The limiting behavior of solutions to p-Laplacian problems with convection and exponential terms

TL;DR

This paper investigates the behavior of solutions to a nonlinear p-Laplacian problem with convection and exponential terms, proving existence, convergence to boundary distance, and analyzing the effects of parameters on solutions.

Contribution

It establishes the existence of positive solutions under certain parameters and demonstrates uniform convergence of solutions to the boundary distance function as p approaches infinity.

Findings

Existence of positive solutions for certain parameter ranges.

Solutions converge uniformly to the boundary distance function as p increases.

New convergence result for problems involving nonlinearities with convection terms.

Abstract

We consider, for $a,l\geq1,$ $b,s,\alpha>0,$ and $p>q\geq1,$ the homogeneous Dirichlet problem for the equation $-\Delta_{p}u=\lambda u^{q-1}+\beta u^{a-1}\left\vert \nabla u\right\vert ^{b}+mu^{l-1}e^{\alpha u^{s}}$ in a smooth bounded domain $\Omega\subset\mathbb{R}^{N}.$ We prove that under certain setting of the parameters $\lambda,$ $\beta$ and $m$ the problem admits at least one positive solution. Using this result we prove that if $\lambda,\beta>0$ are arbitrarily fixed and $m$ is sufficiently small, then the problem has a positive solution $u_{p},$ for all $p$ sufficiently large. In addition, we show that $u_{p}$ converges uniformly to the distance function to the boundary of $\Omega,$ as $p\rightarrow\infty.$ This convergence result is new for nonlinearities involving a convection term.

Equations

$$\left\{\begin{array}[c]{lll}-\Delta_{p}u=\lambda u^{q-1}+\beta u^{a-1}\left|\nabla u\right|^{b}+mu^{l-1}e^{\alpha u^{s}}&\text{in}&\Omega\\ u>0&\text{in}&\Omega\\ u=0&\text{on}&\partial\Omega\end{array}\right.$$

$$a,l\ge 1,b>0,s,\alpha \ge 0,\text{ and }p>q\ge 1.$$

$$d_{\Omega }(x):=\underset{y\in \partial \Omega }{\min }\left|x-y\right|,x\in \overline{\Omega }.$$

$$0<m<m_{\infty }:=(\lambda {\Vert d_{\Omega }\Vert }_{\infty }^{q-1}+\beta {\Vert d_{\Omega }\Vert }_{\infty }^{a-1}){\Vert d_{\Omega }\Vert }_{\infty }^{l-1}{e}^{-\alpha {\Vert d_{\Omega }\Vert }_{\infty }^s}.$$

$$\underset{p\to \infty }{\lim }{u}_p=d_{\Omega }\text{ uniformly in }\overline{\Omega }.$$

$$\left\{\begin{array}[c]{rrll}-\operatorname{div}(\phi_{p}(\left|\nabla u\right|)\nabla u)&=&f(x,u)&\text{in }\Omega\\ u&>&0&\text{in }\Omega,\\ u&=&0&\text{on }\partial\Omega\end{array}\right.$$

$$\left\{\begin{array}[c]{rrll}-\Delta_{p}v&=&1&\text{in }\Omega\\ v&=&0&\text{on }\partial\Omega.\end{array}\right.$$

$${\lambda }_p:=\inf \left\{\frac{{\Vert \nabla v\Vert }_p^p}{{\Vert v\Vert }_p^p}:v\in W_0^{1,p}(\Omega )\setminus \left\{0\right\}\right\}=\frac{{\Vert \nabla e_p\Vert }_p^p}{{\Vert e_p\Vert }_p^p}$$

$$\left\{\begin{array}[c]{rrll}-\Delta_{p}v&=&\lambda_{p}\left|v\right|^{p-2}v&\text{in }\Omega\\ v&=&0&\text{on }\partial\Omega.\end{array}\right.$$

$$k_p:=\underset{w\in {\mathcal{S}}_p}{\sup }{\Vert \nabla w\Vert }_{\infty }$$

$${\mathcal{S}}_p:=\left\{w\in W_0^{1,p}(\Omega ):-{\Delta }_{p}w=g\text{ in }\Omega ,\text{ for some }g\in L^{\infty }(\Omega ),{\Vert g\Vert }_{\infty }=1\right\}.$$

$$\mathcal{E}(M):=\left\{(\lambda ,\beta ,m)\in {\mathbb{R}}_{+}^3:\frac{\lambda A_p}{M^{p-q}}+\frac{\beta B_p}{M^{p-(a+b)}}+\frac{m{A}_p}{M^{p-l}{e}^{-\alpha M^s}}\le 1\right\},$$

$$A_p:={\Vert {\varphi }_p\Vert }_{\infty }^{p-1}\text{ and }{B}_p:=k_p^b{\Vert {\varphi }_p\Vert }_{\infty }^{p-1-b}.$$

$${\left(\frac{\lambda }{{\lambda }_p}\right)}^{1/(p-q)}{e}_p\le u_p\le \frac{M}{{\Vert {\varphi }_p\Vert }_{\infty }}{\varphi }_p\text{ and }{\Vert \nabla u_p\Vert }_{\infty }\le \frac{k_{p}M}{{\Vert {\varphi }_p\Vert }_{\infty }}.$$

$$\left\{\begin{array}[c]{rrll}-\Delta_{p}u&=&g(u)\left|\nabla u\right|^{p}+f(x,u)&\text{in }\Omega\\ u&>&0&\text{in }\Omega,\\ u&=&0&\text{on }\partial\Omega.\end{array}\right.$$

$$\left\{\begin{array}[c]{rrll}-\Delta_{p}u&=&\lambda u^{q-1}+u^{l-1}e^{\alpha u^{s}}&\text{in }\Omega\\ u&>&0&\text{in }\Omega,\\ u&=&0&\text{on }\partial\Omega\end{array}\right.$$

$$\left\{\begin{array}[c]{rrll}-\Delta_{p}u&=&mu^{l-1}e^{\alpha u^{s}}+g(x)&\text{in }\Omega\\ u&=&0&\text{on }\partial\Omega,\end{array}\right.$$

$$\underset{p\to \infty }{\lim }{k}_p=1$$

$${\Vert \nabla w\Vert }_{\infty }\le (c{p}^{\gamma }{)}^{\frac{1}{p-1}}\text{ for all }w\in {\mathcal{S}}_p,p\ge 2,$$

$${\varphi }_p(x)=\frac{p-1}{p}{N}^{-\frac{1}{p-1}}\left(R^{\frac{p}{p-1}}-{\left|x-x_0\right|}^{\frac{p}{p-1}}\right),\left|x-x_0\right|\le R.$$

$${\Vert {\varphi }_p\Vert }_{\infty }=\frac{p-1}{p}{N}^{-\frac{1}{p-1}}{R}^{\frac{p}{p-1}}.$$

$${\Vert {\varphi }_p\Vert }_{\infty }\le M_0:=\frac{p-1}{p}{N}^{-\frac{1}{p-1}}{\left(\frac{\left|\Omega \right|}{{\omega }_N}\right)}^{\frac{p}{N(p-1)}},$$

$${\left|u\right|}_{1,\eta }:={\Vert u\Vert }_{\infty }+{\Vert \nabla u\Vert }_{\infty }+\sum _{i=1}^N{\left[u_{x_i}\right]}_{\eta }$$

$${\left[u_{x_i}\right]}_{\eta }:=\sup \left\{\frac{\left|u_{x_i}(x)-u_{x_i}(y)\right|}{{\left|x-y\right|}^{\eta }}:x,y\in \overline{\Omega }\text{ and }x\ne y\right\}$$

$$\left|u(x)\right|\le {\varphi }_p(x)\forall x\in \overline{\Omega }.$$

$${\left|u\right|}_{1,{\eta }_p}\le K_p.$$

$$0<{\Vert \nabla {\varphi }_p\Vert }_{\infty }\le k_p\le K_p,\text{ whenever }p>1,$$

$${\Vert v\Vert }_{\infty }\le {\Vert d_{\Omega }\Vert }_{\infty }{\Vert \nabla v\Vert }_{\infty }\forall v\in W^{1,\infty }(\Omega )\cap C_0(\overline{\Omega }),$$

$$\frac{{\Vert {\varphi }_p\Vert }_{\infty }}{{\Vert d_{\Omega }\Vert }_{\infty }}\le {\Vert \nabla {\varphi }_p\Vert }_{\infty }\le k_p,\text{ whenever }p>1.$$

$$\left\{\begin{array}[c]{rrll}-\Delta_{p}v&=&g&\text{in }\Omega\\ v&=&0&\text{on }\partial\Omega,\end{array}\right.$$

Full text

2020 MSC: 35B40, 35J92.

keywords: Convection term, distance function, exponential term, gradient estimate.

1 Introduction

In this paper we consider the Dirichlet problem

[TABLE]

where $\Omega$ is a smooth bounded domain $\mathbb{R}^{N},$ $N\geq 2,$ and $\Delta_{p}u:=\operatorname{div}(\left|\nabla u\right|^{p-2}\nabla u)$ is the $p$-Laplacian operator, $p>1$. The parameters $\lambda,$ $\beta$ and $m$ are nonnegative and the constants $q,a,b,l,\alpha$ and $s$ satisfy

[TABLE]

Our main result in this paper is stated as follows, where $d_{\Omega}$ denotes the distance function to the boundary:

[TABLE]

Theorem 1.1

Assume that $\partial\Omega\in C^{1,1}.$ Let $\lambda$ and $\beta$ be arbitrary, but fixed positive real numbers, and let $m$ be such that

[TABLE]

There exists $p_{0}>\max\left\{q,a+b,l\right\}$ such that if $p>p_{0}$ then the Dirichlet problem (P) admits a weak solution $u_{p}\in W_{0}^{1,p}(\Omega).$ Moreover,

[TABLE]

In the last decades, this kind of limiting behavior of solutions to Dirichlet problems of the form

[TABLE]

has been obtained by many authors. We refer to [5], [20], [23] for the $p$-Laplacian, i.e. $\phi_{p}(t)=t^{p-2}$, and [6], [13], [14], [15], [16], [18] for more general functions $\phi_{p}.$

However, up to our acknowledge, this is the first work dealing with the limiting behavior of solutions $u_{p}$ to a $p$-family of Dirichlet problems with convection (i.e. gradient) terms.

The solution $u_{p}$ of Theorem 1.1 is obtained as an application of Theorem 1.2 stated below. To properly state this existence result let us fix the notation, give some definitions and recall some facts.

The standard norm of Lebesgue space $L^{t}(\Omega),$ $1\leq t\leq\infty,$ will be denoted by $\left\|\cdot\right\|_{t}.$

We will denote by $\phi_{p}$ the $p$-torsion function associated with $\Omega,$ that is, the weak solution in the Sobolev space $W_{0}^{1,p}(\Omega)$ to the $p$-torsional creep problem

[TABLE]

The first eigenvalue of the Dirichlet $p$-Laplacian will be denoted by $\lambda_{p}$ whereas $e_{p}$ will denote the positive and $L^{\infty}$-normalized eigenfunction corresponding to $\lambda_{p}$ (so that, $e_{p}>0$ in $\Omega$ and $\left\|e_{p}\right\|_{\infty}=1$). We recall that

[TABLE]

and also that $e_{p}$ is a weak solution to the Dirichlet problem

[TABLE]

Let us define

[TABLE]

where

[TABLE]

(Note that $\phi_{p}\in\mathcal{S}_{p}$.)

For each $M>0$ let $\mathcal{E}(M)$ denote the region of $\mathbb{R}_{+}^{3}$ defined by

[TABLE]

where

[TABLE]

Now, we can state our main existence result.

Theorem 1.2

Assume that $\partial\Omega\in C^{1,\tau}$ for some $\tau\in(0,1)$ and suppose that $(\lambda,\beta,m)\in\mathcal{E}(M)$ for some $M>0.$ Then, the Dirichlet problem (P) admits at least one weak solution $u_{p}\in W_{0}^{1,p}(\Omega)$ satisfying the bounds

[TABLE]

We emphasize that this existence result does not impose any restriction neither to the exponent $s$ in the exponential term nor to the exponent $b$ in the convection term, respectively to the critical values $\frac{N}{N-1}$ (Trudinger-Moser inequality) and $p$ (the natural growth of the gradient).

The proof of Theorem 1.2 is given in Section 2. It is inspired by the approach introduced by Bueno and Ercole [7], which relies on a combination of the sub-super solution method with a version of the Schauder Fixed Point Theorem.

In [10], de Figueiredo, Gossez, Quoirin and Ubilla proved existence results for the following class of $p$-Laplacian problems

[TABLE]

Their results apply to the following particular nonlinearities (in our notation), where $p^{\ast}$ denotes the well-known critical Sobolev exponent:

• (a)

$\beta\left|\nabla u\right|^{p}+u^{l-1}e^{\alpha u},$ with $\beta>0,$ $l>p$ and $0<\alpha<\beta\frac{p^{\ast}-p}{p-1}$ ([10, Example 2.3]);

• (b)

$u^{a-1}\left|\nabla u\right|^{p}+mu^{p-1}e^{\alpha u^{a}},$ with $a>1,$ $0<\alpha<\frac{p^{\ast}-p}{(p-1)a}$ and $0<m<\lambda_{p}$ ([10, Example 2.9]);

• (c)

$\beta\left|\nabla u\right|^{p}+u^{l-1},$ with $\beta>0$  and $l>p$ ([10, Example 2.13]); and

• (d)

$\beta\left|\nabla u\right|^{p}+mu^{l-1}e^{\alpha u},$  with  $\beta>0,$ $0<\alpha<\beta\frac{p^{\ast}-p}{p-1},$ $1\leq l<p$  and $m$ positive, sufficiently small ([10, Theorem 2.17]).

Our Theorem 1.2 complements the existence results for these particular nonlinearities, with a sublinear term $\lambda u^{q-1}$ ($1\leq q<p$) added and with $\left|\nabla u\right|^{b}$ ($b>0$) in the place of $\left|\nabla u\right|^{p}.$ Indeed, items (a) and (c) are complemented by Corollary 2.8, item (b) is complemented by Corollary 2.9, and item (d) is complemented by Corollary 2.6.

In [3] de Araujo and Montenegro considered the following Dirichlet problem (in our notation)

[TABLE]

where $1<q<p,$ $l\geq 1$ and $s>0.$ Using the approach introduced in [7] they proved an existence result for $\lambda$ and $\alpha$ sufficiently small, by assuming $l\not=p.$ Besides including a convection term, our Theorem 1.2 complements Theorem 1.1 of [3].

Still in Section 2 we present two more applications of Theorem 1.2 (see Corollaries 2.7 and 2.8) that extend a recent existence result obtained by de Araujo and Faria in [2].

In Section 3 we prove, as consequence of the Picone’s inequality, a nonexistence result for the Dirichlet problem

[TABLE]

stated in Proposition 3.2, for $g\in L^{\infty}(\Omega),$ $g\geq 0,$ $1\leq l<p$ and $\alpha,s>0.$ This result generalizes Theorem 2.1 by Garcia Alonso and Peral Alonso in [17] and also shows that a restriction for the parameter $m$ in (P) is to be expected when $1\leq l<p$.

In Section 4 we prove Theorem 1.1. To obtain $u_{p}$ we apply Theorem 1.2, and to achieve the limiting behavior (1.2) we show that

[TABLE]

where $k_{p}$ is defined in (1.3). (Recall that $k_{p}$ appears in (1.5) and (1.6)). A crucial step in the proof of this limit comes from the estimate

[TABLE]

where $c$ and $\gamma$ are positive constants independent of $p$ and $w.$ Such an estimate is deduced by applying a version of the global gradient estimate by Cianchi and Maz’ya (see [9]) adapted for the $p$-Laplacian by Ercole in [12].

2 Existence and applications

In this section we assume that $\partial\Omega$ is at least of class $C^{1,\tau}$ for some $\tau\in(0,1).$

We recall that in the particular case where $\Omega$ is a ball centered at $x_{0}\in\mathbb{R}^{N}$ with radius $R>0$ the function $\phi_{p}$ is radially symmetric, radially decreasing and explicitly given by the expression

[TABLE]

It follows from this formula that

[TABLE]

As for a general bounded domain $\Omega,$ one can combine Schwarz symmetrization and (2.1) to derive the upper bound

[TABLE]

where $\left|\Omega\right|$ denotes the volume of $\Omega$ and $\omega_{N}$ denotes the volume of the unit ball.

In sequel $\left|\cdot\right|_{1,\eta}$ will denote the norm of $C^{1,\eta}(\overline{\Omega})$ defined by

[TABLE]

where

[TABLE]

for each $i\in\left\{1,2,\ldots,N\right\}.$

The set $\mathcal{S}_{p}$ that appears in the sequence is defined in (1.4).

Proposition 2.1

If $u\in\mathcal{S}_{p},$ then

[TABLE]

Moreover, there exist positive constants $\eta_{p}$ and $K_{p},$ such that $u\in C^{1,\eta_{p}}(\overline{\Omega})$ and

[TABLE]

Proof. Inequality (2.3) follows directly from the comparison principle applied to $u$ and $-u.$ The remaining assertions follow directly from the well-known regularity result by Lieberman [21, Theorem 1] applied to the $p$-Laplacian by taking into account that (2.2) and (2.3) yield $\left|u\right|\leq M_{0}.$

Remark 2.2

According to [21, Theorem 1] the constants $\eta_{p}$ and $K_{p}$ depend only on $\tau,$ $N,$ $p,$ $M_{0}$ and $\Omega.$

As for the constant $k_{p}$ defined in (1.3) we observe that

[TABLE]

where the latter inequality follows from Proposition 2.1.

Using the well-known fact (see [19]):

[TABLE]

one obtains a lower bound to $k_{p}$ is in terms of $\left\|\phi_{p}\right\|_{\infty}$ and $\left\|d_{\Omega}\right\|_{\infty}$:

[TABLE]

Corollary 2.3

If $g\in L^{\infty}(\Omega)$ and $u\in W_{0}^{1,p}(\Omega)$ is the only weak solution to

[TABLE]

then $u\in C^{1,\eta_{p}}(\overline{\Omega})$ and the following estimates hold

[TABLE]

[TABLE]

and

[TABLE]

Proof. It suffices to consider $\left\|g\right\|_{\infty}\not=0.$ Let

[TABLE]

As $u\in\mathcal{S}_{p}$ (note that $\left\|\widetilde{g}\right\|_{\infty}=1$) the estimates (2.5) and (2.6) follow directly from Proposition 2.1 applied to $v$, and (2.7) follows from the definition of $k_{p}.$

Lemma 2.4

One has

[TABLE]

Proof. As

[TABLE]

and $e_{p}=\phi_{p}=0$ on $\partial\Omega,$ it follows from the comparison principle that

[TABLE]

Hence, $1=\left\|e_{p}\right\|_{\infty}\leq\lambda_{p}^{1/(p-1)}\left\|\phi_{p}\right\|_{\infty}.$

Now, we present our main existence result.

Proof of Theorem 1.2. We recall, from the definition of $\mathcal{E}(M)$ that

[TABLE]

Let us consider the closed, convex and bounded subset $F\subset C^{1}(\overline{\Omega})$ defined by

[TABLE]

Let $T:F\subset C^{1}(\overline{\Omega})\rightarrow C^{1}(\overline{\Omega})$ be the operator that assigns to each $u\in F$ the only function $T(u)\in W_{0}^{1,p}(\Omega)$ satisfying

[TABLE]

Thus, for each $u\in F$ the function $U=T(u)$ is the only weak solution in $W_{0}^{1,p}(\Omega)$ to the Dirichlet problem

[TABLE]

where the nonlinearity $g(x,t)$ is defined from $u$ by the expression

[TABLE]

The uniqueness of $U$ follows from [11] as $g$ is sublinear in the variable $t$ (recall that $p>q$).

Let us define

[TABLE]

We are going to prove the existence of $U$ from the sub-super solution method by showing that: $\overline{u}$ is a supersolution to 2.12, $\underline{u}$ is a subsolution to the same problem, and $\underline{u}\leq\overline{u}$  in $\Omega.$

As $u\in F$ we have that

[TABLE]

and

[TABLE]

Hence,

[TABLE]

Therefore, as

[TABLE]

the above estimates and (2.9) imply that

[TABLE]

since

[TABLE]

As $-\Delta_{p}\phi_{p}=1$ in $\Omega,$ it follows from (2.13) that

[TABLE]

Thus, recalling that $\overline{u}=0$ on $\partial\Omega,$ we conclude that $\overline{u}$ is a supersolution to (2.12).

Using that $e_{p}^{p-q}\leq\left\|e_{p}\right\|_{\infty}^{p-q}\leq 1$ in $\Omega,$ we have that

[TABLE]

Hence, as $\underline{u}=0$ on $\partial\Omega$ we conclude that $\underline{u}$ is a subsolution to (2.12).

In order to prove that $\underline{u}\leq\overline{u}$  in $\Omega$ we first observe from (1.5) and (2.9) that

[TABLE]

so that

[TABLE]

Hence, by using (2.8) we obtain

[TABLE]

since

[TABLE]

It follows from (2.14) that

[TABLE]

and this implies that $\underline{u}\leq\overline{u}$  in $\Omega$ by the comparison principle.

Therefore, we can apply the sub-super solution method to guarantee the existence of a weak solution $U\in W_{0}^{1,p}(\Omega)$ to (2.12) satisfying

[TABLE]

As

[TABLE]

we note from Corollary 2.3 that $U=T(u)\in C^{1,\eta_{p}}(\overline{\Omega}),$

[TABLE]

and

[TABLE]

Combining (2.15) and (2.16) we conclude that $U=T(u)\in F,$ meaning that $T(F)\subset F.$

Using the compactness of the embedding $C^{1,\eta_{p}}(\overline{\Omega})\hookrightarrow C^{1}(\overline{\Omega})$ we can verify that $T:F\rightarrow F$ is compact. Therefore, Schauder’s fixed point theorem guarantees the existence of a fixed point $u_{p}\in F.$ Consequently, $T(u_{p})=u_{p}=u$ in (2.11), so that

[TABLE]

in the weak sense. In addition, as $u_{p}\in F$ the estimates (1.6) hold.

Remark 2.5

We have improved the lower bound in (2.10) with respect to [7] (which was also used in [3]) since we have shown in (2.14) that

[TABLE]

As a simple application of Theorem 1.2 we obtain the following existence result.

Corollary 2.6

Assume that $\partial\Omega\in C^{1,\tau}$ for some $\tau\in(0,1).$ Let $q,$ $a,$ $b,$ $l,$ $\lambda$ and $\beta$ be fixed, with $q,a,l\geq 1,$ $b,\lambda,\beta>0,$ and $s,\alpha\geq 0.$ For each

[TABLE]

there exists a positive constant $M_{p}$ satisfying

[TABLE]

Moreover, if

[TABLE]

then the problem (P) admits a weak solution $u_{p}\in W_{0}^{1,p}(\Omega)$ satisfying (1.6).

Proof. The hypotheses imply that the function

[TABLE]

satisfies $\lim_{t\rightarrow 0^{+}}\varphi(t)=+\infty$ and $\lim_{t\rightarrow\infty}\varphi(t)=0^{+}.$ Consequently, there exists $M_{p}>0$ such that $\varphi(M_{p})=\frac{1}{2},$ which is (2.17). Hence, as $m$ satisfies (2.18) we have

[TABLE]

so that $(\lambda,\beta,m)\in\mathcal{E}(M_{p}).$

Now, we present some more applications of Theorem 1.2 that extends or complements some recent results for problems involving exponential and convection terms.

Corollary 2.7

Assume that $\partial\Omega\in C^{1,\tau}$ for some $\tau\in(0,1).$ Let $q,$ $a,$ $b,$ $l,$ $\alpha,$ s\ and $m$ be fixed, with $p>q\geq 1,$ $m,b>0,$ $\alpha,s\geq 0,$ $a\geq 1,$ and

[TABLE]

There exists a positive constant $M_{p}$ such that if

[TABLE]

then the problem (P) admits a weak solution $u_{p}\in W_{0}^{1,p}(\Omega)$ satisfying (1.6).

Proof. We can write the inequality (2.9) as

[TABLE]

where

[TABLE]

and

[TABLE]

As $\varphi_{2}$ is strictly increasing and

[TABLE]

there exists a unique $M_{p}>0$ such that

[TABLE]

Using such $M_{p}$ we define $\mathcal{D}$ in (2.19) by the inequality

[TABLE]

Thus, if $(\lambda,\beta)\in\mathcal{D}$ we obtain (2.9) from (2.20), (2.21) and (2.22). The existence result follows then from Theorem 1.2.

Geometrically, $\mathcal{D}$ is the region in the quadrant $\mathbb{R}_{+}^{2}$ of the $\lambda\beta$-plane that lies below the line

[TABLE]

In [2], de Araujo and Faria considered the Dirichlet problem

[TABLE]

where $0<r_{1},r_{2}<N-1,$ $a_{1}>0,$ $a_{2}\geq 0,$ and $f:[0,\infty)\rightarrow\mathbb{R}$ is a continuous function satisfying

[TABLE]

They used an approximation scheme to prove the existence of a weak solution $u\in W_{0}^{1,N}(\Omega)$ to (2.23) whenever $\gamma\in(0,\gamma^{\ast}),$ for some $\gamma^{\ast}>0.$

Note that (2.23) is a particular case of (P) with $\lambda=\gamma a_{1},$  $\beta=\gamma a_{2},$  $q=r_{1}+1,$  $a=1,$  $b=r_{2},$  $m=a_{3},$  $l=r_{3}+1,$  and $s=\frac{N}{N-1}.$

We remark that Corollary 2.7 extends the result by de Araujo and Faria in [2] (for the case $f(t)=a_{3}t^{r_{3}}e^{\alpha t^{N/(N-1)}}$) since it admits $p\not=N$ and also allows the convection term to be multiplied by a power of the solution.

The following corollary further extends the result of de Araujo and Faria (for the case $f(t)=a_{3}t^{r_{3}}e^{\alpha t^{N/(N-1)}}$) by admitting $a+b>p$ (in (2.23) this means that $r_{2}>N-1$).

Corollary 2.8

Assume that $\partial\Omega\in C^{1,\tau}$ for some $\tau\in(0,1).$ Let $q,$ $a,$ $b,$ $l,$ $\alpha,$ $s,$ $\beta$ and $m$ be fixed, with $a,l\geq 1,$ $\alpha,s\geq 0$ and $\beta,m>0.$ Suppose that

[TABLE]

There exists a positive constant $M_{p}$ such that if

[TABLE]

then (P) admits a weak solution $u_{p}\in W_{0}^{1,p}(\Omega)$ satisfying (1.6).

Proof. Now, we write the inequality (2.9) as

[TABLE]

where

[TABLE]

and

[TABLE]

As $\varphi_{2}$ is strictly increasing and

[TABLE]

there exists $M_{p}>0$ such that

[TABLE]

Thus, if $\lambda\leq\lambda^{\ast}$ then $\varphi_{1}(M_{p})\leq\frac{1}{2}$ and (2.9) holds. Consequently, we can apply Theorem 1.2 to arrive at the desired result.

In the notation of (2.23) we have

[TABLE]

where $M$ is defined by the equation

[TABLE]

Proceeding as in the two previous proofs we obtain the following result.

Corollary 2.9

Assume that $\partial\Omega\in C^{1,\tau}$ for some $\tau\in(0,1).$ Let $q,$ $a,$ $b,$ $l,$ $\alpha,$ s\ and $\beta$ be fixed, with $b,\beta>0,$ $a\geq 1,$ $\alpha,s\geq 0.$ Suppose that

[TABLE]

If

[TABLE]

and

[TABLE]

then (P) admits a weak solution $u_{p}\in W_{0}^{1,p}(\Omega)$ satisfying (1.6).

3 A nonexistence result

In [17] Garcia Azorero and Peral Alonso proved in Theorem 2.1 that the problem

[TABLE]

does not have a solution if

[TABLE]

In this section, we extends the nonexistence result by Garcia Azorero and Peral Alonso for the more general equation

[TABLE]

The following lemma was proved by Allegretto and Huang (see [1, Theorem 2.4]) as a consequence of Picone’s identity.

Lemma 3.1

Let $h\in L^{\infty}(\Omega)$ be a nonnegative function. The Dirichlet problem

[TABLE]

has a weak solution if and only if $h\equiv 0$ in $\Omega$ and $u=0$ on $\partial\Omega.$ In this case, the solution is a multiple of $e_{p}.$

Proposition 3.2

Suppose that $u_{m}\in W_{0}^{1.p}(\Omega)$ is a positive weak solution to the Dirichlet problem

[TABLE]

where $g\in L^{\infty}(\Omega),$ $g\geq 0,$ $1\leq l<p$ and $\alpha,s>0.$ Then

[TABLE]

Proof. Let us consider the strictly positive function

[TABLE]

A simple calculation shows that the only critical point of $Q$ is

[TABLE]

As

[TABLE]

we have that $t_{m}$ is the only global minimum point. Thus,

[TABLE]

where

[TABLE]

It follows that

[TABLE]

with the equality occurring only if either $u_{m}=t_{m}$ or $u_{m}=0.$

Now, we observe that

[TABLE]

where

[TABLE]

We are going to show that

[TABLE]

which is (3.4). Let us suppose, by contradiction, that

[TABLE]

Owing to (3.6) and Lemma 3.1 this implies that $h\equiv 0$ a.e. in $\Omega.$ Thus,

[TABLE]

Hence, if $mC_{p}>\lambda_{p}$ then (3.7) leads to the absurd

[TABLE]

and if $mC_{p}=\lambda_{p}$, then (3.5) and (3.7) yield

[TABLE]

This implies that $g=0$ and leads to the equality in (3.5) which is absurd, for it means that

[TABLE]

We remark that (3.4) improves the estimate (3.2) when $1<p\leq 1+e.$ In fact, in this case, (3.1) also has no solution if

[TABLE]

4 Asymptotic behavior

In this section we assume a stronger assumption on the regularity of $\Omega$: either $\partial\Omega\in C^{1,1}$ or $\Omega$ convex and $\partial\Omega\in C^{1,\tau}$ ($\tau$ as before).

Our goal is to prove the uniform convergence of $u_{p}$ to $d_{\Omega},$ as $p\rightarrow\infty,$ where $u_{p}\in W_{0}^{1,p}(\Omega)$ is the solution to (P) given by Corollary 2.6. Thus, we consider: $p>\max\left\{q,a+b,l\right\},$ the positive parameters $\lambda$ and $\beta$ arbitrary, and the parameter $m$ restricted to the interval $(0,m_{p}].$ We recall that

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$M_{p}$ is defined by (2.17), and

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To achieve our goal we will make use of the explicit gradient estimates derived by Ercole in [12], They are based on the results by Cianchi and Maz’ya in [9] for a class of operators that includes the $p$-Laplacian as a very particular case.

We recall that the Lorentz space $L^{\theta,1}(\Omega)$ consists of all measurable functions $v:\Omega\rightarrow\mathbb{R}$ such that

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Here, $\theta^{\prime}=\frac{\theta}{\theta-1}$ and $v^{\ast}:[0,\infty)\rightarrow[0,\infty]$ stands for the decreasing rearrangement of $v$, which is defined as

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where

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is the distribution function of $v.$

As it is well known, $L^{\theta,1}(\Omega)$ is a Banach space endowed with the norm

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where $v^{\ast\ast}:(0,\infty)\rightarrow[0,\infty)$ is defined as

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Thus, if $g\in L^{\infty}(\Omega),$ then

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as $\left\|g^{\ast}\right\|_{\infty}=\left\|g\right\|_{\infty}.$

Lemma 4.1

Suppose that $p\geq 2$ and either $\partial\Omega\in C^{1,1}$ or $\Omega$ convex. Let $v\in W_{0}^{1,p}(\Omega)$ be the solution of the Dirichlet problem

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where $g\in L^{\infty}(\Omega).$ Then, there exist positive constants $c$ and $\gamma,$ that are uniform with respect to $p$ and $g,$ such that

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Proof. According to Theorem 1.2 of [12],

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where $C$ is a positive constant that depends at most on $N$ and $\Omega.$ This estimate holds under the following assumptions: $N\geq 3,$ $\partial\Omega\in W^{2}L^{\theta,1},$ for some $\theta>N-1,$ and $f\in L^{N,1}(\Omega).$ Moreover, if the assumption $\partial\Omega\in W^{2}L^{\theta,1}$ is replaced with $\Omega$ convex, then the estimate (4.4) writes as

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Since $C^{1,1}\subset W^{2}L^{\theta,1},$ (4.2) and (4.4) lead to (4.3) with $c:=CN\left|\Omega\right|^{\frac{1}{N}}$ and $\gamma=\frac{5}{2}+\frac{\theta N}{\theta-(N-1)}.$ If $\Omega$ is convex we can take $\gamma=\frac{5}{2}.$

As for $N=2,$ by assuming that $\partial\Omega\in W^{2}L^{\theta,1},$ for some $\theta>1,$ and $f\in L^{q}(\Omega)$, for some $q>2,$ Theorem 1.3 of [12] yields the estimate

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where $C$ depends at most on $\Omega$ and $q.$ Hence, as $\left\|g\right\|_{q}\leq\left\|g\right\|_{\infty}\left|\Omega\right|^{1/p}$ and $C^{1,1}\subset W^{2}L^{\theta,1}$ the estimate (4.5) holds with $c:=C\left|\Omega\right|^{1/p}$ and $\gamma:=\frac{5}{2}+\frac{2\theta}{\theta-1}.$ If $\Omega$ is convex, then (4.5) writes as

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in which case we can take $\gamma=\frac{5}{2}.$

Remark 4.2

Following Cianchi and Maz’ya in [9], the assumption $\partial\Omega\in W^{2}L^{\theta,1}$ means that the boundary of $\Omega$ is locally the subgraph of a function of $N-1$ variables whose second-order distributional derivatives lie on the Lorentz space $L^{\theta,1}.$ The regularity hypothesis $\partial\Omega\in W^{2}L^{N-1,1}$ is the weakest possible integrability assumption on second-order derivatives for the first order derivatives to be continuous, and hence for $\partial\Omega\in C^{1,0}$ [8].

In the sequel we will use some known results that are gathered in the following lemma.

Lemma 4.3

The following convergence results are well known:

$\phi_{p}$ converges uniformly in $\overline{\Omega}$ to $d_{\Omega}$ as $p\rightarrow\infty$ (see [5, 20]). 2. 2.

$\lim_{p\rightarrow\infty}\lambda_{p}^{1/p}=\left\|d_{\Omega}\right\|_{\infty}^{-1}$ (see [19]). 3. 3.

For each sequence $\left(p_{n}\right),$ with $p_{n}\rightarrow\infty,$ there exists a subsequence $\left(p_{n_{j}}\right)$ and a function $e_{\infty}\in W^{1,\infty}(\Omega)\cap C_{0}(\overline{\Omega})$ such that: $\left\|e_{\infty}\right\|_{\infty}=1,$ $e_{p_{n_{j}}}$ converges uniformly in $\overline{\Omega}$ to $e_{\infty},$ and

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Remark 4.4

The strict positiveness of $e_{\infty}$ follows from the Harnack inequality proved in [4, Theorem 1] (see also [22, Corollary 4.5])  since $e_{\infty}$ is $\infty$-superharmonic and not identically zero ($\left\|e_{\infty}\right\|_{\infty}=1$). The equality $e_{\infty}=\frac{d_{\Omega}}{\left\|d_{\Omega}\right\|_{\infty}}$ does not hold for a general bounded domain $\Omega.$ It holds for balls, annuli and stadiums (see [24]), but not for a square, for example (see [19, Proposition 4.1]).

The following result is crucial in our analysis.

Proposition 4.5

One has

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Proof. We observe from (2.4) and item $1$ of Lemma 4.3 that

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According to Lemma 4.1

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Consequently,

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so that

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Lemma 4.6

One has

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and

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where $m_{\infty}$ is defined in (1.1).

Proof. We can write (2.17) as

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where

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It follows from (4.10) that

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so that

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Here we have used the fact that

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(according to item 1 from Lemma 4.3).

Without loss of generality we analyze the case $r\geq q$ (the case $r<q$ is analogous), so that

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for all $p$ sufficiently large. Hence, (4.10) yields

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for all $p$ sufficiently large. Then, using (4.6) and (4.12) we make $p\rightarrow\infty$ in the inequality

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to find

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Combining (4.11) and (4.13) we conclude that

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and then, in view of (4.12), we obtain (4.7).

Now, let us set

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Combining (4.6), (4.12) and (4.14) we obtain from (4.10) the equality

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which leads to (4.8).

Finally, after noticing that

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we obtain (4.9) from (4.7) and (4.8).

Lemma 4.7

If $p_{n}\rightarrow\infty,$ then there exists a subsequence $(u_{p_{n_{j}}})$ converging uniformly in $\overline{\Omega}$ to a function $u_{\infty}\in W^{1,\infty}(\Omega)\cap C_{0}(\overline{\Omega})$ such that

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where $e_{\infty}$ is a positive $\infty$-superharmonic function satisfying $\left\|e_{\infty}\right\|_{\infty}=1.$

Proof. Combining Lemma 4.3 with (4.1), (4.6) and (4.7) we have that

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Therefore, by Arzelá-Ascoli Theorem there exists a subsequence $(u_{p_{n_{j}}})$ converging uniformly in $\overline{\Omega}$ to a function $u_{\infty}\in W^{1,\infty}(\Omega)\cap C_{0}(\overline{\Omega}).$ By item 3 of Lemma 4.3 we can assume that $e_{p_{n_{j}}}$ converges uniformly to a positive $\infty$-superharmonic function satisfying $\left\|e_{\infty}\right\|_{\infty}=1.$

Hence, taking into account item 2 of Lemma 4.3, the inequalities in (4.15) follow after letting $j\rightarrow\infty$ in the estimates

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Proposition 4.8

Assume that $\partial\Omega\in C^{1,1}.$ Let $q,$ $a,$ $b,$ $l,$ $\lambda$ and $\beta$ be fixed, with $q>1,$ $a,l\geq 1,$ and $b,s,\alpha,\lambda,\beta>0.$ For each $p>\max\left\{q,a+b,l\right\}$ let $M_{p},$ $m_{p}$ and $u_{p}\in W_{0}^{1,p}(\Omega)$ be as in Corollary 2.6. Then,

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Proof. It follows from Lemma 4.7 that, up to subsequence, $u_{p}$ converges uniformly in $\overline{\Omega}$ to a function $u_{\infty}\in W^{1,\infty}(\Omega)\cap C_{0}(\overline{\Omega})$ satisfying (4.15).

We recall that $u_{p}$ is also the only weak solution to the Dirichlet problem

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where

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As

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we note that

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We also know that the solution to (4.16) is the only positive minimizer of the functional

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Hence, recalling that $d_{\Omega}\in W_{0}^{1,p}(\Omega)$ and that $\left|\nabla d_{\Omega}\right|=1$ a.e. in $\Omega,$ we obtain from the inequality $I_{p}(u_{p})\leq I_{p}(d_{\Omega})$ that

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It follows from (4.18) that

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As $d_{\Omega}-u_{\infty}\geq 0$, the uniform convergence from $u_{p}$ to $u_{\infty}$ implies that

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Using again that $d_{\Omega}-u_{\infty}\geq 0$ we have

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The uniform convergence from $u_{p}$ to $u_{\infty}$ combined with and (4.17) yields

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since

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Thus, it follows from (4.21) that

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Hence,

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Combining (4.19) and (4.23) we obtain

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In view of (4.20) and (4.22) we arrive at

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Therefore, using once more that $u_{\infty}\leq d_{\Omega}$ we conclude that $u_{\infty}=d_{\Omega}.$

Observing that the limit function is always $d_{\Omega}$ we conclude that $u_{p}$ converges uniformly to $d_{\Omega}$ in $\overline{\Omega}$ (independently of subsequences).

Proof of Theorem 1.1. Let $p_{0}>\max\left\{q,a+b,l\right\}$ be such that $0<m<m_{p}$ for all $p>p_{0}.$ This follows by combining (4.9) with the fact that $m<m_{\infty}.$ Thus, if $p>p_{0}$ the existence of $u_{p}$ follows from Corollary 2.6 and the convergence (1.2) follows from Proposition 4.8.

Acknowledgments

Anderson L. A. de Araujo was partially supported by FAPEMIG/Brazil APQ-02375-21, RED-00133-21 and by CNPq/Brazil 307575/2019-5. Grey Ercole was partially supported by FAPEMIG/Brazil PPM-00137-18, CNPq/Brazil 305578/2020-0 and FAPDF 04/2021.