Existence and uniqueness of solutions to some singular equations with natural growth

Abstract

We study existence and uniqueness of nonnegative solutions to a problem which is modeled by

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta _p u = u^{-\theta }|\nabla u|^p + fu^{-\gamma }& \text {in}\, \Omega , \\ u=0 & \text {on}\ \partial \Omega , \end{array}\right. } \end{aligned}$$

where $\Omega$ is an open bounded subset of ${\mathbb {R}}^N$ ($N\ge 2$), $\Delta _p$ is the p-Laplacian operator ($1<p<N$), $f\in L^1(\Omega )$ is nonnegative and $\theta , \gamma \ge 0$. Examples and extensions are discussed at the end of the paper.

1 Introduction

The aim of the paper is the study of the existence and uniqueness of nonnegative solutions to the following singular elliptic problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -{\text {div}}(a(x,\nabla u)) = g(u)|\nabla u|^p + h(u)f& \text {in}\, \Omega , \\ u=0 & \text {on}\ \partial \Omega , \end{array}\right. } \end{aligned}$$

(1.1)

where $\Omega$ is an open bounded subset of ${\mathbb {R}}^N$ ($N\ge 2$), f is a nonnegative integrable function, a is a nonlinear operator of Leray–Lions type (i.e., it satisfies (2.2), (2.3), (2.4) below) which, at first, can be thought as the p-Laplacian, $1<p<N$. Here, g and h are nonnegative functions which possibly satisfy $g(0)=\infty$ and/or $h(0)=\infty$.

We start giving a brief overview of the mathematical framework related to problem (1.1). Firstly, let us briefly mention that the classical reference for existence of solutions to (1.1) when $g,h\equiv 1$ and $f\in W^{-1,p'}(\Omega )$ is [30]. When f is just a bounded function we refer to [7,8,9] where the above problem has been widely studied in the presence of an absorption term and also with a sub/super solutions argument. The first papers concerning existence and nonnexistence of unbounded solutions to equations having reaction gradient terms with natural growth as in the above equation are [19, 20, 29]. Here, in case p-Laplace operator and $g,h\equiv 1$, it is shown existence of a solution u to (1.1) such that $e^u-1\in W^{1,p}_0(\Omega )$ if $||f||_{L^{\frac{N}{p}}(\Omega )}$ is small enough. Let us stress that the smallness assumption on the norm f is not just technical and, otherwise, nonnexistence phenomenons appear.

When $h\equiv 1$ and g is a continuous and integrable function on ${\mathbb {R}}$, one can show existence of solutions to (1.1) where f can even be a measure (see [36]); only later, in [37], the authors prove that if $f\in L^m(\Omega )$ ($1<m<\frac{N}{p}$) then (1.1) has a solution if there exists a positive $\eta <\frac{N(m-1)}{m}$ such that for all $s\in {\mathbb {R}}$

$$\begin{aligned} M_1\le \frac{e^{|\Gamma (s)|}}{(1+|\Phi (s)|)^{\eta (p-1)}}\le M_2, \end{aligned}$$

for $M_1, M_2$ positive constants, $\Gamma (s) = \int _{0}^{s} |g(t)| \ dt$ and $\Phi (t) = \int _{0}^{s} e^{|\Gamma (t)|} \ dt$. Let us just underline that the previous condition when $m\rightarrow 1$ is simply saying that g needs to be integrable at infinity, regaining the request given in [36]. More references studying different aspects of problem (1.1) in case of a general continuous g are [1, 2, 11, 17, 18, 21, 28, 39].

The study of problem (1.1) when $g(0)=\infty$ is far more limited; in case of a nonlinear operator with linear growth and if $g(s)= Cs^{-\theta }$ for small s with $\theta <1$ and $g(s) \overset{s\rightarrow \infty }{\rightarrow }0$ then existence of solutions is proved in [24] when $f\in L^{\frac{N}{2}}(\Omega )$ is nonnegative. Moreover, the authors also remark that if g is not degenerating at infinity then an existence result can be recovered requiring the norm of f small enough. Let us also underline that in the former paper sign-changing solutions are considered in the presence of f with general sign. In case of a strongly singular g, i.e., $\theta =1$, we refer to [4] where under smallness conditions on C, then one has existence of a positive solution for any nonnegative $f\in L^m(\Omega )$ with $m>1$. We refer the interested reader to [3, 13, 25, 27] for more discussions regarding the case $g(0)=\infty$.

Unsurprisingly uniqueness is a difficult task to achieve for (1.1). In case of the p-Laplace operator $(p\ge 2)$ we refer to [5, 31] where uniqueness of bounded solutions holds when g is nonincreasing. We also refer to [38] where the author shows the uniqueness of a suitable finite energy entropy solution.

We also want to highlight that the following change of variable (recall that $\Gamma (s) = \int _{0}^{s} g(t) \ dt$)

$$\begin{aligned} v= \Phi (u) := \int _{0}^{u} e^{\Gamma (t)} \ dt \end{aligned}$$

formally transforms the equation

$$\begin{aligned} -\Delta u = g(u) |\nabla u|^2 + h(u)f, \end{aligned}$$

into the semilinear one given by

$$\begin{aligned} -\Delta v = h(\Phi ^{-1}(v))e^{\Gamma (\Phi ^{-1}(v))}f, \end{aligned}$$

which has been widely studied in literature. For instance, if one requires that

$$\begin{aligned} \displaystyle \lim _{s\rightarrow \infty } e^{\Gamma (s)}h(s) <\infty , \end{aligned}$$

(1.2)

then one falls into problems as in [10, 12, 15, 22, 23, 26, 32,33,34,35] for which existence and uniqueness (when expected) holds even for measure data. Unfortunately, the change of variable is not always admissible and, even when well defined, it could not take directly to a solution in the presence of a general nonlinear operator.

In the present paper, our aim is giving a complete account of existence and, when possible, uniqueness of a nonnegative solution to (1.1) when $g(s) \le c_1s^{-\theta }$ and $h(s)\le c_2s^{-\gamma }$, with $\theta ,\gamma \ge 0$ for small s and $f \in L^1(\Omega )$ is nonnegative. For what concerns the existence of a solution, we will work by approximation through solutions to regularized problems. We are mostly interested in providing it when g is not necessarily integrable at infinity, namely when the interplay between g and h given by (1.2) is assumed. Let just observe that, if $h\equiv 1$, we recover that g needs to be integrable at infinity, which is coherent with the existing literature. Let us also underline that in the sequel if $\theta <1$ then $\Gamma (s) = \int _{0}^{s} g(t) \ dt$ is well defined. By contrast, when $\theta \ge 1$, (1.2) needs to be meant with a different definition of $\Gamma$ and the change of variable will not work for free.

Clearly, this has strongly implications on the existence methods according to the value of $\theta$; indeed, as $\theta <1$, we will be able to obtain a priori estimates by formally performing the change of variable. Otherwise, since the change of variable is not well defined any more, we will need to work more in order to get the estimates on the approximating sequence. When $\theta <1$ and $\gamma \le 1$ we will show that the solution found has the truncation $T_k(u)$ at any level $k>0$ with finite energy. On the contrary, when $\gamma >1$, this property is preserved just locally in $\Omega$. When $\theta =1$ and the change of variable is lacking, one can show that the same result, with a different technique, holds by requiring a smallness condition on $c_1$ (recall that $g(s) \le c_1s^{-\theta }$ for small s) which is also proven to be sharp. Another important peculiarity of the problem is the positivity of the solution which is fundamental in order to show that $g(u)|\nabla u|^p$ is well posed. This is shown by a suitable application of the strong maximum principle on the solution itself which solves a suitable distributional formulation when $\theta < 1$ and $\gamma \le 1$. Otherwise, when $\gamma >1$, this will not be possible due to the local nature of the solution and this property will be gained at an approximation level and then preserved for the limit solution. Finally, when $\theta =1$, we will actually use the stronger result which is that the approximating sequence is bounded from below on any open set $\omega$ such that $\omega \subset \subset \Omega$; this, roughly speaking, will imply that the change of variable will be well defined on these sets.

Some words on the notion of solution we look for are in order. As already mentioned, when $\theta <1$ and $\gamma \le 1$, our approximation scheme takes us to a solution having the truncation $T_k(u)\in W^{1,p}_0(\Omega )$ at any level k; this suggesting to look for a renormalized solution to (1.1) which, under a suitable monotonicity condition on g, can be proven to be unique. Roughly speaking a renormalized solution is nothing more than looking what kind of formulation is solved by $T_k(u)$; it is more restrictive than the more usual distributional notion and it clearly implies the latter one. For a complete overview on renormalized setting, we refer to [14, 15, 32]. On the other side in case $\theta =1$ and $c_1$ is small enough, one shows the existence of a distributional solution such that $T_k(u)\in W^{1,p}_0(\Omega )$ for any $k>0$. Finally, if $c_1$ is not small enough or if $\theta >1$ and/or $\gamma >1$, we show existence of a distributional solution to (2.1) in which the boundary datum is assumed in a weaker sense than the usual sense of traces.

The paper is organized as follows: in Sect. 2 we set the problem giving the main assumptions and results. In Sect. 3, we present the proof of existence and uniqueness of a solution in the mild singular case $\theta <1$. In Sect. 4, we treat the strongly singular case $\theta =1$. Finally, in Sect. 5, we give some final remarks, a nonexistence result and an extension to a more general equation.

1.1 Notation

For a set E, |E| stands for its N-dimensional Lebesgue measure and we denote by $\chi _{E}$ its characteristic function. For a fixed $k>0$, we use the truncation functions $T_{k}:{{\,{{\mathbb {R}}}\,}}\rightarrow {{\,{{\mathbb {R}}}\,}}$ and $G_{k}:{{\,{{\mathbb {R}}}\,}}\rightarrow {{\,{{\mathbb {R}}}\,}}$ defined by

$$\begin{aligned} T_k(s):&=\max (-k,\min (s,k))\ \ \text { and} \ \ G_k(s):=s- T_k(s). \end{aligned}$$

We also use the following auxiliary functions

$$\begin{aligned} \displaystyle V_{\delta }(s):= {\left\{ \begin{array}{ll} 1 &\quad s\le \delta , \\ \displaystyle \frac{2\delta -s}{\delta } &\quad \delta<s< 2\delta , \\ 0 &\quad s\ge 2\delta , \end{array}\right. } \end{aligned}$$

(1.3)

and

$$\begin{aligned} \pi _\delta (s) = 1-V_\delta (s). \end{aligned}$$

(1.4)

If no otherwise specified, we will denote by C several positive constants whose value may change from line to line and, sometimes, on the same line. These values will only depend on the data but they will never depend on the indexes of the sequences we will gradually introduce. Finally, we underline the use of the convention to not relabel an extracted compact subsequence.

2 Main assumptions and results

We deal with the following problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -{{\,\mathrm{div}\,}}(a(x,\nabla u))= g(u)|\nabla u|^p + h(u)f &\quad \text {in}\;\Omega ,\\ u\ge 0 &\quad \text {in}\;\Omega , \\ u=0 &\quad \text {on}\;\partial \Omega , \end{array}\right. } \end{aligned}$$

(2.1)

where $\displaystyle {a(x,\xi ):\Omega \times {\mathbb {R}}^{N} \rightarrow {\mathbb {R}}^{N}}$ is a Carathéodory function satisfying the classical Leray–Lions assumptions for $1<p<N$, namely

$$\begin{aligned}&a(x,\xi )\cdot \xi \ge \alpha |\xi |^{p}, \quad \alpha >0, \end{aligned}$$

(2.2)

$$\begin{aligned}&|a(x,\xi )|\le \beta |\xi |^{p-1}, \quad \beta >0, \end{aligned}$$

(2.3)

$$\begin{aligned}&(a(x,\xi ) - a(x,\xi ^{'} )) \cdot (\xi -\xi ^{'}) > 0, \end{aligned}$$

(2.4)

for every $\xi \ne \xi ^{'}$ in ${\mathbb {R}}^N$ and for almost every x in $\Omega$.

The nonlinearities $g:[0,\infty )\mapsto [0,\infty ]$ and $h:[0,\infty )\mapsto [0,\infty ]$ are continuous, finite outside the origin with $g(0)\not =0$ and $h(0)\not =0$ and such that

$$\begin{aligned} \displaystyle \exists \;{c_1},\theta ,s_1>0\;\ \text {such that}\;\ g(s)\le \frac{c_1}{s^\theta } \ \ \text {if} \ \ s\le s_1, \end{aligned}$$

(2.5)

and

$$\begin{aligned} \begin{aligned} \displaystyle&\exists \;{c_2},\gamma ,s_2>0\;\ \text {such that}\;\ h(s)\le \frac{c_2}{s^\gamma } \quad \text {if } s\le s_2. \end{aligned} \end{aligned}$$

(2.6)

Finally, the datum $f\in L^1(\Omega )$ is nonnegative. For the entire section, we assume $\theta <1$ which means that the following function is well defined

$$\begin{aligned} \Gamma (s):= \frac{1}{\alpha }\int _{0}^{s} g(t) \ dt, \end{aligned}$$

(2.7)

and we denote by

$$\begin{aligned} \Psi (s) := \int _0^s e^{\frac{\Gamma (t)}{p-1}} \ dt. \end{aligned}$$

(2.8)

We also assume that the following growth relation between g and h is satisfied:

$$\begin{aligned} \limsup _{s \rightarrow \infty } e^{\Gamma (s)}h(s) < \infty . \end{aligned}$$

(2.9)

Remark 2.1

Here, we briefly discuss condition (2.9); for a general h, (2.9) seems to give the suitable interplay between g and h in order to have existence of solutions. Indeed, if $h=1$ then (2.9) requires the integrability of g at infinity which means that we recover classical conditions (see [4, 36,37,38]); let us also consider a different situation when $g(s) \approx s^{-1}$ for s large then the above condition is satisfied once $h(s)\approx s^{-1}$ once again for s large.

We specify the notion of renormalized solution to (2.1).

Definition 2.2

A positive and measurable function u which is almost everywhere finite is a renormalized solution to problem (2.1) if $T_k(u) \in W^{1,p}_0(\Omega )$ for any $k>0$, if

$$\begin{aligned} g(u)|\nabla u|^pS(u)\varphi , h(u)fS(u)\varphi \in L^1(\Omega ), \end{aligned}$$

(2.10)

and if

$$\begin{aligned} \int _{\Omega } a(x,\nabla u) \cdot \nabla \varphi S(u) + \int _{\Omega } a(x,\nabla u) \cdot \nabla u S'(u)\varphi = \int _{\Omega } g(u)|\nabla u|^pS(u)\varphi + \int _\Omega h(u)fS(u)\varphi , \end{aligned}$$

(2.11)

for every $S\in W^{1,\infty }({\mathbb {R}})$ with compact support and for every $\varphi \in W^{1,p}_0(\Omega )\cap L^\infty (\Omega )$. Finally, it holds

$$\begin{aligned} \lim _{t\rightarrow \infty } \frac{1}{t}\int _{\{t<u<2t\}} e^{\Gamma (u)} a(x,\nabla u)\cdot \nabla u =0. \end{aligned}$$

(2.12)

For the sake of clarity, we also fix the concept of distributional solution for (2.1).

Definition 2.3

A positive and measurable function u is a distributional solution to problem (2.1) if $T_k(G_\varepsilon (u)) \in W^{1,p}_0(\Omega )$ for any $k,\varepsilon >0$, if $|a(x,\nabla u)|, g(u)|\nabla u|^p, h(u)f \in L^1_{\mathrm{loc}}(\Omega )$ and if

$$\begin{aligned} \int _{\Omega } a(x,\nabla u) \cdot \nabla \varphi = \int _{\Omega } g(u)|\nabla u|^p\varphi + \int _\Omega h(u)f\varphi , \end{aligned}$$

(2.13)

for every $\varphi \in C^1_c(\Omega )$.

In the sequel we will first treat the case where $\theta <1$ and $\gamma \le 1$; under this assumption we will show that there exists a renormalized solution, which is also unique if some monotonicity condition is satisfied by g and h (see Theorems 2.5 and 2.6 below).

First of all, we show that, in the mild singular case, a renormalized solution turns out to be a distributional one.

Lemma 2.4

Let a satisfy (2.2), (2.3), (2.4) and let g and h satisfy (2.5) and (2.6) with $\theta <1$ and $\gamma \le 1$ with (2.9) in force and let also assume that $f\in L^1(\Omega )$ is nonnegative. Then, a renormalized solution u to (2.1) is also a distributional solution to (2.1).

Proof

Clearly one has that for any $k,\varepsilon$, $T_k(G_\varepsilon (u)) \in W^{1,p}_0(\Omega )$ so that the boundary request is satisfied.

Now, for some $r,k>0$, let us fix $\varphi = e^{\Gamma (T_r(u))}T_k(u)$ in (2.11); using (2.2) one obtains

$$\begin{aligned} \begin{aligned}&\int _{\Omega } a(x,\nabla u) \cdot \nabla T_k(u) e^{\Gamma (T_r(u))} S(u) + \int _{\Omega } a(x,\nabla u) \cdot \nabla u S'(u)e^{\Gamma (T_r(u))}T_k(u)\\&\quad \le \int _{\{u\ge r\}} g(u)|\nabla u|^pS(u)e^{\Gamma (T_r(u))}T_k(u) + \int _\Omega h(u)fS(u)e^{\Gamma (T_r(u))}T_k(u). \end{aligned} \end{aligned}$$

Since S has compact support one can simply take $r\rightarrow \infty$ in the previous, yielding to

$$\begin{aligned} \int _{\Omega } a(x,\nabla u) \cdot \nabla T_k(u) e^{\Gamma (u)} S(u) + \int _{\Omega } a(x,\nabla u) \cdot \nabla u S'(u)e^{\Gamma (u)}T_k(u) \le \int _\Omega h(u)fS(u)e^{\Gamma (u)}T_k(u). \end{aligned}$$

(2.14)

Now, we take $S=V_t$ with $t>k$ ($V_t(s)$ is defined in (1.3)) in (2.14), taking to

$$\begin{aligned} \alpha \int _{\Omega } e^{\Gamma (u)} |\nabla T_k(u)|^p -\frac{1}{t} \int _{\{t<u<2t\}} a(x,\nabla u) \cdot \nabla u e^{\Gamma (u)}T_k(u)\le & \int _\Omega h(u)fV_t(u)e^{\Gamma (u)}T_k(u) \\\le & \int _\Omega h(u)fe^{\Gamma (u)}T_k(u). \end{aligned}$$

(2.15)

Now, let us observe that the second term on the left hand side of (2.15) goes to zero as $t\rightarrow \infty$ thanks to (2.12). This means that

$$\begin{aligned} \alpha \int _{\Omega } e^{\Gamma (u)} |\nabla T_k(u)|^p\le & \int _\Omega h(u)fe^{\Gamma (u)}T_k(u) \le s_1^{1-\gamma } e^{\Gamma (s_1)}||f||_{L^1(\Omega )} \\&+ \sup _{s\in (s_1,\infty )} e^{\Gamma (s)}h(s) \ ||f||_{L^1(\Omega )} k = C(1+k), \end{aligned}$$

(2.16)

where the positive constant C does not depend on k. Hence, estimate (2.16) standardly implies that $|\nabla u|^{p-1} \in L^q(\Omega )$ with $q<\frac{N}{N-1}$ (see [6, Lemma 4.2]).

Now, let us take a nonnegative $\varphi \in C^1_c(\Omega )$ as a test function in (2.11) where $S= V_t$. One gets

$$\begin{aligned} \int _\Omega a(x,\nabla u)\cdot \nabla \varphi V_t(u) = \frac{1}{t}\int _{\{t<u<2t\}} a(x,\nabla u)\cdot \nabla u + \int _\Omega g(u)|\nabla u|^p V_t(u)\varphi + \int _\Omega h(u)fV_t(u)\varphi . \end{aligned}$$

(2.17)

Let us getting rid of the nonnegative first term on the right hand side and applying the Fatou Lemma with respect to t, one obtains that

$$\begin{aligned} \int _\Omega g(u)|\nabla u|^p\varphi + \int _\Omega h(u)f\varphi \le \int _\Omega a(x,\nabla u)\cdot \nabla \varphi \le \beta \int _\Omega |\nabla u|^{p-1}|\nabla \varphi | \end{aligned}$$

and the right hand side of the previous is finite since, as already shown, $|\nabla u|^{p-1} \in L^q(\Omega )$ with $q<\frac{N}{N-1}$. This means that $g(u)|\nabla u|^p, h(u)f\in L^1_{\mathrm{loc}}(\Omega )$. Hence, one can pass to the limit as $t\rightarrow \infty$ by the Lebesgue theorem in all terms of (2.17) except the second one which goes to zero thanks to (2.12). From this easily follows that (2.13) holds. The proof is concluded. $\square$

Hence we have the following existence result:

Theorem 2.5

Let a satisfy (2.2), (2.3), (2.4) and let g and h satisfy (2.5) and (2.6) with $\theta <1$ and $\gamma \le 1$ with (2.9) in force. Finally, assume $f\in L^1(\Omega )$ is nonnegative then there exists a renormalized solution to (2.1) such that $\Psi (u)^{p-1} \in L^q(\Omega )$ for every $q<\frac{N}{N-p}$ and $|\nabla \Psi (u)|^{p-1} \in L^q(\Omega )$ for every $q<\frac{N}{N-1}$.

Here, we state the above cited uniqueness theorem.

Theorem 2.6

Let $a(x,\xi ) = |\xi |^{p-2}\xi$ and let g and h be such that (2.5) and (2.6) are satisfied with $\theta <1$ and $\gamma \le 1$ such that $e^{\Gamma (s)}h(s)$ is nonincreasing with respect to s. There is at most one renormalized solution to (2.1).

Finally, we state the existence of a distributional solution in case of a strongly singular source; namely the case $\gamma >1$ that, in general, will give rise to local solutions which attend the boundary datum in a weak sense.

Theorem 2.7

Let a satisfy (2.2), (2.3), (2.4) and let g and h satisfy (2.5) and (2.6) with $\theta < 1$ with (2.9) in force. Let assume that $f\in L^1(\Omega )$ is nonnegative then there exists a distributional solution to (2.1) such that, for any $\omega \subset \subset \Omega$, $T_k(u)\in W^{1,p}(\omega )$ for any $k>0$, $\Psi (u)^{p-1} \in L^q(\omega )$ for every $q<\frac{N}{N-p}$ and $|\nabla \Psi (u)|^{p-1} \in L^q(\omega )$ for every $q<\frac{N}{N-1}$ and for every $\varepsilon >0$.

Remark 2.8

We highlight that the regularity given on $\Psi (u)$ in Theorems 2.5 and 2.7 holds for u itself. Indeed, by the definition of $\Psi$ one has that both $\Psi (u) \ge u$ and $|\nabla \Psi (u)| \ge |\nabla u|$ holds.

3 Proof of the existence and uniqueness results of Section 2

3.1 A priori estimates

We work by approximating problem (2.1) through

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -{{\,\mathrm{div}\,}}(a(x,\nabla u_n)) + \frac{u_n^{p-1}}{n} = g_n(u_n)|\nabla u_n|^p + h_n(u_n)f_n &\quad \text {in}\;\Omega ,\\ u_n\ge 0 &\quad \text {in}\;\Omega ,\\ u_n=0 & \quad \text {on}\;\partial \Omega , \end{array}\right. } \end{aligned}$$

(3.1)

where $g_n(s) = T_n(g(s))$, $h_n(s) = T_n(h(s))$ and $f_n= T_n(f)$. In order to deal with problem (3.1) we use the following truncations of $\Gamma , \Psi$ defined in (2.7), (2.8):

$$\begin{aligned} \Gamma _{n}(s) = \frac{1}{\alpha }\int _{0}^{s} g_n(t) \ dt, \end{aligned}$$

(3.2)

and

$$\begin{aligned} \Psi _{n}(s) = \int _0^s e^{\frac{\Gamma _n(t)}{p-1}} \ dt. \end{aligned}$$

(3.3)

Now, we briefly sketch how to deduce the existence of a nonnegative solution $u_n \in W^{1,p}_0(\Omega )\cap L^\infty (\Omega )$ of problem (3.1). Firstly, let us observe that it follows from [9] that there exists a nonnegative solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -{{\,\mathrm{div}\,}}(a(x,\nabla w)) + \frac{w^{p-1}}{n} = g_n(w)|\nabla w|^p + h_n(v)f_n &\quad \text {in}\;\Omega ,\\ w=0 & \quad \text {on}\;\partial \Omega , \end{array}\right. } \end{aligned}$$

(3.4)

for any nonnegative v belonging to $L^p(\Omega )$ and such that $||w||_{L^\infty(\Omega)}\le c_n$ for some positive constant $c_n$ which does not depend on $v$. Now, through an application of the Schauder theorem, one can show that the application $T: L^p(\Omega )\mapsto L^p(\Omega )$ such that $T(v)=w$ admits a fixed point. Hence, let us show an invariant ball for T on which the application is both continuous and compact. Indeed, taking $e^{\Gamma _{n}(w)}w$ as a test function in (3.4), one has

$$\begin{aligned}\alpha \int_\Omega |\nabla w|^p \le \int_\Omega h_n(v)f_n e^{\Gamma_{n}(w)} w \le n^2e^{\frac{nc_n}{\alpha}}c_n|\Omega|. \end{aligned}$$

(3.5)

Then, an application of the Poincaré inequality gives that

$$\begin{aligned} ||w||_{L^p(\Omega)} \le \left(\frac{n^2e^{\frac{nc_n}{\alpha}}c_n|\Omega|}{\alpha C_p^p}\right)^{\frac{1}{p}} = r, \end{aligned}$$

where $C_p$ is the Poincaré constant. Therefore the ball of the radius r is invariant for T. Now, let $v_k$ a sequence in the ball of radius r which converges to v in $L^p(\Omega )$ as $k\rightarrow \infty$ and let $w_k=T(v_k)$. Then, in order to show the continuity of T, one needs to prove that $w_k$ converges to $w=T(v)$ in $L^p(\Omega )$ as $k\rightarrow \infty$. To this aim, let us observe that an application of (3.5) gives that $w_k$ is bounded in $W^{1,p}_0(\Omega )$ with respect to k; moreover, it follows from Lemma 2 of [7] that $w_k$ is also bounded in $L^\infty (\Omega )$ with respect to k. Now, under the above assumptions, Lemma 4 of [7] gives that, up to subsequences, $w_k$ converges to a function w in $W^{1,p}_0(\Omega )$. This is sufficient to pass to the limit as $k\rightarrow \infty$ the weak formulation of the equation solved by $w_k$ in order to deduce that $w=T(v)$. For the compactness, it is sufficient to underline that if $v_k$ is bounded in $L^p(\Omega )$ then one can recover that $w_k$ is bounded in $W^{1,p}_0(\Omega )$ with respect to k thanks to (3.5); this implies that, up to subsequences, it converges to a function in $L^p(\Omega )$. Then, we are in position to apply the Schauder theorem in order to deduce the existence of $u_n$.

We also highlight that the zero order term gives us the existence of a solution to (3.4); indeed, in its absence, phenomena of nonnexistence of solutions may appear (see, for instance, [19]). Hence, in all of the next estimates one can get rid of the nonnegative zero order term which will vanish once n will be taken to infinity.

Firstly we prove a priori estimates on $\Psi (u_n)$ in some Sobolev spaces. We underline once again that, since $\Psi _n(u_n) \ge u_n$ and $|\nabla \Psi _n(u_n)| \ge |\nabla u_n|$, one gets that all the next estimates still hold for $u_n$ in place of $\Psi _n(u_n)$.

Lemma 3.1

Let $0\le f\in L^1(\Omega )$, let g and h satisfy (2.5) and (2.6) with $\theta <1$ and $\gamma \le 1$ with (2.9) in force. Let $u_n$ be a solution to (3.1) then

• (i) if $p>2-\frac{1}{N}$, $\Psi _n(u_n)$ is bounded in $W_0^{1,q}(\Omega )$ for every $q<\frac{N(p-1)}{N-1}$ with respect to n;

• (ii) if $1<p\le 2-\frac{1}{N}$, $\Psi _n(u_n)^{p-1}$ is bounded in $L^q(\Omega )$ for every $q<\frac{N}{N-p}$ and $|\nabla \Psi _n(u_n)|^{p-1}$ is bounded in $L^q(\Omega )$ for every $q<\frac{N}{N-1}$ with respect to n.

Moreover $T_k(\Psi _n(u_n)), T_k(u_n)$ are bounded in $W^{1,p}_0(\Omega )$ with respect to n for any $k>0$. Finally $u_n$ converges almost everywhere in $\Omega$ as $n\rightarrow \infty$ to a function u, which is almost everywhere finite.

Proof

Let us observe that $\Psi (s)s^{-\gamma }$ is finite at zero when $\gamma \le 1$ and let us take $e^{\Gamma _n(u_n)}T_k(\Psi _n(u_n)) \ (k>0)$ as a test function in (3.1) yielding to ($\delta <k$)

$$\begin{aligned} \begin{aligned} \int _{\{\Psi _n(u_n)\le k\}} a(x,\nabla u_n)\cdot \nabla u_n e^{\frac{p\Gamma _n(u_n)}{p-1}}&\le \int _\Omega h_n(u_n)f_ne^{\Gamma _n(u_n)}T_k(\Psi _n(u_n))\\&\le \max _{s\in [0,s_2]} [c_2s^{-\gamma }e^{\Gamma (s)}\Psi (s)] ||f||_{L^1(\Omega )}\\&\quad + k \sup _{s\in (s_2,\infty )} [h(s)e^{\Gamma (s)}] ||f||_{L^1(\Omega )}, \end{aligned} \end{aligned}$$

which, by (2.6) and (2.9), implies that

$$\begin{aligned} \int _\Omega |\nabla T_k(\Psi _{n}(u_n))|^p \le C(k+1). \end{aligned}$$

With a similar reasoning one can take $e^{\Gamma _n(u_n)}T_k(u_n) \ (k>0)$ as a test function obtaining that

$$\begin{aligned} \int _\Omega |\nabla T_k(u_n)|^p \le C(k+1). \end{aligned}$$

(3.6)

At this point we can apply Lemmas 4.1, 4.2 of [6] providing that if $p>2-\frac{1}{N}$, $\Psi _n(u_n)$ is bounded in $W^{1,q}_0(\Omega )$ with $q<\frac{N(p-1)}{N-1}$ with respect to n and one also has the existence of an almost everywhere finite function u to which, up to subsequences and as $n\rightarrow \infty$, $u_n$ converges almost everywhere in $\Omega$.

Otherwise, if $1<p\le 2-\frac{1}{N}$, same Lemmas 4.1, 4.2 of [6] give that $\Psi _n(u_n)$ is bounded in the Marcinkiewicz space $M^\frac{N(p-1)}{N-p}(\Omega )$ and that $|\nabla \Psi _n(u_n)|$ is bounded in $M^\frac{N(p-1)}{N-1}(\Omega )$.

From (3.6), up to subsequences, one has that $T_k(u_n)$ is a Cauchy sequence both in $L^p(\Omega )$ and in measure for each $k>0$. Now, in order to show that $u_n$ is a Cauchy sequence in measure let us observe that for all $k,\sigma >0$ and for all $n,m\in {\mathbb {N}}$

$$\begin{aligned} \{|u_n-u_m|>\sigma \}\subseteq \{u_n> k\}\cup \{u_m> k\}\cup \{|T_k(u_n)-T_k(u_m)|>\sigma \}. \end{aligned}$$

(3.7)

Now, if $\varepsilon >0$ is fixed, the Marcinkiewicz estimates imply the existence of ${\overline{k}}>0$ such that

$$\begin{aligned} \left| \{u_n> k\}\right|<\frac{\varepsilon }{3},\;\;\left| \{u_m> k\}\right| <\frac{\varepsilon }{3}\;\forall n,m\in {\mathbb {N}},\;\forall k>{\overline{k}}, \end{aligned}$$

while, since $T_{k}(u_{n})$ is a Cauchy sequence in measure, there exists $\eta _{\varepsilon }>0$ such that

$$\begin{aligned} \left| \{|T_k(u_n)-T_k(u_m)\right|>\sigma \}|<\frac{\varepsilon }{3}\;\forall n,m>\eta _{\varepsilon },\;\forall \sigma >0. \end{aligned}$$

Hence for $k>{\overline{k}}$ one deduces

$$\begin{aligned} \left| \{|u_n-u_m|>\sigma \}\right| <\varepsilon \quad \forall n,m\ge \eta _{\varepsilon },\;\forall \sigma >0, \end{aligned}$$

and so that $u_n$ is a Cauchy sequence in measure. Then, also in this case, there exists a nonnegative function u to which $u_n$ converges almost everywhere in $\Omega$. This concludes the proof. $\square$

Lemma 3.2

Under the assumptions of Lemma 3.1one has that

$$\begin{aligned} \int _\Omega \left( g_n(u_n)|\nabla u_n|^p + h_n(u_n)f\right) S(u_n)\varphi \le C, \end{aligned}$$

(3.8)

for some positive constant C which does not depend on n, for every $S\in W^{1,\infty }({\mathbb {R}})$ with compact support and for every $\varphi \in W^{1,p}_0(\Omega )\cap L^\infty (\Omega )$.

Proof

Let us assume that S has its support contained in $[-M,M]$ and let us take $S(u_n)\varphi$ as a test function in (3.1); then, by the Young inequality one has

$$\begin{aligned} \begin{aligned}&\int _\Omega \left( g_n(u_n)|\nabla u_n|^p + h_n(u_n)f\right) S(u_n)\varphi \\&\quad = \int _{\Omega } a(x,\nabla u_n)\cdot \nabla \varphi S(u_n) + \int _{\Omega } a(x,\nabla u_n)\cdot \nabla u_n S'(u_n) \varphi \\&\qquad + \int _\Omega \frac{u_n^{p-1}}{n}S(u_n)\varphi \\&\quad \le \frac{(p-1)\beta }{p}\int _\Omega |\nabla T_M(u_n)|^{p}S(u_n) + \frac{\beta }{p}\int _\Omega |\nabla \varphi |^{p}S(u_n) + \beta \int _{\Omega } |\nabla T_M(u_n)|^p S'(u_n)\varphi \\&\qquad + \int _\Omega \frac{u_n^{p-1}}{n}S(u_n)\varphi \le C, \end{aligned} \end{aligned}$$

and the constant C does not depend on n by means of Lemma 3.1. $\square$

Lemma 3.3

Under the assumptions of Lemma 3.1let $u_n$ be a solution to (3.1). Then $T_k(u_n)$ strongly converges to $T_k(u)$ in $W^{1,p}_0(\Omega )$ as $n\rightarrow \infty$ for any $k>0$.

Proof

Let us consider the following function

$$\begin{aligned} w_{n,h,k}:= T_{2k}(u_n-T_h(u_n) + T_k(u_n) - T_k(u)), \ \ h>k, \end{aligned}$$

and let us observe that it converges to $T_{2k}(u-T_h(u))$ as $n\rightarrow \infty$, and then, it converges to zero once $h\rightarrow \infty$; both convergences are *-weakly in $L^\infty (\Omega )$. Let us also highlight that $\nabla w_{n,h,k}=0$ if $u_n \ge h+3k$.

We take $e^{\Gamma _n(u_n)} w^+_{n,h,k}$ as test function in the weak formulation of (3.1), deducing

$$\begin{aligned} \int _\Omega e^{\Gamma _n(u_n)} a(x,\nabla u_n) \cdot \nabla w^+_{n,h,k} \le \int _\Omega h_n(u_n)f_ne^{\Gamma _n(u_n)} w^+_{n,h,k}. \end{aligned}$$

(3.9)

Now, we want to take first n and then h to infinity in (3.9). For the right hand side, we take $\delta <k$ and we write

$$\begin{aligned} \int _\Omega h_n(u_n)f_ne^{\Gamma _n(u_n)} w^+_{n,h,k}= & \int _{\{u_n\le \delta \}} h_n(u_n)f_ne^{\Gamma _n(u_n)} (u_n-T_k(u))^+ \\&+ \int _{\{u_n> \delta \}} h_n(u_n)f_ne^{\Gamma _n(u_n)} w^+_{n,h,k}, \end{aligned}$$

(3.10)

and for the first integral we observe that

$$\begin{aligned} h_n(u_n)f_ne^{\Gamma _n(u_n)} (u_n-T_k(u))^+ \le \max _{s\in [0,\delta ]} [h(s)e^{\Gamma (s)}s] f \in L^1(\Omega ), \end{aligned}$$

and that $\chi _{\{u_n\le \delta \}}(u_n-T_k(u))^+$ converges almost everywhere to zero as $n\rightarrow \infty$. Hence, an application of the Lebesgue theorem gives

$$\begin{aligned} \lim _{n\rightarrow \infty } \int _{\{u_n\le \delta \}} h_n(u_n)f_ne^{\Gamma _n(u_n)} (u_n-T_k(u))^+=0. \end{aligned}$$

For the second integral of the right hand side of (3.10) we observe that

$$\begin{aligned} h_n(u_n)f_ne^{\Gamma _n(u_n)} w^+_{n,h,k}\chi _{\{u_n> \delta \}} \le \sup _{s\in [\delta ,\infty )} [h(s)e^{\Gamma (s)}] f_n w^+_{n,h,k}, \end{aligned}$$

which means that by the generalized version of the Lebesgue theorem one has that

$$\begin{aligned} \lim _{n\rightarrow \infty } \int _{\{u_n> \delta \}} h_n(u_n)f_ne^{\Gamma _n(u_n)} w^+_{n,h,k} \le C\int _\Omega fT_{2k}(u-T_h(u))^+, \end{aligned}$$

and taking $h\rightarrow \infty$ in the previous one has

$$\begin{aligned} \lim _{h\rightarrow \infty }\lim _{n\rightarrow \infty } \int _{\{u_n> \delta \}} h_n(u_n)f_ne^{\Gamma _n(u_n)} w^+_{n,h,k} =0. \end{aligned}$$

Hence we have shown that

$$\begin{aligned} \lim _{h\rightarrow \infty }\lim _{n\rightarrow \infty } \int _\Omega h_n(u_n)f_ne^{\Gamma _n(u_n)} w^+_{n,h,k}=0. \end{aligned}$$

For the left hand side of (3.9) one has that

$$\begin{aligned} \begin{aligned} \int _\Omega e^{\Gamma _n(u_n)} a(x,\nabla u_n) \cdot \nabla w^+_{n,h,k}&= \int _{\{u_n\le k\}} e^{\Gamma _n(u_n)} a(x,\nabla u_n) \cdot \nabla (u_n-T_k(u))^+ \\&\quad +\int _{\{k<u_n<h+3k\}} e^{\Gamma _n(u_n)} a(x,\nabla u_n) \cdot \nabla w^+_{n,h,k}\\&\ge \int _{\{u_n\le k\}} e^{\Gamma _n(u_n)} a(x,\nabla u_n) \cdot \nabla (u_n-T_k(u))^+ \\&\quad - \int _{\{k<u_n<h+3k\}} e^{\Gamma _n(u_n)} a(x,\nabla u_n) \cdot \nabla T_k(u). \end{aligned} \end{aligned}$$

We observe that the second term on the right hand side of the previous goes to zero as $n \rightarrow \infty$; moreover, one also has that

$$\begin{aligned} \lim _{n\rightarrow \infty } \int _{\{u_n\le k\}} e^{\Gamma _{n}(u_n)}a(x,\nabla T_k(u)) \cdot \nabla (u_n - T_k(u))^+ =0. \end{aligned}$$

Hence this implies that

$$\begin{aligned} \lim _{n\rightarrow \infty } \int _\Omega (a(x,\nabla T_k(u_n)) - a(x,\nabla T_k(u)))\cdot \nabla (T_k(u_n) - T_k(u))^+ = 0. \end{aligned}$$

A similar reasoning regarding $w^-_{n,h,k}$ allows us to deduce that

$$\begin{aligned} \lim _{n\rightarrow \infty } \int _\Omega (a(x,\nabla T_k(u_n)) - a(x,\nabla T_k(u)))\cdot \nabla (T_k(u_n) - T_k(u)) = 0, \end{aligned}$$

which is known to give the desired result (see Lemma 5 of [7]). $\square$

Remark 3.4

We underline that Lemma 3.3 also gives, through a standard diagonal argument, that $\nabla u_n$ converges almost everywhere to $\nabla u$ in $\Omega$.

3.2 Proof of Theorem 2.5

In this section, we prove the existence of a solution to (2.1) in the mild singular case.

Proof of Theorem 2.5

Let $u_n$ be a solution to (3.1) then by Lemma 3.1 one gets the existence of an almost everywhere finite function u such that $u_n$ converges almost everywhere to u (up to a subsequence) as $n\rightarrow \infty$; moreover, for any $k>0$, $T_k(u)\in W^{1,p}_0(\Omega )$.

We want to show that u is actually a renormalized solution to (2.1). Hence, let $S\in W^{1,\infty }({\mathbb {R}})$ with compact support and $\varphi \in W^{1,p}_0(\Omega )\cap L^\infty (\Omega )$ be both nonnegative. We take $S(u_n)\varphi$ as a test function in (3.1) yielding to

$$\begin{aligned} \begin{aligned}&\int _\Omega a(x,\nabla u_n)\cdot \nabla \varphi S(u_n) + \int _\Omega a(x,\nabla u_n) \cdot \nabla u_n S'(u_n) \varphi +\int _\Omega \frac{u_n^{p-1}}{n}S(u_n)\varphi \\&\quad = \int _\Omega g_n(u_n)|\nabla u_n|^pS(u_n)\varphi + \int _\Omega h_n(u_n)f_nS(u_n)\varphi , \end{aligned} \end{aligned}$$

(3.11)

and we want to pass to the limit in the previous as $n\rightarrow \infty$. It follows from Lemma 3.3, from Remark 3.4 and from the fact that S has compact support in ${\mathbb {R}}$ that $S(u_n)a(x,\nabla u_n)$ converges to $S(u)a(x,\nabla u)$ in $L^\frac{p}{p-1}(\Omega )^N$. This allows to pass to the limit in the first term in (3.11). Moreover, by the strong convergence of $T_k(u_n)$ to $T_k(u)$ in $W^{1,p}_0(\Omega )$ as $n\rightarrow \infty$ we pass to the limit as $n\rightarrow \infty$ the second term in (3.11). It follows from Lemma 3.1 that the third term on the left hand side goes to zero as $n\rightarrow \infty$. For the right hand side, we observe that if both g and h are finite at zero then one standardly passes to the limit in the weak formulation of (3.1) obtaining that u is a solution to (2.1). Hence, from here we suppose that $g(0)=\infty$ and $h(0)=\infty$. If just one between g(0) and h(0) are infinite then the proof will have the obvious simplifications.

Let us take $\delta >0$ such that $\delta \not \in \{\eta : |\{u=\eta \}|>0\}$ and we write for $\varphi \in C^1_c(\Omega )$ nonnegative

$$\begin{aligned} \begin{aligned} \int _{\Omega } \left( g_n(u_n)|\nabla u_n|^p +h_n(u_n)f_n\right) S(u_n)\varphi&= \int _{\{u_n\le \delta \}}\left( g_n(u_n)|\nabla u_n|^p+h_n(u_n)f_n\right) S(u_n)\varphi \\&\quad + \int _{\{u_n> \delta \}}\left( g_n(u_n)|\nabla u_n|^p +h_n(u_n)f_n\right) S(u_n)\varphi . \end{aligned} \end{aligned}$$

(3.12)

and we pass to the limit first as $n\rightarrow \infty$ and then as $\delta \rightarrow 0$. For the second term on right hand side of the previous, we observe that

$$\begin{aligned} g_n(u_n)|\nabla u_n|^pS(u_n)\chi _{\{u_n>\delta \}} \le \sup _{s\in [\delta ,\infty )}[g(s)] |\nabla u_n|^pS(u_n), \end{aligned}$$

and the right hand side of the previous converges in $L^1(\Omega )$ thanks to Lemma 3.3; this allows to apply a generalized version of the Lebesgue theorem in order to pass to the limit as $n\rightarrow \infty$ in this term.

Moreover one has

$$\begin{aligned} h_n(u_n)f_nS(u_n)\chi _{\{u_n>\delta \}} \le \sup _{s\in [\delta ,\infty )}[h(s)S(s)] f \in L^1(\Omega ), \end{aligned}$$

which allows to apply the Lebesgue theorem. Hence, we have shown that

$$\begin{aligned} \lim _{n\rightarrow \infty } \int _{\{u_n> \delta \}} \left( g_n(u_n)|\nabla u_n|^p + h_n(u_n)f_n\right) S(u_n)\varphi = \int _{\{u> \delta \}} \left( g(u)|\nabla u|^p + h(u)f\right) S(u)\varphi . \end{aligned}$$

Now, an application of the Fatou Lemma in (3.8) (recall that here S and $\varphi$ are nonnegative) with respect to n gives that $(g(u)|\nabla u|^p\chi _{\{u>0\}} + h(u)f)S(u)\varphi$ belongs to $L^1(\Omega )$ which also means that, since $h(0)=\infty$,

$$\begin{aligned} \{u=0\} \subset \{f=0\}, \end{aligned}$$

(3.13)

up to a set of zero Lebesgue measure.

Hence from the Lebesgue theorem one gets

$$\begin{aligned} \begin{aligned}&\lim _{\delta \rightarrow 0}\lim _{n\rightarrow \infty } \int _{\{u_n> \delta \}} \left( g_n(u_n)|\nabla u_n|^p + h_n(u_n)f_n\right) S(u_n)\varphi \\&\quad = \int _{\{u> 0\}}\left( g(u)|\nabla u|^p + h(u)f\right) S(u)\varphi \\&\quad {\mathop {=}\limits ^{(3.13)}} \int _{\Omega }\left( g(u)|\nabla u|^p\chi _{\{u> 0\}} + h(u)f\right) S(u)\varphi . \end{aligned} \end{aligned}$$

Now, to deal with the first term of (3.12) we take $S(u_n)V_\delta (u_n) \varphi \ (0\le \varphi \in C^1_c(\Omega )$ and $V_\delta$ is defined in (1.3)) as a test function in (3.1) yielding to

$$\begin{aligned} \begin{aligned}&\int _{\{u_n\le \delta \}} (g_n(u_n)|\nabla u_n|^p + h_n(u_n)f_n)S(u_n)\varphi \\&\quad \le \beta \int _\Omega |\nabla u_n|^{p-1}|\nabla \varphi | S(u_n)V_\delta (u_n)\\&\qquad + \beta \int _\Omega |\nabla u_n|^{p} S'(u_n)V_\delta (u_n)\varphi + \int _\Omega \frac{u_n^{p-1}}{n}S(u_n)V_\delta (u_n)\varphi , \end{aligned} \end{aligned}$$

from which, taking $n \rightarrow \infty$, one has

$$\begin{aligned}&\limsup _{n\rightarrow \infty } \int _{\{u_n\le \delta \}} (g_n(u_n)|\nabla u_n|^p + h_n(u_n)f_n)S(u_n)\varphi \\&\quad \le \beta \int _\Omega |\nabla u|^{p-1}|\nabla \varphi | S(u)V_\delta (u) + \beta \int _\Omega |\nabla u|^{p} S'(u) V_\delta (u) \varphi , \end{aligned}$$

and as $\delta \rightarrow 0$

$$\begin{aligned}&\lim _{\delta \rightarrow 0} \limsup _{n\rightarrow \infty } \int _{\{u_n\le \delta \}} (g_n(u_n)|\nabla u_n|^p + h_n(u_n)f_n)S(u_n)\varphi \\&\quad \le \beta \int _{\{u=0\}} |\nabla u|^{p-1}|\nabla \varphi |S(u) + \beta \int _{\{u=0\}} |\nabla u|^{p} S'(u)\varphi = 0. \end{aligned}$$

Hence, after a simple density argument, we have shown that in a renormalized sense

$$\begin{aligned} -{{\,\mathrm{div}\,}}(a(x,\nabla u)) = g(u)|\nabla u|^p\chi _{\{u>0\}} + h(u)f. \end{aligned}$$

(3.14)

Now, we show that $u>0$ almost everywhere in $\Omega$ which, jointly with the previous formulation, implies that (2.11) holds. We consider the following function

$$\begin{aligned} \displaystyle V_{k,\delta }(s):= {\left\{ \begin{array}{ll} 1 &\quad s\le k, \\ \displaystyle \frac{\delta +k-s}{\delta } \ \ &\quad k<s< k+\delta , \\ 0 &\quad s\ge k+\delta , \end{array}\right. } \end{aligned}$$

and we take $S = V_{k,\delta }$ and $0\le \varphi \in C^1_c(\Omega )$ as a test function in (3.14), yielding to

$$\begin{aligned}&\int _\Omega a(x,\nabla u)\cdot \nabla \varphi V_{k,\delta }(u) -\frac{1}{\delta }\int _{\{k<u<k+\delta \}} a(x,\nabla u)\cdot \nabla u \varphi \\&\quad = \int _\Omega g(u)|\nabla u|^p\chi _{\{u>0\}}V_{k,\delta }(u)\varphi + \int _\Omega h(u)fV_{k,\delta }(u)\varphi . \end{aligned}$$

Now, we can get rid of the nonnegative second term on the left hand side of the previous, and then, we can take $\delta \rightarrow 0$ using the Fatou Lemma for the right hand side; one obtains

$$\begin{aligned} \int _{\{u\le k\}} a(x,\nabla u)\cdot \nabla \varphi \ge \int _{\{u\le k\}} g(u)|\nabla u|^p\chi _{\{u>0\}}\varphi + \int _{\{u\le k\}} h(u)f\varphi , \end{aligned}$$

which means that $-{\text {div}}(a(x,\nabla T_k(u))) \ge 0$ and not identically zero. Hence, we can apply the strong maximum principle (see Theorem 1.2 of [40]) deducing that $u>0$ almost everywhere in $\Omega$. This shows that (2.11) holds.

Now, we are left to show that (2.12) holds.

We take $e^{\Gamma _n(u_n)} \pi _t(u_n)$ ($\pi _t$ is defined in (1.4)) as a test function in (3.1) obtaining

$$\begin{aligned} \begin{aligned}&\frac{1}{t} \int _{\{t<u_n<2t\}} e^{\Gamma _n(u_n)}a(x,\nabla u_n) \cdot \nabla u_n \le \int _{\Omega }h_n(u_n)e^{\Gamma _n(u_n)}f_n\pi _t(u_n). \end{aligned} \end{aligned}$$

Now, thanks to Lemma 3.1, we can simply take $n\rightarrow \infty$ in the previous yielding to

$$\begin{aligned} \begin{aligned}&\frac{1}{t} \int _{\{t<u<2t\}} e^{\Gamma (u)}a(x,\nabla u) \cdot \nabla u \le \int _{\Omega }h(u)e^{\Gamma (u)}f\pi _t(u), \end{aligned} \end{aligned}$$

which gives (2.12) once that $t\rightarrow \infty$. This proves that u is a renormalized solution to (2.1). The regularity follows from Lemma 3.1. The proof is concluded. $\square$

3.3 Uniqueness of the renormalized solution

Here, we need the following truncation of the function $\Gamma (s)$:

$$\begin{aligned} {\tilde{\Gamma }}_{n}(s) = \int _{0}^{s} g(t)\chi _{\{\frac{1}{n}<t<n\}} dt. \end{aligned}$$

Proof of Theorem 2.6

Let us take $e^{{\tilde{\Gamma }}_n(u)}\varphi$ with $\varphi \in W^{1,p}_0(\Omega )\cap L^\infty (\Omega )$ as a test function in (2.1)

$$\begin{aligned}&\int _\Omega e^{{\tilde{\Gamma }}_n(u)}|\nabla u|^{p-2} \nabla u\cdot \nabla \varphi S(u) + \int _\Omega e^{{\tilde{\Gamma }}_n(u)}|\nabla u|^{p} S'(u) \varphi \\&\quad = \int _{\{u\le \frac{1}{n}\}\cup \{u\ge n\}} g(u)|\nabla u|^p e^{\tilde{\Gamma _n}(u)}S(u)\varphi + \int _\Omega h(u) e^{\tilde{\Gamma _n}(u)} f S(u) \varphi . \end{aligned}$$

Now, recalling that S has compact support in ${\mathbb {R}}$, $g(u)|\nabla u|^p e^{\Gamma (u)}S(u)\varphi \in L^1(\Omega )$, and that $u>0$ almost everywhere in $\Omega$, one can simply take $n\rightarrow \infty$ yielding to

$$\begin{aligned} \int _\Omega e^{\Gamma (u)}|\nabla u|^{p-2} \nabla u\cdot \nabla \varphi S(u) + \int _\Omega e^{\Gamma (u)}|\nabla u|^{p} S'(u) \varphi = \int _\Omega h(u) e^{\Gamma (u)} f S(u) \varphi , \end{aligned}$$

(3.15)

for every $\varphi \in W^{1,p}_0(\Omega )\cap L^\infty (\Omega )$.

Let us observe that by taking $T_k(\Psi (u))$ ($\Psi$ is defined in (2.8)) as a test function in (3.15) and $S=V_t$ one has

$$\begin{aligned} \int _\Omega |\nabla T_k(\Psi (u))|^p V_t(u) -\frac{1}{t} \int _\Omega e^{\Gamma (u)}|\nabla u|^{p} T_k(\Psi (u)) = \int _\Omega h(u) e^{\Gamma (u)} f V_t(u) T_k(\Psi (u)), \end{aligned}$$

where one can simply take $t\rightarrow \infty$ (recall that (2.12) holds), yielding to

$$\begin{aligned} \int _\Omega |\nabla T_k(\Psi (u))|^p = \int _\Omega h(u) e^{\Gamma (u)} f T_k(\Psi (u)). \end{aligned}$$

Now, observing that $\Psi (s) \ge s$, one obtains that

$$\begin{aligned} \int _{\Omega } |\nabla T_k(\Psi (u))|^p \le Ck. \end{aligned}$$

(3.16)

Let us now suppose that w, v are renormalized solutions to (2.1); hence, they satisfy (3.15). We take $T_k(\Psi (w)-\Psi (v))V_t(v)$ and $T_k(\Psi (w)-\Psi (v))V_t(w)$ as a test function in the weak formulation (3.15) solved, respectively, by w and v. In both formulation we take $S=V_t$ and we subtract them, yielding to

$$\begin{aligned} \begin{aligned}&\int _\Omega \left( |\nabla \Psi (w)|^{p-2} \nabla \Psi (w)- |\nabla \Psi (v)|^{p-2} \nabla \Psi (v)\right) \cdot \nabla T_k(\Psi (w)-\Psi (v)) V_t(v) V_t(w)\\&\quad - \frac{1}{t}\int _{\{t<v<2t\}} e^{\Gamma (w)}|\nabla w|^{p-2}\nabla w\cdot \nabla v T_k(\Psi (w)-\Psi (v)) V_t(w)\\&\quad + \frac{1}{t}\int _{\{t<w<2t\}} e^{\Gamma (v)}|\nabla v|^{p-2}\nabla v \cdot \nabla w T_k(\Psi (w)-\Psi (v)) V_t(v)\\&\quad - \frac{1}{t}\int _{\{t<w<2t\}} e^{\Gamma (w)}|\nabla w|^{p} T_k(\Psi (w)-\Psi (v)) V_t(v) \\&\quad + \frac{1}{t}\int _{\{t<v<2t\}} e^{\Gamma (v)}|\nabla v|^{p} T_k(\Psi (w)-\Psi (v)) V_t(w)\\&\quad = \int _\Omega \left( h(w) e^{\Gamma (w)}- h(v)e^{\Gamma (v)}\right) f T_k(\Psi (w)-\Psi (v)) V_t(v) V_t(w) \le 0, \end{aligned} \end{aligned}$$

(3.17)

since $h(s) e^{\Gamma (s)}$ is nonincreasing and $\Psi (s)$ is increasing. We now let $t\rightarrow \infty$ in the previous; let us notice that

$$\begin{aligned} \left| \frac{1}{t}\int _{\{t<w<2t\}} e^{\Gamma (w)}|\nabla w|^{p} T_k(\Psi (w)-\Psi (v)) V_t(v)\right| \le \frac{k}{t}\int _{\{t<w<2t\}} e^{\Gamma (w)}|\nabla w|^{p} \end{aligned}$$

and

$$\begin{aligned} \left| \frac{1}{t}\int _{\{t<v<2t\}} e^{\Gamma (v)}|\nabla v|^{p} T_k(\Psi (w)-\Psi (v)) V_t(w)\right| \le \frac{k}{t}\int _{\{t<v<2t\}} e^{\Gamma (v)}|\nabla v|^{p}, \end{aligned}$$

goes both to zero as $t\rightarrow \infty$ by condition (2.9).

Moreover one has that

$$\begin{aligned} \begin{aligned}&\left| \frac{1}{t}\int _{\{t<v<2t\}} e^{\Gamma (w)}|\nabla w|^{p-2}\nabla w\cdot \nabla v T_k(\Psi (w)-\Psi (v)) V_t(w)\right| \\&\quad \le \frac{k}{t}\left( \int _{\Omega } |\nabla T_{2t}(\Psi (w))|^{p}\right) ^\frac{p-1}{p} \left( \int _{\{t<v<2t\}} |\nabla v|^p \right) ^\frac{1}{p}\\&\quad \overset{(3.16)}{\le } C \left( \frac{k}{t}\int _{\{t<v<2t\}} |\nabla v|^p \right) ^\frac{1}{p}, \end{aligned} \end{aligned}$$

which goes to zero as $t \rightarrow \infty$ by (2.12). The same reasoning gives that also the third term on the left hand side of (3.17) goes to zero as $t\rightarrow \infty$. Hence, we have shown that

$$\begin{aligned}&\limsup _{t\rightarrow \infty } \int _{\{|\Psi (w)-\Psi (v)|\le k\}} \left( |\nabla \Psi (w)|^{p-2} \nabla \Psi (w)- |\nabla \Psi (v)|^{p-2} \nabla \Psi (v)\right) \\&\quad \cdot \nabla (\Psi (w)-\Psi (v)) V_t(v) V_t(w)\le 0, \end{aligned}$$

and by (2.4) one can apply the Fatou Lemma, deducing

$$\begin{aligned} \int _{\{|\Psi (w)-\Psi (v)|\le k\}} \left( |\nabla \Psi (w)|^{p-2} \nabla \Psi (w)- |\nabla \Psi (v)|^{p-2} \nabla \Psi (v)\right) \cdot \nabla (\Psi (w)-\Psi (v)) = 0. \end{aligned}$$

Now, one can take $k\rightarrow \infty$ getting

$$\begin{aligned} \int _{\Omega } \left( |\nabla \Psi (w)|^{p-2} \nabla \Psi (w)- |\nabla \Psi (v)|^{p-2} \nabla \Psi (v)\right) \cdot \nabla (\Psi (w)-\Psi (v)) = 0. \end{aligned}$$

which gives that $\nabla \Psi (w) = \nabla \Psi (v)$ that implies $\Psi (w)=\Psi (v)$ in $\Omega$. Clearly, since $\Psi (s)$ is increasing with respect to s, one has that $w=v$ almost everywhere in $\Omega$. This concludes the proof. $\square$

3.4 The strongly singular case

Once again we will work by approximation through problems (3.1). First of all, we show that some a priori estimates locally hold.

Lemma 3.5

Let $0\le f\in L^1(\Omega )$, let g and h satisfy (2.5) and (2.6) with (2.9) in force. Let $u_n$ be a solution to (3.1) then

• (i) if $p>2-\frac{1}{N}$, $\Psi _n(u_n)$ is locally bounded in $W^{1,q}(\Omega )$ for every $q<\frac{N(p-1)}{N-1}$ with respect to n;

• (ii) if $1<p\le 2-\frac{1}{N}$, $\Psi _n(u_n)^{p-1}$ is locally bounded in $L^q(\Omega )$ for every $q<\frac{N}{N-p}$ and $|\nabla \Psi _n(u_n)|^{p-1}$ is locally bounded in $L^q(\Omega )$ for every $q<\frac{N}{N-1}$ with respect to n.

Moreover $T_k(\Psi _n(u_n)), T_k(u_n)$ are locally bounded in $W^{1,p}(\Omega )$ with respect to n for any $k>0$. Finally $u_n$ converges almost everywhere in $\Omega$ as $n\rightarrow \infty$ to a function u, which is almost everywhere finite.

Proof

We first observe that only the behavior at zero is changed for g, h with respect to Lemma 3.1; hence, one can reason as in just cited lemma taking $e^{\Gamma _n(u_n)}T_k(G_\varepsilon (\Psi _n(u_n)))$ ($\varepsilon >0$) as a test function in (3.1) yielding to

$$\begin{aligned} \int _{\Omega }e^{\Gamma _n(u_n)}|\nabla T_k(G_\varepsilon (\Psi _n(u_n)))|^p \le C_\varepsilon (1+k), \end{aligned}$$

which means that, for any $\varepsilon >0$, one can apply Lemmas 4.1, 4.2 of [6] obtaining that, if $p>2-\frac{1}{N}$, $G_\varepsilon (\Psi _n(u_n))$ is bounded in $W^{1,q}_0(\Omega )$ with $q<\frac{N(p-1)}{N-1}$ with respect to n. Otherwise, if $1<p\le 2-\frac{1}{N}$, $G_\varepsilon (\Psi _n(u_n))$ is bounded in $L^q(\Omega )$ with $q<\frac{N}{N-p}$ and $|\nabla G_\varepsilon (\Psi _n(u_n))|^{p-1}$ is bounded in $L^q(\Omega )$ with $q<\frac{N}{N-1}$.

In order to deal with the estimate on the truncation we take $e^{\Gamma _n(u_n)}(T_k(\Psi _n(u_n))- k)\varphi ^p$ ($k>0$ and $0\le \varphi \in C^1_c(\Omega )$) as a test function in the weak formulation of (3.1) yielding to

$$\begin{aligned}&\int _\Omega |\nabla T_k(\Psi _n(u_n))|^p \varphi ^p + p\int _\Omega |\nabla \Psi _n(u_n)|^{p-2}\nabla \Psi _n(u_n) \cdot \nabla \varphi (T_k(\Psi _n(u_n))- k)\varphi ^{p-1} \\&\quad +\int _\Omega e^{\Gamma _n(u_n)}\frac{u_n^{p-1}}{n}(T_k(u_n)- k)\varphi ^{p} \le 0, \end{aligned}$$

which after an application of the Young inequality gives

$$\begin{aligned} \begin{aligned} \int _\Omega |\nabla T_k(\Psi _n(u_n))|^p \varphi ^p&\le pk\int _\Omega |\nabla T_k(\Psi _n(u_n))|^{p-1}|\nabla \varphi | \varphi ^{p-1} + \frac{k^p}{n}\int _\Omega \varphi ^p\\&\le \varepsilon pk\int _\Omega |\nabla T_k(\Psi _n(u_n))|^p \varphi ^{p} + C_\varepsilon pk\int _\Omega |\nabla \varphi |^p + \frac{k^p}{n}\int _\Omega \varphi ^p. \end{aligned} \end{aligned}$$

Fixing $\varepsilon$ small enough one gets that

$$\begin{aligned} \int _\Omega |\nabla T_k(\Psi _n(u_n))|^p \varphi ^p \le C, \end{aligned}$$

for some constant which does not depend on n. Moreover, taking $e^{\Gamma _n(u_n)}(T_k(u_n)- k)\varphi ^p$ ($k>0$ and $0\le \varphi \in C^1_c(\Omega )$) as a test function in the weak formulation of (3.1) and reasoning similarly to what just done one gets that $T_k(u_n)$ is locally bounded in $W^{1,p}(\Omega )$ with respect to n. Finally, reasoning as in Lemma 3.1, one deduces the existence of a function u to which $u_n$ converges almost everywhere in $\Omega$ as $n\rightarrow \infty$. This concludes the proof. $\square$

Here, due to the possibly strongly singular behavior of h, we will not be able to prove that the function u is almost everywhere positive in $\Omega$ as already done in the proof of Theorem 2.5. Hence, we prove a bound from below for the approximating solutions $u_n$ which is independent on n; this assures that u is strictly positive in $\Omega$.

Lemma 3.6

Under the assumptions of Lemma 3.5let $u_n$ be a solution to (3.1) then for every $\omega \subset \subset \Omega$ there exists a constant $c_\omega >0$ which does not depend on n and such that

$$\begin{aligned} u_n\ge c_\omega \ \ \text {a.e. in }\omega . \end{aligned}$$

Proof

We underline that it is possible to construct a function ${\overline{h}}$ which is everywhere finite, nonincreasing and such that ${\overline{h}}(s)\le h_n(s)$ for any $s\ge 0$; see for instance [16] for the explicit construction.

Let us now consider $v\in W^{1,p}_0(\Omega ) \cap L^\infty (\Omega )$ solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -{{\,\mathrm{div}\,}}(a(x,\nabla v)) + v^{p-1} = {\overline{h}}(v)f_1 & \quad \text {in}\;\Omega ,\\ v=0 & \quad \text {on}\;\partial \Omega , \end{array}\right. } \end{aligned}$$

and we observe that by Theorem 1.2 of [40] one has that for any $\omega \subset \subset \Omega$ there exists $c_\omega >0$ such that

$$\begin{aligned} v \ge c_\omega \text { a.e. in } \omega . \end{aligned}$$

(3.18)

Now, let us take $(v-u_n)^+$ as a test function in the difference of weak formulations solved by v and $u_n$; it yields

$$\begin{aligned} \begin{aligned}&\int _{\Omega } (a(x,\nabla v)- a(x,\nabla u_n)) \cdot \nabla (v-u_n)^+ + \int _{\Omega } \left( v^{p-1}-\frac{u^{p-1}_n}{n}\right) (v-u_n)^+\\&\quad = \int _\Omega ({\overline{h}}(v)f_1- h_n(u_n)f_n)(v-u_n)^+ - \int _\Omega g_n(u_n)|\nabla u_n|^p (v-u_n)^+ \le 0, \end{aligned} \end{aligned}$$

which, since the second term on the left hand side is nonnegative, implies

$$\begin{aligned} \begin{aligned}&\int _{\Omega } (a(x,\nabla v)- a(x,\nabla u_n)) \cdot \nabla (v-u_n)^+ \le 0, \end{aligned} \end{aligned}$$

and this gives that $u_n\ge v$ almost everywhere in $\Omega$. Then, the proof is concluded by (3.18). $\square$

We are ready to prove the existence of a solution in this strongly singular setting.

Proof of Theorem 2.7

The proof follows the lines of the one of Theorem 2.5, the only difference given by the boundary datum. Hence, let $u_n$ be a solution to (3.1) then let us take $e^{\Gamma _{n}(u_n)}T_k(G_{\varepsilon }(u_n))$ as a test function in (3.1) yielding to

$$\begin{aligned} \int _{\Omega }e^{\Gamma _{n}(u_n)}|\nabla T_k(G_{\varepsilon }(u_n))|^p \le \int _\Omega h_n(u_n)e^{\Gamma _{n}(u_n)}f_n T_k(G_{\varepsilon }(u_n))\le C_\varepsilon k, \end{aligned}$$

thanks to (2.9) and the fact that $f\in L^1(\Omega )$. The previous implies that $|\nabla T_k(G_\varepsilon (u_n))|^p$ is bounded in $L^1(\Omega )$ with respect to n. Now, by weak lower semicontinuity one gets that $T_k(G_{\varepsilon }(u))\in W^{1,p}_0(\Omega )$ for any $k,\varepsilon >0$. Moreover, by taking $e^{\Gamma _n(u_n)} w^+_{n,h,k}\varphi$ ($\varphi \in C^1_c(\Omega )$) as a test function in (3.1) and reasoning as in Lemma 3.3, one obtains that $T_k(u_n)$ converges to $T_k(u)$ locally in $W^{1,p}(\Omega )$ with respect to n and for any $k>0$. Indeed, the only difference from the proof of Lemma 3.3 is given by the term

$$\begin{aligned} \int _\Omega e^{\Gamma _n(u_n)}a(x,\nabla u_n)\nabla \varphi w^+_{n,h,k}, \end{aligned}$$

which, thanks to the local a priori estimates one has in this case, simply goes to zero as $n,h\rightarrow \infty$.

We are left to show (2.13) which follows in a similar way to what done in Theorem 2.5; hence, we just highlight the main differences. First of all, by taking $S(u_n)\varphi$ as a test function in (3.1) where $\varphi \in C^1_c(\Omega )$ and $S\in W^{1,\infty }({\mathbb {R}})$ with compact support one can show that

$$\begin{aligned} \int _{\Omega } a(x,\nabla u) \cdot \nabla \varphi S(u) + \int _{\Omega } a(x,\nabla u) \cdot \nabla u S'(u)\varphi = \int _{\Omega } g(u)|\nabla u|^pS(u)\varphi + \int _\Omega h(u)fS(u)\varphi , \end{aligned}$$

(3.19)

for every $S\in W^{1,\infty }({\mathbb {R}})$ with compact support and for every $\varphi \in C^1_c(\Omega )$. Here, we also underline the use of Lemma 3.6 which assures that $u>0$ almost everywhere in $\Omega$.

Now, one can take $e^{\Gamma _n(u_n)} \pi _t(u_n)$ ($t>0$) as a test function in (3.1) obtaining

$$\begin{aligned} \begin{aligned}&\frac{1}{t} \int _{\{t<u_n<2t\}} e^{\Gamma _n(u_n)}a(x,\nabla u_n) \cdot \nabla u_n \le \int _{\Omega }h_n(u_n)e^{\Gamma _n(u_n)}f_n\pi _t(u_n). \end{aligned} \end{aligned}$$

Then, by the estimates given by Lemma 3.5 and by condition (2.9) one can pass to the limit as $n\rightarrow \infty$ the previous one, deducing

$$\begin{aligned} \begin{aligned}&\frac{1}{t} \int _{\{t<u<2t\}} e^{\Gamma (u)}a(x,\nabla u) \cdot \nabla u \le \int _{\Omega }h(u)e^{\Gamma (u)}f\pi _t(u). \end{aligned} \end{aligned}$$

Hence, once again thanks to (2.9), one can take $t\rightarrow \infty$ deducing that

$$\begin{aligned} \begin{aligned} \lim _{t\rightarrow \infty }\frac{1}{t} \int _{\{t<u<2t\}} e^{\Gamma (u)}a(x,\nabla u) \cdot \nabla u = 0. \end{aligned} \end{aligned}$$

(3.20)

Now, let just observe that taking a nonnegative $\varphi \in C^1_c(\Omega )$ one obtains that

$$\begin{aligned} \int _\Omega \left( g_n(u_n)|\nabla u_n|^p + h_n(u_n)f_n\right) \varphi \le \beta \int _\Omega |\nabla u_n|^{p-1} |\nabla \varphi | + \frac{1}{n}\int _\Omega u_n^{p-1}\varphi , \end{aligned}$$

and by Lemma 3.5 one has that the right hand side of the previous is bounded by a constant which is independent on n. Hence, an application of the Fatou Lemma with respect to n in the previous gives that (recall that $u>0$ almost everywhere in $\Omega$)

$$\begin{aligned} \int _\Omega \left( g(u)|\nabla u|^p + h(u)f\right) \varphi \le C, \end{aligned}$$

(3.21)

namely $g(u)|\nabla u|^p, h(u)f \in L^1_{\mathrm{loc}}(\Omega )$.

Finally, one can take $S(s) = V_t(s)$ in (3.19), yielding to

$$\begin{aligned} \int _{\Omega } a(x,\nabla u) \cdot \nabla \varphi V_t(u) - \frac{1}{t} \int _{\Omega } a(x,\nabla u) \cdot \nabla u \varphi = \int _{\Omega } g(u)|\nabla u|^pV_t(u)\varphi + \int _\Omega h(u)fV_t(u)\varphi , \end{aligned}$$

and one can take $t\rightarrow \infty$ using (3.20) for the second term on the left hand side of the previous while we use the Lebesgue theorem for the right hand side by using (3.21). The first term simply passes to the limit by Lemma 3.5. This proves that u is a distributional solution to (2.1). The regularity follows from Lemma 3.5. The proof is concluded. $\square$

4 The critical case $\theta =1$

In this section, we treat the case where g satisfies (2.5) with $\theta =1$; here, we need some coerciveness which means that $c_1< \alpha$ (see Section 5.1 below for nonnexistence results when $c_1\ge \alpha$). In this case, we will need the solutions to be bounded away from zero on any open set $\omega$ such that $\omega \subset \subset \Omega$ even if $\gamma \le 1$; this being strictly connected with the fact that $\Gamma (s)$ is blowing up at zero.

Here, we give our main existence result for this section.

Theorem 4.1

Let a satisfy (2.2), (2.3), (2.4) and let g and h satisfy (2.5) and (2.6) with $\theta =1$, $c_1<\alpha$ and $\gamma \le 1$ with (2.9) in force. Finally, assume that $f\in L^1(\Omega )$ is nonnegative then there exists a distributional solution u to (2.1) such that $u^{p-1} \in L^q(\Omega )$ for every $q<\frac{N}{N-p}$ and $|\nabla u|^{p-1} \in L^q(\Omega )$ for every $q<\frac{N}{N-1}$. Moreover $T_k(u)\in W^{1,p}_0(\Omega )$ for any $k>0$.

As already pointed out $\Gamma$ is not even defined in zero; hence, for this section we modify its definition and its approximation. In particular, we set

$$\begin{aligned} \Gamma _{n}(s) := \frac{1}{\alpha }\int _{1}^{s} g_n(t) \ dt. \end{aligned}$$

while

$$\begin{aligned} \Gamma (s) := \frac{1}{\alpha }\int _{1}^{s} g(t) \ dt. \end{aligned}$$

Once again our strategy is using the approximation scheme (3.1); we start with the following lemma.

Lemma 4.2

Let $0\le f\in L^1(\Omega )$, let g and h satisfy (2.5) and (2.6) with $\theta =1$, $c_1<\alpha$ and $\gamma \le 1$ with (2.9) in force. Let $u_n$ be a solution to (3.1) then

• (i) if $p>2-\frac{1}{N}$, $u_n$ is bounded in $W_0^{1,q}(\Omega )$ for every $q<\frac{N(p-1)}{N-1}$ with respect to n;

• (ii) if $1<p\le 2-\frac{1}{N}$, $u_n^{p-1}$ is bounded in $L^q(\Omega )$ for every $q<\frac{N}{N-p}$ and $|\nabla u_n|^{p-1}$ is bounded in $L^q(\Omega )$ for every $q<\frac{N}{N-1}$ with respect to n.

Moreover $T_k(u_n)$ is bounded in $W^{1,p}_{0}(\Omega )$ with respect to n. Finally $u_n$ converges almost everywhere in $\Omega$ a function u, which is almost everywhere finite.

Proof

We take $e^{\Gamma _{n}(u_n)}T_{{\tilde{k}}}(G_{k}(u_n))$ ($k>0$) as a test function in the weak formulation of (3.1). Using (2.2) one has that

$$\begin{aligned} \begin{aligned} \alpha \int _\Omega e^{\Gamma _{n}(u_n)} |\nabla T_{{\tilde{k}}}(G_{k}(u_n))|^p \le \int _\Omega h_n(u_n)f_ne^{\Gamma _{n}(u_n)}T_{{{\tilde{k}}}}(G_{k}(u_n)) \le {{\tilde{k}}} \sup _{s\in [k,\infty )} [e^{\Gamma (s)} h(s)]\int _\Omega f \end{aligned} \end{aligned}$$

(4.1)

and we observe that, for any $k>0$, the right hand side of the previous is finite thanks to condition (2.9). Since in the integral on the left hand side one has $u_n\ge k$ then one gets

$$\begin{aligned} \begin{aligned} \int _\Omega |\nabla T_{{\tilde{k}}}(G_{k}(u_n))|^p \le C_k{{\tilde{k}}}, \end{aligned} \end{aligned}$$

and the previous estimate standardly allows to deduce that, if $p>2-\frac{1}{N}$, $G_{k}(u_n)$ is bounded in $W^{1,q}_0(\Omega )$ with $q<\frac{N(p-1)}{N-1}$ with respect to n. Otherwise if $1<p\le 2-\frac{1}{N}$ one gets $u_n^{p-1}$ bounded in $L^q(\Omega )$ with $q<\frac{N}{N-p}$ and $|\nabla G_{k}(u_n)|^{p-1}$ bounded in $L^q(\Omega )$ with $q<\frac{N}{N-1}$ (see [6]).

Now, we take $T_\delta (G_{k-\delta }(u_n))$ $(k>\delta )$ as a test function in (3.1) obtaining

$$\begin{aligned}&\int _\Omega a(x,\nabla u_n) \cdot \nabla T_{\delta }(G_{k-\delta }(u_n)) +\int _\Omega \frac{u_n^{p-1}T_\delta (G_{k-\delta }(u_n))}{n}\\&\quad = \int _\Omega \left( g_n(u_n)|\nabla u_n|^p + h_n(u_n)f_n\right) T_{\delta }(G_{k-\delta }(u_n)), \end{aligned}$$

which implies

$$\begin{aligned} \begin{aligned} \delta \int _{\{u_n> k\}} \left( g_n(u_n)|\nabla u_n|^p + h_n(u_n)f_n\right)&\le \int _\Omega \left( g_n(u_n)|\nabla u_n|^p + h_n(u_n)f_n\right) T_{\delta }(G_{k-\delta }(u_n))\\&\le \beta \int _\Omega |\nabla T_{\delta }(G_{k-\delta }(u_n))|^p + \int _\Omega \frac{u_n^{p-1}T_\delta (G_{k-\delta }(u_n))}{n} \le C, \end{aligned} \end{aligned}$$

(4.2)

which follows from (4.1) and where C does not depend on n (recall also that $u_n^{p-1}$ is bounded in $L^q(\Omega )$ with $q<\frac{N}{N-p}$).

Now finally, since $\gamma \le 1$, we take $T_k(u_n)$ ($k\le \min (s_1,s_2)$) as a test function in (3.1) obtaining

$$\begin{aligned} \begin{aligned} \alpha \int _\Omega |\nabla T_k(u_n)|^p&\le \int _\Omega \left( g_n(u_n)|\nabla u_n|^p + h_n(u_n)f_n\right) T_k(u_n)\\&= \int _{\{u_n \le k\}}\left( g_n(u_n)|\nabla u_n|^p + h_n(u_n)f_n\right) u_n \\&\quad + k\int _{\{u_n> k\}} \left( g_n(u_n)|\nabla u_n|^p + h_n(u_n)f_n\right) \\&\le \int _{\{u_n \le k\}} g_n(u_n)|\nabla u_n|^pT_k(u_n) + \int _{\{u_n \le k\}} h_n(u_n)f_nT_{k}(u_n)\\&\quad + k\int _{\{u_n > k\}} \left( g_n(u_n)|\nabla u_n|^p + h_n(u_n)f_n\right) \\&\le c_1\int _{\Omega } |\nabla T_k(u_n)|^p + c_2k^{1-\gamma }||f||_{L^1(\Omega )} + C, \end{aligned} \end{aligned}$$

where we have estimated the last term in the previous thanks to (4.2). Hence, one gets

$$\begin{aligned} \begin{aligned} (\alpha - c_1) \int _\Omega |\nabla T_k(u_n)|^p \le c_2k^{1-\gamma }||f||_{L^1(\Omega )} + C. \end{aligned} \end{aligned}$$

Hence, the previous estimate jointly with the one given in (4.1) gives that $T_k(u_n)$ is bounded in $W^{1,p}_0(\Omega )$ with respect to n for any $k>0$. This also proves the uniform estimate on $u_n$ by the decomposition $u_n= T_k(u_n) + G_k(u_n)$.

The existence of an almost everywhere finite function u to which $u_n$ converges almost everywhere in $\Omega$ as $n\rightarrow \infty$ follows as in Lemma 3.1. This concludes the proof. $\square$

Now, we state that the solutions are bounded away from zero.

Lemma 4.3

Under the assumptions of Lemma 4.2let $u_n$ be a solution to (3.1) then for every $\omega \subset \subset \Omega$ there exists a constant $c_\omega >0$ which does not depend on n and such that

$$\begin{aligned} u_n\ge c_\omega \ \ \text {a.e. in }\omega . \end{aligned}$$

(4.3)

Proof

The proof is identical to the one of Lemma 3.6. $\square$

We underline that the previous positivity property guarantees that the function $\Gamma _{n}(u_n)$ is always well defined on any open set $\omega$ such that $\omega \subset \subset \Omega$; this is fundamental in proving the almost everywhere convergence of $\nabla u_n$ towards $\nabla u$ in $\Omega$ as $n\rightarrow \infty$.

Lemma 4.4

Under the assumptions of Lemma 4.2let $u_n$ be a solution to (3.1). Then $T_k(u_n)$ strongly converges to $T_k(u)$ locally in $W^{1,p}(\Omega )$ with respect to n for any $k>0$.

Proof

Once again we consider the following function

$$\begin{aligned} w_{h,k,n}:= T_{2k}(u_n-T_h(u_n) + T_k(u_n) - T_k(u)), \ \ h>k, \end{aligned}$$

but we take $e^{\Gamma _{n}(u_n)} w^+_{n,h,k}\varphi \ (0\le \varphi \in C^1_c(\Omega ))$ as test function in (3.1). deducing

$$\begin{aligned} \begin{aligned}&\int _\Omega e^{\Gamma _{n}(u_n)} a(x,\nabla u_n) \cdot \nabla w^+_{n,h,k} \varphi + \int _\Omega e^{\Gamma _{n}(u_n)} a(x,\nabla u_n) \cdot \nabla \varphi w^+_{n,h,k}\\&\quad \le \int _\Omega h_n(u_n)f_ne^{\Gamma _{n}(u_n)} w^+_{n,h,k} \varphi . \end{aligned} \end{aligned}$$

Since $\varphi$ has compact support in $\Omega$, we can pass to the limit every term of the weak formulation as n, h go to infinity in the previous reasoning as in Lemma 3.3 by using (4.3); the second term, which is the main difference with respect to the case of Lemma 3.3, simply goes to zero as $n,h \rightarrow \infty$. This concludes the proof. $\square$

Once again Remark 3.4 applies and it gives that $\nabla u_n$ converges almost everywhere to $\nabla u$ in $\Omega$.

Proof of Theorem 4.1

let $u_n$ be a solution to (3.1) then by Lemma 4.2$T_k(u_n)$ is bounded in $W^{1,p}_0(\Omega )$ with respect to n; since $u_n$ converges to u (at least) in $L^1(\Omega )$, then by weak lower semicontinuity one gets that $T_k(u)$ belongs to $W^{1,p}_0(\Omega )$ for any $k>0$. This gives the boundary condition. The proof of the weak formulation follows as in the one of Theorem 2.7. This concludes the proof. $\square$

Remark 4.5

Here, we explicitly remark that in the previous proof one can also show that $u_n$ converges to u, which is also a renormalized solution to problem (2.1).

5 Remarks and nonexistence results

5.1 Nonexistence of solutions having the truncation of finite energy

In the previous sections, in case $\theta \le 1$ and $\gamma \le 1$, we have looked for a solution u to (2.1) satisfying that $T_k(u)\in W^{1,p}_0(\Omega )$ for any $k>0$; in particular, when $\theta =1$, we also require a smallness assumption on g. Otherwise, when $\gamma >1$, $T_k(u)$ is just shown to have local finite energy. Here, we show that this is actually a necessary condition since we cannot expect it to have, in general, global finite energy and that our assumptions are in some sense sharp.

We consider a nonnegative, integrable and not identically zero datum f, $g(s)= \lambda s^{-1}$ and $h(s)=s^{-\gamma }$ ($\gamma \ge 0$). Hence, we deal with

$$\begin{aligned} -\Delta _p u = \frac{\lambda |\nabla u|^p}{u} + \frac{f}{u^\gamma }. \end{aligned}$$

(5.1)

and we suppose that $\displaystyle \lambda \ge 1$ and that there exists a positive renormalized solution u to the previous equation which means that $T_k(u)\in W^{1,p}_0(\Omega )$ for any $k>0$. Now, let us take $T_k(u)$ as a test function and $S=V_t$ $(t>k)$ in the renormalized formulation of (5.1) obtaining

$$\begin{aligned} \int _{\{u\le k\}}|\nabla u|^{p}= & \frac{1}{t} \int _{\{t<u<2t\}} |\nabla u|^p T_k(u) + \lambda \int _{\{u\le k\}}|\nabla u|^p + \lambda k\int _{\{u> k\}}\frac{|\nabla u|^pV_t(u)}{u} \\&+ \int _{\Omega } \frac{f T_k(u)V_t(u)}{u^\gamma }. \end{aligned}$$

Hence, by taking $t\rightarrow \infty$, it follows from the Fatou Lemma, from $\lambda \ge 1$ and from the fact that the first term on the right hand side goes to zero, that

$$\begin{aligned} 0 \ge (1-\lambda )\int _{\{u\le k\}}|\nabla u|^p \ge \lambda k\int _{\{u> k\}}\frac{|\nabla u|^p}{u} + \int _{\Omega } \frac{f T_k(u)}{u^\gamma }. \end{aligned}$$

Now, let us observe that having $fu^{-\gamma }\in L^1_{\mathrm{loc}}(\Omega )$ implies that

$$\begin{aligned} \{u=0\} \subset \{f=0\}, \end{aligned}$$

up to a set of zero Lebesgue measure. Hence, the second term on the right hand side of the previous inequality cannot be zero, and this clearly gives a contradiction. The same reasoning holds even for supercritical singular terms, namely $\theta >1$. Indeed, let us take $T_k(u)$ as a test function and $S=V_t$ $(t>k)$ in the renormalized formulation of

$$\begin{aligned} -\Delta _p u = \frac{\lambda |\nabla u|^p}{u^\theta } + \frac{f}{u^\gamma }, \end{aligned}$$

where $\theta >1$ and $\lambda >0$; then, by taking $t\rightarrow \infty$ as before, one obtains

$$\begin{aligned} \left( 1 - \frac{\lambda }{k^{\theta -1}}\right) \int _{\{u\le k\}}|\nabla u|^{p} \ge \lambda k\int _{\{u> k\}}\frac{|\nabla u|^p}{u^\theta } + \int _{\Omega } \frac{f T_k(u)}{u^\gamma }, \end{aligned}$$

which, once again, gives a contradiction if k is small enough. This implies that $T_k(u)$ cannot, in general, belong to $W^{1,p}_0(\Omega )$ for any $k>0$.

5.2 Local solutions beyond the critical values $\theta =1$

As pointed out in the previous section, one cannot expect to have, in general, solutions having any truncation with finite energy. However it is always possible to find solutions to (2.1) as in Definition 2.3 having $T_k(u)\in W^{1,p}_{\mathrm{loc}}(\Omega )$ for any $k>0$ by collecting the ideas for Theorems 2.7 and 4.1. We have the following existence result.

Theorem 5.1

Let a satisfy (2.2), (2.3), (2.4) and let g and h satisfy (2.5) and (2.6) with (2.9) in force. Finally, assume that $f\in L^1(\Omega )$ is nonnegative then there exists a distributional solution u to (2.1) such that $T_k(u)\in W^{1,p}(\omega )$ for any $\omega \subset \subset \Omega$ and for any $k>0$.

Proof

Let $u_n$ be a solution to (2.1). Then, the proof can be carried on as the one of Theorem 4.1 except for the estimate on $T_k(u_n)$. In this case, we follow the strategy already employed in the proof of Theorem 2.7 taking $(T_k(u_n)-k)\varphi ^p$ ($\varphi \in C^1_c(\Omega )$ nonnegative) yielding to $T_k(u_n)$ is locally bounded in $W^{1,p}(\Omega )$ with respect to n and for any $k>0$. $\square$

Here, we want underline that if g does not degenerate at infinity then u solution to (2.1) actually belongs to $W^{1,p}_{\mathrm{loc}}(\Omega )$ even if f is just in $L^1(\Omega )$. Indeed, let us suppose that there exists ${\tilde{s}}>0$ such that $g(s) \ge {\tilde{c}}>0$ for any $s>{\tilde{s}}$ then we simply observe that $g(u)|\nabla u|^p \in L^1_{\mathrm{loc}}(\Omega )$ gives that for any nonnegative $\varphi \in C^1_c(\Omega )$

$$\begin{aligned} \int _{\{u\ge {\tilde{s}}\}} |\nabla u|^p\varphi \le \int _{\Omega } g(u)|\nabla u|^p\varphi < \infty , \end{aligned}$$

which jointly with the fact that $T_k(u)\in W^{1,p}_{\mathrm{loc}}(\Omega )$ gives that $u\in W^{1,p}_{\mathrm{loc}}(\Omega )$.

5.3 The case where h degenerates

Here, we briefly discuss the case in which h touches zero, i.e., there exists ${\tilde{c}}$ such that $h({\tilde{c}})=0$. The aim is to show that there exists a solution to (2.1) which lives (almost everywhere) in the interval $[0,{\tilde{c}}]$. In order to obtain it, we modify the scheme of approximation (3.1) in the following way

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -{{\,\mathrm{div}\,}}(a(x,\nabla u_n)) = g_n(u_n)|\nabla u_n|^p + {\tilde{h}}_n(u_n)f_n & \quad \text {in}\;\Omega ,\\ u_n=0 & \quad \text {on}\;\partial \Omega , \end{array}\right. } \end{aligned}$$

where ${\tilde{h}}_n(s)= T_n({\tilde{h}}(s))$ and

$$\begin{aligned} {\tilde{h}}(s)={\left\{ \begin{array}{ll} h(s) &\quad \text {if}\;s<{\tilde{c}},\\ 0&\quad \text {if}\;s\ge {\tilde{c}}. \end{array}\right. } \end{aligned}$$

Now, we take $e^{\Gamma _{n}(u_n)}G_{{\tilde{c}}}(u_n)$ as a test function in (3.1) yielding to

$$\begin{aligned} \alpha \int _\Omega e^{\Gamma _{n}(u_n)}|\nabla G_{{\tilde{c}}}(u_n)|^p \le 0, \end{aligned}$$

which clearly implies that $u_n\le {\tilde{c}}$ almost everywhere in $\Omega$. Then, one can carry on the proof of the existence as in Theorems 2.5, 2.7 and 4.1 obtaining that there exists a solution $u\in L^\infty (\Omega )$ to

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -{{\,\mathrm{div}\,}}(a(x,\nabla u)) = g(u)|\nabla u|^p + {\tilde{h}}(u)f & \quad \text {in}\;\Omega ,\\ u=0 & \quad \text {on}\;\partial \Omega , \end{array}\right. } \end{aligned}$$

and, since ${\tilde{h}}(s)= h(s)$ for any $0\le s\le {\tilde{c}}$ one gets the existence of a bounded solution u to (2.1).

5.4 More general equations

Here, we remark that the existence result can be found for more general equations of the following type

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -{{\,\mathrm{div}\,}}(a(x,u,\nabla u))= b(x,u,\nabla u) & \quad \text {in}\;\Omega ,\\ u\ge 0 & \quad \text {in}\;\Omega , \\ u=0 & \quad \text {on}\;\partial \Omega , \end{array}\right. } \end{aligned}$$

(5.2)

where $\displaystyle {a(x,s,\xi ):\Omega \times {\mathbb {R}}\times {\mathbb {R}}^{N} \rightarrow {\mathbb {R}}^{N}}$ is a Carathéodory function satisfying ($1<p<N$)

$$\begin{aligned}&a(x,s,\xi )\cdot \xi \ge \alpha |\xi |^{p}, \ \ \ \alpha >0, \end{aligned}$$

(5.3)

$$\begin{aligned}&|a(x,s,\xi )|\le \beta (\eta (x) + |s|^{p-1} + |\xi |^{p-1}), \ \ \ \beta >0, \eta \in L^{\frac{p}{p-1}}(\Omega ), \end{aligned}$$

(5.4)

$$\begin{aligned}&(a(x,s,\xi ) - a(x,s,\xi ^{'} )) \cdot (\xi -\xi ^{'}) > 0, \end{aligned}$$

(5.5)

for every $\xi \ne \xi ^{'}$ in ${\mathbb {R}}^N$ and for almost every x in $\Omega$.

The function $\displaystyle {b(x,s,\xi ):\Omega \times {\mathbb {R}}\times {\mathbb {R}}^{N} \rightarrow {\mathbb {R}}^{N}}$ is a nonnegative Carathéodory function satisfying

$$\begin{aligned} b(x,s,\xi )\le g(s)|\xi |^p + h(s)f, \end{aligned}$$

(5.6)

where g, h satisfy, as previously, (2.5),(2.6) such that (2.9) is fulfilled. Here, we state the result for this section:

Theorem 5.2

Let a satisfy (5.3), (5.4), (5.5) and let assume that b satisfies (5.6) where g and h satisfy (2.5) and (2.6) having (2.9) in force. Finally, assume $f\in L^1(\Omega )$ is nonnegative then:

• (i) if $\theta <1$ and $\gamma \le 1$ then there exists a renormalized solution u to (5.2);

• (ii) if $\theta <1$ and $\gamma > 1$ then there exists a distributional solution u to (5.2);

• (iii) if $\theta =1$ and $c_1<\alpha$ then there exists a distributional solution u to (5.2) such that $T_k(u) \in W^{1,p}_{0}(\Omega )$ for any $k>0$;

• (iv) in the remaining cases there exists a distributional solution u to (5.2).

Proof

The proof follows with slight modifications of the ones of Theorems 2.5, 2.7 and 4.1. $\square$

5.5 Final remarks

Finally, we want to underline that one could study equations (1.1) even if f is a nonnegative Radon measure. Indeed, probably, one could merge the techniques in [15, 32, 36] in order to get an existence of a renormalized/distributional solution to the problem.

One final question regards the uniqueness of a renormalized solution when $\theta =1$; in this case, under a smallness assumption on g, one is able to show the existence of a renormalized solution to the problem but the uniqueness does not work as in the proof of Theorem 2.6; indeed, as the change of variable is not well defined, the proof cannot work that way.