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Existence and regularity of solutions to a quasilinear elliptic problem involving variable sources
Boundary Value Problems volume 2017, Article number: 155 (2017)
Abstract
The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equation \(-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=\frac{f(x)}{u ^{\alpha(x)}}\) with \(f\in L^{m}(\Omega)\) (\(m\geqslant1\)) and \(\alpha(x)>0\). Due to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent \(\alpha(x)\), some classical methods may not directly be applied to our problem. In this paper, we construct a suitable test function and apply the Leray-Schauder fixed point theorem to prove the existence of positive solutions with necessary a priori estimate and compact argument. Furthermore, we also discuss the relationship among the regularity of solutions, the summability of f and the value of \(\alpha(x)\).
1 Introduction
In this paper, we study the existence of solutions for the following quasilinear elliptic problem with nonlinear singular terms and variable exponent:
where Ω is a bounded domain in \(R^{N}\) (\(N\geqslant p\)) with smooth boundary ∂Ω, \(p>2\), \(\alpha(x)\) is a continuous function on Ω̅, \(\alpha(x)>0\), \(\alpha^{+}= \sup_{x\in\overline{\Omega}}\alpha(x)\), \(\alpha^{-}=\inf_{x\in\overline {\Omega}}\alpha(x) \), f is a nonnegative function belonging to the Lebesgue space \(L^{m}(\Omega)\) for some suitable \(m\geqslant1\).
Problem (1.1) has been widely applied in many areas such as the contexts of chemical heterogeneous catalysts, non-Newtonian fluids and also the theory of heat conduction in electrically conducting materials, see [1–4] for detailed discussion.
For constant-exponent cases, Lazer and Mckenna in [5] discussed the case when \(p=2\) and f is a positive regular function in Ω̅. They proved that the solution was in \(H_{0}^{1}( \Omega)\) if and only if \(\alpha<3\), while it was not in \(C^{1}(\overline{ \Omega})\) if \(\alpha>1\).
Lair and Shaker in [6] improved the results of [5]. More specially speaking, they proved that this problem with \(0<\alpha<1\) has a unique weak positive solution in \(H_{0}^{1}( \Omega)\) if \(f(x)\) is a nonnegative nontrivial function in \(L^{2}(\Omega)\).
In 2004, the results of Lair and Shaker were generalized by Zhang and Cheng (see [7]) to the following problem:
where \(g(s)\) is singular near \(s=0\). They proved the existence and uniqueness of classical solutions under the assumption that \(f(x)\in C^{\alpha}(\Omega)\).
Recently, Boccardo and Orsina in [8] studied the existence, regularity and nonexistence of solutions for the following problem:
They discussed how the summability of f and the value of α affected the existence and regularities of solutions to the above problems. For other results of the related problems, the interested readers may refer to [9, 10] and the references therein. Problem (1.3) was discussed and extended to the more general problem of which the right-hand side is \(f(x)/u^{\alpha(x)}\) in [11]. For more related questions, refer to [12–14].
In this paper, we generalize the results in [11] to the case when the left-hand side is a p-Laplace operator. Due to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent \(\alpha(x)\), some classical methods may not directly be applied to our problem. We apply the method of regularization and the Schauder fixed point theorem, construct a suitable test function as well as a necessary compactness argument to overcome the difficulties arising from a variable exponent and a nonlinear differential operator and give an almost complete classification of coefficient m and variable exponent \(\alpha(x)\), then we prove the existence and regularity of solutions.
2 Preliminaries
Firstly, we give the definition of weak solutions to problem (1.1).
Definition 2.1
A function \(u\in W^{1,p}_{0}(\Omega)\) is called a weak solution of problem (1.1) if the following identity holds:
In order to prove our results, we will consider the following approximation problem:
where \(f_{n}(x)=\min\{f(x),n\}\), \(n\in N\).
Lemma 2.1
Problem (2.2) has a nonnegative solution \(u_{n}\) in \(W_{0}^{1,p}( \Omega)\bigcap L^{\infty}(\Omega)\).
Proof
Let \(n\in N\) be fixed, and ω be a function in \(L^{p}(\Omega)\). It is not difficult to prove that the following problem has a unique solution \(v\in W_{0}^{1,p}(\Omega)\bigcap L^{\infty}(\Omega)\) (see [15, 16]):
So, for any \(\omega\in L^{p}(\Omega)\), we define the mapping \(\Gamma: L^{p}(\Omega)\rightarrow L^{p}(\Omega)\) as \(\Gamma( \omega)=v\). Taking v as a test function for (2.3), we have
By the Poincaré inequality (on the left-hand side) and the Sobolev embedding theorem on the right-hand side (\(W^{1,p}(\Omega )\hookrightarrow L^{1}(\Omega)\)), we get that
this implies that
Since the embedding \(W^{1,p}(\Omega)\hookrightarrow L^{p}(\Omega)\) is compact, we obtain that Γ is a compact operator. Moreover, if \(u=\lambda\Gamma u\) for some \(0<\lambda\leqslant1\), then \(\Gamma u=\frac{u}{\lambda}\) and hence \(\Vert u\Vert _{L^{p}(\Omega)}\leqslant \Vert u\Vert _{W^{1,p}(\Omega)}\leqslant C\) for a constant C independent of λ. Then, by Schauder’s fixed point theorem, we know that there exists \(u_{n}\in W_{0}^{1,p}(\Omega)\) such that \(u_{n}=\Gamma(u_{n})\), i.e., problem (2.2) has a solution. Since \(\frac{f_{n}}{(u_{n}+ \frac{1}{n})^{\alpha(x)}}\geqslant0\), the maximum principle in [17, 18] shows that \(u_{n}\geqslant0\), \(u _{n}\in L^{\infty}(\Omega)\). □
Lemma 2.2
The sequence \(\{u_{n}\}\) is increasing with respect to n, \(u_{n}>0\) in Ω, and for every \(\Omega'\subset\subset\Omega \), there exists \(C_{\Omega'}>0\) (independent of n) such that
Proof
Due to \(0\leqslant f_{n}\leqslant f_{n+1}\) and \(\alpha(x)>0\), we have that
so that
Choosing \((u_{n}-u_{n+1})_{+}=\max\{u_{n}-u_{n+1},0\}\) as a test function and observing that
we get that
which implies that \((u_{n}-u_{n+1})_{+}=0\) a.e. in Ω, that is, \(u_{n}\leqslant u_{n+1}\) for every \(n\in N\). Since the sequence \(\{u_{n}\}\) is increasing with respect to n, we only need to prove that (2.4) holds for \(u_{1}\). Using Lemma 2.1, we know that \(u_{1}\in L^{\infty}(\Omega)\), i.e., there exists a constant C (depending only on Ω and N) such that
then
Since \(\frac{f_{1}}{(C+1)^{\alpha(x)}}\geqslant0\), \(\frac{f_{1}}{(C+1)^{ \alpha(x)}}\not\equiv0\), the strong maximum principle implies that \(u_{1}>0\) in Ω and (2.4) holds for \(u_{1}\). Because of the monotonicity of \(u_{n}\), (2.4) holds for \(u_{n}\). □
Remark 2.1
If \(u_{n}\) and \(v_{n}\) are two solutions of (2.2), following the lines of the proof of the first part in Lemma 2.2, we may show that \(u_{n}\leqslant v_{n}\). By symmetry, this implies that the solution of (2.2) is unique.
Lemma 2.3
The solution \(u_{1}\) to problem (2.2) with \(n=1\) satisfies
Proof
By \(\frac{\min\{f(x), 1\}}{(u_{1}+1)^{\alpha(x)}}\leqslant1\) and Lemma 2.2 in [17], we know that there exists \(0<\beta<1\) such that \(u_{1}\in C^{1,\beta}(\overline{\Omega})\) and \(\Vert u_{1}\Vert _{C ^{1,\beta}}\leqslant C\), which implies that the gradient of \(u_{1}\) exists everywhere, then the Hopf lemma in [19] shows that \(\frac{\partial u_{1}(x)}{\partial\nu}>0 \) in Ω̅, where ν is the outward unit normal vector of ∂Ω at x. Moreover, following the lines of the proof of lemma in [5], we get that
□
We know clearly that the estimates on \(u_{n}\) depend on f and \(\alpha(x)\), we will discuss this in different cases.
3 The case \(0<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha^{+}<1\)
In this case, we obtain a priori estimates on \(u_{n}\) in \(H_{0}^{1}( \Omega)\) only if f is more regular than \(L^{1}(\Omega)\). We have the following results.
Lemma 3.1
([20])
Let \(u_{n}\) be the solution of (2.2) with \(0<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha^{+}<1\), and suppose that \(f\in L^{m}(\Omega)\) with \(m=\frac{Np}{Np-N+p+(N-p)\alpha^{-}}= (\frac{p^{*}}{1-\alpha^{-}} )^{\prime}\). Then the sequence \(\{u_{n}\}\) is bounded in \(W_{0}^{1,p}(\Omega)\).
Once we have the boundedness of \(u_{n}\), we can prove an existence result for (1.1).
Theorem 3.1
([20])
Suppose that f is a nonnegative function in \(L^{m}(\Omega)\) (\(f\not\equiv0\)), with \(m=\frac{Np}{Np-N+p+(N-p)\alpha ^{-}}= (\frac{p^{*}}{1-\alpha^{-}} )'\), \(f\not\equiv0\), and let \(0<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha^{+}<1\). Then problem (1.1) has a solution \(u\in W_{0}^{1,p}(\Omega)\) satisfying (2.1).
The summability of u depends on the summability of f, which is proved in the next lemma.
Lemma 3.2
Suppose that \(f\in L^{m}(\Omega)\), \(m\geqslant\frac{Np}{Np-N+p+(N-p) \alpha^{-}}\), and let \(0<\alpha^{-}\leqslant\alpha(x)\leqslant \alpha^{+}<1\). Then the solution u of (1.1) given by Theorem 3.1 is such that:
-
(i)
if \(m>\frac{N}{p}\), then \(u\in L^{\infty}(\Omega)\);
-
(ii)
if \(\frac{Np}{Np-N+p+(N-p)\alpha^{-}}\leqslant m<\frac {N}{p}\), then \(u\in L^{s}(\Omega)\), \(s=\frac{Nm(\alpha^{-}+p-1)}{N-pm}\).
Proof
To prove (i), let \(k>1\) and define \(G_{k}(s)=(s-k)_{+}\). Taking \(G_{k}(u_{n})\) as a test function in (2.2), we obtain
Since \(G_{k}(u_{n})\neq0\), it implies that
Starting from inequality (3.1), Theorem 4.2 in [21] shows that there exists a constant C (independent of n) such that
which implies that u belongs to \(L^{\infty}(\Omega)\).
To prove (ii), noting that if \(m=\frac{Np}{Np-N+p+(N-p)\alpha^{-}}\), \(s=\frac{Np}{N-p}=p^{*}\), since \(u\in W_{0}^{1,p}(\Omega)\), the result when \(m=\frac{Np}{Np-N+p+(N-p)\alpha^{-}}\) is true by the Sobolev embedding theorem. If \(\frac{Np}{Np-N+p+(N-p)\alpha^{-}}< m< \frac{N}{p}\), letting \(\delta>1\) and choosing \(u_{n}^{p\delta-p+1}\) as a test function in (2.2), using Hölder’s inequality, we get that
By the Sobolev inequality (on the left-hand side), we have that
where S is the constant of the Sobolev embedding theorem. Combining with (3.2) and (3.3), we have that
We choose δ in such a way that \(p^{*}\delta=(p\delta-p+1- \alpha^{-})m'\), i.e.,
which yields that \(\delta>1\) if and only if \(\frac{Np}{Np-N+p+(N-p) \alpha^{-}}< m<\frac{N}{p}\), and that \(p^{*}\delta=\frac{Nm(\alpha ^{-}+p-1)}{N-pm}=s\). Therefore, (3.4) becomes
which implies that
Using Young’s inequality on the right-hand side in (3.5), we have that
where \(\varepsilon=\frac{S(p\delta-p+1)}{2\delta^{p}\Vert f\Vert _{L^{m}( \Omega)}}\). Thus, we get that
Therefore, we know that \(u_{n}\) is bounded in \(L^{s}(\Omega)\), so is \(u\in L^{s}(\Omega)\). □
Theorem 3.2
Suppose that \(f\in L^{m}(\Omega)\), \(\frac{(p-1+\alpha^{+})N}{(\alpha ^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}\leqslant m<\frac{Np}{Np-N+p+(N-p) \alpha^{-}}\), and \(0<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha ^{+}<1\). Then problem (1.1) has a solution u in \(W_{0}^{1,q}( \Omega)\), \(q=\frac{Nm(\alpha^{-}+p-1)}{N-m(1-\alpha^{-})}\).
Proof
The lines of our proof are that if we can prove that \(u_{n}\) is bounded in \(W_{0}^{1,q}(\Omega)\) (with q as in the statement), the existence of a solution u in \(W_{0}^{1,q}(\Omega)\) of (1.1) will be proved by passing to the limit in (2.2) as in the proof of Theorem 3.1. To prove that \(u_{n}\) is bounded in \(W_{0}^{1,q}( \Omega)\), we begin by proving that it is bounded in \(L^{s}(\Omega)\), with \(s=\frac{Nm(\alpha^{-}+p-1)}{N-pm}\). To attain this goal, we choose \(u_{n}^{p\delta-p+1}\) as a test function in (2.2) as in the statement of Lemma 3.2, where \(\frac{p-1+\alpha^{+}}{p} \leqslant\delta<1\); however, \(\nabla u_{n}^{p\delta-p+1}\) will be singular at \(u_{n}=0\). Therefore, we choose \((u_{n}+\varepsilon)^{p \delta-p+1}-\varepsilon^{p\delta-p+1}\) as a test function in (2.2), where \(\varepsilon<\frac{1}{n}\) for n fixed. We have that
Since \(f_{n}\leqslant f\), we have that
By the Sobolev embedding theorem (\(W_{0}^{1,p}(\Omega)\hookrightarrow L^{p^{*}}(\Omega)\)) on the left-hand side, it follows that
where S is the best constant of the Sobolev embedding theorem. Combining (3.7) with (3.8), we have that
Using Hölder’s inequality on the right-hand side, we get
Letting \(\varepsilon\rightarrow0\), we get (3.4), i.e.,
where δ is chosen in such a way that \(p^{*}\delta=(p\delta-p+1- \alpha^{-})m'\), i.e.,
If \(m= \frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}\), we choose \(\delta=\frac{p-1+\alpha^{+}}{p}\) in (3.9), and letting \(\varepsilon\rightarrow0\), we have that
Using Hölder’s inequality and Young’s inequality, we get that
where \(\varepsilon=\frac{S(p\delta-p+1)}{2\delta^{p}\Vert f\Vert _{L^{m}( \Omega)}}\). Thus we have that
Therefore we obtain that \(u_{n}\) is bounded in \(L^{\frac{N(p-1+\alpha ^{+})}{N-p}}(\Omega)\), where \(\frac{N(p-1+\alpha^{+})}{N-p}\) is the value of s for \(m=\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+ \alpha^{+})}\).
If \(\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha ^{+})}< m<\frac{Np}{Np-N+p+(N-p) \alpha^{-}}\), it is clear that the inequality on m holds true if and only if \(\frac{p-1+\alpha^{+}}{p}<\delta<1\), starting from (3.4) and arguing as in the proof of Lemma 3.2, we also get that \(u_{n}\) is bounded in \(L^{s}(\Omega)\) with \(s=\frac{Nm( \alpha^{-}+p-1)}{N-pm}\).
The right-hand side of (3.7) is bounded with respect to n (and ε, which we take smaller than 1) by using the estimate on \(u_{n}\) in \(L^{s}(\Omega)\) and the choice of δ.
Since \(\delta<1\),
If \(q=\frac{Nm(\alpha^{-}+p-1)}{N-m(1-\alpha^{-})}< p\), by Hölder’s inequality, we have that
The choice of δ and the value of q are such that \(\frac{pq(1- \delta)}{p-q}=s\), so that the right-hand side of (3.10) is bounded with respect to n and ε. Hence, \(u_{n}\) is bounded in \(W_{0}^{1,q}(\Omega)\). □
Theorem 3.3
Suppose that \(f\in L^{m}(\Omega)\), \(\frac{1}{2-p-\alpha^{+}+p\delta }< m<\frac{(p-1+ \alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}\) (\(\frac{p-1+ \alpha^{-}}{p}<\delta<\frac{p-1+\alpha^{+}}{p}\)), and \(0<\alpha ^{-}\leqslant\alpha(x)\leqslant\alpha^{+}<1\). Then problem (1.1) has a solution u in \(W_{0}^{1,q}(\Omega)\), \(q=\frac{Nm( \alpha^{-}+p-1)}{N-m(1-\alpha^{-})}\).
Proof
The lines of our proof are similar to those in the proof of Theorem 3.2. We also begin by proving that \(u_{n}\) is bounded in \(L^{s}(\Omega)\), with \(s=\frac{Nm(\alpha^{-}+p-1)}{N-pm}\). To this aim, we also choose \((u_{n}+\varepsilon)^{p\delta-p+1}- \varepsilon^{p\delta-p+1}\) as a test function in (2.2), where \(\frac{p-1+\alpha^{-}}{p}<\delta<\frac{p-1+\alpha^{+}}{p}\), \(\varepsilon<\frac{1}{n}\) for n fixed. Since \(f_{n}\leqslant f\), using the Sobolev embedding theorem (\(W_{0}^{1,p}(\Omega )\hookrightarrow L^{p^{*}}(\Omega)\)) on the left-hand side again, we have that
where S is the best constant of the Sobolev embedding theorem.
Using Hölder’s inequality and Lemma 2.3 on the right-hand side, we get that
Letting \(\varepsilon\rightarrow0\), we have that
where δ is chosen in such a way that \(p^{*}\delta=(p\delta-p+1- \alpha^{-})m'\), i.e.,
If \(1< m<\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha ^{+})}\), it is clear that the inequality on m holds true if and only if \(\frac{p-1+\alpha^{-}}{p}<\delta<\frac{p-1+\alpha^{+}}{p}\), and arguing as to the case \(m=\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+ \alpha^{+})}\) in the proof of Theorem 3.2, we also obtain that \(u_{n}\) is bounded in \(L^{s}(\Omega)\), with \(s=\frac{Nm(\alpha ^{-}+p-1)}{N-pm}\).
Since \(\delta<1\),
If \(q=\frac{Nm(\alpha^{-}+p-1)}{N-m(1-\alpha^{-})}< p\), similarly to the proof of Theorem 3.2, we have by Hölder’s inequality that
Due to the choice of δ and the value of q, the right-hand side of the above inequality is bounded with respect to n and ε. Hence, \(u_{n}\) is bounded in \(W_{0}^{1,q}(\Omega)\). □
4 The case \(1<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha^{+}\)
The case \(1<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha^{+}\) has many analogies with the case \(0<\alpha^{-}<\alpha^{+}<1\). In this case, we can also prove that \(u_{n}\) is bounded in \(W_{0}^{1,p}(\Omega)\) only if f is more regular than \(L^{1}(\Omega)\) and \(\alpha^{+}\) and \(\alpha^{-}\) is close to 1. Hence we obtain the existence of problem (1.1).
Lemma 4.1
Suppose that \(f\in L^{m} (\Omega)\) (\(m>1\)), let \(u_{n}\) be the solution of (2.2) with \(1<\alpha^{-}<\alpha^{+}<2-\frac{1}{m}\). Then \(u_{n}\) is bounded in \(W_{0}^{1,p}(\Omega)\).
Proof
Taking \(u_{n}\) as a test function in (2.2), we obtain that
Using Lemma 2.1 and Lemma 2.2, we know that \(u_{n}\geqslant u_{1}\) and there exists a constant \(M>0\) s.t. \(u_{1}\leqslant M\). Hence \((\frac{M}{u_{1}})^{\alpha(x)-1}\leqslant (\frac{M}{u_{1}})^{\alpha^{+}-1}\), and we have that
Using Hölder’s inequality on the right-hand side and Lemma 2.3, we obtain
Therefore, \(u_{n}\) is bounded in \(W_{0}^{1,p}(\Omega)\). □
Once we have the boundedness of \(u_{n}\), we can prove the following existence theorem along the lines of Theorem 3.1.
Theorem 4.1
Suppose that \(f\in L^{m}(\Omega)\) (\(m>1\)), \(f\not\equiv0\) and \(1<\alpha^{-}<\alpha^{+}<2-\frac{1}{m}\). Then problem (1.1) has a solution u in \(W_{0}^{1,p}(\Omega)\).
The summability of u can be proved along the lines of Lemma 3.2 with small changes.
Lemma 4.2
Suppose that \(f\in L^{m}(\Omega)\) (\(m>1\)) and \(1<\alpha^{-}<\alpha^{+}<2- \frac{1}{m}\). Then the solution u of (1.1) given by Theorem 4.1 is such that:
-
(i)
if \(m>\frac{N}{p}\), then \(u\in L^{\infty}(\Omega)\);
-
(ii)
if \(\frac{(p-1+\alpha^{+})N}{(\alpha ^{-}+p-1)(N-p)+p(p-1+\alpha ^{+})}\leqslant m<\frac{N}{p}\), then \(u\in L^{s}(\Omega)\), \(s=\frac{Nm( \alpha^{-}+p-1)}{N-pm}\).
Proof
The proof of (i) is similar to the proof of Lemma 3.2(i), we omit the details here.
To prove (ii), we choose \(u_{n}^{p\delta-p+1}\) as a test function with \(\delta\geqslant\frac{p-1+\alpha^{+}}{p}\) in (2.2). Similarly to the proof of Lemma 3.2, we obtain that
If \(m= \frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}\), choosing \(\delta=\frac{p-1+\alpha^{+}}{p}\) in (4.1), by Hölder’s inequality, we get that
We choose δ in such a way that \(p^{*}\delta=(p\delta-p+1- \alpha^{-})m'\), i.e., \(\delta=\frac{(\alpha^{-}+p-1)m(N-p)}{p(N-mp)}\). Since \(m=\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha ^{+})}\), we get that \(\frac{p}{p^{*}}>\frac{1}{m'}\). Because \(s=p^{*}\delta\), we have the boundedness of \(u_{n}\) in \(L^{\frac{N(p-1+ \alpha^{+})}{N-p}}(\Omega)\), which is the value of s for \(m=\frac{N(p-1+\alpha^{+})}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}\).
If \(\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}< m< \frac{N}{p}\), starting from inequality (4.1), using Hölder’s inequality, we get that
We also choose δ in such a way that \(p^{*}\delta=(p\delta-p+1- \alpha^{-})m'\), which yields that \(\delta>\frac{p-1+\alpha^{+}}{p}\) if and only if \(m>\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+ \alpha^{+})}\), and that \(p^{*}\delta=s\). So, since \(\frac{p}{p^{*}}> \frac{1}{m'}\) being \(m<\frac{N}{p}\), we have the boundedness of \(u_{n}\) in \(L^{s}(\Omega)\), so does \(u\in L^{s}(\Omega)\). □
Moreover, we can prove that a positive power of \(u_{n}\) is bounded in \(W_{0}^{1,p}(\Omega)\) only if f is more regular than \(L^{1}(\Omega )\) and \(\alpha^{+}\) is close to \(\alpha^{-}\), and we only have the boundedness of \(u_{n}\) in \(W_{\mathrm{loc}}^{1,p}(\Omega)\).
Lemma 4.3
Suppose that \(f\in L^{m}(\Omega)\) (\(m>1\)), let \(u_{n}\) be the solution of (2.2) with \(1<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha ^{+}\) and \(\alpha^{+}-\alpha^{-}<1-\frac{1}{m}\). Then \(u_{n}^{\frac{p-1+ \alpha^{-}}{p}}\) is bounded in \(W_{0}^{1,p}(\Omega)\), and \(u_{n}\) is bounded in \(W_{\mathrm{loc}}^{1,p}(\Omega)\) and in \(L^{s}(\Omega)\), with \(s=\frac{N(\alpha^{-}+p-1)}{N-p}\).
Proof
Taking \(u_{n}^{\alpha^{-}}\) as a test function in (2.2), since \(\frac{u_{n}^{\alpha^{-}}}{(u_{n}+\frac{1}{n})^{\alpha^{-}}}\leqslant 1\) and \(f_{n}\leqslant f\), by Hölder’s inequality and Lemma 2.3, we get that
Since
we have that
Thus, we have that \(u_{n}^{\frac{\alpha^{-}+p-1}{p}}\) is bounded in \(W_{0}^{1,p}(\Omega)\).
Applying the Sobolev embedding theorem to \(u_{n}^{ \frac{\alpha^{-}+p-1}{p}}(W_{0}^{1,p}(\Omega)\hookrightarrow L^{p ^{*}}(\Omega))\), we get that
where S is the best constant of the Sobolev embedding theorem. Since the boundedness of \(u_{n}^{\frac{\alpha^{-}+p-1}{p}}\) in \(W_{0}^{1,p}( \Omega)\), we thus have the boundedness of \(u_{n}\) in \(L^{s}(\Omega)\).
To prove the boundedness of \(u_{n}\) in \(W_{\mathrm{loc}}^{1,p}(\Omega)\), we choose \(u_{n}\varphi^{p}\) as a test function in (2.2), where \(\varphi\in C_{0}^{\infty}(\Omega)\), \(\Omega'=\{x\in\Omega, \varphi\neq0\}\). By (2.4), we have that
By Young’s inequality, we have that
Since \(u_{n}\) is bounded in \(L^{s}(\Omega)\) (where \(s\geqslant p\)), by Hölder’s inequality, we obtain that
and hence \(u_{n}\) is bounded in \(W_{\mathrm{loc}}^{1,p}(\Omega)\). □
Once we have the boundedness of \(u_{n}\), we can prove the following existence theorem along the lines of Theorem 3.1.
Theorem 4.2
Suppose that f is a nonnegative function in \(L^{m}(\Omega)\) (\(m>1\)), (\(f \not\equiv0\)), \(1<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha^{+}\) and \(\alpha^{+}-\alpha^{-}<1-\frac{1}{m}\). Then problem (1.1) has a solution u in \(W_{\mathrm{loc}}^{1,p}(\Omega)\). Furthermore, \(u^{\frac{ \alpha^{-}+p-1}{p}}\) belongs to \(W_{0}^{1,p}(\Omega)\).
The summability of u can be proved as the following lemma, the proof is similar to the proof of Lemma 4.2
Lemma 4.4
Suppose that \(f\in L^{m}(\Omega)\), \(1<\alpha^{-}\leqslant\alpha(x) \leqslant\alpha^{+}\) and \(\alpha^{+}-\alpha^{-}<1-\frac{1}{m}\). Then the solution u of (1.1) given by Theorem 4.2 is such that:
-
(i)
if \(m>\frac{N}{p}\), then \(u\in L^{\infty}(\Omega)\);
-
(ii)
if \(\frac{N(p-1+\alpha^{+})}{(\alpha ^{-}+p-1)(N-p)+p(p-1+\alpha ^{+})}\leqslant m<\frac{N}{p}\), then \(u\in L^{s}(\Omega)\), \(s=\frac{Nm( \alpha^{-}+p-1)}{N-mp}\).
5 The case \(0<\alpha^{-}<1<\alpha^{+}\)
If \(0<\alpha^{-}<1<\alpha^{+}\), the boundedness of \(u_{n}\) in \(W_{0}^{1,p}(\Omega)\) can also be obtained only if f is more regular than \(L^{1}(\Omega)\). Furthermore, the existence of problem (1.1) is obtained, the proof has many analogies with the case \(0<\alpha^{-}<\alpha^{+}<1\). We have the following results.
Lemma 5.1
Suppose that \(f\in L^{m}(\Omega)\), with \(m=\frac{Np}{Np-N+p+(N-p) \alpha^{-}}\), and let \(u_{n}\) be the solution of (2.2) with \(0<\alpha^{-}<1<\alpha^{+}<2-\frac{1}{m}\). Then the sequence \(\{u_{n}\}\) is bounded in \(W_{0}^{1,p}(\Omega)\).
Proof
We choose \(u_{n}\) as a test function in (2.2), by Hölder’s inequality and Lemma 2.3, since \(f_{n}\leqslant f\), we have that
Applying the Sobolev embedding theorem on the left-hand side, we get
Combining (5.1) with (5.2) implies that
Let \(p^{*}=(1-\alpha^{-})m'\), it follows that
By Young’s inequality, we get that
Thus, we have that
We choose \(\varepsilon=\frac{S}{2\Vert f\Vert _{L^{m}(\Omega)}}\) to get
So the boundedness of \(u_{n}\) in \(L^{p^{*}}(\Omega)\) is obtained. Using the estimate and (5.1) again, we have the estimate of \(u_{n}\) in \(W_{0}^{1,p}(\Omega)\). □
Once the boundedness of \(u_{n}\) in \(W_{0}^{1,p}(\Omega)\) is obtained, we can prove the following existence theorem.
Theorem 5.1
Suppose that \(f\in L^{m}(\Omega)\) with \(m=\frac{Np}{Np-N+p+(N-p) \alpha^{-}}\), \(f\not\equiv0\), and \(0<\alpha^{-}<1<\alpha^{+}<2- \frac{1}{m}\). Then problem (1.1) has a solution u in \(W_{0}^{1,p}(\Omega)\).
Lemma 5.2
Suppose that \(f\in L^{m}(\Omega)\) with \(m\geqslant\frac{Np}{Np-N+p+(N-p) \alpha^{-}}\), and \(0<\alpha^{-}<1<\alpha^{+}<2-\frac{1}{m}\). Then the solution u of (1.1) given by Theorem 5.1 is such that:
-
(i)
if \(m>\frac{N}{p}\), then \(u\in L^{\infty}(\Omega)\);
-
(ii)
if \(\frac{Np}{Np-N+p+(N-p)\alpha^{-}}\leqslant m<\frac {N}{p}\), then \(u\in L^{s}(\Omega)\), \(s=\frac{Nm(\alpha^{-}+p-1)}{N-mp}\).
Proof
The proof of (i) is similar to that for Lemma 3.2(i), we omit the details here.
To prove (ii), if \(\frac{N(p-1+\alpha^{+})}{(\alpha^{-}+p-1)(N-p)+p(p-1+ \alpha^{+})}\leqslant m<\frac{N}{p}\), the proof is identical to that for Lemma 4.2, we also omit it here.
If \(m=\frac{Np}{Np-N+p+(N-p)\alpha^{-}}\), we can prove the results by the Sobolev embedding theorem.
If \(\frac{Np}{Np-N+p+(N-p)\alpha^{-}}< m<\frac{N(p-1+\alpha^{+})}{( \alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}\), we choose \(1<\delta<\frac{p-1+ \alpha^{+}}{p}\), and use once again \(u_{n}^{p\delta-p+1}\) as a test function in (2.2). Using \(\delta>1>\frac{p-1+\alpha^{-}}{p}\), as well as Hölder’s inequality, the Sobolev embedding theorem, Lemma 2.3, we get that
The choice of δ in such a way that \(p^{*}\delta=(p\delta-p+1- \alpha^{-})m'\) yields that \(1<\delta<\frac{p-1+\alpha^{+}}{p}\) if and only if \(\frac{Np}{Np-N+p+(N-p)\alpha^{-}}< m<\frac{N(p-1+\alpha^{+})}{( \alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}\), and that \(p^{*}\delta=s\). The choice of \(m<\frac{N}{p}\) implies that \(\frac{p}{p^{*}}>\frac{1}{m'}\). Thus we have the boundedness of \(u_{n}\) in \(L^{s}(\Omega)\), and so does the limit u in \(L^{s}(\Omega)\). □
6 Conclusions
In this paper, we study the existence and regularity of solutions to the quasilinear elliptic problem with nonlinear singular terms and variable exponent. Due to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent \(\alpha(x)\), some classical methods may not directly be applied to our problem. We construct a suitable test function and apply the Leray-Schauder fixed point theorem to prove the existence of positive solutions with necessary a priori estimate and compact argument. Furthermore, we prove that the existence and regularity of solutions depend on the summability of f and the value of \(\alpha(x)\).
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Acknowledgements
We are very grateful to the anonymous referees for their valuable suggestions that improved the article. The work is supported by the National Natural Science Foundation of China (11501051, 11626044) and the fund of the “Thirteen Five” Scientific and Technological Research Planning Project of the Department of Education of Jilin Province in China(JJKH20170301KJ).
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Chu, Y., Gao, R. & Sun, Y. Existence and regularity of solutions to a quasilinear elliptic problem involving variable sources. Bound Value Probl 2017, 155 (2017). https://doi.org/10.1186/s13661-017-0888-4
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DOI: https://doi.org/10.1186/s13661-017-0888-4
Keywords
- quasilinear elliptic problem
- nonlinear singular term
- existence
- regularity
- variable exponent