On a weighted elliptic equation of N-Kirchhoff type with double exponential growth

1 Introduction

In this article, we consider the following non-local weighted problem:

where B = B ( 0 , 1 ) is the unit open ball in R N , f ( x , t ) is a radial function with respect to x , and the weight σ ( x ) is given by

and g : R + → R + is a positive continuous function which will be specified later.

In 1883, Kirchhoff studied the following parabolic problem:

for free vibrations of elastic strings. The parameters in equation (3) have the following meanings: L is the length of the string, h is the area of cross-section, E is the Young’s modulus of the material, ρ is the mass density, and P 0 is the initial tension.

These kinds of problems have physical motivations. Indeed, the Kirchhoff operator G ∫ B ∣ ∇ u ∣ 2 d x Δ u also appears in the nonlinear vibration equation, namely,

which have received the attention of several researchers, mainly as a result of the work of Lions [1]. We mention that non-local problems also arise in other areas, e.g., biological systems where the function u describes a process that depends on the average of itself (e.g., population density), see, e.g., [2,3] and references therein.

In the non-weighted case, i.e., when σ ( x ) ≡ 1 and when N = 2 , problem (1) can be seen as a stationary version of the evolution problem (4).

Recently, Xiu et al. [4] studied the following singular nonlocal elliptic problem:

where x ∈ R N , M ( t ) = b ¯ + a ¯ t , a ¯ , b ¯ > 0 , a < N − p p , h ( x ) and H ( x ) are nonnegative function . They proved that this problem has infinitely many solutions by variational methods and the genus theorem.

In order to motivate our study, we begin by giving a brief survey on Trudinger-Moser inequalities. In the past few decades, Moser gives the famous result about the Trudinger-Moser inequality [5,6]; many applications take place as in conformal deformation theory on manifolds, the study of the prescribed Gauss curvature and mean field equations. After that, a logarithmic Trudinger-Moser inequality was used in a crucial way in [7] to study the Liouville equation of the form

where Ω is an open domain of R N , N ≥ 2 , and λ a positive parameter. Equation (5) has a long history and has been derived in the study of multiple condensate solution in the Chern-Simons-Higgs theory [8,9] and also it appeared in the study of Euler flow [10,11, 12,13].

Later, the Trudinger-Moser inequality was improved to weighted inequalities [14,15]. The influence of the weight in the Sobolev norm was studied as the compact embedding [16,17].

When the weight is of logarithmic type, Calanchi and Ruf [18] extended the Trudinger-Moser inequality and gave some applications when N = 2 and for prescribed nonlinearities. After that, Calanchi et al. [19] considered more general nonlinearities and proved the existence of radial solutions.

We point out that recently, in the case g ( t ) = 1 , Deng et al. [20] have proved the existence of a nontrivial solution for the following boundary value problem:

where B is the unit ball in R N , N ≥ 2 , the radial positive weight w ( x ) is of logarithmic type, the function f ( x , u ) is continuous in B × R and has critical double exponential growth.

Also recently, de Figueiredo and Severo [21] studied the following problem:

where Ω is a smooth-bounded domain in R 2 , the nonlinearity f behaves like exp ( α t 2 ) as t → + ∞ , for some α > 0 . The authors proved that this problem has a positive ground state solution. The existence result was proved by combining minimax techniques and Trudinger-Moser inequalities.

Inspired by the last two works, we investigate our problem by adapting weighted Sobolev space setting. We use the Trudinger-Moser inequality to study and prove the existence of solutions of (1).

Let Ω ⊂ R N be a bounded domain and σ ∈ L 1 ( Ω ) be a nonnegative function. We define the weighted Sobolev space as follows:

we will limit our attention to radial functions and then consider the subspace,

equipped with the norm

The choice of the weight and the space W = W 0 , rad 1 , N ( B , σ ) are motivated by the following exponential inequalities.

Let N ′ be the Hölder conjugate of N that is N ′ = N N − 1 ⋅ In view of inequalities (6) and (7), we say that f has subcritical growth at + ∞ , if

and f has critical growth at + ∞ , if there exists some α 0 > 0 , such that

In this article, we consider problem (1) with subcritical or critical growth nonlinearities f ( x , t ) . Furthermore, we suppose that f ( x , t ) satisfies the following hypotheses:

1. The nonlinearity f : B ¯ × R → R is positive, continuous, radial in x , and f ( x , t ) = 0 for t ≤ 0 .

2. There exist t 0 > 0 and M 0 > 0 , such that for all t > t 0 and for all x ∈ B , we have

   0 < F ( x , t ) ≤ M 0 f ( x , t ) ,

   where F ( x , t ) = ∫ 0 t f ( x , s ) d s .

3. For each x ∈ B , f ( x , t ) t 2 N − 1 is increasing for t > 0 .

4. In the critical case, there exists a constant γ 0 with γ 0 > 1 α 0 N − 1 e N g ω N − 1 α 0 N − 1 such that

   lim t → ∞ f ( x , t ) t exp ( N e α 0 ∣ t ∣ N ′ ) ≥ γ 0 uniformly in x ∈ B ¯ .

Also, we have that condition ( H 3 ) leads to

The condition asymptotic ( H 4 ) would be crucial to identify the minmax level of the energy associated with problem (1). We give an example of f . Let f ( t ) = F ′ ( t ) , with F ( t ) = t 2 N + 2 2 N + 2 + t τ exp ( N e α 0 t N ′ ) , with τ > 2 N . A simple calculation shows that f verifies conditions ( H 1 ) , ( H 2 ) , ( H 3 ) , and ( H 4 ) .

We define the function G ( t ) = ∫ 0 t g ( s ) d s . The function g is continuous on R + and verifies

1. There exists g 0 > 0 , such that g ( t ) ≥ g 0 for all t ≥ 0 and

   G ( t + s ) ≥ G ( t ) + G ( s ) ∀ s , t ≥ 0 .

2. g ( t ) t is nonincreasing for t > 0 .

Another consequence of ( G 2 ) is that a simple calculation shows that

So, one has

A typical example of a function g fulfilling conditions ( G 1 ) and ( G 2 ) is given by

Another example is given by g ( t ) = 1 + ln ( 1 + t ) .

The major difficulty in this problem lies in the concurrence between the growths of g and f .

It will be said that u is a solution to problem (1), if u is a weak solution in the following sense.

The energy functional, also known as the Euler-Lagrange functional associated with (1), is defined by J : W → R

It is quite clear that finding weak solutions to problem (1) is equivalent to finding non-zero critical points of the functional J over W .

In the subcritical exponential growth case, we will prove the following result.

In the context of the critical double exponential growth, the study of problem (1) becomes more difficult than in the subcritical case. Our Euler-Lagrange function is losing compactness at a certain level. To overcome this lack of compactness, we choose test functions, which are extremal for the Trudinger-Moser inequality (7). Our result is as follows.

This article is organized as follows. In Section 2, we give some useful lemmas for the compactness analysis. In Section 3, we prove that the functional J satisfies the two geometric properties. Section 4 is devoted to estimate the minmax level of the energy. We conclude with the proofs of Theorems 1.2 and 1.3 in Section 5.

We shall use the notation ‖ u ‖ p for the norm in the Lebesgue space L p ( B ) . We will also use the Sobolev weighted space defined by

equipped with the norm

and where the Lebesgue weighted space,

is endowed with the norm ‖ u ‖ p , σ = ∫ B σ ( x ) ∣ u ∣ p d x 1 p .

2 Preliminaries for the variational formulation

In this section, we will present a number of technical lemmas for our future use. We begin with the radial lemma.

The second important lemma is given as follows:

In an attempt to prove a compactness condition for the energy ℰ , we need a Lions-type result [23] about an improved TM-inequality when we deal with weakly convergent sequences and double exponential case.

3 The mountain pass geometry of the energy

Since the nonlinearity f is critical or subcritical at + ∞ , there exist a , C > 0 positive constants and there exists t 2 > 1 such that

So the functional J given by (11) is well defined and of class C 1 .

In order to prove the existence of nontrivial solution to problem (1), we will prove the existence of nonzero critical point of the functional J by using the theorem introduced by Ambrosetti and Rabinowitz in [25] (mountain pass theorem) without the Palais-Smale condition.

Before starting the proof of the geometric properties for the functional J , it follows from the continuous embedding W ↪ L q ( B ) for all q ≥ 1 , that there exists a constant C > 0 such that ‖ u ‖ N ′ q ≤ c ‖ u ‖ , for all u ∈ W .

In the next lemmas, we prove that the functional J has the mountain pass geometry of Theorem 3.1.

By the following lemma, we prove the second geometric property for the functional J .

4 The minimax estimate of the energy

According to Lemmas 4 and 5, let

and

We are going to estimate the minimax value d of the functional J . The idea is to construct a sequence of functions ( v n ) ∈ W , and estimate max { J ( t v n ) : t ≥ 0 } . For this goal, let us consider the following Moser sequence:

Let v n ( x ) be the function defined by

With this choice of ψ n , the sequence ( v n ) is normalized since

We have the following elementary crucial result.