) : u_{0} (t) < 0}$ is non-empty and of positive measure. Put $\bar{v} (t) = min {0, u_{0} (t)}$ for all $t \in [0, + \infty)$ . Clearly, $\bar{v} \in X$ and one has $\begin{array}{l} K (\int_{0}^{+ \infty} ({| u_{0}^{'} (t) |}^{2} + q (t) {| u_{0} (t) |}^{2}) d t) (\int_{0}^{+ \infty} u_{0}^{'} (t) {\bar{v}}^{'} (t) d t + \int_{0}^{+ \infty} q (t) u_{0} (t) \bar{v} (t) d t) \\ - λ \int_{0}^{+ \infty} f (t, u_{0} (t)) \bar{v} (t) d t + λ \sum_{j = 1}^{p} I_{j} (u_{0} (t_{j})) \bar{v} (t_{j}) + g (u_{0} (0)) \bar{v} (0) = 0 . \end{array}$ Using this fact that $u_{0}$ also is a weak solution of ( P λ f , g ) and by choosing $\bar{v} = u_{0}$ and since f is non-negative and $I_{j}$ is negative for $j = 1, \dots, p$ , for $λ > 0$ , we have $\begin{array}{l} (m_{0} - M_{1}^{2} L) ‖ u_{0} ‖_{A}^{2} & ⩽ K {(\int_{A} ({| u_{0}^{'} (t) |}^{2} + q (t) {| u_{0} (t) |}^{2}) d t)}^{2} + g (u_{0} (0)) u_{0} (0) \\ = λ \int_{A} f (t, u_{0} (t)) u_{0} (t) - λ \sum_{j = 1}^{p} I_{j} (u_{0} (t_{j})) u_{0} (t_{j}) ⩽ 0 . \end{array}$ i.e., $\begin{matrix} (m_{0} - M_{1}^{2} L) ‖ u_{0} ‖_{A}^{2} ⩽ 0 \end{matrix}$ and taking the condition $m_{0} > M_{1}^{2} L$ into account, this contradicts with this fact that $u_{0}$ is a non-trivial weak solution. Hence, our claim is proved.
Remark 3.4.
In Theorem 3.1 we searched for the critical points of the functional $I_{λ}$ naturally associated with the problem ( P λ f , g ). We note that, generally $I_{λ}$ can be unbounded from the below in X. Indeed, for example, if we take $f (ξ) = 1 + | ξ |^{τ - 2} ξ$ for every $ξ \in R$ with $τ > 2$ , for every any fixed $u \in X ∖ {0}$ and $ι \in R$ , we obtain $\begin{array}{l} I_{λ} (ι u) = & Φ (ι u) - λ (\int_{0}^{+ \infty} F (ι u (t)) d t - \sum_{j = 1}^{p} \int_{0}^{ι u (t_{j})} I_{j} (s) d s) \\ ⩽ & ι^{2} \frac{m_{1} + M_{1}^{2} L}{2} ‖ u ‖^{2} - λ ι ‖ u ‖_{L^{1}} - λ \frac{ι^{τ}}{τ} ‖ u ‖_{L^{τ}}^{τ} + ι^{2} k M_{1}^{2} ‖ u ‖^{2} \to - \infty \end{array}$ as $ι \to + \infty$ . Hence, we can not use direct minimization to find critical points of the functional $I_{λ}$ .
Remark 3.5.
For fixed $\bar{θ} ⩾ 0$ let $\begin{matrix} \frac{{\bar{θ}}^{2}}{\int_{0}^{+ \infty} {sup}_{| x | ⩽ \bar{θ}} F (t, x) d t + k {\bar{θ}}^{2}} > \frac{2 M_{1}^{2}}{m_{0} - M_{1}^{2} L} . \end{matrix}$ Then the result of Theorem 3.1 holds with $‖ u_{λ} ‖_{\infty} ⩽ \bar{θ}$ .
Remark 3.6.
We observe that Theorem 3.1 is a bifurcation result in the sense that the pair $(0, 0)$ belongs to the closure of the set $\begin{matrix} {(u_{λ}, λ) \in X \times (0, + \infty) : u_{λ} is a non-trivial weak solution of (P_{λ}^{f, g})} \end{matrix}$ in $X \times R$ . Indeed, by Theorem 3.1 we have that $\begin{matrix} ‖ u_{λ} ‖ \to 0 as λ \to 0 . \end{matrix}$ Hence, there exist two sequences ${u_{j}}$ in X and ${λ_{j}}$ in $R^{+}$ (here $u_{j} = u_{λ_{j}}$ ) such that $\begin{matrix} λ_{j} \to 0^{+} and ‖ u_{j} ‖ \to 0, \end{matrix}$ as $j \to + \infty$ . Moreover, we emphasis that due to the fact that the map $\begin{matrix} (0, λ^{}) ∋ λ \mapsto I_{λ} (u_{λ}) \end{matrix}$ is strictly decreasing, for every $λ_{1}, λ_{2} \in (0, λ^{*})$ , with $λ_{1} \neq λ_{2}$ , the weak solutions $u_{λ_{1}}$ and $u_{λ_{2}}$ ensured by Theorem 3.1 are different.

4. Infinitely many solutions

In this section we discuss the existence of infinitely many weak solutions for the problem ( P λ f , g ).

Put $\begin{matrix} B^{\infty} = \underset{ξ \to + \infty}{lim sup} \frac{\int_{0}^{+ \infty} F (t, ξ e^{- t}) d t + k ξ^{2}}{ξ^{2}} . \end{matrix}$ We formulate our main result as follows.

Theorem 4.1.

Assume that there exist two real sequences ${a_{n}}$ and ${b_{n}}$ with ${lim}_{n \to \infty} b_{n} = + \infty$ , such that

$a_{n} < \sqrt{\frac{2 (m_{0} - M_{1}^{2} L)}{M_{1}^{2} (m_{1} + M_{1}^{2} L) (\overline{q} + 1)}} b_{n}$ for each $n \in N$ ;

$\begin{array}{l} A^{\infty} & = lim_{n \to + \infty} \frac{\int_{0}^{+ \infty} {max}_{| x | ⩽ b_{n}} F (t, x) d t - \int_{0}^{+ \infty} F (t, a_{n} e^{- t}) d t + k (b_{n}^{2} - a_{n}^{2})}{\frac{m_{0} - M_{1}^{2} L}{2 M_{1}^{2}} b_{n}^{2} - \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4} a_{n}^{2}} \\ < \frac{4}{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)} B^{\infty} . \end{array}$

Then, for each

λ \in (\frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4 B^{\infty}}, \frac{1}{A^{\infty}})

, the problem ( P λ f , g ) admits an unbounded sequence of weak solutions.

Proof.

Our goal is to apply Theorem 2.1. Consider the functionals Φ, Ψ and $I_{λ}$ as given in the proof of Theorem 3.1. These functionals satisfy the required hypotheses of Theorem 2.1. To this end, write $\begin{matrix} r_{n} : = \frac{m_{0} - M_{1}^{2} L}{2 M_{1}^{2}} b_{n}^{2} \end{matrix}$ for every $n \in N$ . Moreover, due to (2.1), we have $\begin{matrix} | u (t) | ⩽ ‖ u ‖_{\infty} ⩽ M_{1} ‖ u ‖, \forall t \in [0, + \infty) . \end{matrix}$ Hence, from the first inequality of (3.3) it follows $\begin{matrix} Φ^{- 1} (- \infty, r_{n}) = {u \in X : Φ (u) < r_{n}} \subseteq {u \in X : ‖ u ‖_{\infty} ⩽ b_{n}} . \end{matrix}$ Now, for each $n \in N$ , let $w_{n} (t) = a_{n} e^{- t}$ for all $t \in [0, + \infty)$ . We clearly observe that $w_{n} \in X$ and $\frac{\underline{q} + 1}{2} a_{n}^{2} ⩽ ‖ w_{n} ‖^{2} ⩽ \frac{\overline{q} + 1}{2} a_{n}^{2}$ . Hence, in view of (3.3), we have $\begin{array}{l} \frac{(m_{0} - M_{1}^{2} L) (\underline{q} + 1)}{4} a_{n}^{2} ⩽ Φ (w_{n}) ⩽ \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4} a_{n}^{2} . \end{array}$ From the conditions $(A_{1})$ , we get $Φ (w_{n}) < r_{n}$ . Therefore, for every n large enough, one has $\begin{array}{rcl} φ (r_{n}) & = & inf_{u \in Φ^{- 1} (- \infty, r_{n})} \frac{({sup}_{u \in Φ^{- 1} (- \infty, r_{n})} Ψ (u)) - Ψ (u)}{r_{n} - Φ (u)} \\ ⩽ & \frac{{sup}_{u \in Φ^{- 1} (- \infty, r_{n})} Ψ (u) - \int_{0}^{+ \infty} F (t, u (t)) d t + \sum_{j = 1}^{p} \int_{0}^{u (t_{j})} I_{j} (s) d s}{r_{n} - Φ (u)} \\ (4.1) & ⩽ & \frac{\int_{0}^{+ \infty} {max}_{| x | ⩽ b_{n}} F (t, x) d t - \int_{0}^{+ \infty} F (t, a_{n} e^{- t}) d t + k (b_{n}^{2} - a_{n}^{2})}{\frac{m_{0} - M_{1}^{2} L}{2 M_{1}^{2}} b_{n}^{2} - \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4} a_{n}^{2}} . \end{array}$ Hence, bearing in mind $(A_{2})$ , we obtain $γ ⩽ {lim}_{n \to + \infty} φ (r_{n}) ⩽ A^{\infty} < + \infty$ . Now, we verify that $I_{λ}$ is unbounded from below. First, assume that $B^{\infty} = + \infty$ . Accordingly, fix M such that $M < \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4 λ} + k M_{1}^{2} (\frac{\overline{q} + 1}{2})$ and let ${c_{n}}$ be a sequence of positive numbers, with ${lim}_{n \to + \infty} c_{n} = + \infty$ , such that $\begin{matrix} \int_{0}^{+ \infty} F (t, c_{n} e^{- t}) d t > M c_{n}^{2} \forall n \in N . \end{matrix}$ Thus, if we consider a sequence ${y_{n}}$ in X defined by setting $\begin{matrix} y_{n} (t) = c_{n} e^{- t} for all t \in [0, + \infty), \end{matrix}$ we observe that $y_{n} \in X$ and $\begin{matrix} \frac{(m_{0} - M_{1}^{2} L) (\underline{q} + 1)}{4} c_{n}^{2} ⩽ Φ (y_{n}) ⩽ \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4} c_{n}^{2} . \end{matrix}$ So, $\begin{array}{l} I_{λ} (y_{n}) & ⩽ \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4} c_{n}^{2} - λ \int_{0}^{+ \infty} F (t, c_{n} e^{- t}) d t + λ \sum_{j = 1}^{p} \int_{0}^{c_{n} e^{- t_{j}}} I_{j} (s) d s \\ < c_{n}^{2} (\frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4} - λ M + λ k M_{1}^{2} (\frac{\overline{q} + 1}{2})) \end{array}$ that is, ${lim}_{n \to + \infty} I_{λ} (y_{n}) = - \infty$ . Next, assume that $B^{\infty} < + \infty$ . Since $λ > \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4 B^{\infty}}$ , we can fix $ϵ > 0$ such that $ϵ < B^{\infty} - \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4 λ}$ . Therefore, also calling ${c_{n}}$ a sequence of positive numbers such that ${lim}_{n \to + \infty} c_{n} = + \infty$ and $\begin{matrix} \int_{0}^{+ \infty} F (t, c_{n} e^{- t}) d t > (B^{\infty} - ϵ + k M_{1}^{2} (\frac{\overline{q} + 1}{2})) c_{n}^{2} \forall n \in N, \end{matrix}$ arguing as before and by choosing $y_{n} \in X$ as above, one has $\begin{array}{l} I_{λ} (y_{n}) ⩽ & \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4} c_{n}^{2} - λ \int_{0}^{+ \infty} F (t, c_{n} e^{- t}) d t + λ \sum_{j = 1}^{p} \int_{0}^{c_{n} e^{- t_{j}}} I_{j} (s) d s \\ < & (\frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4} - λ (B^{\infty} - ϵ)) c_{n}^{2} . \end{array}$ So, ${lim}_{n \to + \infty} I_{λ} (y_{n}) = - \infty$ . Hence, in both cases $I_{λ}$ is unbounded from below and the proof is complete. □

Remark 4.2.

If ${a_{n}}$ and ${b_{n}}$ are two real sequences with $a_{n} > 1$ for all $n \in N$ and ${lim}_{n \to + \infty} b_{n} = + \infty$ , such that the assumption $(A_{1})$ in Theorem 4.1 is satisfied. Then, under the conditions $A_{\infty} = 0$ and $B^{\infty} = + \infty$ , Theorem 4.1 assures that for every $λ > 0$ the problem ( P λ f , g ) admits infinitely many weak solutions.

Theorem 4.3.

Assume that

$\begin{array}{l} \underset{ξ \to + \infty}{lim inf} \frac{\int_{0}^{+ \infty} {sup}_{| x | ⩽ ξ} F (t, x) d t + k ξ^{2}}{ξ^{2}} \\ < \frac{2 (m_{0} - M_{1}^{2} L)}{M_{1}^{2} (m_{1} + M_{1}^{2} L) (\overline{q} + 1)} \underset{ξ \to + \infty}{lim sup} \frac{\int_{0}^{+ \infty} F (t, ξ e^{- t}) d t + k ξ^{2}}{ξ^{2}} . \end{array}$

Then, for each

\begin{array}{l} λ \in (\frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4 {lim sup}_{ξ \to + \infty} \frac{\int_{0}^{+ \infty} F (t, ξ e^{- t}) d t + k ξ^{2}}{ξ^{2}}}, \\ \frac{m_{0} - M_{1}^{2} L}{2 M_{1}^{2} {lim inf}_{ξ \to + \infty} \frac{\int_{0}^{+ \infty} {sup}_{| x | ⩽ ξ} F (t, x) d t + k ξ^{2}}{ξ^{2}}}), \end{array}

the problem ( P λ f , g ) has an unbounded sequence of weak solutions.

Proof.

We choose the sequence ${b_{n}}$ of positive numbers such that goes to infinity and $\begin{array}{l} lim_{n \to + \infty} & \frac{\int_{0}^{+ \infty} {sup}_{| x | ⩽ b_{n}} F (t, x) d t + k ξ^{2}}{b_{n}^{2}} = \underset{ξ \to + \infty}{lim inf} \frac{\int_{0}^{+ \infty} {sup}_{| x | ⩽ ξ} F (t, x) d t + k ξ^{2}}{ξ^{2}} . \end{array}$ Now, since $Φ (0) = Ψ (0) = 0$ we can taking $a_{n} = 0$ for every $n \in N$ in (4.1), from Theorem 2.1 the conclusion follows. □

Now, we give an application of Theorem 4.3.

Example 4.4.

Consider the problem $\begin{matrix} (4.2) & \{\begin{matrix} K (\int_{0}^{+ \infty} (| u^{'} (t) |^{2} + | u (t) |^{2}) d t) (- u^{″} (t) + u (t)) \\ = λ f (t, u (t)), t \in [0, + \infty), t \neq t_{1}, \\ Δ (u^{'} (t_{1})) = λ I_{1} (u (t_{1})), \\ u^{'} (0^{+}) = g (u (0)), u^{'} (+ \infty) = 0 \end{matrix} \end{matrix}$ where $K (x) = \frac{3}{2} + \frac{sin (x)}{2}$ for every $x \in [0, + \infty)$ , $I_{1} (ξ) = ξ$ for every $ξ \in R$ , $f (t, x) = 8 e^{- t} x (1 + sin (e^{x})) + 4 e^{- t} x^{2} e^{x} cos (e^{x})$ for every $(t, x) \in [0, + \infty) \times R$ and $g (x) = \frac{1}{2} (1 - cos (x))$ for every $x \in R$ . By choosing $k = 1$ , we observe that (2.2) is fulfilled. By the expression of f we have $F (t, x) = e^{- t} x^{2} (4 + 4 sin (e^{x}))$ for every $(t, x) \in [0, + \infty) \times R$ . According to the above data, we have $m_{0} = 1$ , $m_{1} = 2$ , $M_{1} = 1$ and $L = \frac{1}{2}$ . Since $w (t) = ξ e^{- t} ⩽ ξ$ for all $t \in [0, + \infty)$ , then $\begin{matrix} F (t, ξ e^{- t}) ⩽ F (t, ξ), \end{matrix}$ so, $\begin{matrix} \underset{ξ \to + \infty}{lim sup} \frac{\int_{0}^{+ \infty} F (t, ξ) d t + ξ^{2}}{ξ^{2}} = 9 \end{matrix}$ and $\begin{matrix} \underset{ξ \to + \infty}{lim inf} \frac{\int_{0}^{+ \infty} {sup}_{| x | ⩽ ξ} F (t, x) d t + ξ^{2}}{ξ^{2}} = 1 . \end{matrix}$ We clearly see that all assumptions of Theorem 4.3 are satisfied then, the problem (4.2) for every $λ \in] \frac{5}{36}, \frac{1}{4} [$ has an unbounded sequence of weak solutions in X.

Here, we point out two simple consequences of Theorems 4.1 and 4.3, respectively.

Corollary 4.5.

Assume that there exist two real sequences ${a_{n}}$ and ${b_{n}}$ with ${lim}_{n \to + \infty} b_{n} = + \infty$ , such that the assumption $(A_{1})$ in Theorem 4.1 holds, $A_{\infty} < 1$ and $B_{\infty} > \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4}$ . Then, the problem $\begin{matrix} (4.3) & \{\begin{matrix} K (\int_{0}^{+ \infty} (| u^{'} (t) |^{2} + q (t) | u (t) |^{2}) d t) (- u^{″} (t) + q (t) u (t)) \\ = f (t, u (t)), t \in [0, + \infty), t \neq t_{j}, \\ Δ (u^{'} (t_{j})) = I_{j} (u (t_{j})), \\ u^{'} (0^{+}) = g (u (0)), u^{'} (+ \infty) = 0 \end{matrix} \end{matrix}$ has an unbounded sequence of weak solutions.

Corollary 4.6.

Assume that $B_{\infty} > \frac{m_{1} (1 + \overline{q}) + 2 M_{1}^{2} L}{4}$ and $\begin{matrix} \underset{ξ \to + \infty}{lim inf} \frac{\int_{0}^{+ \infty} {sup}_{| x | ⩽ ξ} F (t, x) d t + k ξ^{2}}{ξ^{2}} < \frac{2 M_{1}^{2}}{m_{0} - M_{1}^{2} L} . \end{matrix}$ Then, the problem ( 4.3 ) has an unbounded sequence of weak solutions.

We here give the following two consequences of the main result.

Corollary 4.7.

${lim}_{n \to + \infty} \frac{\int_{0}^{+ \infty} {sup}_{| x | ⩽ b_{n}} F_{1} (t, x) d t - \int_{0}^{+ \infty} F_{1} (t, a_{n} e^{- t}) d t + k (b_{n}^{2} - a_{n}^{2})}{\frac{m_{0} - M_{1}^{2} L}{2 M_{1}^{2}} b_{n}^{2} - \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4} a_{n}^{2}} < + \infty$ ;

${lim sup}_{ξ \to + \infty} \frac{\int_{0}^{+ \infty} F_{1} (t, ξ e^{- t}) d t}{ξ^{2}} = + \infty$ .

Then, for every function

f_{i} \in C ([0, + \infty) \times R, R)

, denoting

F_{i} (t, x) = \int_{0}^{x} f_{i} (t, ξ) d ξ

for all

x \in R

for

2 ⩽ i ⩽ n

, satisfying

\begin{matrix} max {sup_{ξ \in R} F_{i} (t, ξ); 2 ⩽ i ⩽ n} ⩽ 0 \end{matrix}

and

\begin{matrix} min {\underset{ξ \to + \infty}{lim inf} \frac{F_{i} (t, ξ)}{ξ^{2}}; 2 ⩽ i ⩽ n} > - \infty, \end{matrix}

for each

\begin{matrix} λ \in (0, \frac{1}{{lim}_{n \to + \infty} \frac{\int_{0}^{+ \infty} {sup}_{| x | ⩽ b_{n}} F_{1} (t, x) d t - \int_{0}^{+ \infty} F_{1} (t, a_{n} e^{- t}) d t + k (b_{n}^{2} - a_{n}^{2})}{\frac{m_{0} - M_{1}^{2} L}{2 M_{1}^{2}} b_{n}^{2} - \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4} a_{n}^{2}}}), \end{matrix}

the problem

\begin{matrix} \{\begin{matrix} K (\int_{0}^{+ \infty} (| u^{'} (t) |^{2} + q (t) | u (t) |^{2}) d t) (- u^{″} (t) + q (t) u (t)) \\ = λ \sum_{i = 1}^{n} f_{i} (t, u), t \in [0, + \infty), t \neq t_{j}, \\ Δ (u^{'} (t_{j})) = λ I_{j} (u (t_{j})), \\ u^{'} (0^{+}) = g (u (0)), u^{'} (+ \infty) = 0 \end{matrix} \end{matrix}

has an unbounded sequence of weak solutions.

Proof.

Set $F (t, ξ) = \sum_{i = 1}^{n} F_{i} (t, ξ)$ for all $ξ \in R$ . Assumption $(A_{4})$ along with the condition $\begin{matrix} min {\underset{ξ \to + \infty}{lim inf} \frac{F_{i} (t, ξ)}{ξ^{2}}; 2 ⩽ i ⩽ n} > - \infty \end{matrix}$ ensures $\begin{array}{l} \underset{ξ \to + \infty}{lim sup} \frac{\int_{0}^{+ \infty} F (t, ξ e^{- t}) d t}{ξ^{2}} = \underset{ξ \to + \infty}{lim sup} \frac{\sum_{i = 1}^{n} \int_{0}^{+ \infty} F_{i} (t, ξ e^{- t}) d t}{ξ^{2}} = + \infty . \end{array}$ Moreover, Assumption $(A_{4})$ together with the condition $\begin{matrix} max {sup_{ξ \in R} F_{i} (t, ξ); 2 ⩽ i ⩽ n} ⩽ 0, \end{matrix}$ implies $\begin{array}{l} lim_{n \to + \infty} \frac{\int_{0}^{+ \infty} {sup}_{| x | ⩽ b_{n}} F (t, x) d t - \int_{0}^{+ \infty} F (t, a_{n} e^{- t}) d t + k (b_{n}^{2} - a_{n}^{2})}{\frac{m_{0} - M_{1}^{2} L}{2 M_{1}^{2}} b_{n}^{2} - \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4} a_{n}^{2}} \\ ⩽ lim_{n \to + \infty} \frac{\int_{0}^{+ \infty} {sup}_{| x | ⩽ b_{n}} F_{1} (t, x) d t - \int_{0}^{+ \infty} F_{1} (t, a_{n} e^{- t}) d t + k (b_{n}^{2} - a_{n}^{2})}{\frac{m_{0} - M_{1}^{2} L}{2 M_{1}^{2}} b_{n}^{2} - \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4} a_{n}^{2}} < + \infty . \end{array}$ Hence, the conclusion follows from Theorem 4.1. □

Now put $\begin{matrix} B^{0} = \underset{ξ \to 0^{+}}{lim sup} \frac{\int_{0}^{+ \infty} F (t, ξ e^{- t}) d t + k ξ^{2}}{ξ^{2}} . \end{matrix}$ Arguing as in the proof of Theorem 4.1, but using conclusion (c) of Theorem 2.1 instead of (b), one establishes the following result.

Theorem 4.8.

Assume that there exist two real sequences ${d_{n}}$ and ${e_{n}}$ with ${lim}_{n \to + \infty} e_{n} = 0$ , such that

$d_{n} < \sqrt{\frac{2 (m_{0} - M_{1}^{2} L)}{M_{1}^{2} (m_{1} + M_{1}^{2} L) (\overline{q} + 1)}} e_{n}$ ;

$\begin{array}{l} A_{0} & : = lim_{n \to + \infty} \frac{\int_{0}^{+ \infty} {max}_{| x | ⩽ e_{n}} F (t, x) d t - \int_{0}^{+ \infty} F (t, d_{n} e^{- t}) d t + k (e_{n}^{2} - d_{n}^{2})}{\frac{m_{0} - M_{1}^{2} L}{2 M_{1}^{2}} e_{n}^{2} - \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4} d_{n}^{2}} \\ < \frac{4}{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)} B^{0} . \end{array}$

Then, for each

λ \in (λ_{3}, λ_{4})

with

λ_{3} : = \frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4 B^{0}}

and

λ_{4} : = \frac{1}{A_{0}}

, the problem ( P λ f , g ) has a sequence of pairwise distinct weak solutions which strongly converges to 0 in X.

Theorem 4.9.

Assume that

$\begin{array}{l} \underset{ξ \to 0^{+}}{lim inf} \frac{\int_{0}^{+ \infty} {sup}_{| x | ⩽ ξ} F (t, x) d t + k ξ^{2}}{ξ^{2}} \\ < \frac{2 (m_{0} - M_{1}^{2} L)}{M_{1}^{2} (m_{1} + M_{1}^{2} L) (\overline{q} + 1)} \underset{ξ \to 0^{+}}{lim sup} \frac{\int_{0}^{+ \infty} F (t, ξ e^{- t}) d t + k ξ^{2}}{ξ^{2}} . \end{array}$

Then, for each

\begin{array}{l} λ \in (\frac{(m_{1} + M_{1}^{2} L) (\overline{q} + 1)}{4 {lim sup}_{ξ \to 0^{+}} \frac{\int_{0}^{+ \infty} F (t, ξ e^{- t}) d t + k ξ^{2}}{ξ^{2}}}, \\ \frac{m_{0} - M_{1}^{2} L}{2 M_{1}^{2} {lim inf}_{ξ \to 0^{+}} \frac{\int_{0}^{+ \infty} {sup}_{| x | ⩽ ξ} F (t, x) d t + k ξ^{2}}{ξ^{2}}}), \end{array}

the problem ( P λ f , g ) has a sequence of pairwise distinct weak solutions which strongly converges to 0 in X.

We now exhibit an example in which the hypotheses of Theorem 4.9 are satisfied.

Example 4.10.

We consider the problem $\begin{matrix} (4.4) & \{\begin{matrix} K (\int_{0}^{+ \infty} (| u^{'} (t) |^{2} + q (t) | u (t) |^{2}) d t) (- u^{″} (t) + q (t) u (t)) \\ = λ f (t, u (t)), t \in [0, + \infty), t \neq t_{1}, \\ Δ (u^{'} (t_{1})) = λ I_{1} (u (t_{1})), \\ u^{'} (0^{+}) = g (u (0)), u^{'} (+ \infty) = 0 \end{matrix} \end{matrix}$ where $K (x) = 2 + cos (x)$ for every $x \in [0, + \infty)$ , $q (t) = \frac{1}{1 + t^{2}} + 1$ for every $t \in [0, + \infty)$ , $I_{1} (ξ) = 3 (1 - cos (ξ))$ for every $ξ \in R$ , $\begin{matrix} f (t, x) = \{\begin{matrix} 0, & if x \in] - \infty, 0], \\ 2 e^{- t} x (1 + 27 {cos}^{2} (ln \frac{1}{x}) - 27 cos (ln \frac{1}{x}) sin (ln \frac{1}{x})), & if x \in] 0, \infty [ \end{matrix} \end{matrix}$ and $g (x) = \frac{\sqrt{2}}{2} ln (1 + x^{2})$ for every $x \in R$ . By choosing $k = 2$ , (2.2) is satisfied. By the expression of f we have $\begin{matrix} F (t, x) = \{\begin{matrix} 0, & if x \in] - \infty, 0], \\ e^{- t} x^{2} (1 + 27 {cos}^{2} (ln \frac{1}{x})), & if x \in] 0, \infty [. \end{matrix} \end{matrix}$ According to the above data, we have $m_{0} = 1$ , $m_{1} = 3$ , $M_{1} = \frac{1}{2^{\frac{1}{4}}}$ and $L = \frac{\sqrt{2}}{2}$ . Since $w (t) = ξ e^{- t} ⩽ ξ$ for all $t \in [0, + \infty)$ , then $\begin{matrix} F (t, ξ e^{- t}) ⩽ F (t, ξ), \end{matrix}$ so, $\begin{matrix} \underset{ξ \to 0^{+}}{lim sup} \frac{\int_{0}^{+ \infty} F (t, ξ) d t + 2 ξ^{2}}{ξ^{2}} = 30 \end{matrix}$ and $\begin{matrix} \underset{ξ \to 0^{+}}{lim inf} \frac{\int_{0}^{+ \infty} {sup}_{| x | ⩽ ξ} F (t, x) d t + 2 ξ^{2}}{ξ^{2}} = 3, \end{matrix}$ then all conditions in Theorem 4.9 are satisfied. Therefore, it follows that for each $λ \in] \frac{21}{240}, \frac{\sqrt{2}}{12} [$ the problem (4.4) has an unbounded sequence of weak solutions in X.

Remark 4.11.

Applying Theorem 4.8, results similar to Remark 4.2 and Corollaries 4.5, 4.6 and 4.7 can be obtained.

Footnotes

Acknowledgement

We would like to show our great thanks to Professor Marek Galewski for his valuable suggestions and comments, which improved the former version of this paper and made us to rewrite the paper in a more clear way.

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