Elliptic equations in dimension 2 with double exponential nonlinearities

A boundary value problem on the unit disk in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R^2$$\end{document} is considered, involving an elliptic operator with a singular weight of logarithmic type and nonlinearities which are subcritical or critical with respect to the associated gradient norm. The existence of non-trivial solutions is proved, relying on variational methods. In the critical case, the associated energy functional is non-compact. A suitable asymptotic condition allows to avoid the non-compactness levels of the functional.


Introduction
In this article we study the solvability of problems of the form where B is the unitary disk in R 2 , and the function f (|x|, s) has a maximal growth in s with respect to the weighted gradient norm, where the radial positive weight w(x) is of logarithmic type and will be specified below.
In some recent papers, the influence of weights on limiting inequalities of Trudinger-Moser type (for short, TM-inequalities) has been explored in some detail. In [3][4][5][6], the interest is devoted in particular to the impact of weights in the Sobolev norm.
More precisely, let Ω ⊂ R N be a bounded domain. If w ∈ L 1 (Ω) is a non-negative function, we introduce the weighted Sobolev space A general embedding theory for such weighted Sobolev spaces has been developed in Kufner [16]. It turns out that for weighted Sobolev spaces of form (1.2) logarithmic weights have a particular significance, since they concern limiting situations of such embeddings. However, to obtain interesting results, one needs to restrict attention to radial functions. So let us consider the subspace of radial functions, i.e. On the space of radial functions H 1 0,rad (B, w), the weight has a relevant impact on the corresponding embedding inequalities. In fact, the well-known Trudinger-Moser growth e u 2 is drastically increased: in [4] the following double exponential inequality was proved.
Theorem A. ) Let w be given by (1.3).
We remark that if we consider the supremum in (1.5) on the whole Sobolev space H 1 0 (B), then the weight w has no effect and we remain with the standard Trudinger-Moser growth (see [19,20,22]). Indeed, as a consequence of the standard TM-inequality and [4, Proposition 8], we have: Returning to equation ( Therefore the choice of β > 0 in the previous formulas has no influence. For convenience, we choose β = 2. Problems as (1.1) in the non-weighted case have been widely studied by several authors, see. e.g. [1,2,[7][8][9][12][13][14]18,21]. To our knowledge, the case with weighted Sobolev norm has not been considered, except for some particular applications of the above inequalities which can be found in [4,6].
More precisely, in this paper we consider nonlinearities which have subcritical and critical growth in the sense of (1.6) and (1.7). We make the following assumptions throughout this paper: (E1) f : B × R → R is continuous, radial in x, and f (x, t) = 0 for t ≤ 0 (E2) There exist t 0 > 0 and M > 0 such that Remark 2. Apparently, Remark 1 seems to downplay the importance of the critical exponent β 0 = 2. However, its crucial role is restored by assumption (E5) which is the counterpart, in our weighted framework, of the growth condition [9, (H7)].
Then f satisfies the hypotheses of Theorem 1.2, with α 0 = 1 and γ 0 > 1 e 2 . The proof in both cases uses a variational approach and follows the schemes of [9]. In order to apply the classical Mountain Pass theorem of Ambrosetti-Rabinowitz, we first need to prove some geometric estimates. Theorem 1.1 then follows in a standard way, since we can prove compactness due to the subcritical growth. The proof of Theorem 1.2 is more delicate, since compactness is lost due to the critical growth of the nonlinearity. First, we show that the non-compactness levels of the functional are "quantized", i.e. they occur at discrete levels, and we determine the first such level. Then we prove, using a logarithmic concentrating sequence (Moser sequence) that the mountain-pass level of the functional avoids this non-compactness level, which requires new and delicate estimates. In this step the crucial assumption (E5) is used.
We recall that in [10] and [11] (see also [15]) existence of solutions for elliptic equations with TM type nonlinearities were proved in the (slightly) supercritical regime, using the Lyapunov-Schmidt reduction method. We expect that with such methods solutions may be found also for the double exponential equation (1.1) in non-symmetric domains Ω ⊂ R 2 , near the non-compactness level.

Preliminaries on the variational formulation
Let us consider the space and note that · comes from the inner product Let Φ : H → R be the functional defined by This functional is of class C 1 , since the hypothesis on the growth of f (subcritical or critical growth) ensures the existence of positive constants c and C such that First, we prove that the functional Φ has a mountain-pass geometry.
To complete the geometric requirements of the Mountain Pass Theorem we need the following Proposition: From (E1) and (E2), by integration, there exists a constant C such that In particular, for p > 2, there exists C such that Therefore and the result easily follows.

The compactness levels
In this section we prove a compactness result (Proposition 3.2). We first need a Lions' type result [17] concerning an improved TM-inequality for weakly convergent sequences, adapted to the double exponential case.
Proof. First, we recall the following elementary inequality which is a direct consequence of Young's inequality. In fact, Also, we may estimate Therefore, for any p > 1, we have for some ε > 0 and q > 1. We assume u < 1, the proof in the case u = 1 is similar. When u < 1, for and for any ε > 0 there exists k = k(ε) ≥ 1 such that Then, for q = 1 + ε with ε := 3 √ 1 + δ − 1 > 0, and for any k ≥ k and this yields (3.3) in the case u < 1.
Next, we identify the first non-compactness level of the functional Φ.

Proposition 3.2. The functional Φ satisfies the Palais-Smale condition at level c, i.e. the (P S) c -condition, for any
and where ε n → 0, as n → +∞. From (E2) it follows that for any ε > 0 there exists t ε > 0 such that Then, for any ε > 0, where we also used (3.5) in the first inequality and (3.6) in the last one. Therefore so that {u n } is bounded in H, and hence u n → u weakly in H and strongly in L q (B) for any q ≥ 1. Moreover, from (3.5), (3.6) and (3.8) and applying Lemma 2.1 of [9] f (x, u n ) → f (x, u), in L 1 (B) as n → +∞, (3.8) we may apply the generalized Lebesgue dominated convergence Theorem to conclude that and u is a weak (and strong, via standard regularity results) solution of the problem. In particular, testing the equation with v = u, We will distinguish three cases according to Case (I) c = 0. We prove that u n → u strongly in H. Indeed, f (x, u n )u n dx = 2c We show that this case cannot happen.
We begin by assuming that there exists q > 1 such that where q and q are conjugate exponents with q > 1. The strong convergence u n → 0 in H leads to c = 0, which contradicts the assumption c > 0.
We have to estimate this second integral.
Case (III) c > 0 and u = 0. We will prove that Φ(u) = c and this yields To prove that Φ(u) = c, note that We argue by contradiction assuming that Φ(u) < c, or equivalently Let v n = un un and v = u Since v n v weakly in H, v = 0, v n = 1 and v < 1, we may apply the Lions-type Lemma (Lemma 3.1) to obtain sup n B e 2e p2πv 2 n dx < +∞, for 1 < p < 1 1 − v 2 Next, we estimate the L q -norm of {f (x, u n )} with q > 1. We have, using (3.13) as above, Elliptic equations in dimension 2 Page 11 of 18 29 provided that α 0 (1 + ε) u n 2 < 2πp, for some 1 < p < 1 1− v 2 . Indeed, note that by the definition of v and, to obtain the desired estimate, it is enough to show that we can choose ε > 0 sufficiently small such that Since Φ(u) ≥ 0, and c < π α0 , it is indeed possible to choose ε > 0 such that (3.15) is valid.
From the boundedness of {f (x, u n )} in L q (B) for q > 1, we will deduce that u n → u strongly in H. In fact which contradicts (3.14).

Proof of Theorems 1.1 and 1.2.
The proof in the subcritical case of Theorem 1.1 follows easily from an application of the Mountain Pass Theorem, since in this case the functional satisfies the (P S) c -condition for all c ∈ R.
To conclude the proof of Theorem 1.2 we have to show that the mountainpass level c satisfies c < π α0 , and for this it is sufficient to show that there exists w ∈ H, with w = 1, such that To this aim, we consider the functions w k = w k (x) defined by means of the identity where {ψ k } k is the Moser-type sequence introduced in [4]. More precisely, is a normalized sequence which tends pointwise and weakly to zero in H [and likewise for ψ k (s)], and which blows up in 0 (at +∞, respectively). In the following Lemma 4.1 we calculate the limit in (ii), which will be crucial in the proof of Theorem 1.2. It also shows that the integral in (ii) (and hence in (1.5)) is not weakly continuous in 0, since in w = 0 the value of the integral in (ii) is πe 2 , while the following Lemma 4.1 shows that the limit in (ii) is 3πe 2 . Proof. Performing the changes of variable s = 1 + t and j = k + 1, we may rewrite Next, we will study the limit of the integrals on [ √ j, j− √ j] and [j− √ j, j], respectively. To this aim, we begin by computing and we observe that η j ( j) ≤ − j, ∀j ≥ 2 4 (4.4) Also, and we conclude that for any ε ∈ (0, 1) there exists j ε ≥ 1 such that η j (j − j) ≤ −2(1 − ε) j, ∀j ≥ j ε (4.5) Let us fix j ≥ 1 and assume j is sufficiently large. A qualitative study of η j on [1, +∞) shows that there exists a unique s j ∈ (1, j) such that η j (s j ) = 0 and hence