Abstract
This paper uses the Galerkin method to investigate the existence of positive solution to a class of singular elliptic problems given by
where \(\Omega \subset \mathbb{R}^{2}\) is a bounded smooth domain, \(0<\beta _{0}\), \(\gamma _{0} \leq 1\), \(\alpha _{0} \in [0,2)\), \(h_{0}(x)\geq 0\), \(h_{0}\neq 0\), \(h_{0}\in L^{\infty}(\Omega )\), \(0<\|h_{0}\|_{\infty} < \lambda _{0} < \Lambda _{0}\), and \(f_{0}\) are continuous functions. More precisely, \(f_{0}\) has a critical exponential growth, that is, the nonlinearity behaves like \(\exp (\overline{\Upsilon}s^{2})\) as \(|s| \to \infty \), for some \(\overline{\Upsilon}>0\).
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1 Introduction
The purpose in the first part of this article is to prove the existence of solution to the following class of singular problems
where is a bounded smooth domain \(0<\beta _{0}\), \(\gamma _{0}\leq 1\), \(0< \|h_{0}\|_{\infty}< \lambda _{0}< \Lambda _{0}\), and \(\alpha _{0} \in [0,2)\) are real parameters.
Elliptic problems with singularities are a class of partial differential equations (PDEs) that arise in various fields of science and engineering. These problems are characterized by the presence of singularities in the coefficients or boundary conditions of the elliptic PDE, which can have a significant impact on the behavior of the solutions. Singularities can occur at specific points or along curves, and they often represent physical phenomena or geometrical features that require special treatment in the mathematical modeling of the problem.
The presence of singularities in elliptic problems can complicate the analysis and numerical solution of these equations. Understanding and handling these singularities is crucial for obtaining accurate and physically meaningful solutions in practical applications. Researchers in various fields, including mathematics, physics, engineering, and computational science, have explored numerous approaches and techniques for tackling elliptic problems with singularities.
To delve deeper into this topic and explore the methods and theories used in the analysis of elliptic problems with singularities, one may refer to the key references [6, 9, 12], and [13].
On the other hand, nonlinear elliptic problems with gradient-dependent growth are a class of partial differential equations involving a nonlinear term in the equation, which depends on the gradient (or derivative) of the solution. These problems are typically expressed as follows:
Understanding and analyzing such problems is crucial in many areas of science and engineering, including nonlinear elasticity, porous media flow, phase transition, and optimal control problems with pointwise constraints. For more information on physical motivation for this class of problems, one can refer to [4, 19], and [21].
We are going back to our problem to enunciate the hypotheses on the functions \(h_{0}\) and \(f_{0}\). More precisely, the functions are continuous satisfying the following properties:
- \((f_{1})\):
-
There exists \(\overline{\Upsilon}> 0\) such that the exponential growth conditions at infinity are given by:
$$ \lim _{t\rightarrow \infty} \frac{f_{0}(t)}{\exp \left (\Upsilon _{0}\vert t\vert ^{2}\right )}=0 \ \text{ for } \Upsilon _{0}>\overline{\Upsilon}\ \text{ and }\ \lim _{t \rightarrow \infty} \frac{f_{0}(t)}{\exp \left (\Upsilon _{0}\vert t\vert ^{2}\right )}= \infty \ \text{ for }\ \ 0< \Upsilon _{0}< \overline{\Upsilon}. $$ - \((f_{2})\):
-
The growth condition at the origin:
$$ \lim _{t\rightarrow 0^{+}}\frac{f_{0}(t)}{t}=0. $$ - \((h_{1})\):
-
\(h_{0}\in L^{\infty}(\Omega )\), \(h_{0}(x) \geq 0\) and \(h_{0}(x)\neq 0\).
Since we are looking for positive solution, in this paper, we consider \(f_{0}(t)=0\) for all \(t<0\).
The main result is:
Theorem 1.1
Assume that conditions \((f_{1})-(f_{2})\) and \((h_{1})\) hold. Then, there exists \(\lambda ^{*}>0\) such that the problem (1.1) has a positive weak solution for every \(\Lambda _{0} \in (0,\lambda ^{*})\).
Problems involving singularities, exponential growth, or dependence on the solution’s gradient have been extensively studied in recent years. For example, after the excellent article by Adimurthi and Sandeep [2], which introduced the version of the Trudinger-Moser inequality with a singular term, several articles have emerged discussing this topic. In [1], the authors demonstrated the existence of two solutions to the problem
where \(0< p \in L^{\infty}(\Omega )\), \(0< \lambda \) and \(0<\delta <3\).
In [10], the authors investigated the following problem
The authors established some existence results for a class of critical elliptic problems with singular exponential nonlinearities, but they do not assume any global sign conditions on the nonlinearity, which makes our results new even in the nonsingular case.
In [8], the authors apply minimax methods to obtain existence and multiplicity of weak solutions to singular and nonhomogeneous elliptic equation of the following form
where f has the maximal growth on s.
In [3], the authors use the Galerkin method to demonstrate the existence of solutions to the problem
where h has sublinear and singular terms, and g is a continuous function with \(0 \leq \lambda \).
Using the nonlinear domain decomposition method, in [5], the authors provide a sufficient condition for the problem
The version for \(\mathbb{R}^{N}\) of this class of problems was studied in [11]. More precisely, the authors investigated the problem
where \(0< a<1\), and p is a positive weight. Under the hypothesis that f is a nondecreasing function with sublinear growth and g is decreasing and unbounded around the origin, they established the existence of a ground state solution vanishing at infinity. The arguments used by the authors rely essentially on the maximum principle.
Before finishing the introduction part, we would like to mention the study with other techniques of some problems related to the ones we are studying, such as [17, 18], and [22].
Below, we list what we believe are the main contributions of our paper:
-
1)
We complete the results in [1] because, in our article, we consider the convection term and the critical exponential growth with singularity.
-
2)
Unlike what was studied in [8] and [10], in our article, we are considering the convection term and singularity in u.
-
3)
We also complete the results in [3, 5], and [11] because we are considering exponential growth with singularity, which makes the estimates more delicate.
The plan of the paper is the following: In Sect. 2, we recall some preliminary results. In Sect. 3, we study an auxiliary problem. We show the existence of solution to the auxiliary problem in Sect. 4, where we prove Theorem 1.1.
2 Preliminary results
Let us consider the Sobolev space \(W_{0}^{1,2}(\Omega )\) endowed with the norm
We say that \(u\in W_{0}^{1,2}(\Omega )\) is a weak solution to the problem (1.1) if \(u>0\) in Ω and it verifies
for all \(\phi \in W_{0}^{1,2}(\Omega )\).
First, we recall some important results by Adimurthi-Sandeepr [2] and Hardy-Sobolev [14].
Theorem 2.1
(A singular Trudinger-Moser inequality) Let Ω be a bounded domain in \(\mathbb{R}^{2}\), \(u\in W_{0}^{1,2}(\Omega )\) and \(\Upsilon >0\), then
and there exists a constant \(M>0\) such that
if, only if, \(\Upsilon \leq 2\pi (2-\alpha )\).
Theorem 2.2
(Hardy-Sobolev inequality) If \(u\in C^{1}(\overline{\Omega})\cap W_{0}^{1,2}(\Omega )\), then \(\dfrac{u}{Cd^{\tau}}\in L^{r}(\Omega )\), for \(\frac{1}{r}=\frac{1}{2}-\frac{1-\tau}{2}\), \(0< \tau \leq 1\) and
where \(d(x)=dist(x,\partial \Omega )\), and C is a positive constant independent of x.
We observe that, from \((f_{1})\)–\((f_{2})\), for all \(\delta >0\) and for all \(\alpha >\alpha _{0}\), there exists \(C_{\delta}>0\) such that
for all \(q_{0}\geq 0\). In this paper, we will use \(q_{0}>2\).
3 An auxiliary problem
For each \(\varepsilon >0\), we consider the following auxiliary problem
where the function \(h_{0}\), \(f_{0}\) satisfies the hypotheses of the Theorem 1.1.
To prove Theorem 1.1, we first show the existence of a solution to problem (3.1). For this, we will use the Galerkin method together with the following fixed point theorem, see [20] and [16, Theorem 5.2.5].
Lemma 3.1
Let be a continuous function such that \(\langle G(\xi ),\xi \rangle \geq 0\) for every with \(\vert \xi \vert = r\) for some \(r>0\). Then, there exists \(z_{0}\in \overline{B}_{r}(0)\) such that \(G(z_{0})=0\).
The main result in this section is the following:
Lemma 3.2
For each \(0<\varepsilon <1\), there exists \(\lambda ^{*}>0\) such that the problem (3.1) has a positive weak solution for every \(\lambda _{0}, \Lambda _{0} \in (0,\lambda ^{*})\).
Proof
Let \(B=\{e_{1},e_{2},\ldots ,e_{m},\ldots \}\) be a Schauder basis of \(W_{0}^{1,2}(\Omega )\). For each , let
be the finite-dimensional space generated by \(\{e_{1},e_{2},\ldots ,e_{m}\}\). Note that the spaces \((W_{m},\Vert \cdot \Vert _{m})\) and are isometrically isomorphic by natural mapping
given by
where
Moreover,
In the rest of the text, we will identify \(u \in W_{m}\) with \(\xi \in \mathbb{R}^{m}\) via T isometry. For each , define the function such that
where ,
\(j=1,2,\ldots ,m\) and \(u=\displaystyle \sum _{j=1}^{m}\xi _{j}e_{j}\in W_{m}\). Therefore,
Note that
Moreover,
Using the Hölder inequality for \(\frac{3}{\gamma _{0}}\) and \(\frac{3-\gamma _{0}}{3}\), we have
Using (2.1), (3.5), and the Sobolev embedding, there exists positive constant \(C_{1}\) such that
It follows from (3.3), (3.4), and (3.6) that
for some \(\widetilde{C}>0\). Since \(0<\|h_{0}\|_{\infty}< \lambda _{0} < \Lambda _{0}\), we get
Using Hölder’s inequality with \(s,s'>1\) with s sufficiently close to 1, such that \(\dfrac{1}{s}+\dfrac{1}{s'}=1\), we get
Since \(q_{0}>2\) and \(s'>1\), by the Sobolev embedding, there exists \(\widetilde{C_{1}}>0\) such that
Then, it follows from (3.8) and (3.9) that
Assume now that \(\Vert u\Vert =r\) for some \(0< r<1\) to be chosen later. We have
and applying Theorem 2.1, we impose that
Therefore, there exists \(M>0\) such that
and hence,
Now, it is necessary to choose r such that
in others words,
Thus, considering \(r\leq \min \left \lbrace 1, \left [ \frac{2\pi (2-s\alpha _{0})}{\Upsilon s}\right ]^{1/2},\left ( \frac{(1/2-\delta C_{1})}{C_{\delta}\widetilde{C_{1}} M^{\frac{1}{s}}} \right )^{\frac{1}{q_{0}-2}}\right \rbrace \), we get
Furthermore, choosing
we obtain
By virtue of Lemma 3.1, for every , there exists with \(\vert y\vert _{s}\leq r<1\) such that \(G(y)=0\). Thus, from (3.2), there exists \(u_{m}\in W_{m}\) satisfying
such that
Multiplying equality (3.14) by any constant \(\sigma _{j}\), for each \(j=1,2,\ldots ,m\), and adding them, we conclude
which shows that \(u_{m}\) is an approximate weak solution to problem (3.1).
Since r is independent of m and \(W_{m}\subset W_{0}^{1,2}(\Omega )\), for all , then \((u_{m})\) is a bounded sequence in \(W_{0}^{1,2}(\Omega )\). Thus, for some subsequence, there exists \(u\in W_{0}^{1,2}(\Omega )\) such that
Fix and consider \(m\geq k\), then \(W_{k}\subset W_{m}\) and
Since \(\phi _{k}\in W_{k}\), note that
Using (3.16), we have
Therefore, we use [7, Theorem 4.2] to obtain that
Note that \(|\nabla u_{m}(x)|^{\gamma _{0}}\to |\nabla u(x)|^{\gamma _{0}}\) a.e in Ω. Moreover, \(|\nabla u_{m}|^{\gamma _{0}} \in L^{2/\gamma _{0}}(\Omega )\). Hence, from the Brezis-Lieb lemma [15, Lemma 4.8], we have that
for all \(\phi \in L^{2/(2-\gamma _{0})}(\Omega )\). In particular, for all \(\phi \in W_{k}\).
Now, since \(f_{0}\) is a continuous function, using (3.16), we have
Using (2.1), we get
We will need to prove that the function defined by
is convergent in \(L^{1}(\Omega )\). Indeed, we invoke (3.16) to obtain
and
Arguing as (3.5) and using (3.16), we conclude that
Furthermore, from (3.16), we get
Now, considering \(s,s'>1\) such that \(\dfrac{1}{s}+\dfrac{1}{s'}=1\), with s sufficiently close to 1, we use (3.16) and the fact that \(q_{0}>2\) to obtain
Moreover, by (3.12), we have
Hence, by (3.25), (3.26), and Hölder’s inequality, we get
We use (3.24), (3.27), and [15, Theorem 4.8] to conclude that
It follows from (3.28) that, for all \(\phi _{k} \in W_{k}\), we get
Therefore, by (3.23) and (3.29), we prove that
which shows the identity
Since is dense in \(W_{0}^{1,2}(\Omega )\), we have
Therefore, since \(\phi \in W_{0}^{1,2}(\Omega )\) is arbitrary, it follows from (3.30) that
for all \(\phi \in W_{0}^{1,2}(\Omega )\), which shows that u is a weak solution to problem (3.1).
Furthermore, \(u> 0\) in Ω. In fact, since \(f_{0}(t)=0\), \(\forall t<0\), we use \(\phi =u^{-}\) in (3.31) to obtain
which implies that \(u^{-}=0\) and then \(u=u^{+}\geq 0\). However, due to [3, Lemma 3.1], \(u\in C^{2}(\Omega )\cap C^{1}(\overline{\Omega})\) and the maximum principle, we get \(u>0\) in Ω. □
4 Proof of Theorem 1.1
For each , let \(\varepsilon =\dfrac{1}{n}\) and \(u_{\frac {1}{n}}=u_{n}\), where \(u_{n}\) is a solution of auxiliary problem (3.1)
obtained by Lemma 3.2. Note that, from \((f_{3})\), we get
Then,
Considering \(v\in W^{1,2}_{0}(\Omega )\) the unique positive solution to the problem
and using comparison principle, we conclude that
which implies that \(u_{n}(x)\nrightarrow 0\), for each \(x\in \Omega \).
Now, from (3.16), we get
and it follows from (3.13) that
Therefore, r does not depend on n, which shows that \((u_{n})\) is a bounded sequence in \(W^{1,2}_{0}(\Omega )\). Thus, since \(W_{0}^{1,2}(\Omega )\) is a reflective Banach space, for some subsequence, there exists \(u\in W_{0}^{1,2}(\Omega )\) such that
Recall from (3.31) that
By same computation in (3.19) and (3.29), we obtain
and
Note that, from (4.3), we get
Since \(v\in C^{1}(\overline{\Omega})\) and Ω is a smooth bounded domain in \(\mathbb{R}^{N}\), by the maximum principle that \(u_{n}(x)\geq v(x)>Cd(x)>0\), where \(d(x)=dist(x,\partial \Omega )\), and C is a positive constant that does not depend on x. Thus,
Hence, we invoke Theorem 2.2 to obtain \(\left \vert \frac{\phi}{Cd(x)^{\beta _{0}}} \right \vert \in L^{r}( \Omega )\). Therefore, by (4.7) and [7, Theorem 4.2], we get
Letting \(n\rightarrow +\infty \) in (4.4), we use (4.5), (4.6), and (4.8) to conclude that
which proves that \(u\in W_{0}^{1,2}(\Omega )\) is a weak solution to problem (1.1).
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This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-RP23014).
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Figueiredo wrote the main manuscript text and Baraket and Ben Ghorbal reviewed the manuscript.
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Baraket, S., Ben Ghorbal, A. & Figueiredo, G.M. Existence of positive solutions for a class of singular elliptic problems with convection term and critical exponential growth. Bound Value Probl 2024, 91 (2024). https://doi.org/10.1186/s13661-024-01897-5
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DOI: https://doi.org/10.1186/s13661-024-01897-5