The role of boundary data on the solvability of some equations involving non-autonomous nonlinear differential operators

governed by nonlinear differential operators such as the classical p-Laplacian or its generalizations. Various types of differential operators, even singular or non-surjective, have been considered due to many applications in different fields. We now quote just some of the papers devoted to this study, such as for the scalar case Bereanu and Mawhin [1, 2], Cabada and Pouso [3, 4], Cabada and Cid [5], Cid and Torres [6], Calamai [7], Garcia-Huidobro et al. [8], Dang and Oppenheimer [9], Ferracuti and Papalini [10], O’Regan [11], Papageorgiou and Papalini [12]. In [13] Manásevich and Mawhin treated systems of equations with periodic boundary conditions. Finally, in the framework of differential inclusions, we quote [14] and the papers by Kyritsi, Matzakos and Papageorgiou [15, 16] for systems of differential inclusions involving maximal monotone operators and with various boundary conditions.

has assumed a certain interest.

In [18] a periodic problem on a compact interval for a vectorial inclusion with a differential operator of the type ${(a(x){\parallel x^{\prime }\parallel }^{p-2}{x}^{\prime })}^{\prime }$ is studied, where $a:\mathbb{R}\to \mathbb{R}$ is a positive, continuous function. Moreover, in [16] a Dirichlet problem driven by a more general differential operator, having the structure ${(A(x,x^{\prime }))}^{\prime }$, is investigated.

have been studied in [19], where existence and non-existence of solutions was put in relation to the behavior of Φ and $f(t,x,\cdot )$ at 0 and $f(\cdot ,x,y)$ at infinity, while the presence of the function a does not influence the existence of solutions. Subsequently, in [20] a critical case was considered in which also the dependence on the state variable x of the functions a and f and the value of the boundary data are relevant for the solvability of the boundary value problem.

with ${\nu }^{-}<{\nu }^{+}$ given constants, where $\mathrm{\Phi }:\mathbb{R}\to \mathbb{R}$ is a general increasing homeomorphism, with $\mathrm{\Phi }(0)=0$, and a is a positive, continuous function, but with possibly null infimum. It was shown that also the dependence on t of the function a plays a central role for the existence and non-existence of solutions and some sufficient criteria for the existence and non-existence of solutions were established. However, in [21] the case when $|f(t,x,y)/\mathrm{\Phi }(y)|\to 0$ as $y\to 0$ was considered, and in this setting neither the behavior with respect to x, nor the boundary data influence the existence or non-existence of solutions.

The aim of this paper is to complete this study, investigating the critical case $|f(t,x,y)|\sim |\mathrm{\Phi }(y)|$ as $y\to 0$ for problem (1.1) governed by non-autonomous differential operators.

We provide sharp sufficient conditions guaranteeing the solvability of problem (1.1) together with conditions implying the non-existence of solutions, closely related to the former ones, involving the asymptotic behaviors of $a(\cdot ,x)$ and $f(\cdot ,x,y)$ as $|t|\to +\mathrm{\infty }$, the asymptotic behaviors of Φ and $f(t,x,\cdot )$ as $y\to 0$, and the maxima/minima of the functions $a(t,\cdot )$, $f(t,\cdot ,y)$ in the interval $[{\nu }^{-},{\nu }^{+}]$ defined by the boundary data.

We present general existence and non-existence results (see Theorems 2.1, 2.2 and 2.3) together with operative criteria (see Propositions 3.1-3.6) useful when the functions a and f appearing in the differential equation have a product structure. Some examples of application complete the paper.

Throughout the paper, Φ is a general increasing homeomorphism on ℝ such that $\mathrm{\Phi }(0)=0$, $a:{\mathbb{R}}^2\to \mathbb{R}$ is a positive continuous function and $f:{\mathbb{R}}^3\to \mathbb{R}$ is a Carathéodory function.

Of course, $M^{\ast }(t)\ge M(t)\ge m(t)\ge m^{\ast }(t)>0$ for every $t\in \mathbb{R}$, with ${inf}_{t\in \mathbb{R}}m(t)$ possibly null.

As we have mentioned in Introduction, in the present paper, we treat problems for which, roughly speaking, $|f(t,x,\cdot )|\sim |\mathrm{\Phi }(y)|$ as $y\to 0$. But also the rate of growth of Φ at ∞ has a great relevance, and we separately consider the case of superlinear growth from that of linear or sublinear growth.

We first state an existence result for differential operators growing at most linearly at infinity.

(with $\frac{1}{q}=0$ if $q=+\mathrm{\infty }$).

Proof The scheme of the proof is the same as in [[21], Theorem 3.1]. We sketch now the main points and prove in detail the parts which differ from that proof. Notice that there are only two differences between the present statement and that of [[21], Theorem 3.1]; that is, we here take $\gamma =1$ and modify the definition of the auxiliary function $N_C$.

Finally, for every $x\in \mathbb{R}$, put $w(x):=max\{x-{\nu }^{+},0\}+min\{x-{\nu }^{-},0\}$.

Following the same argument in the proof of [[21], Theorem 3.1], it is possible to prove that problem (2.11) admits a solution $u_n$ for every $n>L$, such that ${\nu }^{-}\le u_n(t)\le {\nu }^{+}$ for all $t\in I_n$. Moreover, $u_n$ is increasing in $[-n,-L]$ and in $[L,n]$ and if $u_n^{\prime }(t_0)=0$ for some $|t_0|>L$, then $u_n^{\prime }(t)=0$ whenever $|t|>|t_0|$ (see Steps 1-2 in the proof of [[21], Theorem 3.1]). Finally, as in Step 3 of the same proof, one can show that there exists a suitable constant C such that $|u_n^{\prime }(t)|<C\le N_C(t)$ for every $t\in [-L,L]$. Notice that till this point in the proof of [[21], Theorem 3.1], the definition of $N_C$, or the fact that $\gamma >1$, were not used.

Now our goal is to show that $|u_n^{\prime }(t)|\le N_C(t)$ also for every $t\in I_n\setminus [-L,L]$.

implying that $u_n^{\prime }(t)<N_C(t)$ for every $t\in [L,\hat{t}]$ (see (2.2), (2.3) and (2.8)), a contradiction when $\hat{t}<n$. So, $\hat{t}=n$ and the claim is proved. The same argument works in the interval $[-n,-L]$ too.

Now, following the same argument as in [[21], Theorem 3.1], it is possible to prove that the sequence ${({\tilde{u}}_n)}_n$ of the functions $u_n$, continued in a constant way in the whole ℝ, converges to a solution x of problem (1.1), satisfying all the properties stated in the assertion. □

Similarly to what was done in [21], one can prove a result for differential operators having superlinear growth at infinity, provided that condition (2.7) is strengthened requiring that the Nagumo function has sublinear growth at infinity, as the following result states, whose proof is just the same as that of [[21], Theorem 3.2], taking account of the modifications due to the different auxiliary function $N_C$, we showed in the proof of Theorem 2.1.

Then the assertion of Theorem  2.1 follows.

Of course, the operators here considered as having superlinear growth are quite general and extend the classical p-Laplacian. Nevertheless, when dealing just with the p-Laplacian, the results can be slightly improved by using the positive homogeneity of the operator, as we will show in a forthcoming paper.

The key tools in the previous existence theorems is the summability of function $N_C(t)$ (condition (2.8)) joined with assumption (2.9). Such conditions are not improvable in the sense that if (2.9) is satisfied with the reversed inequality and $N_C$ is not summable, then problem (1.1) does not admit solutions, as the following result states.

does not belong to $L^1(\mathbb{R})$.

Then problem (1.1) does not admit solutions such that ${\nu }^{-}\le x(t)\le {\nu }^{+}$, that is, no function $x\in C^1(\mathbb{R})$, with $t\mapsto a(t,x(t))\mathrm{\Phi }(x^{\prime }(t))$ almost everywhere differentiable, exists satisfying the conditions of problem (1.1).

Proof Also this proof follows the scheme of that of [[21], Theorem 3.3]. More in detail, it is possible to show that if $x\in C^1(\mathbb{R})$ with ${\nu }^{-}\le x(t)\le {\nu }^{+}$ and $a(t,x(t))\mathrm{\Phi }(x^{\prime }(t))$ almost everywhere differentiable (not necessarily belonging to $W^{1,1}(\mathbb{R})$) is a solution of problem (1.1), then the function x is monotone increasing in $[L,+\mathrm{\infty })$ and in $(-\mathrm{\infty },-L]$ with ${lim}_{t\to \pm \mathrm{\infty }}{x}^{\prime }(t)=0$.

Then if $T<+\mathrm{\infty }$, necessarily we have $x^{\prime }(T)=0$ in contradiction with the above inequality. Therefore, $T=+\mathrm{\infty }$ and again, by the above inequality, we deduce $x(+\mathrm{\infty })=+\mathrm{\infty }$ since by (2.15) the function on the right-hand side in not summable by assumption. Therefore, $x^{\prime }(t^{\ast })=0$, implying that $t^{\ast }=0$, $x^{\prime }(t)=0$ for every $t\ge 0$ and, consequently, $x(0)={\nu }^{+}$.

Similarly, using (2.14) one can show that $x(0)={\nu }^{-}$, a contradiction. □

We will highlight how the local behaviors of $c(x,\cdot )$ at $y=0$ and of $b(\cdot ,x)$, $\alpha (\cdot )$ at infinity, related to the maximum and the minimum of the functions β, g in the interval $[{\nu }^{-},{\nu }^{+}]$, play a relevant role for the existence or non-existence of solutions.

where recall that $m(t):={min}_{x\in [{\nu }^{-},{\nu }^{+}]}a(t,x)$ and $M(t):={max}_{x\in [{\nu }^{-},{\nu }^{+}]}a(t,x)$.