Abstract
In this paper, we are concerned with the problem
where \(\Omega \subset {\mathbb {R}}^{2}\) is a bounded smooth domain and \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a superlinear continuous function with critical exponential growth. We first make a truncation on the prescribed mean curvature operator and obtain an auxiliary problem. Next, we show the existence of positive solutions of this auxiliary problem by using the Nehari manifold method. Finally, we conclude that the solution of the auxiliary problem is a solution of the original problem by using the Moser iteration method and Stampacchia’s estimates.
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1 Introduction
In this paper, we are concerned with the existence of positive solutions for the following prescribed mean curvature problem with Dirichlet boundary condition
where \(\Omega \subset {\mathbb {R}}^{2}\) is a bounded smooth domain. The function f is given, and we seek a solution u satisfying (P).
Since the left-hand side is the mean curvature of the graph of u, problem (P) is a prescribed mean curvature equation whose prescription depends on the location of the graph. Problems of this type have been studied starting with the pioneering contributions of Gethardt [15] and Miranda [25] who constructed \(H^{1,1}\) solutions, respectively BV solutions of the prescribed mean curvature equation with Dirichlet boundary condition. We also refer to the seminal paper by [18], where there are established necessary and sufficient conditions for the existence of solutions in a particular case but without boundary conditions. Moreover, prescribed mean curvature equation has been the object of extensive studies in the past due to arises from some problems associated with differential geometry and physics such as combustible gas dynamics [4,5,6,7, 10, 14, 19, 31] and also due to the close connection with the capillary problem. For example, radial solution of (P) in \({\mathbb {R}}^{N}\) when f(u) is replaced by \(\kappa u\) has been studied in the context of the analysis of capillary surfaces, as can be seen in [9, 13, 16, 20, 22, 28] and [32].
Recently, by using variational methods, Obersnel and Omari [29] have considered the existence and multiplicity of positive solutions to problem (P) with respect to the behavior of the nonlinearity near the origin and at infinity. In the references of [29], the reader will find different contributions to the study of the prescribed mean curvature equation.
To state our main result, we need some hypotheses. The hypotheses on the continuous function f are the following:
- \({(f_{1}})\):
-
There exists \(\alpha _0 > 0\) such that the function f(t) satisfies
$$\begin{aligned}\displaystyle \lim _{t \rightarrow \infty }\frac{f(t)}{\exp (\alpha |t|^{2})} = 0 \,\,\, \text{ for } \,\,\, \alpha > \alpha _{0} \ \ \text{ and } \ \ \displaystyle \lim _{t \rightarrow \infty }\frac{f(t)}{\exp (\alpha |t|^{2})} = \infty \,\,\, \text{ for } \,\,\, \alpha < \alpha _{0}. \end{aligned}$$ - \({(f_{2}})\):
-
The following limit holds:
$$\begin{aligned}\displaystyle \lim _{t \rightarrow 0^+} \frac{f(t)}{ t} =0.\end{aligned}$$Moreover, \(f(t)=0\) for all \(t\le 0\).
- \({(f_{3}})\):
-
The function \(t \mapsto \displaystyle \frac{f(t)}{t}\) is increasing in \((0, +\infty )\).
- \({(f_{4}})\):
-
There exist \(r>{\frac{32}{7}\sqrt{2}}\) and \(\tau >\tau ^*\) such that
$$\begin{aligned} f(t) \ge \tau t^{r-1}, \end{aligned}$$for all \(t \ge 0\), where
$$\begin{aligned} \tau >\tau ^* := \max \biggl \{\biggl [\frac{16r\sqrt{2}}{{7r-32\sqrt{2}}}\frac{c_r \alpha _0}{\pi }\biggl ]^{r-2/2}, \frac{K_2}{\delta } \biggl [\frac{32r\sqrt{2}}{{7r-32\sqrt{2}}}c_r\biggl ]^{r-2/2}, {1}\biggl \} \end{aligned}$$and the constant \(c_r\) will appear in the Sect. 4, \(K_2>0\) will appear in Lemma 5.1, and \(\delta >0\) will appear in (3.4).
- \({(f_{5}})\):
-
The following inequality holds:
$$\begin{aligned} 0< r F(t)\le f(t)t , \end{aligned}$$for all \(t > 0\), where \(F(t):=\int _0^tf(s)\mathrm{d}s\).
The main result of this paper establishes the following existence and regularity property.
Theorem 1.1
Assume that conditions \((f_{1})- (f_{5})\) hold. Then, problem (P) has a positive solution \(u \in C^{1}(\overline{\Omega })\).
Hypothesis \({(f_{1}})\) is closely related to the Trudinger–Moser inequality and establishes that the function f has an exponential critical growth in \({{\mathbb {R}}}^2\).
We would like to highlight that our theorem can be applied for the model nonlinearity
where \(\tau\) and r are the constants in \((f_4)\) and \(\alpha _0\) is the constant in \((f_1)\).
Nonlinear problems with exponential growth have been considered recently by Alves and de Freitas [1], Alves and Santos [2], Ambrosio [3], Figueiredo and Severo [12], Li, Santos and Yang [23], Medeiros, Severo and Silva [24], etc.
There are some recent papers to prescribe mean curvature problem in two-dimensional case. In [27] the authors studied the prescribed mean curvature problem with nonhomogeneous boundary condition. More precisely, the authors investigate the boundary behavior of variational solutions of problem (P) at smooth boundary points where certain boundary curvature conditions are satisfied. In [11] the authors show a nonexistence result. To the best of our knowledge, the main result in this paper is the first work on the problem of medium curvature in dimension two and non-linearity with critical exponential growth.
The plan of the paper is as follows. We first associate to problem (P) a related nonhomogeneous auxiliary problem with Dirichlet boundary condition. In Sect. 3 we study the variational structure of this auxiliary nonlinear problem and we establish several qualitative properties of the associated energy functional. The key abstract tools in these arguments are the Trudinger–Moser inequality and the Nehari manifold method. Next, minimizing the energy function on the Nehari manifold, we prove the existence of solutions to the auxiliary problem. In the final section of this paper, we prove that the solution of the auxiliary problem is a solution of the original problem. This is essentially done by using the Moser iteration method and Stampacchia’s estimates. We refer to the recent monograph by Papageorgiou, Rădulescu and Repovš [30] for some of the abstract methods used in this paper.
2 An auxiliary problem
Consider the following auxiliary problem
where
Lemma 2.1
The function \(a:{\mathbb {R}}_{+}\rightarrow {\mathbb {R}}_{+}\) is decreasing and of \(C^{1}\) class. Moreover, it satisfies the following conditions:
- \((a_{1})\) :
-
\(\displaystyle \frac{7}{8\sqrt{2}}\le a(s^{2}) \le 1, \ \ \text{ for } \text{ all } \ \ s\ge 0.\)
- \((a_{2})\) :
-
\(\displaystyle a'(s)s\le 0< a(s), \ \ \text{ for } \text{ all } \ \ s\ge 0.\)
- \((a_{3})\) :
-
The function
$$\begin{aligned} s\mapsto A(s^{2}) \ \ \text{ is } \text{ convex } \text{ for } \ \ s\ge 0, \end{aligned}$$where \(A(s)=\displaystyle \int ^{s}_{0}a(t)\mathrm{d}t\).
Proof
Since
the items \((a_1)\) and \((a_2)\) follow by straightforward computation. For item \((a_3)\) note that
If we define \(b(s):=a(s)+2s a'(s)\), we can prove that b is strictly decreasing in \([0,\frac{6}{5}]\), strictly increasing in \([\frac{6}{5}, 2]\) and constant in \([2, +\infty )\). Then,
for all \(s \ge 0\) and this completes the proof. \(\square\)
In this section, we prove some auxiliary results which will be very useful throughout the paper.
Lemma 2.2
If \((a_{1})-(a_2)\) are true, then:
- (i):
-
The function \(s\mapsto a(s^2) s\) is increasing.
- (ii):
-
For all \(x,y \in {\mathbb {R}}^{2}\), we have
$$\begin{aligned} \langle a(|x|^{2})x- a(|y|^{2})y, x-y\rangle \ge \frac{7}{8\sqrt{2}} |x-y|^{2}. \end{aligned}$$(2.1)
Proof
In order to prove (i), note that using \((a_{1})-(a_2)\), we get
Let us prove (ii). Firstly, note that for \(z\in {\mathbb {R}}^{2}\), we have
where we have denoted \(\delta _{ij}\) by the Kronecker delta. Hence, for all \(z, \xi \in {\mathbb {R}}^{2}\) we get
Since
we have
Thus, using (2.2) we deduce that
From \((a_2)\), we have that
Hence,
Now for \(z=y+t(x-y)\), \(t \in [0,1]\) and \(\xi =x-y\), we have
Finally, using (2.4) we get (2.1). \(\square\)
The next result due to Stampacchia [33] will be useful in the arguments used in this paper.
Lemma 2.3
Let \(B(\eta )\) be a given \(C^1\) vector field in \({\mathbb {R}}^2\) and f(x, s) a bounded Carathéodory function in \(\Omega \times {\mathbb {R}}\). Let \(u \in H^1_0(\Omega )\) be a solution of
that is,
Assume that there exists \(0< \nu < K\) such that
for all \(i,j = 1, 2\) and \(\xi \in {\mathbb {R}}^2\). Then, \(u \in W^{2,p}(\Omega ) \cap C^{1,\alpha }(\overline{\Omega })\), for all \(\alpha \in (0,1)\) and for all \(p \in (1,\infty )\). Moreover,
In the following result, we show that the differential operator involved in (Aux) verifies conditions (2.5).
Lemma 2.4
Assume that hypotheses \((a_{1})-(a_2)\) are fulfilled. Then, for all \(u \in H^{1}_{0}(\Omega )\), the second-order differential operator \(B(\nabla u)= a(|\nabla u|^{2})\nabla u\) satisfies (2.5) of Lemma 2.3.
Proof
Note that
and then
Now, observing (2.2), we can repeat the reasoning of the proof of Lemma 2.2, and using \((a_{1})-(a_2)\), we conclude that
On the other hand, using \((a_{1})-(a_2)\), we get
for some positive constant \(K>0\). The proof is now complete. \(\square\)
3 The variational framework and some technical lemmas
Note that, by the hypothesis \((a_{1})\), we have that the functional \(I: H_0^{1}(\Omega )\rightarrow {\mathbb {R}}\) given by
is well defined, where \(F(t)=\displaystyle \int ^{t}_{0}f(s) \mathrm{d}s\).
Moreover, we have
for all \(\phi \in H_0^{1}(\Omega )\). Thus, I is a \({{\mathcal {C}}}^1\) functional on \(H_0^{1}(\Omega )\) and its critical points are weak solution of problem (Aux).
The Nehari manifold associated to the functional I is given by
where \(J(u)=I'(u)u\) for \(u \in H^{1}_{0}(\Omega )\). Let us start with the following important result due to Trudinger [34] and Moser [26].
Theorem 3.1
For every \(u \in H^{1}_{0}(\Omega )\) and \(\alpha >0\)
and there is a constant \(M>0\) such that
for every \(\alpha \le 4\pi .\)
Moreover, if \(\alpha > 4\pi\), then
Note that, from \((f_{2})\), for any \(\varepsilon >0\), there exists \(\delta >0\) such that
and
for all \(0< t\le \delta\).
Furthermore, from \((f_{1})\), given \(\alpha >\alpha _0\), there exists \(K>0\) such that
for all \(t\ge K\). In particular, we get
with implies
where \(C_{\varepsilon }=\frac{\varepsilon }{K^{q-2}}\). Moreover, from (3.6), we get
for all \(t\ge K\), for all \(\alpha >\alpha _0\), for all \(q\ge 0\) with \(\tilde{C_{\varepsilon }}= \frac{\varepsilon }{\alpha _0 K^{q+1}}\).
Consequently, using (3.4), (3.5), (3.7) and (3.8), for all \(\varepsilon >0\) and for all \(\alpha >\alpha _0\), there exists \(C_{\varepsilon }>0\) such that
and
for all \(u \in H^{1}_{0}(\Omega )\). In particular, in this paper, we will use \(q>2\).
In the next result, we prove that \(\mathcal {N}\) is not empty and that I restricted to \(\mathcal {N}\) is bounded from below.
Lemma 3.1
For each \(u \in H^{1}_{0}(\Omega ) \setminus \{0\}\), there exists a unique \(t>0\) such that \(tu \in \mathcal {N}\). Moreover, \(I(u)> 0\) for every \(u \in \mathcal {N}\).
Proof
Given \(u \in H^{1}_{0}(\Omega ) \setminus \{0\}\), let \(\mathcal {T}_u(t)=I(tu)\) for \(t>0\). Then, \(tu \in \mathcal {N}\) if and only if \(\mathcal {T}_{u}'(t)=0\). Note that, taking \(\varepsilon >0\) sufficiently small in (3.9) and using \((a_{1})\) and Sobolev embedding, there exists \(C>0\) such that
Using Hölder’s inequality with \(s',s>1\), we get
Choosing \(\alpha > \alpha _0\) and \(t_1>0\) such that \(\alpha s\Vert t_1 u\Vert ^{2}< 4\pi\), using (3.2) we obtain
for some \(D_1,D_2>0\) and for all \(0\le t\le t_1\). Thus, since \(2<q\), there exists \(0<t^{*}\le t_1\) such that \(\mathcal {T}_u(t)>0\) for all \(0<t< t^{*}\le t_1\).
Now, from \((a_1)\) and \((f_{4})\), we have
where \(\Omega ^{+}_{u}=\{x\in \Omega : u(x)>0\}\). Therefore, since \(r>2\), we conclude \(\displaystyle \lim _{t \rightarrow +\infty } \mathcal {T}_u(t)=- \infty\). Consequently, there exists at least one \(t(u)>0\) such that \(\mathcal {T}_u'(t(u))=0\), that is, \(t(u)u \in \mathcal {N}\).
Suppose, by contradiction, that there are \(t>0\) and \(\widetilde{t}>0\) such that
and
Then,
If \(t>\widetilde{t}\), from Lemma 2.1 and \((f_{3})\), we have
which is a contradiction. In the same way, we obtain that we cannot have the case \(t<\widetilde{t}\). We conclude that there is a unique parameter \(t>0\) such that \(t(u)u \in \mathcal {N}\). Note, in particular, that t(u) is a global maximum point of \(\mathcal {T}_u\) and \(\mathcal {T}_u(t(u))>0\), i.e. \(I(t(u)u)>0\). Since \(t(u)=1\) if \(u \in \mathcal {N}\), we deduce that \(I(u)>0\) for every \(u \in \mathcal {N}\). \(\square\)
In the next result we prove that sequences in \(\mathcal {N}\) cannot converge to 0.
Lemma 3.2
There exists a constant \(C>0\) such that \(0<C\le \Vert u\Vert\), for every \(u \in \mathcal {N}\).
Proof
Suppose, by contradiction, that there is \((u_n) \subset \mathcal {N}\) such that
Then, using (3.10), we have
Now, from \((a_1)\) we get
Using Sobolev embedding, there exists \(C>0\) such that
Using Hölder’s inequality with \(s',s>1\), we have
Note that by (3.11), there is \(n_0 \in \mathbb {N}\) such that
for all \(n \ge n_0\) and for some \(\alpha >\alpha _0\). Then, from (3.2) and Sobolev embedding again, we have
This inequality implies
Since \(q>2\), the above inequality contradicts (3.11) and the lemma is proved. \(\square\)
We set \(c:=\inf _{\mathcal {N}}I\), and in the next result we will prove that minimizing sequence for c are bounded.
Lemma 3.3
If \((u_n) \subset \mathcal {N}\) is a minimizing sequence for c, then \((u_n)\) is bounded.
Proof
Suppose, by contradiction, that up to a subsequence, \(\Vert u_{n}\Vert \rightarrow \infty\) and consider \(v_{n}=\frac{u_{n}}{\Vert u_{n}\Vert }\rightharpoonup v_{0}\). If \(v_{0}=0\), then for all \(t>0\), from \((a_1)\) we obtain
where \(o_{n}(1)\rightarrow 0\) as \(n\rightarrow +\infty\). Since \(v_n \rightarrow 0\) in \(L^{q}(\Omega )\) and \(\Vert v_n\Vert < 4\pi\), using (3.10), Hölder inequality as in (3.12) and Theorem 3.1, we get
But this last convergence implies
which is a contradiction.
Suppose now that \(v_{0}\ne 0\). Then,
where \(o_{n}(1)\rightarrow 0\) as \(n\rightarrow +\infty\). Hence, using \((a_1)\) and \((f_4)\), we get
Since \(v_n \rightarrow v\) in \(L^{r}(\Omega )\) and \(\Vert u_n\Vert \rightarrow +\infty\), we have a contradiction. \(\square\)
To end up this section, let us prove that if the minimum of I over \(\mathcal {N}\) is achieved in some \(u_0 \in \mathcal {N}\), then in fact \(u_0\) is a critical point of I. This follows from some arguments used in [8, Proof of Theorem 1.1 (completed)].
Lemma 3.4
If \(u_0 \in \mathcal {N}\) is such that
then \(I'(u_0) = 0\).
Proof
Suppose, by contradiction, that c is achieved by \(u_0\) and this one is not a critical point of I. Then, there exists \(\phi \in H^{1}_{0}(\Omega )\) such that
By the continuity of \(I'\), we can find \(\widetilde{\varepsilon }, \widetilde{\delta } >0\) small such that
Moreover, since the map \(t\mapsto I(tu_0)\) attains its maximum at \(t=1\) as shown in the proof of Lemma 3.1, we have
Then, again by the continuity of \(I'\), there exists \(\overline{\sigma } \in (0,\widetilde{\delta })\) such that
i.e. \(\mathcal {T}_{u+\overline{\sigma }\phi }^{\prime}(1-\widetilde{\varepsilon })>0>\mathcal {T}_{u+\overline{\sigma }\phi }'(1+\widetilde{\varepsilon })\). It follows that
From (3.13), we have
so that
which contradicts (3.14) [also, because \(I(u_0)=\min _{\mathcal {N}}I\)]. Therefore, \(I'(u_0)=0\) and the proof is complete. \(\square\)
4 Existence of solution to the auxiliary problem
In this section, in order to prove the existence of result in the exponential critical case, we consider the auxiliary problem
where r is the constant that appear in the hypothesis \((f_4)\).
The energy functional associated to problem (A) is defined by
We also define the Nehari manifold
Since the embedding \(H^{1}_{0}(\Omega )\hookrightarrow L^{r}(\Omega )\) is compact, using the mountain pass theorem and the classical maximum principle, we can prove that there exists a positive solution to problems (A) given by \(w_r \in H^{1}_{0}(\Omega )\) such that
and
where \(c_r=\displaystyle \inf _{\mathcal {N}_r}I_r\). The next result is an estimate to \(c=\displaystyle \inf _{\mathcal {N}}I\).
Lemma 4.1
The value \(c=\displaystyle \inf _{\mathcal {N}}I\) satisfies
Proof
Note that, by the hypotheses \((a_1)\) and \((f_4)\), we have
This inequality implies that \(I'(w_r)w_r \le 0\). Then, from \((f_3)\), there exists \(\beta \in (0,1{]}\) such that \(\beta w_r \in \mathcal {N}\). Using \((a_1)\) and \((f_4)\) again, we obtain
Since \(I'_r(w_r)=0\), we conclude that
Using (4.1), we have
The proof is now complete. \(\square\)
Lemma 4.2
If \((u_n)\subset \mathcal {N}\) is a minimizing sequence for c, then
Proof
Note that
From \((a_1)\) and \((f_5)\), we get
Since \(r> \frac{32\sqrt{2}}{7}\), we obtain
By the estimate on c in Lemma 4.1, we find
Since \(\tau > \tau ^{*}\) in \((f_4)\), then
and the result follows. \(\square\)
The next result establishes some compactness properties of minimizing sequences.
Lemma 4.3
If \((u_{n})\subset \mathcal {N}\) is a minimizing sequence for c, then there exists \(u\in H^{1}_{0}(\Omega )\) such that
and
Proof
It is enough to prove the first limit, since the second one is analogous. By Lemma 4.2, we have
and, up to a subsequence, then there exists \(u\in H^{1}_{0}(\Omega )\) such that
and
Now it is sufficient to prove that there is \(g:{\mathbb {R}} \rightarrow {\mathbb {R}}\) such that \(|f(u_n)u_n|\le g(u_n)\) with \((g(u_n))\) convergent in \(L^{1}(\Omega )\).
Note that by the inequality (3.9) we have
We will prove that \((g(u_n))\) is convergent in \(L^{1}(\Omega )\). First, note that
Considering \(s,s'>1\) such that \(\frac{1}{s} + {\frac{1}{s'}} =1\) and s close to 1, we get
Now choosing \(\alpha >\alpha _0\) but close to 1, we have that
Using Lemma 4.2, we can conclude that
Then, it follows by Theorem 3.1 that there is \(M>0\) such that
Since
we use [21, Lemma 4.8] and conclude that
Now using (4.4), (4.5) and [21, Lemma 4.8] again, we conclude
The proof is complete. \(\square\)
Theorem 4.1
The auxiliary problem has a nonnegative solution \(v_0 \in H^{1}_{0}(\Omega )\).
Proof
Consider \((u_{n}) \subset \mathcal {N}\) a minimizing sequence for c. Then, by Lemma 3.3, \((u_{n})\) is bounded in \(H^{1}_{0}(\Omega )\) and, up to a subsequence,
We claim that \(u_0 \not \equiv 0\). Indeed, if \(u_0 \equiv 0\), then, from \((a_{1})\) and Lemma 4.3, we get
which implies
contradicting Lemma 3.2. Let \(t_{0}>0\) such that \(v_{0}:=t_{0}u_{0} \in \mathcal {N}\). Since by \((a_3)\) the function \(s\mapsto A(s^{2})\) is convex, we get \(\displaystyle \int _{\Omega }A(|\nabla tu_0|^{2}) \mathrm{d}x\le \displaystyle \liminf _{n\rightarrow \infty }\displaystyle \int _{\Omega }A(|\nabla tu_n|^{2}) \mathrm{d}x\). From Lemma 4.3, we infer that \(\displaystyle \int _{\Omega }F(tu_0) \mathrm{d}x= \displaystyle \lim _{n\rightarrow \infty }\displaystyle \int _{\Omega }F(tu_n) \mathrm{d}x\). Then,
From Lemma 3.1, we conclude that
The equality \(I'(v_0)=0\) is a consequence of Lemma 3.4. Since \(f(t)=0\), for all \(t\le 0\), we get \(v_0\ge 0\) in \(\Omega\). \(\square\)
5 Proof of Theorem 1.1
We first establish some estimates on solutions of the auxiliary problem from which the existence of positive solution of problem (P) will be deduced. Let us point out that the classical elliptic regularity theory [17] cannot be applied immediately because the coefficients in the differential operator are not necessarily continuous. Throughout this section, we assume that \(\gamma =7/(8\sqrt{2})\) and \(\Gamma =1\).
Lemma 5.1
If \(v_0 \in H^{1}_{0}(\Omega )\) is a solution of the auxiliary problem, then \(v_0 \in L^{\infty }(\Omega )\) and there exists \(K_2>0\) not depending on \(v_0\) such that
Proof
Since \(I(v_0)=c\) and \(I'(v_0)=0\), arguing as Lemma 4.2, we have
Considering \(\tau > \tau ^{*}\) in \((f_4)\), then
In what follows, let \(R>R_{1}>0\) with \(R>1\) and take a cutoff function \(\eta _{R} \in C^{\infty }_{0}(\Omega )\) such that \(0 \le \eta _R \le 1\), \(\eta _{R} \equiv 0\) in \(B_{R}^{c}\), \(\eta _R \equiv 1\) in \(B_{R_{1}}\) and \(|\nabla \eta _{R}|\le C/R\), where \(B_{R}(0)\subset \Omega\) is a ball in \({\mathbb {R}}^2\) and \(C>0\) is a constant.
Define for \(L > 0\),
with \(\sigma > 1\) to be determined later. Taking \(z_{L,0}\) as a test function we obtain
In other words,
By (3.9), (5.2) and Theorem 3.1, we obtain
Using \(z_{L,0}\) and \((a_1)\), we obtain
The definition of \(v_{L,0}\) implies
Thus, by \((a_1)\) again
Taking \(\widetilde{\tau }>0\) and using Young’s inequality, we obtain
Choosing \(\widetilde{\tau }\) sufficient small, it follows that
On the other hand, we get
where \(S_\Upsilon\) is the best Sobolev constant of \(H^{1}_{0}(\Omega )\) in \(L^{\Upsilon }(\Omega )\) and \(\Upsilon .1\) that will fix after. But
and therefore,
From this estimate and (5.3),
for every \(\sigma >1\).
The above expression, the properties of \(\eta _{R}\) and \(v_{L,0} \le v_{0}\) imply that
Taking
then we can apply Hölder’s inequality with exponents \(t/(t-1)\) and t in (5.5) to get
Since \(\eta _{R}\) is constant on \(B_{R_1} \cup B_{R}^c\) and \(|\nabla \eta _{R}| \le C/R\), we conclude that
We have used \(R>1\) and \(2t>q > 2\) in the last inequality.
Considering
we can use (5.7) and (5.8) to conclude that
Since
we can apply Fatou’s lemma in the variable L and Sobolev embedding to obtain
Here, \(C_7\) is a positive constant independent on R. Iterating this process, for each \(k\in \mathbb {N}\), it follows that
Since \(\Omega\) can be covered by a finite number of balls \(B_{R_{1}}^j\), we have that
Using (5.1) and since \(\sigma >1\), we let \(k \rightarrow \infty\) to get \(K_{2}>0\) such that
The proof is now complete. \(\square\)
5.1 Proof of Theorem 1.1 completed
By Lemma (5.1) and \((f_4)\), we obtain
Now from (3.4), given \(\epsilon >0\), we get
Using Lemma 2.3, then \(v_{0} \in C^{1,\alpha }(\Omega )\), and for \(\epsilon >0\) sufficient small, we have
The proof of Theorem 1.1 is now complete. \(\square\)
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Acknowledgements
The authors thank a knowledgeable referee for very rigorous enlightening comments, which considerably improved the initial version of this paper. Giovany M. Figueiredo was partially supported by CNPq, Capes and Fap-DF - Brazil. The work of Vicenţiu D. Rădulescu was supported by a grant of the Romanian Ministry of Education and Research, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2020-0068, within PNCDI III.
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Figueiredo, G.M., Rădulescu, V.D. Positive solutions of the prescribed mean curvature equation with exponential critical growth. Annali di Matematica 200, 2213–2233 (2021). https://doi.org/10.1007/s10231-021-01077-7
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DOI: https://doi.org/10.1007/s10231-021-01077-7
Keywords
- Prescribed mean curvature problem
- Critical exponential growth
- Nehari manifold method
- Moser iterations
- Stampacchia estimates