Positive solutions of the prescribed mean curvature equation with exponential critical growth

In this paper, we are concerned with the problem -div∇u1+|∇u|2=f(u)inΩ,u=0on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\text{ div } \left( \displaystyle \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) = f(u) \ \text{ in } \ \Omega , \ \ u=0 \ \text{ on } \ \ \partial \Omega , \end{aligned}$$\end{document}where Ω⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^{2}$$\end{document} is a bounded smooth domain and f:R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:{\mathbb {R}}\rightarrow {\mathbb {R}}$$\end{document} 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.


Introduction
In this paper, we are concerned with the existence of positive solutions for the following prescribed mean curvature problem with Dirichlet boundary condition − div where Ω ⊂ 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 [4-7, 10, 14, 19, 31] and also due to the close connection with the capillary problem. For example, radial solution of (P) in ℝ N when f(u) is replaced by 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 0 > 0 such that the function f(t) satisfies (f 2 ) The following limit holds: Moreover, f (t) = 0 for all t ≤ 0.
(f 4 ) There exist r > 32 7 √ 2 and > * such that for all t ≥ 0 , where and the constant c r will appear in the Sect. 4, K 2 > 0 will appear in Lemma 5.1, and > 0 will appear in (3.4).
The following inequality holds: The main result of this paper establishes the following existence and regularity property.
Hypothesis (f 1 ) is closely related to the Trudinger-Moser inequality and establishes that the function f has an exponential critical growth in ℝ 2 .
We would like to highlight that our theorem can be applied for the model nonlinearity where and r are the constants in (f 4 ) and 0 is the constant in (f 1 ).
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.

An auxiliary problem
Consider the following auxiliary problem 1 3

Lemma 2.1
The function a ∶ ℝ + → ℝ + is decreasing and of C 1 class. Moreover, it satisfies the following conditions: 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) + 2sa � (s) , we can prove that b is strictly decreasing in [0, 6 5 ] , strictly increasing in [ 6 5 , 2] and constant in [2, +∞) . Then, for all s ≥ 0 and this completes the proof. ◻ In this section, we prove some auxiliary results which will be very useful throughout the paper.

3
Finally, using (2.4) we get (2.1). ◻ The next result due to Stampacchia [33] will be useful in the arguments used in this paper.

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. ◻

The variational framework and some technical lemmas
Note that, by the hypothesis (a 1 ) , we have that the functional Moreover, we have for all ∈ H 1 0 (Ω) . Thus, I is a C 1 functional on H 1 0 (Ω) 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 ∈ H 1 0 (Ω) . Let us start with the following important result due to Trudinger [34] and Moser [26]. Moreover, if > 4 , then Note that, from (f 2 ) , for any > 0 , there exists > 0 such that and for all 0 < t ≤ .
Furthermore, from (f 1 ) , given > 0 , there exists K > 0 such that for all t ≥ K . In particular, we get with implies where C = K q−2 . Moreover, from (3.6), we get for all t ≥ K , for all > 0 , for all q ≥ 0 with C = 0 K q+1 . Consequently, using (3.4), (3.5), (3.7) and (3.8), for all > 0 and for all > 0 , there exists C > 0 such that and for all u ∈ H 1 0 (Ω) . In particular, in this paper, we will use q > 2. In the next result, we prove that N is not empty and that I restricted to N is bounded from below.
Note that, taking > 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 > 0 and t 1 > 0 such that s‖t 1 u‖ 2 < 4 , using (3.2) we obtain Consequently, there exists at least one t(u) > 0 such that T � u (t(u)) = 0 , that is, t(u)u ∈ N . Suppose, by contradiction, that there are t > 0 and � t > 0 such that and Then, If t > � 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 < � t . We conclude that there is a unique parameter t > 0 such that t(u)u ∈ N . Note, in particular, that t(u) is a global maximum point of T u and T u (t(u)) > 0 , i.e. I(t(u)u) > 0 . Since t(u) = 1 if u ∈ N , we deduce that I(u) > 0 for every u ∈ N . ◻ In the next result we prove that sequences in N cannot converge to 0. Since q > 2 , the above inequality contradicts (3.11) and the lemma is proved. ◻ We set c ∶= inf N I , and in the next result we will prove that minimizing sequence for c are bounded.

Lemma 3.3 If (u n ) ⊂ N is a minimizing sequence for c, then (u n ) is bounded.
Proof Suppose, by contradiction, that up to a subsequence, ‖u n ‖ → ∞ and consider v n = u n ‖u n ‖ ⇀ v 0 . If v 0 = 0 , then for all t > 0 , from (a 1 ) we obtain where o n (1) → 0 as n → +∞ . Since v n → 0 in L q (Ω) and ‖v n ‖ < 4 , 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 ≠ 0 . Then, where o n (1) → 0 as n → +∞ . Hence, using (a 1 ) and (f 4 ) , we get Since v n → v in L r (Ω) and ‖u n ‖ → +∞ , we have a contradiction. 16 (1).

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 (Ω) ↪ L r (Ω) 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 ∈ H 1 0 (Ω) such that and where c r = inf N r I r . The next result is an estimate to c = inf N I.

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 = 7∕(8 √ 2) and Γ = 1. In what follows, let R > R 1 > 0 with R > 1 and take a cutoff function R ∈ C ∞ 0 (Ω) such that Define for L > 0, with > 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 � > 0 and using Young's inequality, we obtain Choosing ̃ sufficient small, it follows that On the other hand, we get where S Υ is the best Sobolev constant of H 1 0 (Ω) in L Υ (Ω) and Υ.1 that will fix after. But and therefore, From this estimate and (5.3), for every > 1.
The above expression, the properties of R and v L,0 ≤ 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 R is constant on B R 1 ∪ B c R and |∇ R | ≤ 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 ∈ ℕ , it follows that Since Ω can be covered by a finite number of balls B j R 1 , we have that Using (5.1) and since > 1 , we let k → ∞ to get K 2 > 0 such that The proof is now complete. ◻

Proof of Theorem 1.1 completed
By Lemma (5.1) and (f 4 ) , we obtain Now from (3.4), given > 0 , we get Using Lemma 2.3, then v 0 ∈ C 1, (Ω) , and for > 0 sufficient small, we have The proof of Theorem 1.1 is now complete. ◻