Existence of quasilinear elliptic equations with prescribed limiting behavior

Abstract

We consider quasilinear elliptic equations $Δ_{p} u + f (u) = 0$ in the quarter-plane Ω, with zero Dirichlet data. For some general nonlinearities f, we prove the existence of a positive solution with a prescribed limiting profile. The question is motivated by the result in (Adv. Nonlinear Stud. 13(1) (2013) 115–136), where the authors establish that for solutions $u (x_{1}, x_{2})$ of the preceding Dirichlet problem, ${lim}_{x_{1} \to \infty} u (x_{1}, x_{2}) = V (x_{2})$ , where V is a solution of the corresponding one-dimensional problem with $V (+ \infty) = z$ and z is a root of f. Starting with such a profile V and a carefully selected z, the authors of this paper apply Perron’s method in order to prove the existence of a solution u with limiting profile V.

The work in this paper is similar in spirit to that in (Math. Methods Appl. Sci. 39(14) (2016) 4129–4138), where the authors compare the sub and the super solutions by using arguments based on the strong maximum principle for semilinear equations. However, for the quasilinear case, such a maximum principle is lacking. This difficulty is overcome by employing a less classical weak sweeping principle that requires a careful boundary analysis.

Keywords

asymptotic behavior positive solution

1. Introduction and main result

In this paper we deal with the study of existence of positive solutions of the following quasilinear elliptic problem $\begin{matrix} (1.1) & \{\begin{matrix} Δ_{p} u + f (u) = 0 & in Ω, \\ u = 0 & on \partial Ω \end{matrix} \end{matrix}$ where $p > 1$ , $Ω = {(x_{1}, x_{2}) \in R^{2} : x_{1} > 0 and x_{2} > 0}$ and $Δ_{p} u : = div (| \nabla u |^{p - 2} \nabla u)$ is the usual p-Laplace operator. The function $f : R_{+} \to R$ is assumed to be locally Lipschitz, $f (0) = 0$ , and $\begin{matrix} (1.2) & Z_{f} : = {z \in [0, \infty [: f (z) = 0} is a discrete set . \end{matrix}$ Motivated by the asymptotic behavior of solutions of (1.1) on unbounded domains, particularly by the work of [2], we provide a method of construction of a positive solution that converges to a prescribed root z of f. In other words, we ask and examine the same question raised in [2] but in reverse. This work can be considered as a sequel to the papers [6,9] where a similar result was obtained for the special case $p = 2$ in quarter planes then extended to sectorial domains. Indeed, the study of the semilinear case $p = 2$ is based on a careful construction of a subsolution $\underline{u}$ (using a minimization technique) and supersolution $\overline{u}$ (using ODE methods); comparing the two solutions and then using Perron’s method to conclude. However, even if one managed to show the existence of sub- and supersolutions, the situation in the quasilinear case is much more involved. In fact, when working with the p-Laplace operator, both the weak and the strong comparison principles might no longer exist. These principles rely crucially on the assumption that $p = 2$ and hence on the exploitation of the usual arguments and tricks related to the linear character of the Laplace operator. To overcome this difficulty will take two different approaches.

The first one involves the use of the weak sweeping principle (see [1,4,5]) to replace the sliding method, which is based on the comparison principle for the Laplacian case and was effectively used in [9] to show that $\underline{u} ⩽ \overline{u}$ . The sliding method and the weak sweeping principle utilize a similar machinery where, in order to compare sub- and supersolutions of autonomous equations as (1.1) on a domain, we first find a suitable subdomain where the comparison holds (usually at the asymptotic level) then we use the translation invariance of the solution to maintain the comparison on the whole domain.

The second approach is surprisingly similar to the Laplacian case where we exploit a recent result (see [7,8]) of strong comparison principle for quasilinear elliptic equations in bounded regular domains $\begin{matrix} D \subseteq R^{2} \end{matrix}$ under the hypothesis that $\begin{matrix} (1.3) & p > 3 / 2 and f (u) > 0 in \overline{D} . \end{matrix}$

Let us mention that, although our first approach, which is similar to that used in [2], covers (1.3), the exhausting technical difficulties we encounter when using the weak sweeping principle on the one hand, and the wide class of nonlinearities f and values of p that are covered by (1.3) on the other hand, makes the second approach worth investigating and even original as it largely simplifies the arguments in [2]. The second approach somehow makes the analysis of the quasilinear problem almost identical to the semilinear one studied in [9].

We also want to emphasize that, although the core analytical part will focus on the first approach, we will be providing the essential and simplified ideas that take place when dealing with (1.3).

Throughout this paper, the function F denotes a primitive of f: $\begin{matrix} F (s) = \int_{0}^{s} f (t) d t, \end{matrix}$ and the roots that we consider are assumed to be elements of $Z_{f}^{*} \neq \emptyset$ with $\begin{matrix} (1.4) & Z_{f}^{*} = \{z \in Z_{f} ∖ {0} : | \begin{matrix} F (s) < F (z) for all s \in [0, z [, \\ \underset{s ↗ z}{lim sup} \frac{f (s)}{{(z - s)}^{p - 1}} < \infty \end{matrix}\} . \end{matrix}$ Before stating our main result, we recall some definitions and preliminary results.

Definition 1.1 (Weak solution).

We say that $u \in W^{1, p} (D)$ is a weak subsolution of $\begin{matrix} (1.5) & \{\begin{matrix} Δ_{p} u + f (u) = 0 & in D \\ u = 0 & on \partial D, \end{matrix} \end{matrix}$ if $u ⩽ 0$ (in the sense of trace) and $\begin{matrix} \int_{D} | \nabla u |^{p - 2} \nabla u \cdot \nabla v d x ⩽ \int_{D} f (u) v d x \forall v \in W_{0}^{1, p} (D), v ⩾ 0 a.e. in D . \end{matrix}$ Similarly, we say that $u \in W^{1, p} (D)$ is a weak supersolution of (1.5) if $u ⩾ 0$ and $\begin{matrix} \int_{D} | \nabla u |^{p - 2} \nabla u \cdot \nabla v d x ⩾ \int_{D} f (u) v d x \forall v \in W_{0}^{1, p} (D), v ⩾ 0 a.e. in D . \end{matrix}$ Finally, a solution $u \in W_{0}^{1, p} (D)$ is both a sub- and supersolution.

Remark 1.2.
Notice that, by the elliptic regularity for the p-Laplacian equations (see [11,12]), a weak solution of (1.5) also belongs to $C^{1, α} (\overline{D})$ . In this case, we may slightly modify the above definition of weak sub- and supersolution by considering boundary inequalities in a classical sense and test functions $v \in C_{0}^{\infty} (D)$ .

1.1. Preliminary comparison results

We now recall, in a form that is convenient to our work, the statements of the weak sweeping principle and the strong comparison/maximum principle for the p-Laplacian problem.

Theorem 1.3 (Weak sweeping principle [1]).

Let $u_{t}$ and $v_{t}$ , $t \in [0, 1]$ , be functions in $W^{1, p} (D) \cap C (\overline{D})$ and satisfy in the weak sense, for some $ζ > 0$ , $\begin{array}{l} (1.6) & - Δ_{p} u_{t} ⩾ f (u_{t}), - Δ_{p} v_{t} ⩽ f (v_{t}) - ζ in D, \forall t \in [0, 1], \\ (1.7) & u_{t} ⩾ v_{t} + ζ on \partial D, \forall t \in [0, 1] . \end{array}$ Suppose that $u_{t_{0}} ⩾ v_{t_{0}}$ in $D$ for some $t_{0} \in [0, 1]$ and $t \to u_{t}$ , $t \to v_{t}$ are continuous from $[0, 1]$ to $C (\overline{D})$ . Then $\begin{matrix} (1.8) & u_{t} ⩾ v_{t} on D, \forall t \in [0, 1] . \end{matrix}$

Theorem 1.4 (Strong comparison principle [7]).

Let $u, v \in C^{1} (\overline{D})$ be two weak solutions to $\begin{matrix} Δ_{p} w + f (w) = 0 in D, \end{matrix}$ where $p > \frac{3}{2}$ . Assume that $\begin{matrix} (1.9) & either f (u) or f (v) does not change sign (positive or negative) in \overline{D} . \end{matrix}$ Suppose furthermore that $\begin{matrix} (1.10) & u ⩽ v in D . \end{matrix}$ Then $u \equiv v$ in $D$ unless $\begin{matrix} (1.11) & u < v in D . \end{matrix}$

Theorem 1.5 (Strong maximum principle [13]).

Let $u \in C^{1} (D)$ be such that $Δ_{p} u \in L_{loc}^{2} (D)$ , $u ⩾ 0$ a.e. in $D$ , $Δ_{p} u ⩽ γ (u)$ a.e. in $D$ with $γ : [0, \infty [\to R$ continuous, nondecreasing, $γ (0) = 0$ , $γ (s) > 0$ for all $s > 0$ , and $\begin{matrix} \int_{0}^{1} {(j (s))}^{- 1 / p} d s = \infty, \end{matrix}$ where $j (s) = \int_{0}^{s} γ (t) d t$ . Then if u does not vanish identically on $D$ it is positive everywhere in $D$ . Moreover, if $u \in C^{1} (D \cup {x_{0}})$ for an $x_{0} \in \partial D$ that satisfies an interior sphere condition and $u (x_{0}) = 0$ then $\begin{matrix} \frac{\partial u}{\partial ν} (x_{0}) > 0 \end{matrix}$ where ν in an interior normal at $x_{0}$ .

1.2. Main result

In order to construct a solution of (1.1) that converges (in a sense to be made precise later) to $z \in Z_{f}^{*}$ (see (1.4)), we first need to build suitable sub- and supersolutions. On the one hand, the supersolution $\overline{u}$ of (1.1) relies on the existence of the corresponding one-dimensional problem: $\begin{matrix} (1.12) & \{\begin{matrix} {(| V^{'} |^{p - 2} V^{'})}^{'} + f (V) = 0 & in (0, \infty), \\ V (0) = 0 < V (t) < z = V (\infty) & for all t > 0 \\ V^{'} (t) > 0 & for all t ⩾ 0 . \end{matrix} \end{matrix}$ This ODE has a regular solution (see Proposition 3.4) $\begin{matrix} V_{z} \in C^{1} ([0, \infty [) \cap C^{2} (0, \infty), \end{matrix}$ and the supersolution is thus defined as follows $\begin{matrix} (1.13) & \overline{u} = V_{z} (d) in Ω, \end{matrix}$ where $\begin{matrix} d = d (x) = d (x, \partial Ω) = ‖ x ‖_{\infty} . \end{matrix}$ On the other hand, the subsolution $\underline{u}$ of (1.1) relies on the existence, using variational arguments, of (1.5) with $\begin{matrix} D = B_{R}, \end{matrix}$ where $B_{R} = B_{R} (0)$ is the open ball of center 0 radius $R > 0$ which is considered to be sufficiently large. We also require $\begin{matrix} \underline{u} \to z as d \to \infty . \end{matrix}$ where Theorems 1.3 and 1.4 are then utilized to show that $\begin{matrix} (1.14) & \underline{u} ⩽ \overline{u}, \end{matrix}$ and the solution u is thus given by the classical Perron’s method for quasilinear equations (see for example [3]). We are now ready to state our main theorem.

Theorem 1.6.
Let $z \in Z_{f}^{}$ . Then there exists a solution u of (* 1.1 ) satisfying $\begin{matrix} (1.15) & 0 < u ⩽ V_{z} (d) in Ω and (V_{z} (d) - u) \to 0 as d \to \infty . \end{matrix}$

2. Approximate subsolution

In order to find a suitable subsolution $\underline{u}$ , we first aim to construct an approximate subsolution of (1.1) taking a constant value (sufficiently close to z) on the set $\begin{matrix} Ω_{R} = {x \in Ω : d (x) > R} \end{matrix}$ for sufficiently large $R > 0$ . In particular, we prove

Proposition 2.1.
Let $z \in Z_{f}^{}$ . Then, for every* $ε > 0$ small enough, there exists $R = R (ε) > 0$ and a subsolution ${\underline{u}}_{ε}$ of ( 1.1 ) such that $\begin{matrix} (2.1) & 0 < z - {\underline{u}}_{ε} < ε on Ω_{R} . \end{matrix}$

The proof is based on the existence of radial solutions $w = w (| x |)$ of the following Dirichlet problem on the ball $B_{R} = B_{R} (0) \subset R^{2}$ : $\begin{matrix} (2.2) & \{\begin{matrix} Δ_{p} w + g_{ε} (w) = 0 & in B_{R}, \\ 0 ⩽ w ⩽ z_{ε} & in \overline{B_{R}} \\ w = 0 & on \partial B_{R} \\ w (0) = {max}_{\overline{B_{R}}} w > z_{ε} - ε, \end{matrix} \end{matrix}$ where $\begin{matrix} (2.3) & g_{ε} (s) = f (s) - ε s if f changes sign in (0, z), \end{matrix}$ or $\begin{matrix} (2.4) & g_{ε} (s) = f (s) if f is positive in (0, z) and p > 3 / 2 . \end{matrix}$ Here $z_{ε}$ is a root (to be made precise in the proof of Lemma 2.4) of $g_{ε}$ with ${lim}_{ε \to 0} z_{ε} = z$ . We simply take $z_{ε} = z$ in the case $g_{ε} = f$ . The modification of f is made for the purpose of using the weak sweeping principle as illustrated in the introduction. Indeed, by using the function $g_{ε}$ given by (2.3), we will be creating the ζ-gap so that the conditions (1.6) and (1.7) in Theorem 1.3 are met. However, when $g_{ε}$ is given by (2.4), there is no need for this gap as we will directly use the strong comparison principle announced in Theorem 1.4.
Remark 2.2.
Since $g_{ε} (s) ⩽ f (s)$ for $s ⩾ 0$ then a solution of (2.2) is a subsolution of the following problem $\begin{matrix} (2.5) & \{\begin{matrix} Δ_{p} w + f (w) = 0 & in B_{R}, \\ w = 0 & on \partial B_{R} . \end{matrix} \end{matrix}$
Remark 2.3.
Since the first equation of (2.5) does not explicitly depend on the variables $x_{1}$ and $x_{2}$ , a translation of the solution in any direction is again a subsolution of (2.5) on the resulting translated ball. The idea then is to move the ball completely inside the quarter plane Ω and extend the solution by zero, thus obtaining a subsolution of (1.1). By allowing the ball to be displaced in all of Ω, we are lead to a family of subsolutions whose supremum is the required ${\underline{u}}_{ε}$ . The set $Ω_{R}$ is thus the set of all centers of all displaced balls inside Ω (see Fig. 1).
Fig. 1.
Geometric interpretation of the set $Ω_{R}$ .

To prove Proposition 2.1 we first show the following lemma.
Lemma 2.4.
Let $z \in Z_{f}^{}$ and let* $g_{ε}$ be given by ( 2.3 ). Then, for every $ε > 0$ small enough, there exists $R = R (ε) > 0$ and a solution $u_{ε, R}$ of ( 2.2 ) with $\begin{matrix} (2.6) & 0 < z_{ε} < z, g_{ε} (z_{ε}) = 0 and lim_{ε \to 0} z_{ε} = z . \end{matrix}$
Proof.
Since the zeros of f are discrete, $f (z) = 0$ and $\begin{matrix} (2.7) & s ⟶ \int_{s}^{z} f (t) d t > 0, \forall s \in [0, z), \end{matrix}$ then there exists $z^{}$ such that $0 < z^{} < z$ with $f > 0$ on $[z^{}, z)$ . For sufficiently small $ε > 0$ , we have $\begin{matrix} g_{ε} (z^{}) g_{ε} (z) < 0 \end{matrix}$ leading to the existence of $z_{ε} \in (z^{}, z)$ such that $\begin{matrix} (2.8) & g_{ε} (z_{ε}) = 0, g_{ε} > 0 on [z^{}, z_{ε}), \end{matrix}$ and $\begin{matrix} (2.9) & lim_{ε \to 0} z_{ε} = z . \end{matrix}$ We set $\begin{matrix} G_{ε} (s) = \int_{s}^{z_{ε}} g_{ε} (t) d t, \end{matrix}$ and it is straightforward to see that $\begin{matrix} (2.10) & G_{ε} (s) ⟶ \int_{s}^{z} f (t) d t uniformly in [0, z] as ε \to 0 . \end{matrix}$ From (2.7), (2.8) and (2.10), we deduce, for sufficiently small $ε > 0$ , that $\begin{matrix} \{\begin{matrix} G_{ε} > 0 & on [0, z^{}], \\ G_{ε} ↘ 0 & on [z^{}, z_{ε}], \end{matrix} \end{matrix}$ hence $\begin{matrix} G_{ε} > 0 in [0, z_{ε}), \end{matrix}$ and therefore, choosing $z_{ε} - ε > 0$ and by the compactness of the interval $[0, z_{ε} - ε]$ , we can find $α = α (ε) > 0$ such that $\begin{matrix} (2.11) & G_{ε} ⩾ α on [0, z_{ε} - ε] . \end{matrix}$ Let ${\tilde{g}}_{ε}$ be the function defined over $R$ by: $\begin{matrix} {\tilde{g}}_{ε} (t) = \{\begin{matrix} g_{ε} (t) & if 0 ⩽ t ⩽ z_{ε} \\ 0 & elsewhere \end{matrix} \end{matrix}$ and set $\begin{matrix} {\tilde{G}}_{ε} (s) = \int_{s}^{z_{ε}} {\tilde{g}}_{ε} (t) d t . \end{matrix}$ Clearly ${\tilde{G}}_{ε}$ is Lipschitz continuous and ${\tilde{G}}_{ε} (s) ⩾ 0$ for all $s \in R$ . Let r be any positive real number and define $\begin{matrix} E_{r} (v) = \frac{1}{p} \int_{B_{r}} | \nabla u |^{p} + \int_{B_{r}} {\tilde{G}}_{ε} (v), \end{matrix}$ for all $v \in W_{0}^{1, p} (B_{r})$ . This functional is well-defined and is coercive and thus, by standard arguments, we know it is has a minimizer $v_{r}$ in $W_{0}^{1, p} (B_{r})$ . It is also well-known that a critical point of $E_{r} (v)$ corresponds to a weak solution of $\begin{matrix} \{\begin{matrix} Δ_{p} v_{r} + {\tilde{g}}_{ε} (v_{r}) = 0 & in B_{r} \\ v_{r} = 0 & on \partial B_{r} . \end{matrix} \end{matrix}$ Since ${\tilde{g}}_{ε} = 0$ on $(- \infty, 0] \cup [z_{ε}, \infty)$ , it follows from the weak maximum principle that $\begin{matrix} (2.12) & 0 ⩽ v_{r} ⩽ z_{ε} in \overline{B_{r}} . \end{matrix}$ Consequently ${\tilde{g}}_{ε} (v_{r}) = g_{ε} (v_{r})$ and therefore $v_{r}$ is a solution of $\begin{matrix} \{\begin{matrix} Δ_{p} v_{r} + g_{ε} (v_{r}) = 0 & in B_{r} \\ v_{r} = 0 & on \partial B_{r} . \end{matrix} \end{matrix}$ Moreover, by elliptic regularity for p-Laplacian equations ([11,12]) we know that such a solution belongs to $C^{1, α} (\overline{B_{r}})$ . Also, by rearrangement theory (see for instance [10]) this solution must be radially symmetric and decreasing away from the center of the domain. Thus $\begin{matrix} 0 ⩽ v_{r} (| x |) ⩽ v_{r} (0) = max_{\overline{B_{r}}} v_{r} ⩽ z_{ε} in \overline{B_{r}} . \end{matrix}$ To finish the proof, it suffices to find $r > 0$ large enough such that $v_{r} (0) > z_{ε} - ε$ . Assume otherwise that $v_{r} ⩽ z_{ε} - ε$ for all $r > 0$ . Then, by (2.11), we obtain $\begin{matrix} E_{r} (v_{r}) ⩾ \int_{B_{r}} G_{ε} (v_{r}) ⩾ \int_{B_{r}} α = α π r^{2}, \forall r > 0 . \end{matrix}$ On the other hand, for $r > 1$ , define the test function $w_{r} \in W_{0}^{1, p} (B_{r})$ by $\begin{matrix} w_{r} (x) = \{\begin{matrix} z_{ε} & for | x | < r - 1, \\ z_{ε} (r - | x |) & for r - 1 ⩽ | x | ⩽ r . \end{matrix} \end{matrix}$ As $w_{r} \equiv z_{ε}$ for $| x | < r - 1$ and since $G_{ε} (z_{ε}) = 0$ , we deduce that $| \nabla w_{r} |^{p}$ and $G_{ε} (w_{r})$ are supported on the annulus ${r - 1 ⩽ | x | ⩽ r}$ . Thus, for some constant C independent of r, we get $\begin{matrix} E_{r} (w_{r}) ⩽ C (2 r - 1), \forall r > 1 . \end{matrix}$ But since $v_{r}$ is a minimizer of $E_{r}$ , we have $E_{r} (v_{r}) ⩽ E_{r} (w_{r})$ and thus $\begin{matrix} α r^{2} ⩽ \frac{C}{π} (2 r - 1), \forall r > 1 . \end{matrix}$ By noticing that $α > 0$ , we may deduce that the above inequality does not hold for large r. Therefore, there exists $r = R > 0$ such that $v_{R} (0) > z_{ε} - ε$ . Finally set $\begin{matrix} u_{ε, R} = v_{R} \end{matrix}$ and the proof of the lemma is complete. □

We now present the
Proof of Proposition 2.1.
Before delving into the details of the proof, let us clarify one technical point related to (2.1). To fix the idea, we denote $ε^{'} : = ε$ in Lemma 2.4 then the solution $u_{ε^{'}, R}$ satisfies (see (2.2)), $\begin{matrix} (2.13) & z_{ε^{'}} - ε^{'} < u_{ε^{'}, R} (0) ⩽ z_{ε} \end{matrix}$ and $\begin{matrix} lim_{ε^{'} \to 0} (z_{ε^{'}} - ε^{'}) = z . \end{matrix}$ Thus, by fixing $0 < ε < z$ in Proposition 2.1, we may always find $ε^{'} > 0$ such that $\begin{matrix} 0 < z - ε < z_{ε^{'}} - ε^{'} < z_{ε^{'}} < z, \end{matrix}$ and, in such a case, we directly (see (2.13)) get: $\begin{matrix} (2.14) & 0 < z - ε < z_{ε^{'}} - ε^{'} < u_{ε^{'}, R} (0) ⩽ z_{ε^{'}} < z, \end{matrix}$ leading eventually to (2.1) as will be explained in the rest of the proof. Due to this, and although we keep the ε-notation for the sake of clarity, it should be understood that, in all what follows, this particular $ε^{'}$ is the one used whenever Lemma 2.4 is involved.

We now set $\begin{matrix} {\underline{u}}_{ε, R} = \{\begin{matrix} u_{ε, R} & in \overline{B_{R}}, \\ 0 & otherwise, \end{matrix} \end{matrix}$ where $u_{ε, R}$ is given by Lemma 2.4. Thanks to Remark 2.2 and the regularity of $u_{ε, R}$ on $B_{R}$ , we can assert that ${\underline{u}}_{ε, R}$ is a subsolution of the first equation of (1.5) with $D = R^{2}$ . In fact, the function $u_{ε, R} \in C^{1, α} (B_{R})$ being radially symmetric and decreasing away from the center of $B_{R}$ provides a sign to $u_{ε, R}^{'}$ and thus improves the regularity to $u_{ε, R} \in C^{2} (B_{R})$ . This allows to use similar arguments as in [6] to show our aforementioned assertion.

Fix $y \in Ω_{R}$ and consider ${\underline{u}}_{ε, R}^{y}$ ; the translation of ${\underline{u}}_{ε, R}$ by the vector y, $\begin{matrix} {\underline{u}}_{ε, R}^{y} (x) = {\underline{u}}_{ε, R} (x - y) . \end{matrix}$ Since the space variable x appears only through the solution u in equation (1.1), it directly follows that ${\underline{u}}_{ε, R}^{y} (x)$ is a subsolution this equation. However, in order to obtain a subsolution satisfying (2.1), we take (see for instance [3]), $\begin{matrix} (2.15) & {\underline{u}}_{ε} = sup_{y \in Ω_{R}} {\underline{u}}_{ε, R}^{y} |_{Ω} . \end{matrix}$ Notice that ${\underline{u}}_{ε}$ is constant on $Ω_{R}$ . More precisely, $\begin{matrix} {\underline{u}}_{ε} \equiv u_{ε, R} (0) in Ω_{R} . \end{matrix}$ Inequality (2.1) then directly follows from (2.14). □

It is worth mentioning that the functions ${{\underline{u}}_{ε, R}^{y} : y \in Ω_{R}}$ are the building blocks of ${\underline{u}}_{ε}$ and therefore we may consider ${\underline{u}}_{ε, R}$ as the fundamental part of the subsolution ${\underline{u}}_{ε}$ of (1.1). As it was already mentioned at the beginning of this section, we have modified the nonlinearity (while showing the existence of ${\underline{u}}_{ε, R}$ ), from f in (1.1) to $g_{ε}$ in (2.2), for the only purpose of using Theorem 1.3 in order to compare sub- and supersolutions and then to get (1.14). This will be made clear in the proof Theorem 1.6 in the next section. However, if one has another tool to maintain a comparison (e.g. Theorem 1.4) that would do the job with the nonlinearity f, then the proof of Lemma 2.4 can be largely simplified as we will show in the next corollary. Corollary 2.5.
Let $z \in Z_{f}^{}$ . Then, for every* $ε > 0$ small enough, there exists $R = R (ε) > 0$ and a solution $u_{ε, R}$ of $\begin{matrix} (2.16) & \{\begin{matrix} Δ_{p} w + f (w) = 0 & in B_{R}, \\ 0 ⩽ w < z & in \overline{B_{R}} \\ w = 0 & on \partial B_{R} \\ w (0) = {max}_{\overline{B_{R}}} w > z - ε . \end{matrix} \end{matrix}$
Proof.
The idea of the proof follows that of Lemma 2.4 with $G_{ε}$ directly replaced by the function given by (2.7). No need then to consider $z_{ε}$ , and no need for the uniform convergence to get an estimates like (2.11) of $G_{ε}$ . The only point left to check is (2.12) that is now rewritten: $\begin{matrix} 0 ⩽ v_{r} ⩽ z in \overline{B_{r}} . \end{matrix}$ Indeed, inequality (2.1) requires a stronger estimate on $v_{r}$ than the one mentioned above, namely $v_{r} < z$ , that would finally boils down to $\begin{matrix} (2.17) & v_{r} (0) < z . \end{matrix}$ Note that, in the ε-case, this issue has been taken care of through (2.14). To show (2.17), we argue by contradiction and we assume $v_{r} (0) = z$ . Let $\begin{matrix} B^{'} = {x \in B_{r} : v_{r} (x) = z} \neq \emptyset, \end{matrix}$ then, from the qualitative properties of $v_{r}$ and as $v_{r} (0) = z$ , we know that $B^{'} = B_{r^{'}} (0)$ is actually a ball centered at 0 with a radius $0 ⩽ r^{'} < r$ . We also know that there exists $x_{0} \in B_{r} ∖ B^{'}$ and $r^{″} > 0$ small enough such that $B^{″} = B_{r^{″}} (x_{0}) \subseteq B_{r}$ (see the below figure), $\begin{matrix} v_{r} < z in B^{″} \end{matrix}$ and $\begin{matrix} \partial B^{'} \cap \partial B^{″} \neq \emptyset . \end{matrix}$ Let $x^{} \in \partial B^{'} \cap \partial B^{″}$ then $\begin{matrix} v_{r} (x^{}) = z and v_{r}^{'} (| x^{} |) = 0 . \end{matrix}$

Due to the continuity of $v_{r}$ at $x^{}$ , and thanks to (1.4), in particular that ${lim sup}_{s ↗ z} \frac{f (s)}{{(z - s)}^{p - 1}} < \infty$ , we have (by possibly choosing a smaller $r^{″}$ ): $\begin{matrix} (2.18) & f (v_{r} (x)) ⩽ c {(z - v_{r} (x))}^{p - 1} for some constant c and all x \in B^{″} . \end{matrix}$ We set $ψ : = z - v_{r}$ (also a radial function as $v_{r}$ ) then $ψ (x^{}) = 0$ and $ψ > 0$ in $B^{″}$ . Moreover, by (2.18), $\begin{matrix} Δ_{p} ψ = f (v_{r}) ⩽ c ψ^{p - 1} in B^{″} . \end{matrix}$ Therefore, by applying Theorem 1.5 with $\begin{matrix} γ (s) = c s^{p - 1}, \end{matrix}$ we conclude that $ψ^{'} (| x^{} |) > 0$ and so $v_{r}^{'} (| x^{} |) < 0$ , a contradiction to $v_{r}^{'} (| x^{} |) = 0$ . □

3. Supersolution

We study the existence and extendibility of the solutions of the one-dimensional p-Laplacian initial value problem $\begin{matrix} (3.1) & \{\begin{matrix} {(| u^{'} |^{p - 2} u^{'})}^{'} + f (u) = 0 in R, \\ u (0) = α, u^{'} (0) = β, \end{matrix} \end{matrix}$ where $f : R \to R$ is a continuous, $p > 1$ and α, β are real values. By a solution of $\begin{matrix} (3.2) & {({| u^{'} |}^{p - 2} u^{'})}^{'} + f (u) = 0 \end{matrix}$ in an interval $I \subseteq R$ we mean a function $u : I \to R$ such that u and $| u^{'} |^{p - 2} u^{'}$ are continuously differentiable on I and satisfy (3.2). For the ease of notation, consider the function $φ_{p} : R \to R$ defined by $φ_{p} (s) = | s |^{p - 2} s$ so that equation (3.2) becomes ${(φ_{p} (u^{'}))}^{'} + f (u) = 0$ . It is worth noticing that $\begin{matrix} φ_{p}^{- 1} = φ_{p^{'}} with \frac{1}{p} + \frac{1}{p^{'}} = 1 . \end{matrix}$

Proposition 3.1.
Suppose that there is a $k > 0$ such that $\begin{matrix} (3.3) & | f (x) | ⩽ k | x |^{p - 1} \end{matrix}$ for all $x \in R$ , then ( 3.1 ) has a solution defined in $R$ .
Proof.
We first show local existence in an interval $[- δ, δ]$ for sufficiently small $δ > 0$ . Denote by $X_{δ}$ the Banach space of all continuous functions $u : [- δ, δ] \to R$ , with corresponding sup-norm $‖ u ‖_{δ} = {sup}_{[- δ, δ]} | u |$ . Integrating the first of (3.1) from 0 to τ, we get $\begin{matrix} (3.4) & φ_{p} (u^{'} (τ)) = φ_{p} (β) - \int_{0}^{τ} f (u (s)) d s \end{matrix}$ then $\begin{matrix} u^{'} (τ) = φ_{p}^{- 1} [φ_{p} (β) - \int_{0}^{τ} f (u (s)) d s], \end{matrix}$ where, upon integrating from 0 to t, we obtain $\begin{matrix} (3.5) & u (t) = α + \int_{0}^{t} φ_{p}^{- 1} [φ_{p} (β) - \int_{0}^{τ} f (u (s)) d s] d τ . \end{matrix}$ Define the operator $T : X_{δ} \to X_{δ}$ by $\begin{matrix} T (u) (t) = α + \int_{0}^{t} φ_{p^{'}} [φ_{p} (β) - \int_{0}^{τ} f (u (s)) d s] d τ, \end{matrix}$ then T is a completely continuous operator and, from (3.5), we infer that problem (3.1) is equivalent to the fixed point problem $\begin{matrix} (3.6) & u = T (u) . \end{matrix}$ We solve (3.6) using Schauder’s fixed point theorem by showing that, for some sufficiently small δ and sufficiently large $r > 0$ , we have $\begin{matrix} T (B_{X_{δ}} [0, r]) \subseteq B_{X_{δ}} [0, r] \end{matrix}$ where $B_{X_{δ}} [0, r]$ is the closed ball of center 0 and radius r in $X_{δ}$ . For $| t | ⩽ δ$ , and noting that $| φ_{q} (s) | = | s |^{q - 1}$ , $q > 1$ , we have, thanks to (3.3), $\begin{matrix} (3.7) & | T (u) (t) | ⩽ | α | + δ {(| β |^{p - 1} + k δ ‖ u ‖_{δ}^{p - 1})}^{p^{'} - 1} . \end{matrix}$ However, by choosing $0 < δ < \frac{1}{k}$ and $r > | β |$ , we get for $u \in B_{X_{δ}} [0, r]$ , $\begin{matrix} {‖ T (u) ‖}_{δ} ⩽ | α | + δ {(2 r^{p - 1})}^{p^{'} - 1} = | α | + δ 2^{p^{'} - 1} r, \end{matrix}$ and, again, by adjusting the values of δ and r, namely $\begin{matrix} δ < \frac{1}{2^{p^{'} - 1}} and r > \frac{| α |}{1 - δ 2^{p^{'} - 1}}, \end{matrix}$ we thus obtain $‖ T (u) ‖_{δ} ⩽ r$ whenever $‖ u ‖_{δ} ⩽ r$ . To sum up, by choosing $\begin{matrix} 0 < δ < min (\frac{1}{k}, \frac{1}{2^{p^{'} - 1}}) \end{matrix}$ and $\begin{matrix} r > max (| β |, \frac{| α |}{1 - δ 2^{p^{'} - 1}}), \end{matrix}$ the existence of a solution of (3.6) follows from Schauder’s fixed point theorem and thus (3.1) has a solution u defined in $[- δ, δ]$ .

Next let us prove that we can extend u to $[0, \infty [$ and hence to $R$ . Suppose that u is defined on a maximal interval $[0, a [$ . If $a < \infty$ then $| u (t) | + | u^{'} (t) | \to \infty$ as $t ↗ a$ and, to show that this is not the case, let us define the function $\begin{matrix} w (t) = \int_{0}^{t} ({| u (s) |}^{p} + {| u^{'} (s) |}^{p}) d s for t \in [0, a [. \end{matrix}$ Using (3.4) with $τ : = t$ , and (3.3), we apply Holder’s inequality to obtain $\begin{matrix} (3.8) & {| u^{'} (t) |}^{p - 1} ⩽ c_{1} + c_{2} {(\int_{0}^{t} {| u (s) |}^{p} d s)}^{\frac{p - 1}{p}}, \end{matrix}$ where $c_{1}$ , $c_{2}$ are positive generic constants that may depend on a (may also change from line to line). Since $\frac{p}{p - 1} > 1$ , we rely on the convexity of $x \in [0, \infty [\to x^{\frac{p}{p - 1}}$ to get our estimate on the term $| u^{'} (t) |^{p}$ that reads: $\begin{matrix} (3.9) & {| u^{'} (t) |}^{p} ⩽ c_{1} + c_{2} \int_{0}^{t} {| u (s) |}^{p} d s . \end{matrix}$ On the other hand, the classical identity $\begin{matrix} u (t) = α + \int_{0}^{t} u^{'} (s) d s, t \in [0, a [ \end{matrix}$ together with Holder’s inequality and the convexity of $x \in [0, \infty [\to x^{p}$ , give our estimate on the term $| u (t) |^{p}$ that reads: $\begin{matrix} (3.10) & {| u (t) |}^{p} ⩽ c_{3} + c_{4} \int_{0}^{t} {| u^{'} (s) |}^{p} d s, \end{matrix}$ with $c_{3}, c_{4} > 0$ are constants that may also depend on a. Adding (3.9) and (3.10), we finally deduce that $\begin{matrix} w (t) ⩽ A e^{B t}, t \in [0, a [, \end{matrix}$ where $A, B > 0$ are constants. This inequality and a standard argument imply that the local solution of (3.1) can be extended to $[0, \infty [$ . □
Remark 3.2.
If f satisfies (3.3) then $f (0) = 0$ .
Corollary 3.3.
Let $p > 1$ and let f be a bounded Lipschitz function with $f (0) = 0$ then ( 3.1 ) has a solution on $R$ .
Proof.
Straightforward adaptation of the proof of Proposition 3.1. □

We are now ready to show the existence of the supersolution $\overline{u}$ . Proposition 3.4 (Existence of supersolution).

Let $z \in Z_{f}^{*}$ , then there exists a regular solution $V_{z}$ of ( 1.12 ). Moreover, the function $\overline{u}$ defined by ( 1.13 ) is a supersolution of ( 1.1 ).

Proof.
As a first step, we will apply the result of Proposition 3.1 with an appropriate velocity $V^{'} (0)$ . Notice that a solution of (1.12) is eventually bounded between 0 and z and since $f (0) = f (z) = 0$ then we may, without loss of generality, consider $f \equiv 0$ outside $[0, z]$ . The function f can then be assumed bounded. Also notice that, for as long as V is sufficiently regular and $V^{'} > 0$ on some interval I, we may multiply the first equation of (1.12) by $V^{'}$ to obtain: $\begin{matrix} (3.11) & \frac{p - 1}{p} {(V^{'} (t))}^{p} + F (V (t)) = c, \forall t \in I, \end{matrix}$ where c is a constant. If $0 \in I$ then (3.11) becomes $\begin{matrix} (3.12) & \frac{p - 1}{p} {(V^{'} (t))}^{p} + F (V (t)) = \frac{p - 1}{p} {(V^{'} (0))}^{p}, \forall t \in I \end{matrix}$ and if we are allowed to let $t \to \infty$ in I then, knowing that $V (\infty) = z$ , we are lead to $\begin{matrix} V^{'} (0) = {(\frac{p}{p - 1} F (z))}^{1 / p} . \end{matrix}$ This suggests to apply Proposition 3.1 with $α = 0$ and $β = {(\frac{p}{p - 1} F (z))}^{1 / p}$ so we have a solution $V_{z}$ of (3.1). In order to show that $V_{z}$ satisfies (1.12), it remains to show $V_{z}^{'} > 0$ and $V_{z} (\infty) = z$ . Since $F (z) > 0$ then $V_{z}^{'} (0) > 0$ and thus $V_{z}^{'} (t) > 0$ for sufficiently small t. We claim that this is true for all $t ⩾ 0$ . Otherwise, there exists $t_{0} > 0$ such that $\begin{matrix} V_{z}^{'} > 0 in [0, t_{0}) and V_{z}^{'} (t_{0}) = 0 . \end{matrix}$ It is worth mentioning that since $V_{z}^{'} > 0$ in $(0, t_{0})$ then $V_{z}$ is $C^{2}$ there and so equation (3.12) can be reformulated as $\begin{matrix} (3.13) & \frac{p - 1}{p} {(V_{z}^{'} (t))}^{p} + F (V_{z} (t)) = F (z), \forall t \in [0, t_{0}] . \end{matrix}$ By substituting $t = t_{0}$ in (3.13) we get that $F (V_{z} (t_{0})) = F (z)$ and hence, from (1.4), we get $\begin{matrix} V_{z} (t_{0}) ⩾ z . \end{matrix}$ However, If $V_{z} (t_{0}) > z$ then by the intermediate value property, there exits $0 < t^{} < t_{0}$ such that $\begin{matrix} V_{z} (t^{}) = z and V_{z}^{'} (t^{}) > 0 . \end{matrix}$ This is not possible since $V_{z}^{'} (t^{}) = 0$ by a direct substitution in (3.13). Consequently $\begin{matrix} V_{z} (t_{0}) = z . \end{matrix}$ Thanks to (1.4), in particular that ${lim sup}_{s ↗ z} \frac{f (s)}{{(z - s)}^{p - 1}} < \infty$ , and thanks to the continuity of $V_{z}$ at $t_{0}$ , we can find a small $δ > 0$ such that $\begin{matrix} (3.14) & f (V_{z} (t)) ⩽ c {(z - V_{z} (t))}^{p - 1} for some constant c and all t \in (t_{0} - δ, t_{0}) . \end{matrix}$ We set $W : = z - V_{z}$ then $W (t_{0}) = 0$ and $W > 0$ in $(t_{0} - δ, t_{0})$ . Moreover, by (3.14), $\begin{matrix} {({| W^{'} |}^{p - 2} W^{'})}^{'} = f (V_{z}) ⩽ c W^{p - 1} in (t_{0} - δ, t_{0}) . \end{matrix}$ Therefore, by applying Theorem 1.5 with $\begin{matrix} γ (s) = c s^{p - 1}, \end{matrix}$ we conclude that $W^{'} (t_{0}) < 0$ and so $V_{z}^{'} (t_{0}) > 0$ , a contradiction to $V_{z}^{'} (t_{0}) = 0$ . From all what precedes, $\begin{matrix} V_{z}^{'} > 0 and lim_{t \to \infty} V_{z} (t) = ℓ \in (0, z] . \end{matrix}$ However, if $ℓ < z$ , we again reach a contradiction from (3.13) by letting $t \to \infty$ . Finally $V_{z} (\infty) = z$ .

We now show that $\overline{u}$ given by (1.13) is a supersolution of (1.1). Remark that $\begin{matrix} (3.15) & \overline{u} (x) = min {V_{z} (x_{1}), V_{z} (x_{2})}, x = (x_{1}, x_{2}) \in Ω . \end{matrix}$ Since $V_{z}$ is a solution of (1.12), it is easily seen that $V_{z} (x_{1})$ and $V_{z} (x_{2})$ are both supersolutions of (1.1) and so is (see [3]) the minimum $\overline{u}$ . □

4. Proof of Theorem 1.6

This final section is devoted to the proof of the maim result. Our next proposition compares ${\underline{u}}_{ε}$ and $\overline{u}$ in Ω by using an adapted sliding argument.

Proposition 4.1.

Let ${\underline{u}}_{ε}$ be the subsolution of ( 1.1 ) obtained by Proposition 2.1 , and let $\overline{u}$ be the supersolution of ( 1.1 ) obtained by Proposition 3.4 . Then $\begin{matrix} (4.1) & {\underline{u}}_{ε} ⩽ \overline{u} in \overline{Ω} . \end{matrix}$

Proof.

By the definition (3.15) of $\overline{u}$ over Ω, it suffices to show that $\begin{matrix} {\underline{u}}_{ε} (x) ⩽ V_{z} (x_{1}), \forall x = (x_{1}, x_{2}) \in \overline{Ω} . \end{matrix}$ Since ${\underline{u}}_{ε}$ is generated (see (2.15)) by ${\underline{u}}_{ε, R}^{y}$ , we may further simplify the analysis since the above inequality would now be obtained by: $\begin{matrix} {\underline{u}}_{ε, R}^{y} (x) ⩽ V_{z} (x_{1}), \forall x = (x_{1}, x_{2}) \in \overline{Ω}, \forall y \in Ω_{R} . \end{matrix}$ However, $\begin{matrix} {\underline{u}}_{ε, R}^{y} = 0 on \overline{Ω} ∖ B_{R} (y), \end{matrix}$ thus, to establish the proof, it is sufficient to show that, for every $y \in Ω_{R}$ , $\begin{matrix} (4.2) & u_{ε, R}^{y} (x) ⩽ V_{z} (x_{1}), \forall x = (x_{1}, x_{2}) \in B_{R} (y), \end{matrix}$ where $\begin{matrix} (4.3) & u_{ε, R}^{y} (x) = u_{ε, R} (x - y) . \end{matrix}$ The comparison (4.2) on a varying ball $B_{R} (y)$ is somehow incompatible with Theorem 1.3 (weak sweeping principle) as it demands comparison on an unmoving domain $D$ . Nevertheless, having a closer look at (4.3) ( $u_{ε, R}^{y}$ is just a translation of $u_{ε, R}$ ) suggests that we compare $u_{ε, R}$ , defined on $B_{R}$ , with translations of $V_{z}$ on a fixed domain (to be made precise later). Moreover, since $V_{z}$ depends on the $x_{1}$ variable, we may only consider translations in the $x_{1}$ -direction. For that reason, we set $\begin{matrix} V_{z}^{μ} (s) = V_{z} (s + μ), \end{matrix}$ and the proof terminates by showing $\begin{matrix} (4.4) & u_{ε, R} ⩽ V_{z}^{μ} in B_{R}, \forall μ ⩾ R, \end{matrix}$ with the aid of Theorem 1.3. Notice that (4.4) holds true for sufficiently large μ as revealed through the following arguments. Since $u_{ε, R} < z$ on the compact $\overline{B_{R}}$ then there exists small $β > 0$ such that $u_{ε, R} < z - β$ in $\overline{B_{R}}$ . This, together with the fact that $V_{z}$ is increasing and $V_{z} (\infty) = z$ , enables us to find $μ_{0} > 0$ large enough so that (see the below figure), $\begin{matrix} (4.5) & u_{ε, R} ⩽ V_{z}^{μ} in B_{R}, \forall μ ⩾ μ_{0} . \end{matrix}$

For any $τ > 0$ small enough, let $\begin{matrix} μ_{1} : = R + τ < μ_{0}, \end{matrix}$ and set $\begin{matrix} μ_{t} : = (1 - t) μ_{0} + t μ_{1}, t \in [0, 1] . \end{matrix}$ In what follows, and for simplicity sake, we may use the notation x to replace a point $(x, 0)$ . Since $\begin{matrix} μ_{t} ⩾ μ_{1} > R for all t \in [0, 1], \end{matrix}$ and since $V_{z}$ is continuous and positive on the compact $⋃_{t \in [0, 1]} \overline{B_{R} (μ_{t})}$ then there exists $δ > 0$ (can be selected small if necessary) such that $\begin{matrix} V_{z} ⩾ δ in ⋃_{t \in [0, 1]} \overline{B_{R} (μ_{t})} . \end{matrix}$ Dealing with translations of $V_{z}$ , the above inequality becomes: $\begin{matrix} (4.6) & V_{z}^{μ_{t}} ⩾ δ in B_{R}, \forall t \in [0, 1] . \end{matrix}$ Let $n ⩾ 2$ be an integer. Note that if $u_{ε, R} = 0$ on an annular region ${0 < R_{1} ⩽ | x | ⩽ R}$ then (4.4) is immediately satisfied there. As a result, we may assume $u_{ε, R} > 0$ on $B_{R}$ . This assumption together with the properties of $u_{ε, R}$ allow us to find $\begin{matrix} (4.7) & 0 < R_{n} < R with lim_{n \to \infty} R_{n} = R, \end{matrix}$ such that $\begin{matrix} (4.8) & inf_{| x | < R_{n}} u_{ε, R} = inf_{| x | = R_{n}} u_{ε, R} = u_{ε, R} (R_{n}) = δ / n . \end{matrix}$ We set $\begin{matrix} D = B_{R_{n}} (0), \end{matrix}$ and we aim to apply Theorem 1.3 with $\begin{matrix} u_{t} = V_{z}^{μ_{t}}, v_{t} = u_{ε, R} and ζ = \frac{ε δ}{n} . \end{matrix}$ By a straightforward computation using (4.6) and (4.8), we have $\begin{matrix} - Δ_{p} u_{ε, R} = f (u_{ε, R}) - ε u_{ε, R} ⩽ f (u_{ε, R}) - ζ in D, \end{matrix}$ and (assuming without loss of generality that $ε < 1$ ), $\begin{matrix} u_{ε, R} + ζ = \frac{δ}{n} + \frac{ε δ}{n} ⩽ \frac{2}{n} δ ⩽ δ ⩽ V_{z}^{μ_{t}} on \partial D . \end{matrix}$ Moreover, from (4.5), we have $u_{ε, R} ⩽ V_{z}^{μ_{0}}$ on $D$ . Thus we can apply Theorem 1.3 to conclude that $\begin{matrix} u_{ε, R} ⩽ V_{z}^{μ_{t}} in D, \forall t \in [0, 1], \end{matrix}$ and this comparison can be extended from $D$ to the whole ball $B_{R}$ by relying on (4.7). As a result we get $\begin{matrix} (4.9) & u_{ε, R} ⩽ V_{z}^{μ} in B_{R}, \forall μ ⩾ μ_{1} = R + τ . \end{matrix}$ It remains to show that (4.9) holds for $μ = R$ . This is obtained using the continuity of $V_{z}$ . Indeed, for every $x = (x_{1}, x_{2}) \in B_{R}$ , we have $\begin{matrix} V_{z}^{R} (x) = V_{z}^{R} (x_{1}) = V_{z} (x_{1} + R) = lim_{τ \to 0} V_{z} (x_{1} + R + τ) = lim_{τ \to 0} V_{z}^{R + τ} (x_{1}) = lim_{τ \to 0} V_{z}^{R + τ} (x) ⩾ u_{ε, R} (x) . \end{matrix}$ This ends the proof. □

In the following proposition, a similar result is obtained when f, p are as in (2.4), and $u_{ε, R}$ is given by Corollary 2.5. As was explained right after the proof of Proposition 2.1, this choice of f and p somehow simplifies the overall proof arguments of our main result, since we may now drop the gap from f and use the more traditional strong comparison principle.

Proposition 4.2.

Let f, p be as in ( 2.4 ) and let $u_{ε, R}$ be the solution of ( 2.16 ) given by Corollary 2.5 . Let ${\underline{u}}_{ε}$ be the subsolution of ( 1.1 ) obtained by Proposition 2.1 , and let $\overline{u}$ be the supersolution of ( 1.1 ) obtained by Proposition 3.4 . Then ( 4.1 ) holds.

Proof.

The proof is basically similar to the one of the previous proposition. We only outline the modifications which are needed. Again, the whole point revolves around showing (4.2) where, to do that now, we keep our $V_{z}$ fixed and consider translations of $u_{ε, R}$ . Note that this is indeed more natural and more in accordance with (4.2). Let $μ > R$ and set $\begin{matrix} u_{μ} : = u_{ε, R}^{y} with y = (μ, 0) . \end{matrix}$ This function is understood to be defined on the ball $\overline{B_{R} (y)}$ . Thanks to (2.16) we have $\begin{matrix} u_{μ} < z in \overline{B_{R} (y)}, \end{matrix}$ and consequently, there exists small $β > 0$ such that $u_{μ} < z - β$ there. Since $V_{z} (\infty) = z$ then there exists $μ_{0} > R$ large enough such that $\begin{matrix} (4.10) & u_{μ} < V_{z} for all μ ⩾ μ_{0} . \end{matrix}$

Let us show that (4.10) holds for all $μ > R$ . Indeed, if this is not true, then there is some value $R < μ^{*} < μ_{0}$ with $\begin{matrix} (4.11) & u_{μ^{*}} ⩽ V_{z} in \overline{B_{R} (μ^{*}, 0)}, \end{matrix}$ and equality at some point $x^{*} \in \overline{B_{R} (μ^{*}, 0)}$ . The point $x^{*}$ can not be on the boundary $\partial B_{R} (μ^{*}, 0)$ of the ball because, as $R < μ^{*}$ , we have $V_{z} > 0$ on $\partial B_{R} (μ^{*}, 0)$ , while $u_{μ^{*}} = 0$ there. Consequently $\begin{matrix} (4.12) & u_{μ^{*}} (x^{*}) = V_{z} (x^{*}), x^{*} \in B_{R} (μ^{*}, 0) . \end{matrix}$ Moreover, and again as $R < μ^{*}$ , we in fact have $\begin{matrix} (4.13) & 0 < V_{z} < z in \overline{B_{R} (μ^{*}, 0)} \end{matrix}$ then thanks to (2.4), we deduce that $\begin{matrix} f (V_{z}) > 0 in \overline{B_{R} (μ^{*}, 0)} . \end{matrix}$ This, together with the fact that $u_{μ^{*}}$ and $V_{z}$ are both solutions of the equation $\begin{matrix} Δ_{p} w + f (w) = 0 in B_{R} (μ^{*}, 0), \end{matrix}$ enable us to make use of Theorem 1.4 with $u = u_{μ^{*}}$ and $v = V_{z}$ . By applying that theorem we deduce, from (4.11) and (4.12), that $\begin{matrix} u_{μ^{*}} \equiv V_{z} in \overline{B_{R} (μ^{*}, 0)}, \end{matrix}$ which is in direct contradiction with (4.13). □

We are now ready to present the proof of our theorem.

Proof of Theorem 1.6.

Let ${\underline{u}}_{ε}$ be the subsolution of (1.1) obtained by Proposition 2.1. From Proposition 4.1 (or Proposition 4.2), we know that $\begin{matrix} (4.14) & {\underline{u}}_{ε} ⩽ \overline{u} . \end{matrix}$ Let $\begin{matrix} (4.15) & \underline{u} = sup {\underline{u}}_{ε} . \end{matrix}$ From (4.14) we deduce that $\underline{u}$ is finite and again a subsolution of (1.1) with $\begin{matrix} \{\begin{matrix} \underline{u} ⩽ \overline{u} in Ω, \\ \underline{u} |_{\partial Ω} = \overline{u} |_{\partial Ω} = 0 . \end{matrix} \end{matrix}$ Using Perron’s method, we get the existence of a solution u of (1.1) satisfying: $\begin{matrix} \underline{u} ⩽ u ⩽ \overline{u}, \end{matrix}$ hence $\begin{matrix} 0 < u ⩽ V_{z} (d) in Ω . \end{matrix}$ To complete the proof, we need to show the convergence in (1.15). Indeed, for any $ε > 0$ small enough, Proposition 2.1 ensures the existence of $R = R (ε) > 0$ such that $\begin{matrix} z - {\underline{u}}_{ε} < ε for d > R, \end{matrix}$ and consequently, thanks to (4.15) and the fact that $V_{z} < z$ , we get $\begin{matrix} V_{z} (d) - u < ε for d > R . \end{matrix}$ This terminates the proof. □

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Existence of quasilinear elliptic equations with prescribed limiting behavior

Abstract

Keywords

1. Introduction and main result

Definition 1.1 (Weak solution).

Theorem 1.3 (Weak sweeping principle [1]).

Theorem 1.4 (Strong comparison principle [7]).

Theorem 1.5 (Strong maximum principle [13]).

1.2. Main result

Theorem 1.6. Let z ∈ Z f ∗ . Then there exists a solution u of ( 1.1 ) satisfying (1.15) 0 < u ⩽ V z ( d ) in Ω and ( V z ( d ) − u ) → 0 as d → ∞ . 2. Approximate subsolution

References

Theorem 1.6.
Let $z \in Z_{f}^{}$ . Then there exists a solution u of (* 1.1 ) satisfying $\begin{matrix} (1.15) & 0 < u ⩽ V_{z} (d) in Ω and (V_{z} (d) - u) \to 0 as d \to \infty . \end{matrix}$

2. Approximate subsolution