# Solvability of some boundary value problems involving p-Laplacian and non-autonomous differential operators

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## Abstract

*a*is a positive, continuous function and

*g*is a Caratheódory nonlinear function.

We prove an existence result, underlying the relationship between the behavior of \(g(t,x,\cdot )\) as \(y\to 0\) related to that of \(g(\cdot ,x,y)\) and \(h(\cdot ,x) \) as \(|t|\to +\infty \).

## Keywords

Boundary value problems Unbounded domains Heteroclinic solutions Nonlinear differential operators*p*-Laplacian operator

*Φ*-Laplacian operator

## MSC

34B40 34C37 34B15 34L30## 1 Introduction

Differential equations involving the *p*-Laplacian operator and its generalization, the so-called *Φ*-Laplacian, have been widely studied due to several applications in various sciences. Indeed, many models in non-Newtonian fluid theory, diffusion of flows in porous media, nonlinear elasticity, and theory of capillary surfaces can be expressed in terms of such differential operators.

*Φ*-Laplacian operator is

More recently, other types of differential operators, governed by an increasing function *Φ*, possibly singular and not necessarily surjective, have been considered. The theory on this subject can be found in [2] for operators having a bounded domain and in [1] for non-surjective operators.

*Φ*is a general increasing homeomorphism in \(\mathbb{R}\). The generality of the differential operator

*Φ*required a rather strong growth assumption on the right-hand side \(g(t,x,x')\) with respect to \(x'\) (see conditions 3.3 and 3.4 in [20, Theorem 3.1]). For instance, when \(g(t,x,y)=a(t)b(x)c(y)\) and

*Φ*has a superlinear growth at infinity, then the condition \(c(x')/|\varPhi (x')|\to 0\) as \(|x'|\to \infty \) is needed (among others) in order to obtain the existence of solutions (see [20, Corollary 4.13]). So, in the special case of the

*p*-Laplacian operator, the function \(g(t,x,x')\) has to grow less than \(p-1\) as \(|x'|\to +\infty \).

*p*-Laplacian operator, that is, when one has the following equation:

*g*considered in [20] can be improved and the class of solvable problems can be widened. More in detail, when \(g(t,x,y)=a(t)b(x)c(y)\), the assumption \(c(x')/|\varPhi (x')|\to 0\) as \(|x'|\to \infty \) can be replaced by \(c(y)=O(|y|^{p})\) (see condition (3.26)). To our knowledge, the existence result here presented is new also for the classical case \(p=2\). For instance, as an application of our results, we have that the differential equation

*a*,

*b*with \(a< b\), provided that

*m*is odd and \(m>2n+1\) (see Example 3.7). We underline that the previous equation can not be treated by means of the results in [20], since in this case \(p=2\) and the growth of

*f*with respect to \(x'\) is greater than \(p-1\).

## 2 Existence and non-existence theorem

In the whole paper we will consider a positive continuous function \(h:\mathbb{R}\times [a,b]\to \mathbb{R}\) and a Carathéodory function \(g:\mathbb{R}^{3}\to \mathbb{R}\).

Our main results are the following general existence and non-existence theorems.

### Theorem 2.1

*Suppose that*

*and that there exist constants*\(C_{1},C_{2}>0\),

*a continuous function*\(\mu :\mathbb{R}^{+}\to \mathbb{R}^{+}\),

*and a function*\(\lambda \in L^{q}([-C_{1},C_{1}])\)

*with*\(1\le q \le \infty \)

*such that*

*with*\(\frac{q-1}{q(p-1)}= \frac{1}{p-1}\)

*if*\(q=+\infty \)).

*At last*,

*suppose that there exists a constant*\(\gamma > 1\)

*such that*,

*for every*\(C>0\),

*there exists a function*\(\varLambda _{C} \in L^{1}_{ \mathrm{loc}}([0, +\infty ))\),

*null in*\([0,C_{1}]\)

*and positive in*\((C_{1},+\infty )\),

*such that*:

*and there exists a function*\(\eta _{C}\in L^{1}(\mathbb{R})\)

*such that putting*

*for every*\(x \in [a, b]\)

*and every*\(|y|\le N_{C}(t)\),

*we have*

*Then there exists a function*\(x \in C^{1}(\mathbb{R})\)

*such that*\(t\mapsto h(t,x(t))|x'(t)|^{p-1}x'(t))\)

*belongs to*\(W^{1,1}( \mathbb{R})\)

*and*

### Proof

*C*such that

By the same argument developed in the proof of [20, Theorem 3.1], it is possible to prove that problem (2.13) admits a solution \(u_{n}\) for every \(n >C_{1}\) such that \(a\le u_{n}(t) \le b\) for all \(t \in I_{n}\), so that \(U_{u_{n}}(t)=u_{n}(t)\) for every \(t\in I_{n}\). Moreover, \(u_{n}\) is increasing in \([-n,-C_{1}]\) and in \([C_{1},n]\) and if \(u_{n}'(t_{0})=0\) for some \(C_{1}<|t_{0}|<n\), then \(u_{n}'(t)=0\) whenever \(|t|>|t_{0}|\) (see Steps 1–2 in the proof of [20, Theorem 3.1]).

Now our goal is to prove that \(|u_{n}'(t)|\le N_{C}(t)\) for every \(t\in I_{n}\), so that also \(V_{u_{n}}(t)= u_{n}'(t)\) in \(I_{n}\) and consequently \(u_{n}\) is a solution of equation (2.1) too.

We now prove that \(u_{n}'(t)\le N_{C}(t)\) for every \(t\in I_{n}\setminus [-C_{1},C_{1}]\).

*x*of problem (2.10), satisfying all the properties stated in the assertion. □

The main tool in the previous existence theorem is the summability of function \(N_{C}(t)\) (condition (2.6)) combined with assumption (2.8). Such conditions are not improvable in the sense that if (2.8) is satisfied with the reversed inequality and \(N_{C}\) is not summable, then problem \((P)\) does not admit solutions, as stated in the following result.

### Theorem 2.2

*Suppose that there exist three constants*\(C_{1}\ge 0\), \(\rho >0\), \(\gamma >1\)

*and a positive function*\(\varLambda \in L^{1}_{\mathrm{loc}}([C _{1},+\infty ))\)

*such that one of the following pairs of conditions holds*:

*or*

*and for every constant*

*C*,

*the function*

*does not belong to*\(L^{1}([1,+\infty ))\).

*Moreover*,

*assume that*

*and there exist two constants*\(k, C_{2}>0\)

*such that*

*Then each possible solution**x**of problem*\((P)\)*is constant in*\([C_{1},+\infty )\) (*when* (2.15) *holds*) *or constant in*\((-\infty , -C_{1}]\) (*when* (2.16) *holds*).

*Therefore*, *if both* (2.15) *and* (2.16) *hold and*\(C_{1}=0\), *then problem*\((P)\)*does not admit solutions*, *that is*, *there exists no function*\(x\in C^{1}(\mathbb{R})\)*such that*\(t\mapsto h(t,x(t))|x'(t)|^{p-2}x'(t)\)*is almost everywhere differentiable*, *satisfying the conditions of problem*\((P)\).

### Remark 2.1

Of course, (2.19), (2.20) are satisfied (for \(C_{1}=0\)) when \(h(t,x)=h_{1}(t) h_{2}(x)\), provided that \(h_{1}(t)\) is decreasing in \((-\infty ,0)\) and increasing in \((0,+\infty )\), or \(h_{1}\) is uniformly continuous in \(\mathbb{R}\) and \({\inf_{t\in \mathbb{R}}} \,h_{1}(t)>0 \), or \(h_{1}(t)\sim |t|^{-k}\) as \(|t|\to + \infty \) for some \(k>0\), since \(h_{2}(x)>0\) on \([a,b]\).

## 3 Asymptotic criteria

We emphasize the link between the local behavior of \(g_{2}(x,\cdot )\) at \(y=0\) and of \(g_{1}(\cdot , x)\), \(h_{1}(\cdot )\) at infinity, which is crucial for the existence or non-existence of solutions.

The following existence theorems are application of Theorem 2.1.

### Proposition 3.1

*Suppose that*,

*for some*\(C_{1}>0\),

*we have*

*and there exists a function*\(\lambda \in L^{q}_{\mathrm{loc}}( \mathbb{R})\), \(1\le q \le + \infty \),

*such that*

*Moreover*,

*assume that there exist real constants*

*σ*,

*δ*,

*and*\(\gamma >1\)

*satisfying one of the following pairs of conditions*:

*such that*,

*for every*\(x\in [a, b]\),

*we have*

*for certain positive constants*\(h_{1}\), \(h_{2}\), \(k_{1}\), \(k_{2}\),

*ρ*, \(C_{2}\).

*Then problem*\((P)\)*admits solutions*.

### Proof

It is not restrictive to assume \(C_{2}>\max \{C_{1},\frac{b - a}{2C _{1}}\}\). Put \(\mu (r):=k_{2} (\frac{r}{m^{*}(C_{1})} ) ^{\frac{qp-1}{q(p-1)}}\) for \(r>0\) (see (2.2)), from (3.3) and (3.9) the validity of conditions (2.4) and (2.5) follows.

*Λ*is positive. Observe that by (3.10) it follows that

Therefore, we can apply Theorem 2.1 and obtain the assertion of the present result.

The case of (3.5) is similar.

*Λ*is the function defined above, by the first inequality in (3.5) we get that the integral function \(t\mapsto \int _{0}^{t} \frac{\varLambda (|s|)}{M(s)^{\gamma }}\, \mathrm{d} s \) is bounded. So, the second condition in (3.5) implies the validity of assumption (2.6). Moreover, if \(\eta _{C}\) is defined as above, then

### Remark 3.1

We underline that conditions (3.9) and (3.10) are compatible with each other for large \(|y|\) only if \(q>1\) and \(\gamma \le \frac{p-\frac{1}{q}}{p-1}\). However, if \(m_{\infty }>0\) (see (3.1)), condition (3.10) can be improved requiring that it holds only for \(|y|\) small enough, as the following result states.

### Proposition 3.2

### Proof

Notice that \(N_{C}(t)\le \varGamma _{C}\) for every \(t\ge C_{1}\); moreover \(g_{2}(x,y)\ge h_{C} |y|^{\gamma (p-1)}\) whenever \(|y|\le \varGamma _{C}\). So, condition (2.8) holds whenever \(|y|<\varGamma _{C}\) and the proof proceeds as that of Proposition 3.1, replacing everywhere \(k_{1}\) with \(h_{C}\). □

We state now two non-existence results, obtained applying Theorem 2.2.

### Proposition 3.3

*Suppose that*

*and let there exist real constants*

*δ*, \(\gamma >1\), \(\varLambda > 0\)

*and a positive function*\(\ell (t)\in L^{1}([0,\varLambda ])\)

*such that*

*for some positive constants*\(\lambda _{1}\), \(\lambda _{2}\),

*ρ*.

*Moreover*,

*assume that*(3.6)

*holds for some constants*\(h_{1}\), \(h_{2}\),

*σ*

*such that one of the following pairs of conditions is satisfied*:

*At last*,

*suppose that there exist two constants*\(\epsilon , C_{2}>0\)

*such that*

*Then problem*\((P)\)*does not admit solutions*.

### Proof

*Λ*is a positive function belonging to \(L^{1}_{\text{loc}}([0,+\infty ))\) and one can easily verify that conditions (3.15), (3.16), and (3.17) ensure the validity of (2.15) and (2.16) with \(C_{1}=0\). Moreover, by (3.6) and the very definition of

*Λ*, one has

*C*, we have

The case of (3.19) is similar.

By using the same notations, notice that under the first condition in (3.19) the integral function \(\rho (t) = \int _{0}^{t} \frac{\varLambda (\tau )}{m(\tau )^{\gamma }} \, \mathrm{d} \tau \) is bounded, hence \(N_{C}(t)^{p-1} \ge \text{ Const. } t^{- \sigma } \), implying that \(N_{C}(t) \ge \text{ Const. } t^{-\sigma /(p-1)}\). Therefore \(N_{C}\) is not summable at infinity owing to the second condition in assumption (3.19), and the assertion follows from Theorem 2.2, applied for \(C_{1}=0\). □

As an immediate application of the previous theorems, the following criteria hold.

### Corollary 3.4

*Let*\(g(t,x,y)=g_{1}(t)g_{2}(x) g_{3}(y)\), *where*\(g_{1}\in L^{q}_{ \mathrm{loc}}(\mathbb{R})\)*for some*\(1\le q\le +\infty \), \(g_{3}\)*is continuous in*\(\mathbb{R}\), *and*\(g_{2}\)*is continuous and positive in*\([a,b]\).

*Assume that*\(g_{3}(y)>0\)

*for*\(y\ne 0\); \(t\cdot g_{1}(t)\le 0\)

*for every*

*t*

*and suppose that there exist constants*\(c_{1},\dots ,c_{3}>0\)

*such that*

*with*

*Then*, *if conditions* (3.20), (3.21) *hold and*\(p \le \beta + \sigma -\delta \), \((P)\)*has no solution*.

*Vice versa*,

*if*\(p >\beta + \sigma -\delta \)

*and*

*then*\((P)\)

*admits solutions*.

### Proof

The assertion is an immediate consequence of Propositions 3.1 and 3.3 taking \(\gamma =\beta /(p-1)\). □

Taking into account what we have observed in Remark 3.1, the following result holds in the particular case \(m_{\infty }>0\).

### Corollary 3.5

*Let all the assumption of Corollary*3.4*hold*, *with the exception of* (3.27). *Then*, *if*\(m_{\infty }>0\), *problem*\((P)\)*admits solutions*.

When assumption (3.25) is not satisfied, we can use the following result, the consequence of Propositions 3.1 and 3.3.

### Corollary 3.6

*Let all the assumptions of Corollary*3.4

*be satisfied*,

*with the exception of*(3.25),

*that is*,

*assume that*

*Then*, *if conditions* (3.20), (3.21) *hold and*\(\sigma +1 \le p < \beta +1 \), \((P)\)*has no solution*.

*Vice versa*, *if*\(p < \sigma +1 \), \(p<\beta +1\), *and we further assume* (3.26) *and* (3.27), *then*\((P)\)*admits solutions*.

At last, a result analogous to Corollary 3.6 holds when condition (3.27) is removed, provided that \(m_{\infty }>0\), as in Corollary 3.5.

### Example 3.7

*β*,

*g*are generic positive continuous functions. By virtue of Corollary 3.4 taking \(q=\infty \), \(\sigma =n\), \(\delta =m\), \(s=2\), with \(m>2n-1\), we deduce that the differential equations admit solutions satisfying \(x(-\infty )=a\), \(x(+\infty )=b\), for any pair of ordered data

*a*,

*b*.

## Notes

### Acknowledgements

Not applicable.

### Availability of data and materials

Not applicable.

### Authors’ contributions

All authors read and approved the final manuscript.

### Funding

Not applicable.

### Competing interests

The author declares that they have no competing interests.

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