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Existence of solutions for a class of porous medium type equations with lower order terms
Journal of Inequalities and Applications volume 2015, Article number: 294 (2015)
Abstract
This paper deals with a class of degenerate quasilinear elliptic equations of the form \(-\operatorname{div}(a(x,u,\nabla u))+F(x,u,\nabla u)=f \), where \(a(x,u,\nabla u)\) is allowed to degenerate with the unknown u. Under some hypothesis on a, F, and f, we obtain the existence of bounded solutions \(u\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\). For the case \(f\in L^{1}(\Omega)\), we also prove that there exists at least one renormalized solution.
1 Introduction
This paper concerns the following degenerate problem:
where Ω is a bounded domain of \(\mathbb{R}^{N}\) (\(N\geq2\)), \(f\in L^{q}(\Omega)\) with \(q\geq1\) and \(a(x,s,\xi)\) is a Carathéodory function. Furthermore, we assume that there exists a continuous function α from \(\mathbb{R}^{+}\) into \(\mathbb{R}^{+}\) such that \(\alpha(0)=0\) and \(a(x,s,\xi)\xi\geq\alpha(|s|)|\xi|^{p}\) for any \(s\in\mathbb{R}\), \(\xi \in \mathbb{R}^{N}\), and almost every x in Ω. Thus problem (\(\mathscr{P}\)) degenerates for the subset \(\{x\in\Omega:u(x)=0\}\).
Problem (\(\mathscr{P}\)) has important and extensive applications to the fluid dynamics in porous media, in hydrology and in petroleum engineering (see [1, 2]). The simplest model is the stationary case of the porous media equation with zero Dirichlet boundary condition:
which has been widely studied in the literature (see [3–6] and references therein).
For the case \(\alpha\equiv\mathrm{constant}>0\), the existence of bounded solutions to problem (\(\mathscr{P}\)) is proved in [7], when the data f is small in a suitable norm.
Concerning the case that α is a positive function, Porretta and Segura de León investigated the existence results to problem (\(\mathscr{P}\)); see [8]. We remark that in [8], no sign condition is imposed on F, but the growth of F at infinity need to be controlled. We also point out that a variational inequality related to problem (\(\mathscr{P}\)) was studied in [9], and similar results can be found in [10] and [11].
In the case \(\alpha(0)=0\), \(f\in W^{-1,r}(\Omega)\cap L^{1}(\Omega)\) with \(r\geq p'\), \(r>\frac{N}{p-1} \), Rakotoson proved the existence of a bounded weak solution to problem (\(\mathscr{P}\)) (see [12]), provided that F satisfies a sign condition. As \(F=0\) and \(f\in W^{-1,r}(\Omega)\), the existence of solutions to problem (\(\mathscr{P}\)) has been discussed in [13]. We point out that the parabolic version of [13] has been studied in [14].
As \(f\in L^{q}(\Omega)\) with \(q\geq\max\{1,\frac{N}{p}\}\), we shall give a direct method to prove the existence of bounded weak solutions to problem (\(\mathscr{P}\)) in the standard sense, i.e. \(u\in W_{0}^{1,p}(\Omega)\). The main difficulty comes from the facts that its modulus of ellipticity vanishes when the solution u vanishes. To overcome this difficulty, we shall firstly establish the \(L^{\infty}\) estimate for solution u, by the technique of rearrangement which is differs from the usual Stampacchia \(L^{\infty}\) regularity procedure. Then, by constructing suitable approximate problems, and using a priori estimates and a test function method, we shall finish the proof of this existence results.
Furthermore, we will study the case when \(f\in L^{1}(\Omega)\). Since no growth conditions are required for ω and β (see (H2)), it is not obvious that the term \(-\operatorname{div}(a(x, u, \nabla u))\) makes sense even as a distribution. To overcome this difficulty, we shall use the concept of renormalized solutions, which is introduced by Diperna and Lions (see [15]). This notion was adapted by many authors to study partial differential equations with measurable data, especially for \(L^{1}\) data (see [16–18] for example). We remark that an equivalent notion called entropy solutions, was introduced independently by Bénilan et al. [19].
The main ideas and methods come from [8, 10, 12, 20]. This paper is organized as follows: in Section 2 we give some preliminaries and state the main results; in Section 3, we study the existence of bounded solution to problem (\(\mathscr{P}\)); in Section 4, we prove the existence of renormalized solution.
2 Some preliminaries and the main results
2.1 Properties of the relative rearrangement
Let Ω be a bounded open subsets of \(\mathbb{R}^{N}\), we denote by \(|E|\) the Lebesgue measure of a set E. Assume that \(u: \Omega \rightarrow\mathbb{R}\) be a measurable function, we define the distribution function \(\mu_{u}(t)\) of u as follows:
The decreasing rearrangement \(u_{\ast}\) of u is defined as the generalized inverse function of \(\mu_{u}(t)\), i.e.
We recall also that u and \(u_{\ast}\) are equi-measurable, i.e.
which implies that for any non-negative Borel function ψ we have
and if \(E\subset\Omega\) be a measurable subset, then
Using the Fleming-Rishel formula, Hölder’s inequality, and the isoperimetric inequality, we can get the following result (see [7, 9, 12]).
Lemma 2.1
For any non-negative function \(u\in W_{0}^{1,1}(\Omega)\), the following chain of inequalities holds:
where \(C_{N}\) denotes the measure of the unit ball in \(\mathbb{R}^{N}\).
For more details as regards the theory of rearrangement, we just refer to [21] and the references therein.
2.2 Assumptions and the main results
Let Ω be an open bounded set of \(\mathbb{R}^{N}\) (\(N\geq2\)) and \(p>1\), we make the following assumptions.
- (H1):
-
\(a: \Omega\times\mathbb{R}\times\mathbb{R}^{N}\rightarrow \mathbb{R}^{N}\) is a Carathéodory vector function satisfying: there exists a continuous function α from \(\mathbb{R}_{+}\) into \(\mathbb {R}_{+}\) such that \(\alpha(0)=0\) and \(\alpha(s)>0\) if \(s>0\) and
$$\begin{aligned}& a(x,s,\xi)\xi\geq\alpha\bigl(\vert s\vert \bigr)|\xi|^{p},\quad \forall s\in R, \mbox{a.e. } x\in \Omega, \forall\xi\in\mathbb{R}^{N}, \\& \int_{0}^{+\infty}\alpha^{\frac{1}{p-1}}(s)\, \mathrm{d}s=\int_{0}^{+\infty }\frac{1}{\alpha(s)}\, \mathrm{d}s =+\infty \end{aligned}$$and
$$ \frac{1}{\alpha}\in L^{1}(0,b) \quad \mbox{for any given }b>0. $$ - (H2):
-
There exists a Carathéodory vector function ā such that for a.e. \(x\in\Omega\), \(\forall s\in\mathbb{R}\), \(\forall\xi, \xi'\in \mathbb{R}^{N}\) with \(\xi\neq\xi'\):
-
(i)
\(a(x,s,\xi)=\alpha(|s|)\bar{a}(x,s,\xi)\).
-
(ii)
\([\bar{a}(x,s,\xi)-\bar{a}(x,s,\xi')][\xi-\xi']>0\).
-
(iii)
There exist an increasing function ω from \(\mathbb{R}^{+}\) into \(\mathbb{R}^{+}\) and a non-negative function \(\bar{\omega}\in L^{p'}(\Omega)\) such that
$$\bigl\vert \bar{a}(x,s,\xi)\bigr\vert \leq\omega\bigl(\vert s\vert \bigr)\bigl[|\xi|^{p-1}+\bar{\omega}(x)\bigr]. $$ -
(iv)
The function ā is a positively homogeneous of degree \((p-1)\) with respect to the variable ξ, i.e.
$$\bar{a}(x,s,t\xi)=t^{p-1}\bar{a}(x,s,\xi),\quad \forall t\geq 0. $$
-
(i)
- (H3):
-
\(F:\Omega\times\mathbb{R}\times\mathbb{R}^{N}\rightarrow \mathbb{R}\) is a Carathéodory function, for which there exists an increasing function β from \([0,+\infty)\) into \([0,+\infty)\) vanishing and continuous at zero such that for a.e. \(x\in\Omega\), \(\forall s\in\mathbb{R}\) and \(\forall\xi\in\mathbb {R}^{N}\):
$$\bigl\vert F(x,s,\xi)\bigr\vert \leq\beta\bigl(\vert s\vert \bigr)| \xi|^{p}. $$ - (H4):
-
\(f\in L^{q}(\Omega)\) with \(q> \max\{1,\frac{N}{p}\}\).
- (H5):
-
\(\lim_{s\rightarrow \infty}\frac{e^{\gamma(|s|)}}{(1+\phi(|s|))^{p-1}}=0\), where γ and ϕ are defined as follows:
$$ \gamma(s)=\int_{0}^{s} \frac{\beta(|\sigma|)}{\alpha(|\sigma|)}\, \mathrm {d}\sigma;\qquad \phi(s)=\int_{0}^{s} \bigl(\alpha\bigl(|\sigma|\bigr)\bigr)^{\frac{1}{p-1}}e^{\frac{\gamma (|s|)}{p-1}}\, \mathrm{d}\sigma. $$(2.1)
Remark 2.1
Assumption (H1) allows us to consider the porous medium operators \(\triangle(|u|^{m-1}u)=\operatorname {div}(m|u|^{m-1}\nabla u)\). In this case, it yields \(\alpha (|s|)=|s|^{m-1}\), so that the conditions \(\alpha(0)=0\) and \(\frac {1}{\alpha}\in L^{1}(0,b)\) indicate \(1< m<2\). Thus, in this case, the porous medium equation becomes a slow diffusion equation.
We now introduce several auxiliary functions by
As usual, the usual truncation function \(T_{\theta}\) at level ±θ is defined as \(T_{\theta}(s)=\max\{-\theta, \min\{\theta,s\} \}\). Throughout this paper, we use \(C(\theta_{1},\theta_{2},\ldots,\theta _{m})\) to denote positive constants depending only on specified quantities \(\theta_{1}, \theta_{2},\ldots, \theta_{m}\).
Now we give the definition of weak solutions of problem (\(\mathscr{P}\)).
Definition 2.1
A measurable function \(u\in W_{0}^{1,p}(\Omega)\) is called a weak solution to problem (\(\mathscr {P}\)), if \(a(\cdot, u, \nabla u)\in L^{p'}(\Omega)\) and \(F(\cdot, u, \nabla u)\in L^{1}(\Omega)\) such that
For the existence of weak solutions, our result is stated as follows.
Theorem 2.1
If assumptions (H1)-(H5) hold, then there exists at least one bounded weak solution \(u\in L^{\infty}(\Omega)\) to problem (\(\mathscr{P}\)) in the sense of Definition 2.1.
As we have said before, when dealing with the case \(f\in L^{1}(\Omega)\), we shall use the notion of renormalized solution.
Definition 2.2
A measurable function \(u: \Omega\rightarrow \mathbb{R}\) is a renormalized solution of problem (\(\mathscr{P}\)) if
and if for any \(h\in W^{1,\infty}(\Omega)\) with compact support and \(\upsilon\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\), u satisfies
The existence result for \(L^{1}\) data is stated as follows.
Theorem 2.2
Assume that (H1) to (H3) hold and \(\frac{\beta}{\alpha}\in L^{1}(\mathbb{R_{+}})\). If \(f\in L^{1}(\Omega)\), then problem (\(\mathscr{P}\)) admits at least one renormalized solution.
Remark 2.2
In Theorem 2.1, the conditions (H4) and (H5) are only needed in proving the \(L^{\infty}(\Omega)\) estimate of u. Therefore in Theorem 2.2, we do not need these assumptions. But instead, we need the condition \(\frac{\beta}{\alpha}\in L^{1}(\mathbb {R_{+}})\) as in [11]. Moreover, by the result of [22], the solution obtained in Theorem 2.2 belongs to \(W_{0}^{1,r}(\Omega)\), provided \(2-\frac{1}{N}< p< N\).
3 Existence of weak solution to problem (\(\mathscr{P}\))
To prove Theorem 2.1, we first establish the \(L^{\infty}\) estimate of solutions to problem (\(\mathscr{P}\)).
Lemma 3.1
Assume that (H1) to (H5) hold. If \(u\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\) is a weak solution to problem (\(\mathscr{P}\)), then u satisfies the following estimate:
where M is a constant which depends only on N, p, q, α, β, \(\|f\|_{L^{q}(\Omega)}\).
Proof of Lemma 3.1
For \(t>0\), \(h>0\), let \(S_{t, h}\) be a real function defined by
It is easy to see that \(S_{t, h}(\phi(u))\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\) and so \(S_{t, h}(\phi (u))e^{\gamma_{\theta}(|u|)}\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\), where ϕ and \(\gamma _{\theta}\) are defined as in (2.1) and (2.3). Taking \(v=e^{\gamma_{\theta}(|u|)}S_{t, h}(\phi(u))\) as a test function in (2.5), we have
Then letting \(\theta\rightarrow0\), we obtain
where γ is defined as in (2.1). Notice that \(|S_{t, h}(\phi(u))|\leq1\), by (H1), (H3), and applying Hölder’s inequality, we deduce from (3.3) that
where \(\omega=|\phi(u)|=\phi(|u|)\). Let h tend to zero, we find that
Setting
since ϕ is strictly increasing and \(\lim_{s\rightarrow \pm\infty}\phi(s)=0\), we have
Concerning the term \((\int_{\{\omega> t\}}|e^{\gamma(|u|)}|^{q'}\,\mathrm {d}x)^{\frac{1}{q}}\), we have
By (3.4), (3.6), and Lemma 2.1, it follows that
which indicates that, for \(0<\theta<\theta+h<|\Omega|\),
Then we employ (1.15) of [9] to get
Then letting h tend to zero, we deduce that, for almost \(\theta\in [0,|\Omega|]\),
which leads, after applying Young’s inequality, to
Since \(q>\frac{N}{p}\), we have \(q_{0}=\frac{p'}{pq'}+\frac {p'}{N}-p'+1>0\). From (3.5), we deduce that there exists \(t_{0}>0\) such that
Hence, upon integration over \([0,\mu_{\omega}(t_{0})]\), inequality (3.8) gives
which implies that \(\|u\|_{L^{\infty}(\Omega)}\leq\phi^{-1}(1+2t_{0})\). We observe that \(t_{0}\) only depends on p, q, N, \(|\Omega|\), α, β, thus the proof of Lemma 3.1 is finished. □
To prove Theorem 2.1, we shall consider suitable approximate problems. First of all, we recall the following lemma, proved in [12].
Lemma 3.2
There exists a function \(g\in C^{1}(\mathbb{R})\) such that g is odd, strictly increasing, and
For a.e. \(x\in\Omega\), \(\forall s\in\mathbb{R}\), and \(\forall\xi\in \mathbb{R}^{N}\), we define for fixed \(\varepsilon>0\):
For any fixed \(\varepsilon>0\), we introduce the approximate problem
where \(\{f_{\varepsilon}\}\) satisfy
The existence result to problem (\(\mathscr{P}_{\varepsilon}\)) is stated as follows.
Theorem 3.1
Problem (\(\mathscr{P}_{\varepsilon}\)) admits at least a solution \(u_{\varepsilon}\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega )\) with \(\|g(u_{\varepsilon})\|_{L^{\infty}(\Omega)}\leq M_{0}\), where \(M_{0}\) is a positive constant depending on M (see Lemma 3.1) and the behavior of function g.
Proof of Theorem 3.1
For any \(l>0\), let us consider the following truncated problem:
By the classic result (see [23]), problem (\(\mathscr {P}_{\varepsilon l}\)) admits a solution \(u_{\varepsilon}\in W_{0}^{1,p}(\Omega)\in L^{\infty}(\Omega)\). Then using the same argument of Lemma 3.1, we conclude
In view of Lemma 3.2, it is easy to see that \(g^{-1}\) is defined well and strictly increasing in \(\mathbb{R}\).
Now choosing \(l>g^{-1}(M)\), we obtain
Thus we have \(T_{l}(u_{\varepsilon})=u_{\varepsilon}\), which implies that \(u_{\varepsilon}\) is a weak solution of (\(\mathscr{P}_{\varepsilon}\)). The proof is finished. □
Proof of Theorem 2.1
Taking \(e^{\tilde{\gamma}_{\theta}(|u_{\varepsilon}|)}u_{\varepsilon}\) as a test function in problem (\(\mathscr{P}_{\varepsilon}\)), we have
where \(\tilde{\gamma}_{\theta}\) is defined as in (2.4), and g is defined as in Lemma 3.2. Then letting θ tend to zero, using assumptions (H1)-(H4) and Theorem 3.1 we get
where γ̃ is defined as in (2.4).
In view of Theorem 3.1, (H1), and (H2), the above estimate gives
Now denoting \(\bar{u}_{\varepsilon}=g(u_{\varepsilon})\), estimates (3.11) and (3.12) imply that \(\bar{u}_{\varepsilon}\) is bounded uniformly in \(W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\). As a consequence, there exist a subsequence (still denoted by \(\{\bar{u}_{\varepsilon}\}\)) and a measurable function \(\bar{u}\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\) such that
In the following, the rest of the proof is divided into several steps.
Step 1: To deal with the difficulty that α vanishes at zero, we define the following truncation function near the origin:
where \(k>0\) is a fixed constant. Then we easily get
Now taking \(\rho_{\theta}^{\varepsilon}=e^{\gamma_{\theta}(\bar {u}_{\varepsilon})}[ \zeta_{k}(\bar{u}_{\varepsilon}) -\zeta_{k}(\bar{u})]_{+}\) as a test function in problem (\(\mathscr {P}_{\varepsilon}\)), by (H1) we have
It is easy to see that the fourth term of (3.17) is non-negative. So letting θ tend to zero, the above inequality leads to
where
Now we estimate all the terms of (3.18).
Estimate of \(I_{2}(\varepsilon)\). Using (3.11), (3.13), and the Hölder inequality, we conclude that
Hence, by (3.12) we easily get
Estimate of \(I_{3}(\varepsilon)\). By (3.11), (3.14), and the Lebesgue dominated convergence theorem, we infer that
Estimate of \(I_{1}(\varepsilon)\). Since \(a(x,s,0)=0\) for a.e. \(x\in\Omega\) and every \(s\in\mathbb{R}\), we obtain
where
For the term \(\bar{I}_{11}(\varepsilon)\), we can write
Collecting (3.11), (3.13), (3.14), and (3.16), it is easy to verify that
Using (3.22), (3.23), (H1), and (H2), we find that
where we have used the fact \(a(x,s,0)=0\) for a.e. \(x\in\Omega\).
For the term \(\bar{I}_{12}(\varepsilon)\), it is easy to get
The above two convergence results show that
Substituting (3.19), (3.20), and (3.24) into (3.18), we conclude
Now choosing \(\rho_{\theta}^{\varepsilon}=-e^{\gamma_{\theta}(\bar {u}_{\varepsilon})}[ \zeta_{k}(\bar{u}_{\varepsilon}) -\zeta_{k}(\bar{u})]_{+}\) as a test function in problem (\(\mathscr {P}_{\varepsilon}\)), by the same arguments as in the proof of (3.25) we arrive at
As a consequence of (3.25) and (3.26), we have
Then, arguing as in [24], we derive that
Step 2: For any fixed \(k>0\), let us define
Proceeding as in Step 1, taking \(\rho_{\theta}^{\varepsilon}=e^{\gamma _{\theta}(\bar{u}_{\varepsilon})}[ \bar{\zeta}_{k}(\bar{u}_{\varepsilon}) -\bar{\zeta}_{k}(\bar{u})]_{+}\) and \(\rho_{\theta}^{\varepsilon}=-e^{-\gamma _{\theta}(\bar{u}_{\varepsilon})}[ \bar{\zeta}_{k}(\bar{u}_{\varepsilon}) -\bar{\zeta}_{k}(\bar{u})]_{-}\) as two test functions in problem (\(\mathscr {P}_{\varepsilon}\)), we obtain
By (3.27) and (3.28), it follows that
In the following, we prove that u is a weak solution to problem (\(\mathscr{P}\)).
Since \(u_{\varepsilon}\) is a weak solution to problem (\(\mathscr{P}\)), it follows that
Concerning the third term on the left-hand side of (3.30), we rewrite it as
To take the limits in \(I_{1\varepsilon}\), we next show that
Indeed, by (3.14) and (3.29), we already know that \(F(x,t,\bar{u}_{\varepsilon},\nabla\bar{u}_{\varepsilon})\chi_{\{|\bar {u}_{\varepsilon}|>k\}}\rightarrow F(x,t,\bar{u}, \nabla\bar{u})\chi_{\{ |\bar{u}|>k\}}\) almost everywhere in Ω, it suffices to prove the equi-integrability of this sequence and then apply Vitali’s convergence theorem. Using Theorem 3.1 and (H3), we get
where \(C_{0}\) is a positive constant independent of ε and k. Then the equi-integrability of \(|\nabla\bar{u}_{\varepsilon}|^{p} \chi _{\{|\bar{u}_{\varepsilon}|>k\}}\), which follows from (3.29), indicates that of \(F(x,\bar{u}_{\varepsilon},\nabla\bar{u}_{\varepsilon})\chi_{\{|\bar{u}_{\varepsilon}|>k\}}\). Therefore, (3.32) is proved.
As a conclusion, we have
so that
Moreover, by assumption (H3) and (3.12) we get
where \(C_{1}\) is a positive constant independent of ε and k. Therefore,
since β is a continuous function from \([0,+\infty)\) into \([0,+\infty)\) and \(\beta(0)=0\).
It follows from (3.31), (3.33), and (3.34) that
Similarly, we have
Furthermore, the same argument as (3.19) shows that
Finally, it is easy to see that
Now letting ε tend to zero, from (3.36)-(3.38), we deduce that ū satisfies (2.5), with u replaced by ū. Thus, the proof is finished. □
4 Existence of renormalized solution to problem (\(\mathscr{P}\))
Proof of Theorem 2.2
By the proof of Theorem 3.1, we deduce that there exists at least one weak solution \(u_{\varepsilon}\) satisfying \(u_{\varepsilon}\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\) such that
where \(f_{\varepsilon}\) satisfy
As before, set \(\bar{u}_{\varepsilon}=g(u_{\varepsilon})\). For any given \(l>s_{0}\) and \(\bar{l}=g^{-1}(l)\), let us take \(v=e^{\tilde{\gamma}_{\theta}(|u_{\varepsilon}|)}T_{\bar{l}}(u_{\varepsilon})\) in (4.1), where \(s_{0}\) is defined as in the proof of Theorem 3.1. Then sending θ tend to zero, using (H1)-(H3) and the fact \(\frac{\beta}{\alpha}\in L^{1}(0,+\infty)\), it follows that
where C is a positive constant independent of ε.
Hence, by the Sobolev space embedding theorem, there exist a measurable function ū and a subsequence (still denoted by \(\{\bar{u}_{\varepsilon}\}\)), such that
and
Step 4.1. In this step, we prove the following result:
For any integer \(n>1\), define \(\rho_{n}\) by
Obviously, we have
Taking \(v=e^{\gamma_{\theta}(|\bar{u}_{\varepsilon}|)}\rho_{n}(\bar {u}_{\varepsilon})\) in (4.1), we get
Passing to the limit as θ tend to zero in (4.7), it follows from (H1) and (H3) that
Let \(\varepsilon\rightarrow0\) and then \(n\rightarrow\infty\) in (4.8). Recalling that \(\frac{\beta}{\alpha}\in L^{1}(\mathbb{R}_{+})\), using (4.6) we get
It is easy to check that \(\lim_{n\rightarrow\infty}\int_{\Omega} f e^{\gamma(|\bar{u}|)}\rho_{n}(\bar{u})\,\mathrm{d}x=0 \). Thus, passing to the limit as \(n\rightarrow\infty\) in (4.9), the desired result (4.5) follows immediately.
Step 4.2. For any fixed \(k>0\) and \(l>\max\{k,s_{0}\}\), we denote
Then we have, in view of (4.3) and (4.4),
Let λ be a positive number to be determined, denote
and
where \(\gamma_{\theta}\) is defined as in (2.3). We now choose a sequence of increasing function \(S_{n}\in C^{\infty}(\mathbb {R})\) such that
Taking \(v=S_{n}(\bar{u}_{\varepsilon})\rho_{\theta}^{\varepsilon}\) in (4.1), we obtain
where
Limit behaviors of \(\hat{I}_{2}(\theta,\varepsilon,n)\), \(\hat {I}_{5}(\theta,\varepsilon,n)\), and \(\hat{I}_{8}(\theta,\varepsilon,n)\). Thanks to (4.11), we have
and thus
where \(C_{1}\) is a positive constant independent of ε. Therefore, using (4.2) we get
Similarly, we have
and
Limit behaviors of \(\hat{I}_{3}(\theta,\varepsilon,n)\) and \(\hat {I}_{6}(\theta,\varepsilon,n)\). Since
we get
As far as \(\hat{I}_{6}(\theta,\varepsilon,n)\) is concerned, we have
Limit behavior of \(\hat{I}_{4}(\theta,\varepsilon,n)\). From (4.5) and (4.11), it follows that
Limit behavior of \(\hat{I}_{7}(\theta,\varepsilon,n)\). For the term \(\hat{I}_{7}(\theta,\varepsilon,n)\), we have
where
and
Combining (4.3) with (4.4), we infer that
and
Substituting (4.20) and (4.21) into (4.19), we obtain
Limit behavior of \(\hat{I}_{9}(\theta,\varepsilon,n)\). It is straightforward that
Limit behavior of \(\hat{I}_{1}(\theta,\varepsilon,n)\). Note that \(a(x,s,0)=0\) for a.e. \(x\in\Omega\) and every \(s\in\mathbb{R}\), and we get
where
Using (4.3), (4.4), and (4.11), it is clear that
and
Note that \(a(x,s,0)=0\) for a.e. \(x\in\Omega\) and every \(s\in\mathbb{R}\), the term \(\hat {I}_{21}(\varepsilon)\) can be rewritten as follows:
where
By (4.3), (4.4), and (4.10), we find that
As a direct consequence of (4.24)-(4.27), we have
Choosing \(\lambda=2\max_{s\in[k,l]}\frac{\beta(|s|)}{ \alpha(|s|)}\) in the definition of φ, and then combining the limit behaviors of \(\hat{I}_{1}(\theta,\varepsilon,n)\)-\(\hat{I}_{9}(\theta ,\varepsilon,n)\), we get
which yields
Step 4.3. Choosing \(v=-S_{n}(\bar{u}_{\varepsilon })e^{-\gamma_{\theta}(\bar{u}_{\varepsilon}) +\gamma_{\theta}(\zeta^{l}_{k}(\bar{u}_{\varepsilon}))} \varphi((\zeta^{l}_{k}(\bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar{u}))_{-}) \) as a test function in (4.1), then arguing as before, we have
It follows from (4.29) and (4.30) that
Taking into account that \(S_{n}'(\bar{u}_{\varepsilon}) a(x,\zeta^{l}_{k}(\bar {u}_{\varepsilon}) ,\nabla\zeta^{l}_{k}(\bar{u}_{\varepsilon}))=a(x,\zeta^{l}_{k}(\bar {u}_{\varepsilon}) ,\nabla\zeta^{l}_{k}(\bar{u}_{\varepsilon}))\) for \(n>l\), using (4.31) we get
which yields
Then, arguing as in [24], we derive
Step 4.4. For any fixed \(l>k>0\), we denote
Choosing \(v=S_{n}(\bar{u}_{\varepsilon})e^{\gamma_{\theta}(\bar {u}_{\varepsilon}) -\gamma_{\theta}(\bar{\zeta}^{l}_{k}(\bar{u}_{\varepsilon}))} \varphi((\bar{\zeta}^{l}_{k}(\bar{u}_{\varepsilon})-\bar{\zeta}^{l}_{k}(\bar {u}))_{+})\) as a test function in (4.1), arguing as before we obtain
Next choosing \(v=-S_{n}(\bar{u}_{\varepsilon})e^{\gamma_{\theta}(\bar{\zeta }^{l}_{k}(\bar{u}_{\varepsilon}))- \gamma_{\theta}(\bar{u}_{\varepsilon})} \varphi((\bar{\zeta}^{l}_{k}(\bar{u}_{\varepsilon})-\bar{\zeta}^{l}_{k}(\bar {u}))_{-})\) as a test function in (4.1), applying the same argument we get
Proceeding as in Step 4.3, we infer that
As a consequence of (4.33) and (4.34), we have
Step 4.5. In this step we prove that ū satisfies (2.7), where u is replaced by ū.
For any fixed \(m>k\), one has
Thus, passing to the limit as ε tends to zero in (4.36), we deduce that, for fixed \(m>k\geq0\),
Taking the limit as m tends to +∞ in (4.37) and using (4.5), we conclude that ū satisfies (2.7).
In the following, we prove that ū satisfies (2.8). Indeed, by (4.1), we have
for any given \(\upsilon\in W^{1,\infty}(\Omega)\) and \(h\in W^{1,\infty }(\mathbb{R})\) such that \(\operatorname{supp}h\subseteq[-l,l]\) for some \(l>0\).
Now we first analyze the fifth term on the left-hand side of (4.38). Recall that \(\operatorname{supp}h\subseteq[-l,l]\), we get
Therefore, for any k satisfying \(0< k< l\), one has
Similarly to the proof of (3.33) and (3.34), using (4.3) and (4.35) we obtain
and
which imply that
Similarly, we have
and
As far as the second term of the left-hand side of (4.38) is concerned, by (4.1) we easily get
thus
Reasoning as in (4.45), one has
Finally, it is clear that
Then, letting ε tend to zero in (4.38), we conclude from (4.42)-(4.47) that ū satisfies (2.8). Hence, ū is a renormalized solution to problem (\(\mathscr{P}\)). □
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This work is supported by the NSFC of China (No. 11461048), the Natural Science Foundation of Jiangxi Province of China (No. 20132BAB211006), the foundation of Jiangxi Educational Committee (No. GJJ14546).
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Zou, W. Existence of solutions for a class of porous medium type equations with lower order terms. J Inequal Appl 2015, 294 (2015). https://doi.org/10.1186/s13660-015-0799-9
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DOI: https://doi.org/10.1186/s13660-015-0799-9