Abstract
We consider the following anisotropic problem, with singular nonlinearity having a variable exponent
where \(\Omega \) is a bounded regular domain in \({\mathbb {R}}^{N}\) and \(\gamma (x)>0\) is a smooth function, having a convenient behavior near \(\partial \Omega .\) f is assumed to be a non negative function belonging to a suitable Lebesgue space \(L^{m}\left( \Omega \right) .\) We will also assume without loss of generality that \(2\le p_{1}\le p_{2}\le \cdots \le p_{N}.\) Using approximation techniques, we obtain existence and regularity of positive solutions to the considered problem.
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1 Introduction
We consider in this paper, the following problem
where
\(\gamma (x)>0\) is assumed to be a regular function, say for example \(\gamma (x)\in C(\overline{\Omega })\), and \(\Omega \) is a bounded regular domain in \({\mathbb {R}}^{N}\). We will assume without loss of generality that \(2\le p_{1}\le p_{2}\le \cdots \le p_{N}\) and that f is a non negative function belonging to a suitable Lebesgue space \(L^{m}\left( \Omega \right) .\)
When the differential operator is assumed to be semilinear, and \(\gamma (x)=\gamma \), Boccardo and Orsina in their leading work [2], obtained existence and regularity of the solution, and this was generalized to the case of the p-laplacian in [7], and to the the case of the anisotropic operator L in [14].
In the very recent work [3] the authors consider a singular semilinear elliptic problem with variable exponent \(\gamma (x)\), they obtained existence and regularity of the solution, under some conditions on the behavior of the function \(\gamma (x)\) near the boundary of \(\Omega \).
There exists a huge literature, devoted to the study of the anisotropic operator L, as it has many applications in fluid dynamics, and physical phenomena with anisotropic diffusion, we cite for example [8–11], and the references therein.
When a singular nonlinearity is considered in interaction with different types of differential operators as the laplacian or the p-laplcian, we invite the reader to see the works [1, 4–6, 10, 12, 15, 16, 18].
Problem (1) is associated to the following anisotropic Sobolev spaces
and
endowed by the usual norm
Definition 1.1
We will say that \(u\in W_{0}^{1,(p_{i})}\left( \Omega \right) \) is an “energy” solution to (1) if and only if
and we will say that u is a “weak” solution to (1) if \(\partial _{i}u^{p_{i}-1}\in L^1(\Omega )\), \(\frac{f}{u^{\gamma (x) }} \in L^1_{loc}(\Omega )\), and one has the identity
We will also very often use the following indices
and
The following Theorem states some anisotropic Sobolev type inequalities, for more details we refer to the early works [13, 17, 20].
Theorem 1.2
There exists a positive constant C, depending only on \(\Omega \), such that for every \( v\in W_{0}^{1,(p_{i})}\left( \Omega \right) ,\) we have
and \(\forall v\in W_{0}^{1,(p_{i})}\left( \Omega \right) \cap L^{\infty }\left( \Omega \right) ,\overline{p}<N\)
for every r and \(t_{j}\) chosen in such a way to have
In the whole paper, C will denote a constant that may change from line to line.
2 Approximation problems
All the results obtained in this section, are direct consequences of the ones presented in [2, 14], but for the reader convenience we present them in details.
Let us first consider the following approximation problems
where \(f_{n}=T_{n}(f).\)
Recalling that
Lemma 2.1
The problem (5) has a solution in \(W_{0}^{1,(p_{i})}\left( \Omega \right) .\)
Proof
We will follow the same reasoning as in [2].
Fix \(n\in \mathbb {N~}\), and let \(v\in L_{\overline{p}^{*}}\left( \Omega \right) .\) Consider the equation
it is clear that the previous problem has a unique solution whenever the right hand side belongs to \(L^{s}\left( \Omega \right) \) with \(s\ge p_{\infty }^{\prime }\) see for instance [8, 9]. Denoting \( w=S(v),\)
using w as test function in (6), we obtain
by Sobolev inequality (2),
by Hölder inequality
Hence
and then
which means that the ball of radius \(R_{N}\) in \(L^{\overline{p}^{*}}\left( \Omega \right) \) is invariant by S, and so by Sobolev embedding and Schauder’s fixed point theorem we conclude that the approximation problem (5) has a solution in \(W_{0}^{1,(p_{i})}\left( \Omega \right) \), for every fixed n.
\(\square \)
Lemma 2.2
The sequence \(\left\{ u_{n}\right\} _{n}\) is increasing with respect to n.
Proof
We recall that \(f_{n}=T_{n}\left( f\right) \) and so \(0\le f_{n}\le f_{n+1}\)
as
and so one has that
using \(\left( u_{n}-u_{n+1}\right) ^{+}\) as test function in the last inequality, the right hand side gives
Now, taking into account the problems associated to \(u_n\) and to \(u_{n+1}\), it follows that
Thus
Integrating over the subset of \(\Omega \) where \(u_{n}\ge u_{n+1}\) and using the following inequality for \(p_{i}\ge 2\)
we reach that
Hence
which allows us to conclude that \(\left\{ u_{n}\right\} _{n}\) is increasing with respect to n.
\(\square \)
Remark 2.3
We limit ourselves to the case \(p_{i}\ge 2\) because (at our knowledge), the operator L verify a strong maximal principle only in the case \(p_{i}\ge 2\) see for instance [8], maximal principle that will be necessary in the sequel.
Lemma 2.4
For all \(n\in {\mathbb {N}}\), \(u_{n}\) the solution to the approximation problem (5), is such that \(u_{n}\in L^{\infty }\left( \Omega \right) \) and for all \(K\subset \subset \Omega \), \(u_{n}\ge C_{K}>0.\)
Proof
By some modifications in the theory of Leray-Lions operators theory one can show the existence of solution to
and so
the strong maximum principle, and the monotonicity of \(\left\{ u_{n}\right\} _{n}\) give that \(u_{n}\ge C_{K}>0.\) The \(L^{\infty }\left( \Omega \right) \) estimate of \(\left\{ u_{n}\right\} _{n},\) is a direct consequence of Stampachia result [19], as done in [2]. \(\square \)
3 Passage to the limit
For fixed \(\delta \), let \(\Omega _{\delta }=\left\{ x\in \Omega ,dist(x,\partial \Omega )<\delta \right\} \)
Theorem 3.1
Let \(s=\dfrac{N\overline{p}}{N\left( \overline{p}-1\right) +\overline{p}}\) and \(f\in L^{s}\left( \Omega \right) ,\) assume that there exists a \(\delta >0\) such that \(\gamma (x)\le 1\) in \(\Omega _{\delta }\), then the sequence \(\left\{ u_{n}\right\} _{n}\) of solutions to (5), is bounded in \(W_{0}^{1,\left( p_{i}\right) }\left( \Omega \right) .\)
Proof
Put \( \omega _{\delta } = \Omega \backslash \overline{{\Omega _{\delta } }} \), by the previous results we know that \(u_{n}\ge C_{\omega _{\delta }}>0.\) Now using \(u_{n}\) as test function in (5) we obtain
Using Hölder and Sobolev inequalities, we then obtain
which implies that
where C is a constant independant of n. \(\square \)
Theorem 3.2
Let \(s=\dfrac{N\overline{p}}{N\left( \overline{p}-1\right) +\overline{p}}\) and \(f\in L^{s}\left( \Omega \right) ,\) assume that there exists a \(\delta >0 \) such that \(\gamma (x)\le 1\) in \(\Omega _{\delta }\), then problem (1) posses a solution \(u\in W_{0}^{1,\left( p_{i}\right) }\left( \Omega \right) . \)
Proof
By the previous proposition \(\left\{ u_{n}\right\} _{n}\) is bounded in \( W_{0}^{1,\left( p_{i}\right) }\left( \Omega \right) \), thus (up to a subsequence) \(\left\{ u_{n}\right\} _{n}\) converges weakly to some u in \( W_{0}^{1,\left( p_{i}\right) }\left( \Omega \right) .\) On the other hand, \( \left\{ u_{n}\right\} _{n}\) converges strongly in \(L^{\theta }\left( \Omega \right) \) for \(\theta <\overline{p}^{*}\), thus \(\left\{ u_{n}\right\} _{n}\) converges to u almost everywhere in \(\Omega .\) So one has that for every \(\varphi \in C_{0}^{1}\left( \Omega \right) \)
By the fact that
for every \(\varphi \in C_{0}^{1}\left( \Omega \right) \), whenever \(\varphi \ne 0\) and on the set where \(u_{n}\ge C_{\omega },\) \(\omega \) being the support of \(\varphi ; \) the dominated Lebesgue’s theorem permits us to conclude that
by the sequel, the limit u of the sequence \(\left\{ u_{n}\right\} _{n}\) verify
\(\square \)
Theorem 3.3
Assume that for some \(\gamma ^{*}>1\) and some \( \delta >0\) we have \(\left\| \gamma \right\| _{L^{\infty }\left( \Omega \right) }\le \gamma ^{*}.\) Provided that \(f\in L^{s}\left( \Omega \right) \) with \(s=\dfrac{N\left( \gamma ^{*}-1+\overline{p} \right) }{N\left( \overline{p}-1\right) +\overline{p}\gamma ^{*}},\) problem (1) has a solution u in \(L^{\alpha }\left( \Omega \right) \) with \(\alpha =\dfrac{N\left( \gamma ^{*}-1+\overline{p}\right) }{\left( N-\overline{p}\right) },\) belonging to \(W_{loc}^{1,\left( p_{i}\right) }\left( \Omega \right) .\)
Proof
Let us use \(u_{n}^{\gamma ^{*}}\) as test function in (5), so we obtain for every \(i=1,2,...,N\)
with \(\beta =\dfrac{N\left( \gamma ^{*}-1+\overline{p}\right) }{\left( N- \overline{p}\right) \gamma ^{*}},\) and so
thus
which implies that
with the following choice of exponents
Sobolev inequality (4) gives
and so
by the fact that
we conclude that \(\{u_{n}\}_{n}\) is bounded in \(L^{\alpha }\left( \Omega \right) \) with \(\alpha =\dfrac{N\left( \gamma ^{*}-1+\overline{p}\right) }{\left( N-\overline{p} \right) }\) and by the monotone convergence theorem, \(\{u_{n}\}_n\) converges strongly to \(u \in L^{\alpha }\left( \Omega \right) . \)
On the other side using \(u_{n}^{\gamma ^{*}}\) as test function in (5) we get
by strong maximum principle we have for every compact \(K\subset \subset \Omega \)
thus we obtain weak convergence of \(\left\{ u_{n}\right\} _{n}\) to u in \( W_{loc}^{1,\left( p_{i}\right) }\left( \Omega \right) .\)
To complete the proof, we follow the same steps as in the previous Proposition. \(\square \)
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Miri, S.EH. On an anisotropic problem with singular nonlinearity having variable exponent. Ricerche mat 66, 415–424 (2017). https://doi.org/10.1007/s11587-016-0309-5
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DOI: https://doi.org/10.1007/s11587-016-0309-5