# Quasilinear boundary value problem with impulses: variational approach to resonance problem

## Abstract

This paper deals with the resonance problem for the one-dimensional p-Laplacian with homogeneous Dirichlet boundary conditions and with nonlinear impulses in the derivative of the solution at prescribed points. The sufficient condition of Landesman-Lazer type is presented and the existence of at least one solution is proved. The proof is variational and relies on the linking theorem.

MSC:34A37, 34B37, 34F15, 49K35.

## 1 Introduction

Let $p>1$ be a real number. We consider the homogeneous Dirichlet boundary value problem for one-dimensional p-Laplacian

(1)

where $\lambda \in \mathbb{R}$ is a spectral parameter and $f\in {L}^{{p}^{\prime }}\left(0,1\right)$, $\frac{1}{p}+\frac{1}{{p}^{\prime }}=1$, is a given right-hand side.

Let $0={t}_{0}<{t}_{1}<\cdots <{t}_{r}<{t}_{r+1}=1$ be given points and let ${I}_{j}:\mathbb{R}\to \mathbb{R}$, $j=1,2,\dots ,r$, be given continuous functions. We are interested in the solutions of (1) satisfying the impulse conditions in the derivative

${\mathrm{\Delta }}_{p}{u}^{\prime }\left({t}_{j}\right):={|{u}^{\prime }\left({t}_{j}^{+}\right)|}^{p-2}{u}^{\prime }\left({t}_{j}^{+}\right)-{|{u}^{\prime }\left({t}_{j}^{-}\right)|}^{p-2}{u}^{\prime }\left({t}_{j}^{-}\right)={I}_{j}\left(u\left({t}_{j}\right)\right),\phantom{\rule{1em}{0ex}}j=1,2,\dots ,r.$
(2)

For the sake of brevity, in further text we use the following notation:

$\phi \left(s\right):={|s|}^{p-2}s,\phantom{\rule{1em}{0ex}}s\ne 0;\phantom{\rule{2em}{0ex}}\phi \left(0\right):=0.$

For $p=2$ this problem is considered in [1] where the necessary and sufficient condition for the existence of a solution of (1) and (2) is given. In fact, in the so-called resonance case, we introduce necessary and sufficient conditions of Landesman-Lazer type in terms of the impulse functions ${I}_{j}$, $j=1,2,\dots ,r$, and the right-hand side f. They generalize the Fredholm alternative for linear problem (1) with $p=2$.

In this paper we focus on a quasilinear equation with $p\ne 2$ and look just for sufficient conditions. We point out that there are principal differences between the linear case ($p=2$) and the nonlinear case ($p\ne 2$). In the linear case, we could benefit from the Hilbert structure of an abstract formulation of the problem. It could be treated using the topological degree as a nonlinear compact perturbation of a linear operator. However, in the nonlinear case, completely different approach must be chosen in the resonance case. Our variational proof relies on the linking theorem (see [2]), but we have to work in a Banach space since the Hilbert structure is not suitable for the case $p\ne 2$.

It is known that the eigenvalues of

$\begin{array}{r}-{\left(\phi \left({u}^{\prime }\left(x\right)\right)\right)}^{\prime }-\lambda \phi \left(u\left(x\right)\right)=0,\\ u\left(0\right)=u\left(1\right)=0\end{array}$
(3)

are simple and form an unbounded increasing sequence $\left\{{\lambda }_{n}\right\}$ whose eigenspaces are spanned by functions $\left\{{\varphi }_{n}\left(x\right)\right\}\subset {W}_{0}^{1,p}\left(0,1\right)\cap {C}^{1}\left[0,1\right]$ such that ${\varphi }_{n}$ has $n-1$ evenly spaced zeros in $\left(0,1\right)$, ${\parallel {\varphi }_{n}\parallel }_{{L}^{p}\left(0,1\right)}=1$, and ${\varphi }_{n}^{\prime }\left(0\right)>0$. The reader is invited to see [[3], p.388], [[4], p.780] or [[5], pp.272-275] for further details. See also Example 1 below for more explicit form of ${\lambda }_{n}$ and ${\varphi }_{n}$.

Let $\lambda \ne {\lambda }_{n}$, $n=1,2,\dots$ , in (1). This is the nonresonance case. Then, for any $f\in {L}^{{p}^{\prime }}\left(0,1\right)$, there exists at least one solution of (1). In the case $p=2$, this solution is unique. In the case $p\ne 2$, the uniqueness holds if $\lambda \le 0$, but it may fail for certain right-hand sides $f\in {L}^{{p}^{\prime }}\left(0,1\right)$ if $\lambda >0$. See, e.g., [6] (for $2) and [7] (for $1).

The same argument as that used for $p=2$ in [[1], Section 3] for the nonresonance case yields the following existence result for the quasilinear impulsive problem (1), (2).

Theorem 1 (Nonresonance case)

Let $\lambda \ne {\lambda }_{n}$, $n=1,2,\dots$ , ${I}_{j}:\mathbb{R}\to \mathbb{R}$, $j=1,2,\dots ,r$, be continuous functions which are $\left(p-1\right)$-subhomogeneous at ±∞, that is,

$\underset{|s|\to \mathrm{\infty }}{lim}\frac{{I}_{j}\left(s\right)}{{|s|}^{p-2}s}=0.$

Then (1), (2) has a solution for arbitrary $f\in {L}^{{p}^{\prime }}\left(0,1\right)$.

Variational approach to impulsive differential equations of the type (1), (2) with $p=2$ was used, e.g., in paper [8]. The authors apply the mountain pass theorem to prove the existence of a solution for $\lambda <{\lambda }_{1}$. Our Theorem 1 thus generalizes [[8], Theorem 5.2] in two directions. Firstly, it allows also $\lambda >{\lambda }_{1}$ ($\lambda \ne {\lambda }_{n}$, $n=2,3,\dots$) and, secondly, it deals with quasilinear equations ($p\ne 2$), too.

Let $\lambda ={\lambda }_{n}$ for some $n\in \mathbb{N}$. This is the resonance case. Contrary to the linear case ($p=2$), there is no Fredholm alternative for (1) in the nonlinear case ($p\ne 2$). If $\lambda ={\lambda }_{1}$, then

$f\in {\varphi }_{1}^{\mathrm{\perp }}:=\left\{h\in {L}^{\mathrm{\infty }}\left(0,1\right):{\int }_{0}^{1}h\left(x\right){\varphi }_{1}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x=0\right\}$

is the sufficient condition for solvability of (1), but it is not necessary if $p\ne 2$. Moreover, if $f\notin {\varphi }_{1}^{\mathrm{\perp }}$ but f is ‘close enough’ to ${\varphi }_{1}^{\mathrm{\perp }}$, problem (1) has at least two distinct solutions. The reader is referred to [3] or [9] for more details. It appears that the situation is even more complicated for $\lambda ={\lambda }_{n}$, $n\ge 2$ (see, e.g., [10]).

In the presence of nonlinear impulses which have certain asymptotic properties (to be made precise below), we show that the fact $f\in {\varphi }_{n}^{\mathrm{\perp }}$ might still be the sufficient condition for the existence of a solution to (1) (with $\lambda ={\lambda }_{n}$) and (2). For this purpose we need some notation. Let $0<{x}_{1}<{x}_{2}<\cdots <{x}_{n-1}<1$ denote evenly spaced zeros of ${\varphi }_{n}$, let ${\mathcal{I}}_{+}=\left(0,{x}_{1}\right)\cup \left({x}_{2},{x}_{3}\right)\cup \cdots$ and ${\mathcal{I}}_{-}=\left({x}_{1},{x}_{2}\right)\cup \left({x}_{3},{x}_{4}\right)\cup \cdots$ denote the union of intervals where ${\varphi }_{n}>0$ or ${\varphi }_{n}<0$, respectively. We arrange ${t}_{j}$, $j=1,2,\dots ,r$, into three sequences: $0<{\tau }_{1}<{\tau }_{2}<\cdots <{\tau }_{{r}_{+}}<1$, ${\tau }_{i}\in {\mathcal{I}}_{+}$, $i=1,2,\dots ,{r}_{+}$; $0<{\sigma }_{1}<{\sigma }_{2}<\cdots <{\sigma }_{{r}_{-}}<1$, ${\sigma }_{j}\in {\mathcal{I}}_{-}$, $j=1,2,\dots ,{r}_{-}$; ${\xi }_{k}\in \left\{{x}_{1},{x}_{2},\dots ,{x}_{n-1}\right\}$, $k=1,2,\dots ,{r}_{0}$. Obviously, we have ${r}_{+}+{r}_{-}+{r}_{0}=r$ and ${r}_{0}\le n-1$. Assume that ${r}_{+}+{r}_{-}>0$, i.e., ${r}_{0}. The impulse condition (2) can be written in an equivalent form

$\begin{array}{r}{\mathrm{\Delta }}_{p}{u}^{\prime }\left({\tau }_{i}\right)={I}_{i}^{\tau }\left(u\left({\tau }_{i}\right)\right),\phantom{\rule{1em}{0ex}}i=1,2,\dots ,{r}_{+},\\ {\mathrm{\Delta }}_{p}{u}^{\prime }\left({\sigma }_{j}\right)={I}_{j}^{\sigma }\left(u\left({\sigma }_{j}\right)\right),\phantom{\rule{1em}{0ex}}j=1,2,\dots ,{r}_{-},\\ {\mathrm{\Delta }}_{p}{u}^{\prime }\left({\xi }_{k}\right)={I}_{k}^{\xi }\left(u\left({\xi }_{k}\right)\right),\phantom{\rule{1em}{0ex}}k=1,2,\dots ,{r}_{0}.\end{array}$
(4)

We assume that ${I}_{i}^{\tau },{I}_{j}^{\sigma },{I}_{k}^{\xi }:\mathbb{R}\to \mathbb{R}$, $i=1,2,\dots ,{r}_{+}$; $j=1,2,\dots ,{r}_{-}$; $k=1,2,\dots ,{r}_{0}$, are continuous, bounded functions and there exist limits ${lim}_{s\to ±\mathrm{\infty }}{I}_{i}^{\tau }\left(s\right)={I}_{i}^{\tau }\left(±\mathrm{\infty }\right)$, ${lim}_{s\to ±\mathrm{\infty }}{I}_{j}^{\sigma }\left(s\right)={I}_{j}^{\sigma }\left(±\mathrm{\infty }\right)$. We consider the following Landesman-Lazer type conditions: either

$\begin{array}{rl}\sum _{i=1}^{{r}_{+}}{I}_{i}^{\tau }\left(-\mathrm{\infty }\right){\varphi }_{n}\left({\tau }_{i}\right)+\sum _{j=1}^{{r}_{-}}{I}_{j}^{\sigma }\left(+\mathrm{\infty }\right){\varphi }_{n}\left({\sigma }_{j}\right)& <{\int }_{0}^{1}f\left(x\right){\varphi }_{n}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ <\sum _{i=1}^{{r}_{+}}{I}_{i}^{\tau }\left(+\mathrm{\infty }\right){\varphi }_{n}\left({\tau }_{i}\right)+\sum _{j=1}^{{r}_{-}}{I}_{j}^{\sigma }\left(-\mathrm{\infty }\right){\varphi }_{n}\left({\sigma }_{j}\right)\end{array}$
(5)

or

$\begin{array}{rl}\sum _{i=1}^{{r}_{+}}{I}_{i}^{\tau }\left(+\mathrm{\infty }\right){\varphi }_{n}\left({\tau }_{i}\right)+\sum _{j=1}^{{r}_{-}}{I}_{j}^{\sigma }\left(-\mathrm{\infty }\right){\varphi }_{n}\left({\sigma }_{j}\right)& <{\int }_{0}^{1}f\left(x\right){\varphi }_{n}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ <\sum _{i=1}^{{r}_{+}}{I}_{i}^{\tau }\left(-\mathrm{\infty }\right){\varphi }_{n}\left({\tau }_{i}\right)+\sum _{j=1}^{{r}_{-}}{I}_{j}^{\sigma }\left(+\mathrm{\infty }\right){\varphi }_{n}\left({\sigma }_{j}\right).\end{array}$
(6)

Our main result is the following.

Theorem 2 (Resonance case)

Let $\lambda ={\lambda }_{n}$ for some $n\in \mathbb{N}$ in (1). Let the nonlinear bounded impulse functions ${I}_{j}:\mathbb{R}\to \mathbb{R}$, $j=1,2,\dots ,r$, and the right-hand side $f\in {L}^{{p}^{\prime }}\left(0,1\right)$ satisfy either (5) or (6). Then (1), (2) has a solution.

The result from Theorem 2 is illustrated in the following special example.

Example 1 It follows from the first integral associated with the equation in (3) that the eigenvalues and the eigenfunctions of (3) have the form

${\lambda }_{n}=\left(p-1\right){\left(n{\pi }_{p}\right)}^{p},\phantom{\rule{2em}{0ex}}{\varphi }_{n}\left(x\right)=\frac{{sin}_{p}\left(n{\pi }_{p}x\right)}{{\parallel {sin}_{p}\left(n{\pi }_{p}x\right)\parallel }_{{L}^{p}\left(0,1\right)}},$

where ${\pi }_{p}=\frac{2\pi }{psin\frac{\pi }{p}}$ and $x={\int }_{0}^{{sin}_{p}x}\frac{\mathrm{d}s}{{\left(1-{s}^{p}\right)}^{\frac{1}{p}}}$, $x\in \left[0,\frac{{\pi }_{p}}{2}\right]$, ${sin}_{p}x={sin}_{p}\left({\pi }_{p}-x\right)$, $x\in \left[\frac{{\pi }_{p}}{2},{\pi }_{p}\right]$, ${sin}_{p}x=-{sin}_{p}\left(2{\pi }_{p}-x\right)$, $x\in \left[{\pi }_{p},2{\pi }_{p}\right]$, see [[3], p.388]. Let us consider $\lambda ={\lambda }_{2}$ in (1) and ${t}_{1}=\frac{{\pi }_{p}}{4}$, ${t}_{2}=\frac{3{\pi }_{p}}{4}$, ${I}_{j}\left(s\right)=arctans$, $s\in \mathbb{R}$, $j=1,2$, in (2). Since $sin\frac{{\pi }_{p}}{2}=\frac{1}{p-1}$, $sin\frac{3{\pi }_{p}}{2}=-\frac{1}{p-1}$, condition (6) reads as follows:

$-\frac{\pi }{p-1}<{\int }_{0}^{1}f\left(x\right){sin}_{p}2{\pi }_{p}x\phantom{\rule{0.2em}{0ex}}\mathrm{d}x<\frac{\pi }{p-1}.$

## 2 Functional framework

We say that u is the classical solution of (1), (2) if the following conditions are fulfilled:

• $u\in C\left[0,1\right]$, $u\in {C}^{1}\left({t}_{j},{t}_{j+1}\right)$, $\phi \left({u}^{\prime }\left(\cdot \right)\right)$ is absolutely continuous in $\left({t}_{j},{t}_{j+1}\right)$, $j=0,1,\dots ,r$;

• the equation in (1) holds a.e. in $\left(0,1\right)$ and $u\left(0\right)=u\left(1\right)=0$;

• one-sided limits ${u}^{\prime }\left({t}_{j}^{+}\right)$, ${u}^{\prime }\left({t}_{j}^{-}\right)$ exist finite and (2) holds.

We say that $u\in {W}_{0}^{1,p}\left(0,1\right)$ is a weak solution of (1), (2) if the integral identity

${\int }_{0}^{1}\phi \left({u}^{\prime }\left(x\right)\right){v}^{\prime }\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\lambda {\int }_{0}^{1}\phi \left(u\left(x\right)\right)v\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\sum _{j=1}^{r}{I}_{j}\left(u\left({t}_{j}\right)\right)v\left({t}_{j}\right)={\int }_{0}^{1}f\left(x\right)v\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x$
(7)

holds for any function $v\in {W}_{0}^{1,p}\left(0,1\right)$.

Integration by parts and the fundamental lemma in calculus of variations (see [[11], Lemma 7.1.9]) yields that every weak solution of (1), (2) is also a classical solution and vice versa. Indeed, let u be a weak solution of (1), (2), $v\in \mathcal{D}\left({t}_{j},{t}_{j+1}\right)$ (the space of smooth functions with a compact support in $\left({t}_{j},{t}_{j+1}\right)$, $j=0,1,\dots ,r$), $v\equiv 0$ elsewhere in $\left(0,1\right)$, then

${\int }_{{t}_{j}}^{{t}_{j+1}}\left(\phi \left({u}^{\prime }\left(x\right)\right)+{\int }_{0}^{x}\left[\lambda \phi \left(u\left(\tau \right)\right)+f\left(\tau \right)\right]\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau \right){v}^{\prime }\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x=0.$

Since v is arbitrary, we have $\phi \left({u}^{\prime }\left(x\right)\right)+{\int }_{0}^{x}\left[\lambda \phi \left(u\left(\tau \right)\right)+f\left(\tau \right)\right]\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau =0$ for a.e. $x\in \left({t}_{j},{t}_{j+1}\right)$. Then $\phi \left({u}^{\prime }\left(\cdot \right)\right)$ is absolutely continuous in $\left({t}_{j},{t}_{j+1}\right)$ and

$-{\left(\phi \left({u}^{\prime }\left(x\right)\right)\right)}^{\prime }-\lambda \phi \left(u\left(x\right)\right)=f\left(x\right)$
(8)

for a.e. $x\in \left({t}_{j},{t}_{j+1}\right)$, $j=0,1,\dots ,r$. Taking now $v\in {W}_{0}^{1,p}\left(0,1\right)$ arbitrary, integrating by parts in the first integral in (7) and using (8), we get

$\sum _{j=1}^{r}\left[\phi \left({u}^{\prime }\left({t}_{j}^{+}\right)\right)-\phi \left({u}^{\prime }\left({t}_{j}^{-}\right)\right)\right]v\left({t}_{j}\right)=\sum _{j=1}^{r}{I}_{j}\left(u\left({t}_{j}\right)\right)v\left({t}_{j}\right),$

and hence also (2) follows. Similarly, we show that every classical solution is a weak solution at the same time.

Let $X:={W}_{0}^{1,p}\left(0,1\right)$ with the norm $\parallel u\parallel ={\left({\int }_{0}^{1}{|{u}^{\prime }\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\right)}^{\frac{1}{p}}$, ${X}^{\ast }$ be the dual of X and $〈\cdot ,\cdot 〉$ be the duality pairing between ${X}^{\ast }$ and X. For $u\in X$, we set

$\begin{array}{c}A\left(u\right):=\frac{1}{p}{\int }_{0}^{1}{|{u}^{\prime }\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x,\phantom{\rule{2em}{0ex}}B\left(u\right):=\frac{1}{p}{\int }_{0}^{1}{|u\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x,\hfill \\ F\left(u\right)={\int }_{0}^{1}f\left(x\right)u\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x,\phantom{\rule{2em}{0ex}}J\left(u\right):=\sum _{j=1}^{r}{\int }_{0}^{u\left({t}_{j}\right)}{I}_{j}\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s.\hfill \end{array}$

Then, for $u,v\in X$, we have

$\begin{array}{c}〈{A}^{\prime }\left(u\right),v〉={\int }_{0}^{1}\phi \left({u}^{\prime }\left(x\right)\right){v}^{\prime }\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x,\phantom{\rule{2em}{0ex}}〈{B}^{\prime }\left(u\right),v〉={\int }_{0}^{1}\phi \left(u\left(x\right)\right)v\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x,\hfill \\ 〈{F}^{\prime },v〉={\int }_{0}^{1}f\left(x\right)v\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x,\phantom{\rule{2em}{0ex}}〈{J}^{\prime }\left(u\right),v〉=\sum _{j=1}^{r}{I}_{j}\left(u\left({t}_{j}\right)\right)v\left({t}_{j}\right).\hfill \end{array}$

Lemma 1 The operators ${A}^{\prime },{B}^{\prime },{J}^{\prime }:X\to {X}^{\ast }$ have the following properties:

1. (A)

${A}^{\prime }$ is $\left(p-1\right)$-homogeneous, odd, continuously invertible, and ${\parallel {A}^{\prime }\left(u\right)\parallel }_{\ast }={\parallel u\parallel }^{p-1}$ for any $u\in X$.

2. (B)

${B}^{\prime }$ is $\left(p-1\right)$-homogeneous, odd and compact.

3. (J)

${J}^{\prime }$ is bounded and compact.

By the linearity of $F:X\to \mathbb{R}$, ${F}^{\prime }\in {X}^{\ast }$ is a fixed element.

Proof See [[12], Lemma 10.3, p.120]. □

With this notation in hands we can look for (classical) solutions of (1), (2) either as for solutions $u\in X$ of the operator equation

${A}^{\prime }\left(u\right)-\lambda {B}^{\prime }\left(u\right)+{J}^{\prime }\left(u\right)={F}^{\prime }$
(9)

or, alternatively, as for critical points of the functional $\mathcal{F}:X\to \mathbb{R}$,

$\mathcal{F}\left(u\right):=A\left(u\right)-\lambda B\left(u\right)+J\left(u\right)-F\left(u\right).$
(10)

As mentioned already above, in the nonresonance case ($\lambda \ne {\lambda }_{n}$, $n\in \mathbb{N}$), we can use the Leray-Schauder degree argument and prove the existence of a solution of the equation (9) exactly as in [[1], proof of Thm. 1]. Note that the $\left(p-1\right)$-subhomogeneous condition on ${I}_{j}$ is used here instead of the sublinear condition imposed on ${I}_{j}$ in [1] and the proof of Theorem 1 follows the same lines. For this reason we skip it and concentrate on the resonance case ($\lambda ={\lambda }_{n}$ for some $n\in \mathbb{N}$) in the next section.

## 3 Resonance problem, variational approach

We use the following definition of linked sets and the linking theorem (cf. [13]).

Definition 1 Let be a closed subset of X and let Q be a submanifold of X with relative boundary ∂Q. We say that and ∂Q link if

1. (i)

$\mathcal{E}\cap \partial Q=\mathrm{\varnothing }$ and

2. (ii)

for any continuous map $h:X\to X$ such that $h|\partial Q=\mathrm{id}$, there holds $h\left(Q\right)\cap \mathcal{E}\ne \mathrm{\varnothing }$.

(See [[14], Def. 8.1, p.116].)

Suppose that $\mathcal{F}\in {C}^{1}\left(X\right)$ satisfies the Palais-Smale condition. Consider a closed subset $\mathcal{E}\subset X$ and a submanifold $Q\subset X$ with relative boundary ∂Q, and let $\mathrm{\Gamma }:=\left\{h\in {C}^{0}\left(X,X\right):h|\partial Q=\mathrm{id}\right\}$. Suppose that and ∂Q link in the sense of Definition  1, and

$\underset{u\in \mathcal{E}}{inf}\mathcal{F}\left(u\right)>\underset{u\in \partial Q}{sup}\mathcal{F}\left(u\right).$

Then $\beta ={inf}_{h\in \mathrm{\Gamma }}{sup}_{u\in Q}\mathcal{F}\left(h\left(u\right)\right)$ is a critical value of .

(See [[14], Thm. 8.4, p.118].)

The purpose of the following series of lemmas is to show that the hypotheses of Theorem 3 are satisfied provided that either (5) or (6) holds. From now on we assume that $\lambda ={\lambda }_{n}$ (for some $n\in \mathbb{N}$) in (1).

Lemma 2 If either (5) or (6) is satisfied, then satisfies the Palais-Smale condition.

Proof Suppose that $\left\{{u}_{k}\right\}\in X$ such that $|\mathcal{F}\left({u}_{k}\right)|\le c$ and ${\mathcal{F}}^{\prime }\left({u}_{k}\right)\to 0$ in ${X}^{\ast }$. We must show that $\left\{{u}_{k}\right\}$ has a subsequence that converges in X. We prove first that $\left\{{u}_{k}\right\}$ is a bounded sequence. We proceed via contradiction and suppose that $\parallel {u}_{k}\parallel \to \mathrm{\infty }$ and consider ${v}_{k}:=\frac{{u}_{k}}{\parallel {u}_{k}\parallel }$. Without loss of generality, we can assume that there is ${v}_{0}\in X$ such that ${v}_{k}⇀{v}_{0}$ (weakly) in X (X is a reflexive Banach space). Since

$0←{\mathcal{F}}^{\prime }\left({u}_{k}\right)={A}^{\prime }\left({u}_{k}\right)-{\lambda }_{n}{B}^{\prime }\left({u}_{k}\right)+{J}^{\prime }\left({u}_{k}\right)-{F}^{\prime },$

dividing through by ${\parallel {u}_{k}\parallel }^{p-1}$, we have

${A}^{\prime }\left({v}_{k}\right)-{\lambda }_{n}{B}^{\prime }\left({v}_{k}\right)+\frac{{J}^{\prime }\left({u}_{k}\right)}{{\parallel {u}_{k}\parallel }^{p-1}}-\frac{{F}^{\prime }}{{\parallel {u}_{k}\parallel }^{p-1}}\to 0.$

By the boundedness of ${J}^{\prime }$ we know that $\frac{{J}^{\prime }\left({u}_{k}\right)}{{\parallel {u}_{k}\parallel }^{p-1}}\to 0$. We also have $\frac{{F}^{\prime }}{{\parallel {u}_{k}\parallel }^{p-1}}\to 0$. By the compactness of ${B}^{\prime }$ we get ${B}^{\prime }\left({v}_{k}\right)\to {B}^{\prime }\left({v}_{0}\right)$ in ${X}^{\ast }$. Thus ${v}_{k}\to {v}_{0}={\left({A}^{\prime }\right)}^{-1}\left({\lambda }_{n}{B}^{\prime }\left({v}_{0}\right)\right)$ in X by Lemma 1(A). It follows that ${v}_{0}=±\frac{1}{{\lambda }_{n}^{\frac{1}{p}}}{\varphi }_{n}$.

We assume ${v}_{0}=\frac{1}{{\lambda }_{n}^{\frac{1}{p}}}{\varphi }_{n}$ and remark that a similar argument follows if ${v}_{0}=-\frac{1}{{\lambda }_{n}^{\frac{1}{p}}}{\varphi }_{n}$. Next we estimate

$p\mathcal{F}\left({u}_{k}\right)-〈{\mathcal{F}}^{\prime }\left({u}_{k}\right),{u}_{k}〉=pJ\left({u}_{k}\right)-〈{J}^{\prime }\left({u}_{k}\right),{u}_{k}〉+\left(1-p\right){\int }_{0}^{1}f\left(x\right){u}_{k}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x.$
(11)

Our assumption $|\mathcal{F}\left({u}_{k}\right)|\le c$ yields

$-cp\le p\mathcal{F}\left({u}_{k}\right)\le cp$
(12)

and the Cauchy-Schwarz inequality implies

$-\parallel {u}_{k}\parallel {\parallel {\mathcal{F}}^{\prime }\left({u}_{k}\right)\parallel }_{\ast }\le -〈{\mathcal{F}}^{\prime }\left({u}_{k}\right),{u}_{k}〉\le \parallel {u}_{k}\parallel {\parallel {\mathcal{F}}^{\prime }\left({u}_{k}\right)\parallel }_{\ast },$
(13)

where ${\parallel \cdot \parallel }_{\ast }$ denotes the norm in ${X}^{\ast }$. It follows from (11)-(13) that

$\begin{array}{rl}-cp-\parallel {u}_{k}\parallel {\parallel {\mathcal{F}}^{\prime }\left({u}_{k}\right)\parallel }_{\ast }& \le pJ\left({u}_{k}\right)-〈{J}^{\prime }\left({u}_{k}\right),{u}_{k}〉+\left(1-p\right){\int }_{0}^{1}f\left(x\right){u}_{k}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le cp+\parallel {u}_{k}\parallel {\parallel {\mathcal{F}}^{\prime }\left({u}_{k}\right)\parallel }_{\ast }.\end{array}$

Dividing through by $\parallel {u}_{k}\parallel$ and writing $\frac{{\int }_{0}^{{u}_{k}\left({t}_{j}\right)}{I}_{j}\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s}{\parallel {u}_{k}\parallel }={\stackrel{ˆ}{I}}_{j}\left({u}_{k}\left({t}_{j}\right)\right){v}_{k}\left({t}_{j}\right)$, where

$j=0,1,\dots ,r$, we get

$\begin{array}{r}|p\sum _{j=1}^{r}{\stackrel{ˆ}{I}}_{j}\left({u}_{k}\left({t}_{j}\right)\right){v}_{k}\left({t}_{j}\right)-\sum _{j=1}^{r}{I}_{j}\left({u}_{k}\left({t}_{j}\right)\right){v}_{k}\left({t}_{j}\right)+\left(1-p\right){\int }_{0}^{1}f\left(x\right){v}_{k}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x|\\ \phantom{\rule{1em}{0ex}}\le \frac{cp}{\parallel {u}_{k}\parallel }+{\parallel {\mathcal{F}}^{\prime }\left({u}_{k}\right)\parallel }_{\ast }\to 0.\end{array}$
(14)

Since ${\int }_{0}^{1}f\left(x\right){v}_{k}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\to \frac{1}{{\lambda }_{n}^{\frac{1}{p}}}{\int }_{0}^{1}f\left(x\right){\varphi }_{n}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x$ as $k\to \mathrm{\infty }$, we obtain from (14):

$\underset{k\to \mathrm{\infty }}{lim}\sum _{j=1}^{r}\left(p{\stackrel{ˆ}{I}}_{j}\left({u}_{k}\left({t}_{j}\right)\right)-{I}_{j}\left({u}_{k}\left({t}_{j}\right)\right)\right){v}_{k}\left({t}_{j}\right)=\frac{p-1}{{\lambda }_{n}^{\frac{1}{p}}}{\int }_{0}^{1}f\left(x\right){\varphi }_{n}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x.$
(15)

Recall that X embeds compactly in $C\left[0,1\right]$, so, without loss of generality, we assume that ${v}_{k}\left({t}_{j}\right)\to \frac{1}{{\lambda }_{n}^{\frac{1}{p}}}{\varphi }_{n}\left({t}_{j}\right)$, $j=0,1,\dots ,r$, as $k\to \mathrm{\infty }$. Hence, ${u}_{k}\left({t}_{j}\right)\to ±\mathrm{\infty }$ for ${t}_{j}\in {\mathcal{I}}_{±}$, which implies ${I}_{j}\left({u}_{k}\left({t}_{j}\right)\right)\to {I}_{j}\left(±\mathrm{\infty }\right)$ as well as ${\stackrel{ˆ}{I}}_{j}\left({u}_{k}\left({t}_{j}\right)\right)\to {I}_{j}\left(±\mathrm{\infty }\right)$ as $k\to \mathrm{\infty }$ by an application of the l’Hospital rule to $\frac{{\int }_{0}^{\sigma }{I}_{j}\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s}{\sigma }$. Notice that by the boundedness of ${I}_{j}$ we have

if ${t}_{j}$ is a zero point of ${\varphi }_{n}$ for some $j\in \left\{1,2,\dots ,r\right\}$. Thus, passing to the limit in (15) as $k\to \mathrm{\infty }$, we get

$\sum _{i=1}^{{r}_{+}}{I}_{i}^{\tau }\left(+\mathrm{\infty }\right){\varphi }_{n}\left({\tau }_{i}\right)+\sum _{j=1}^{{r}_{-}}{I}_{j}^{\sigma }\left(-\mathrm{\infty }\right){\varphi }_{n}\left({\sigma }_{j}\right)={\int }_{0}^{1}f\left(x\right){\varphi }_{n}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x,$

which contradicts (5) or (6). Hence $\left\{{u}_{k}\right\}$ is bounded.

By compactness there is a subsequence such that ${B}^{\prime }\left({u}_{k}\right)$ and ${J}^{\prime }\left({u}_{k}\right)$ converge in ${X}^{\ast }$ (see Lemma 1(B), (J)). Since ${\mathcal{F}}^{\prime }\left({u}_{k}\right)\to 0$ by our assumption, we also have that ${A}^{\prime }\left({u}_{k}\right)$ converges in ${X}^{\ast }$. Finally, ${u}_{k}={\left({A}^{\prime }\right)}^{-1}\left({A}^{\prime }\left({u}_{k}\right)\right)$ converges in X by Lemma 1(A). The proof is finished. □

With the Palais-Smale condition in hands, we can turn our attention to the geometry of the functional . To this end we have to find suitable sets which link in the sense of Definition 1. Actually, we use the sets constructed in [13] and explain that they fit with the hypotheses of Theorem 3 if either (5) or (6) is satisfied.

Consider the even functional

and the manifold

$\mathcal{S}:=\left\{u\in {W}_{0}^{1,p}\left(0,1\right):B\left(u\right)=1\right\}.$

For any $n\in \mathbb{N}$, let , where ${\mathcal{S}}^{n-1}$ represents the unit sphere in ${\mathbb{R}}^{n}$. Next we define

${\lambda }_{n}:=\underset{\mathcal{A}\in {\mathcal{F}}_{n}}{inf}\underset{u\in \mathcal{A}}{sup}E\left(u\right),\phantom{\rule{1em}{0ex}}n\in \mathbb{N}.$
(16)

It is proved in [[15], Section 3] that $\left\{{\lambda }_{n}\right\}$ is a sequence of eigenvalues of homogeneous problem (3). It then follows from the results in [16] that this sequence exhausts the set of all eigenvalues of (3) with the properties described in Section 1.

Now consider the functions ${\varphi }_{n,i}={\chi }_{\left[\frac{i-1}{n},\frac{i}{n}\right]}{\varphi }_{n}$ for $i=1,2,\dots ,n$, where ${\chi }_{\left[\frac{i-1}{n},\frac{i}{n}\right]}$ is a characteristic function of the interval $\left[\frac{i-1}{n},\frac{i}{n}\right]$, and let

Observe that ${\mathrm{\Lambda }}_{n}$ is symmetric and is homeomorphic to the unit sphere in ${\mathbb{R}}^{n}$. Moreover, for $u\in {\mathrm{\Lambda }}_{n}$, we have

$\begin{array}{rl}B\left(u\right)& =B\left({\alpha }_{1}{\varphi }_{n,1}+\cdots +{\alpha }_{n}{\varphi }_{n,n}\right)=B\left({\alpha }_{1}{\varphi }_{n,1}\right)+\cdots +B\left({\alpha }_{n}{\varphi }_{n,n}\right)\\ ={|{\alpha }_{1}|}^{p}B\left({\varphi }_{n,1}\right)+\cdots +{|{\alpha }_{n}|}^{p}B\left({\varphi }_{n,n}\right)=1.\end{array}$

Notice that the second equality holds thanks to the fact

$\left\{x:{\varphi }_{n,i}\left(x\right)\ne 0\right\}\cap \left\{x:{\varphi }_{n,j}\left(x\right)\ne 0\right\}=\mathrm{\varnothing }$

for $i\ne j$, $i,j=1,2,\dots ,n$, while the third one follows from the p-homogeneity of B. Thus ${\mathrm{\Lambda }}_{n}\subset \mathcal{S}$ and so ${\mathrm{\Lambda }}_{n}\in {\mathcal{F}}_{n}$. A similar computation then shows that $E\left(u\right)=A\left(u\right)={\lambda }_{n}$ for all $u\in {\mathrm{\Lambda }}_{n}$. For a given $T>0$, we let

${Q}_{n,T}:=\left\{su:0\le s\le T,u\in {\mathrm{\Lambda }}_{n}\right\}.$

Then ${Q}_{n,T}$ is homeomorphic to the closed unit ball in ${\mathbb{R}}^{n}$. For a given $c\in \mathbb{R}$, we denote by

${\mathcal{E}}_{c}:=\left\{u\in X:A\left(u\right)\ge cB\left(u\right)\right\}=\left\{u\in X\mathrm{\setminus }\left\{0\right\}:E\left(u\right)\ge c\right\}\cup \left\{0\right\}$

a super-level set, and

${\mathcal{K}}_{c}:=\left\{u\in X\mathrm{\setminus }\left\{0\right\}:E\left(u\right)=c,{E}^{\prime }\left(u\right)=0\right\}.$

The existence of a pseudo-gradient vector field with the following properties is proved in [[13], Lemma 6] (cf. [[14], pp.77-79] and [[2], p.55]).

Lemma 3 For $\epsilon , there is $\stackrel{˜}{\epsilon }\in \left(0,\epsilon \right)$ and a one-parameter family of homeomorphisms $\eta :\left[-1,1\right]×\mathcal{S}\to \mathcal{S}$ such that

1. (i)

$\eta \left(t,u\right)=u$ if $E\left(u\right)\in \left(-\mathrm{\infty },{\lambda }_{n}-\epsilon \right]\cup \left[{\lambda }_{n}+\epsilon ,\mathrm{\infty }\right)$ or if $u\in {\mathcal{K}}_{{\lambda }_{n}}$;

2. (ii)

$E\left(\eta \left(t,u\right)\right)$ is strictly decreasing in t if $E\left(u\right)\in \left({\lambda }_{n}-{\stackrel{˜}{\epsilon }}_{n},{\lambda }_{n}+{\stackrel{˜}{\epsilon }}_{n}\right)$ and $u\notin {\mathcal{K}}_{{\lambda }_{n}}$;

3. (iii)

$\eta \left(t,-u\right)=-\eta \left(t,u\right)$;

4. (iv)

$\eta \left(0,\cdot \right)=\mathrm{id}$.

An important fact is that the flow η ‘lowers’ ${Q}_{n,T}$ and ‘raises’ ${\mathcal{E}}_{{\lambda }_{n}}$ if we modify them as follows:

${\stackrel{˜}{\mathcal{E}}}_{{\lambda }_{n}}:=\left\{su:s\in \mathbb{R},u\in \eta \left(-1,{\mathcal{E}}_{{\lambda }_{n}}\cap \mathcal{S}\right)\right\}$

and

${\stackrel{˜}{Q}}_{n,T}:=\left\{su:0\le s\le T,u\in \eta \left(1,{\mathrm{\Lambda }}_{n}\right)\right\}.$

Then, by Lemma 3 and the definition of ${\mathcal{E}}_{{\lambda }_{n}}$, we have

$A\left(u\right)-{\lambda }_{n}B\left(u\right)\ge 0$

for $u\in {\stackrel{˜}{\mathcal{E}}}_{{\lambda }_{n}}$ with equality if and only if $u=c{\varphi }_{n}$ for some $c\in \mathbb{R}$. Similarly,

$A\left(u\right)-{\lambda }_{n}B\left(u\right)\le 0$

for $u\in {\stackrel{˜}{Q}}_{n,T}$ with equality if and only if $u=c{\varphi }_{n}$ for some $c\in \mathbb{R}$.

It is proved in [[13], Lemma 7] that the couple $\mathcal{E}:={\mathcal{E}}_{{\lambda }_{n+1}}$ and $Q:={\stackrel{˜}{Q}}_{n,T}$ satisfies condition (ii) from Definition 1. It is also proved in [[13], Lemma 8] that the couple $\mathcal{E}:={\stackrel{˜}{\mathcal{E}}}_{{\lambda }_{n}}$ and $Q:={Q}_{n-1,T}$ satisfies the same condition. To show that also other hypotheses of Theorem 3 are satisfied, we need some technical lemmas.

Lemma 4 If (6) is satisfied, then there exist $R>0$ and $\delta >0$ such that $〈{\mathcal{F}}^{\prime }\left(su\right),u〉\le -\delta$ for any $s\ge R$ and $u\in \eta \left(1,{\mathrm{\Lambda }}_{n}\right)$.

Proof We proceed via contradiction and assume that there exist ${s}_{k}\to \mathrm{\infty }$ and ${u}_{k}\in \eta \left(1,{\mathrm{\Lambda }}_{n}\right)$ such that

$\underset{k\to \mathrm{\infty }}{lim sup}〈{\mathcal{F}}^{\prime }\left({s}_{k}{u}_{k}\right),{u}_{k}〉\ge 0.$
(17)

Since $\eta \left(1,{\mathrm{\Lambda }}_{n}\right)$ is compact, we may assume, without loss of generality, that ${u}_{k}\to {u}_{0}$ in $\eta \left(1,{\mathrm{\Lambda }}_{n}\right)$ for some ${u}_{0}\in \eta \left(1,{\mathrm{\Lambda }}_{n}\right)$.

If ${u}_{0}\ne ±{p}^{\frac{1}{p}}{\varphi }_{n}$, then there exists $\epsilon >0$ such that

${\int }_{0}^{1}{|{u}_{0}^{\prime }\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-{\lambda }_{n}{\int }_{0}^{1}{|{u}_{0}\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le -\epsilon .$

Hence, there exists ${k}_{\epsilon }\in \mathbb{N}$ such that for any $k\ge {k}_{\epsilon }$ we have

${\int }_{0}^{1}{|{u}_{k}^{\prime }\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-{\lambda }_{n}{\int }_{0}^{1}{|{u}_{k}\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le -\frac{\epsilon }{2}.$

This implies

$〈{\mathcal{F}}^{\prime }\left({s}_{k}{u}_{k}\right),{u}_{k}〉\le -\frac{\epsilon }{2}{s}_{k}^{p-1}+\sum _{j=1}^{r}{I}_{j}\left({s}_{k}{u}_{k}\left({t}_{j}\right)\right){u}_{k}\left({t}_{j}\right)-{\int }_{0}^{1}f\left(x\right){u}_{k}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x$

for $k\ge {k}_{\epsilon }$. However, this contradicts (17).

If ${u}_{0}={p}^{\frac{1}{p}}{\varphi }_{n}$, we still have

${\int }_{0}^{1}{|{u}_{k}^{\prime }\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-{\lambda }_{n}{\int }_{0}^{1}{|{u}_{k}\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le 0,$

and so

$〈{\mathcal{F}}^{\prime }\left({s}_{k}{u}_{k}\right),{u}_{k}〉\le \sum _{j=1}^{r}{I}_{j}\left({s}_{k}{u}_{k}\left({t}_{j}\right)\right){u}_{k}\left({t}_{j}\right)-{\int }_{0}^{1}f\left(x\right){u}_{k}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x$

for all $k\in \mathbb{N}$. The boundedness of ${I}_{j}$, $j=1,2,\dots ,r$, and uniform convergence ${u}_{k}\to {p}^{\frac{1}{p}}{\varphi }_{n}$ as $k\to \mathrm{\infty }$ (due to continuous embedding $X↪C\left[0,1\right]$) then yield

$\begin{array}{rl}\underset{k\to \mathrm{\infty }}{lim}〈{\mathcal{F}}^{\prime }\left({s}_{k}{u}_{k}\right),{u}_{k}〉& \le {p}^{\frac{1}{p}}\left(\sum _{i=1}^{{r}_{+}}{I}_{i}^{\tau }\left(+\mathrm{\infty }\right){\varphi }_{n}\left({\tau }_{i}\right)+\sum _{j=1}^{{r}_{-}}{I}_{j}^{\sigma }\left(-\mathrm{\infty }\right){\varphi }_{n}\left({\sigma }_{j}\right)-{\int }_{0}^{1}f\left(x\right){\varphi }_{n}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\right)\\ <0\end{array}$

by the first inequality in (6). This contradicts (17) again. Notice that by the boundedness of ${I}_{j}$ we have

if ${t}_{j}$ is a zero point of ${\varphi }_{n}$ for some $j\in \left\{1,2,\dots ,r\right\}$. The case ${u}_{0}=-{p}^{\frac{1}{p}}{\varphi }_{n}$ is proved similarly using the second inequality in (6). □

Lemma 5 If (6) is satisfied, then there exists $T>0$ such that

$\underset{u\in {\mathcal{E}}_{{\lambda }_{n+1}}}{inf}\mathcal{F}\left(u\right)>\underset{u\in \partial {\stackrel{˜}{Q}}_{n,T}}{sup}\mathcal{F}\left(u\right).$
(18)

Proof There exists $\alpha \in \mathbb{R}$ such that for any $u\in {\mathcal{E}}_{{\lambda }_{n+1}}$ we have

$\mathcal{F}\left(u\right)\ge \frac{1}{p}\left({\lambda }_{n+1}-{\lambda }_{n}\right){\parallel u\parallel }_{{L}^{p}\left(0,1\right)}^{p}+\sum _{j=1}^{r}{\int }_{0}^{u\left({t}_{j}\right)}{I}_{j}\left(\zeta \right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}\zeta -{\int }_{0}^{1}f\left(x\right)u\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x>\alpha .$

By Lemma 4 there exists $c\in \mathbb{R}$ such that for all $s>R$ and $u\in \eta \left(1,{\mathrm{\Lambda }}_{n}\right)$ we have

$\mathcal{F}\left(su\right)=\mathcal{F}\left(Ru\right)+\mathcal{F}\left(su\right)-\mathcal{F}\left(Ru\right)=\mathcal{F}\left(Ru\right)+{\int }_{R}^{s}〈{\mathcal{F}}^{\prime }\left(\zeta u\right),u〉\phantom{\rule{0.2em}{0ex}}\mathrm{d}\zeta \le c-\delta \left(s-R\right).$

Thus there exists $T>R$ such that

$\mathcal{F}\left(su\right)\le c-\delta \left(s-R\right)<\alpha$

for all $s\ge T$, $u\in \eta \left(1,{\mathrm{\Lambda }}_{n}\right)$. In particular, $\mathcal{F}\left(u\right)<\alpha$ for all $u\in \partial {\stackrel{˜}{Q}}_{n,T}$ and (18) is proved. □

Now we can finish the proof of Theorem 2 under assumption (6). Indeed, it follows from (18) that ${\mathcal{E}}_{{\lambda }_{n+1}}\cap \partial {\stackrel{˜}{Q}}_{n,T}=\mathrm{\varnothing }$ and thus the hypotheses of Theorem 3 hold with $\mathcal{E}:={\mathcal{E}}_{{\lambda }_{n+1}}$ and $Q:={\stackrel{˜}{Q}}_{n,T}$. It then follows that has a critical point and hence (1), (2) has a solution.

Next we show that the sets $\mathcal{E}:={\stackrel{˜}{\mathcal{E}}}_{{\lambda }_{n}}$ and $Q:={Q}_{n-1,T}$ satisfy the hypotheses of Theorem 3 if (5) is satisfied.

The principal difference consists in the fact that, in contrast with $\eta \left(1,{\mathrm{\Lambda }}_{n}\right)$, the set $\eta \left(-1,{\mathcal{E}}_{{\lambda }_{n}}\cap \mathcal{S}\right)$ is not compact. That is why one more technical lemma is needed.

Lemma 6 For any ${\epsilon }^{\prime }>0$, there exists $\delta >0$ such that

$E\left(u\right)\ge {\lambda }_{n}+\delta$
(19)

for $u\in \eta \left(-1,{\mathcal{E}}_{{\lambda }_{n}}\cap \mathcal{S}\right)\setminus {B}_{{\epsilon }^{\prime }}\left(±{\varphi }_{n}\right)$. (Here ${B}_{{\epsilon }^{\prime }}\left(±{\varphi }_{n}\right)$ is the ball in X centered at $±{\varphi }_{n}$ with radius ${\epsilon }^{\prime }$.)

Proof We note that the pseudo-gradient flow η from Lemma 3 is constructed as a solution of the initial value problem $\frac{d}{dt}\eta \left(t,u\right)=-\stackrel{˜}{v}\left(\eta \left(t,u\right)\right)$, $\eta \left(0,\cdot \right)=\mathrm{id}$, where

$v\left(u\right)$ is a locally Lipschitz continuous symmetric pseudo-gradient vector field associated with E on $\stackrel{˜}{\mathcal{S}}$ and $\psi ⟶\left[0,1\right]$ is a smooth function such that $\psi \left(u\right)=1$ for u satisfying ${\lambda }_{n}-\stackrel{˜}{\epsilon }\le E\left(u\right)\le {\lambda }_{n}+\stackrel{˜}{\epsilon }$ and $\psi \left(u\right)=0$ for u satisfying $E\left(u\right)\le {\lambda }_{n}-\epsilon$ or ${\lambda }_{n}+\epsilon \le E\left(u\right)$.

Let ${\epsilon }^{\prime }>0$ and $u\in \eta \left(-1,{\mathcal{E}}_{{\lambda }_{n}}\cap \mathcal{S}\right)\setminus {B}_{{\epsilon }^{\prime }}\left(±{\varphi }_{n}\right)$. Without loss of generality, we may assume that $E\left(u\right)\le {\lambda }_{n}+\stackrel{˜}{\epsilon }$. Let ${u}_{0}\in {\mathcal{E}}_{{\lambda }_{n}}\cap \mathcal{S}$ be such that $u=\eta \left(-1,{u}_{0}\right)$. Observe that there is a constant $M>0$ such that for $t\in \left[-1,1\right]$ we have

$\parallel \frac{d}{dt}\eta \left(t,{u}_{0}\right)\parallel \le \parallel \stackrel{˜}{v}\left(\eta \left(t,{u}_{0}\right)\right)\parallel \le dist\left(\eta \left(t,{u}_{0}\right),{\mathcal{K}}_{{\lambda }_{n}}\right)\parallel \stackrel{˜}{v}\left(\eta \left(t,{u}_{0}\right)\right)\parallel

Hence $\eta \left(t,{u}_{0}\right)\notin {B}_{\frac{{\epsilon }^{\prime }}{2}}\left(±{\varphi }_{n}\right)$ for $t\in \left[-1,-1+\frac{{\epsilon }^{\prime }}{2M}\right]$. Since E satisfies the Palais-Smale condition on $\mathcal{S}$ (see [[13], Lemma 2]), there exists $\rho >0$ such that ${\parallel {E}^{\prime }\left(u\right)\parallel }_{\ast }\ge \rho$ for all $u\in \left\{w\in \mathcal{S}:{\lambda }_{n}\le E\left(w\right)\le {\lambda }_{n}+\stackrel{˜}{\epsilon }\right\}\setminus {B}_{\frac{{\epsilon }^{\prime }}{2}}\left(±{\varphi }_{n}\right)$. Then

$\begin{array}{rl}\parallel \frac{d}{dt}E\left(\eta \left(t,{u}_{0}\right)\right)\parallel & =\parallel 〈{E}^{\prime }\left(\eta \left(t,{u}_{0}\right)\right),\frac{d}{dt}\eta \left(t,{u}_{0}\right)〉\parallel \\ =\parallel \psi \left(\eta \left(t,{u}_{0}\right)\right)dist\left(\eta \left(t,{u}_{0}\right),{\mathcal{K}}_{{\lambda }_{n}}\right)〈{E}^{\prime }\left(\eta \left(t,{u}_{0}\right)\right),v\left(\eta \left(t,{u}_{0}\right)\right)〉\parallel \\ \ge 1\cdot \frac{{\epsilon }^{\prime }}{2}\cdot min\left\{\parallel {E}^{\prime }\left(\eta \left(t,{u}_{0}\right)\right)\parallel ,1\right\}\parallel {E}^{\prime }\left(\eta \left(t,{u}_{0}\right)\right)\parallel \ge \frac{{\epsilon }^{\prime }}{2}{\rho }^{2}\end{array}$

for all $t\in \left[-1,-1+\frac{{\epsilon }^{\prime }}{2M}\right]$. The last but one inequality holds due to the following property of $v\left(u\right)$:

$〈{E}^{\prime }\left(u\right),v\left(u\right)〉>min\left\{\parallel {E}^{\prime }\left(u\right)\parallel ,1\right\}\parallel {E}^{\prime }\left(u\right)\parallel$

(see [14] and [2]). We also used the fact that $\psi \left(\eta \left(t,{u}_{0}\right)\right)\equiv 1$ for $t\in \left[-1,0\right]$. Hence

$\begin{array}{rl}E\left(u\right)& =E\left(\eta \left(-1,{u}_{0}\right)\right)=E\left(\eta \left(-1+\frac{{\epsilon }^{\prime }}{2M},{u}_{0}\right)\right)+{\int }_{-1+\frac{{\epsilon }^{\prime }}{2M}}^{-1}\frac{\mathrm{d}}{\mathrm{d}t}E\left(\eta \left(t,{u}_{0}\right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\\ \ge E\left(\eta \left(-1+\frac{{\epsilon }^{\prime }}{2M},{u}_{0}\right)\right)+\frac{{\epsilon }^{\prime }}{2}{\rho }^{2}\cdot \frac{{\epsilon }^{\prime }}{2M}\ge {\lambda }_{n}+\delta \end{array}$

with $\delta =\frac{{\left({\epsilon }^{\prime }\rho \right)}^{2}}{4M}$. □

The following lemma is a counterpart of Lemma 4 in the case of condition (5).

Lemma 7 If (5) is satisfied, then there exist $R>0$ and $\delta >0$ such that $〈{\mathcal{F}}^{\prime }\left(su\right),u〉\ge \delta$ for any $s\ge R$ and $u\in \eta \left(-1,{\mathcal{E}}_{{\lambda }_{n}}\cap \mathcal{S}\right)$.

Proof We proceed via contradiction and assume that there exist ${s}_{k}\to \mathrm{\infty }$ and ${u}_{k}\in \eta \left(-1,{\mathcal{E}}_{{\lambda }_{n}}\cap \mathcal{S}\right)$ such that

$\underset{k\to \mathrm{\infty }}{lim sup}〈{\mathcal{F}}^{\prime }\left({s}_{k}{u}_{k}\right),{u}_{k}〉\le 0.$
(20)

If there is ${\epsilon }^{\prime }>0$ such that ${u}_{k}\in \eta \left(-1,{\mathcal{E}}_{{\lambda }_{n}}\cap \mathcal{S}\right)\setminus {B}_{{\epsilon }^{\prime }}\left(±{\varphi }_{n}\right)$ for all k large enough, then Lemma 6 leads to the estimate

$〈{\mathcal{F}}^{\prime }\left({s}_{k}{u}_{k}\right),{u}_{k}〉\ge \delta {s}_{k}^{p-1}+\sum _{j=1}^{r}{I}_{j}\left({s}_{k}{u}_{k}\left({t}_{j}\right)\right){u}_{k}\left({t}_{j}\right)-{\int }_{0}^{1}f\left(x\right){u}_{k}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x$

contradicting (20). Thus it must be ${u}_{k}\to ±{p}^{\frac{1}{p}}{\varphi }_{n}$ as $k\to \mathrm{\infty }$. If ${u}_{k}\to {p}^{\frac{1}{p}}{\varphi }_{n}$ as $k\to \mathrm{\infty }$, we still have

${\int }_{0}^{1}{|{u}_{k}^{\prime }\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-{\lambda }_{n}{\int }_{0}^{1}{|{u}_{k}\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\ge 0,$

and so

$〈{\mathcal{F}}^{\prime }\left({s}_{k}{u}_{k}\right),{u}_{k}〉\ge \sum _{j=1}^{r}{I}_{j}\left({s}_{k}{u}_{k}\left({t}_{j}\right)\right){u}_{k}\left({t}_{j}\right)-{\int }_{0}^{1}f\left(x\right){u}_{k}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x$

for all $k\in \mathbb{N}$. Similar arguments as in the proof of Lemma 4 lead to

$\begin{array}{rl}\underset{k\to \mathrm{\infty }}{lim}〈{\mathcal{F}}^{\prime }\left({s}_{k}{u}_{k}\right),{u}_{k}〉& \ge {p}^{\frac{1}{p}}\left(\sum _{i=1}^{{r}_{+}}{I}_{i}^{\tau }\left(+\mathrm{\infty }\right){\varphi }_{n}\left({\tau }_{i}\right)+\sum _{j=1}^{{r}_{-}}{I}_{j}^{\sigma }\left(-\mathrm{\infty }\right){\varphi }_{n}\left({\sigma }_{j}\right)-{\int }_{0}^{1}f\left(x\right){\varphi }_{n}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\right)\\ >0\end{array}$

by the second inequality in (5). This contradicts (20) again. The case ${u}_{k}\to -{p}^{\frac{1}{p}}{\varphi }_{n}$ as $k\to \mathrm{\infty }$ is proved similarly but using the first inequality in (5). □

Lemma 8 If (5) is satisfied, then there exists $T>0$ such that

$\underset{u\in {\stackrel{˜}{\mathcal{E}}}_{{\lambda }_{n}}}{inf}\mathcal{F}\left(u\right)>\underset{u\in \partial {Q}_{n-1,T}}{sup}\mathcal{F}\left(u\right).$
(21)

Proof By Lemma 7 there exists $d\in \mathbb{R}$ such that for all $s>R$ and $u\in \eta \left(-1,{\mathcal{E}}_{{\lambda }_{n}}\cap \mathcal{S}\right)$ we have

$\mathcal{F}\left(su\right)=\mathcal{F}\left(Ru\right)+\mathcal{F}\left(su\right)-\mathcal{F}\left(Ru\right)=\mathcal{F}\left(Ru\right)+{\int }_{R}^{s}〈{\mathcal{F}}^{\prime }\left(\zeta u\right),u〉\phantom{\rule{0.2em}{0ex}}\mathrm{d}\zeta \ge d+\delta \left(s-R\right).$

Hence, there exists $\alpha \in \mathbb{R}$ such that for any $u\in {\stackrel{˜}{\mathcal{E}}}_{{\lambda }_{n}}$ we have

$\mathcal{F}\left(u\right)>\alpha .$

On the other hand, for any $s>0$ and $u\in {\mathrm{\Lambda }}_{n-1}$, we get

$\begin{array}{rl}\mathcal{F}\left(su\right)& =\frac{1}{p}\left({\lambda }_{n-1}-{\lambda }_{n}\right){\parallel su\parallel }_{{L}^{p}\left(0,1\right)}^{p}+\sum _{j=1}^{r}{\int }_{0}^{su\left({t}_{j}\right)}{I}_{j}\left(\zeta \right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}\zeta -s{\int }_{0}^{1}f\left(x\right)u\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\left({\lambda }_{n-1}-{\lambda }_{n}\right){s}^{p}+\sum _{j=1}^{r}{\int }_{0}^{su\left({t}_{j}\right)}{I}_{j}\left(\zeta \right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}\zeta -s{\int }_{0}^{1}f\left(x\right)u\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x.\end{array}$

Thus, there exists $T>0$ such that, for $u\in \partial {Q}_{n-1,T}$,

$\mathcal{F}\left(u\right)<\alpha$

and (21) is proved. □

It follows that the sets $\mathcal{E}:={\stackrel{˜}{\mathcal{E}}}_{{\lambda }_{n}}$ and $Q:={Q}_{n-1,T}$ satisfy the hypotheses of Theorem 3 if (5) is satisfied. The proof of Theorem 2 is thus completed.

Final remark Reviewers of our manuscript suggested to include some recent references on impulsive problems. Variational approach to impulsive problems can be found, e.g., in [1721]. The last reference deals with the p-Laplacian with the variable exponent $p=p\left(t\right)$. Singular impulsive problems are treated in [2224]. Impulsive problems are still ‘hot topic’ attracting the attention of many mathematicians and the bibliography on that topic is vast.

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## Acknowledgements

This research was supported by Grant 13-00863S of the Grant Agency of Czech Republic and by the European Regional Development Fund (ERDF), project ‘NTIS - New Technologies for the Information Society’, European Centre of Excellence, CZ.1.05/1.1.00/02.0090.

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Correspondence to Martina Langerová.

### Competing interests

The authors declare that they have no competing interests.

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Drábek, P., Langerová, M. Quasilinear boundary value problem with impulses: variational approach to resonance problem. Bound Value Probl 2014, 64 (2014). https://doi.org/10.1186/1687-2770-2014-64