On singularly perturbed (p, N)-Laplace Schrödinger equation with logarithmic nonlinearity

1 Introduction

In this article, we deal with the following singularly perturbed (p, N)-Laplace Schrödinger equation

where

with N ≥ 2. Further, we assume that 1 < p < N and ɛ is a very small positive parameter. The operator Δ _t u = div(|∇u| ^t−2∇u) with t ∈ {p, N} is the standard t-Laplace operator and the scalar potential V : R N → R is a continuous function. The nonlinearity f : R → R has critical exponential growth at infinity, i.e., it behaves like exp ( α | u | N N − 1 ) when |u| → ∞ for some α > 0, which means that there exists a positive constant α ₀ such that the following condition holds:

Throughout the paper, we suppose the following assumptions on the scalar potential V : R N → R :

1. V ∈ C ( R N ; R ) and there exists a constant V ₀ > 0 such that inf x ∈ R N V ( x ) ≥ V 0 .

2. There exists an open and bounded set Λ ⊂ R N such that

   V 0 = inf x ∈ Λ V ( x ) < min ∂ Λ V ( x ) .

for δ > 0 small enough such that M _δ ⊂ Λ. Moreover, the nonlinearity f : R → R is supposed to satisfy the following conditions:

1. Let f ∈ C 1 ( R , R ) be an odd function such that f(0) = 0, f(s) < 0 for all s < 0 and f(s) > 0 for all s > 0. Further, there exists a constant α ₀ ∈ (0, α) with the property that for all τ > 0, there exists κ _τ > 0 such that for all s ∈ R , we have

   | f ( s ) | ≤ τ | s | N − 1 + κ τ Φ α 0 | s | N ′   with    Φ ( t ) = exp ( t ) − ∑ j = 0 N − 2 t j j !   and   N ′ = N N − 1 .

2. There exists μ > N such that

   s f ( s ) − μ F ( s ) ≥ 0   for all  s ∈ R ,  where   F ( s ) = ∫ 0 s f ( t ) d t   for all  s ∈ R .

3. The mapping s ↦ f ( s ) | s | N − 2 s is increasing for all s > 0 and decreasing for all s < 0.

4. There exists a constant γ > 0 such that f(s) ≥ γs ^μ−1 for all s ≥ 0.

In order to familiarize the reader with the special behaviors of the classical Sobolev spaces, it is worth pointing out that the space W 1 , p ( R N ) can be distinguished in three different ways, namely:

The Sobolev embedding theorem says that for p < N, there holds W 1 , p ( R N ) ↪ L q ( R N ) for any q ∈ [p, p*], where p * = N p N − p is the critical Sobolev exponent to p. In this scenario, to study variational problems, the nonlinearity cannot exceed the polynomial of degree p*. In contrast to this, for the Sobolev limiting case commonly known as the Trudinger-Moser case, one can notice that p* converges to ∞ as p converges to N and thus, we might except that W 1 , N ( R N ) is continuously embedded in L ∞ ( R N ) . This is, however, wrong for N > 1. In order to see this, let φ ∈ C c ∞ ( R N , [ 0,1 ] ) be such that φ ≡ 1 in B ₁(0) and φ ≡ 0 in B 2 c ( 0 ) , then the function u ( x ) = φ ( x ) log log 1 + 1 | x | belongs to W 1 , N ( R N ) but not to L ∞ ( R N ) . Moreover, in this situation, every polynomial growth is allowed. To fill this gap, it is fairly natural to look for the maximal growth of a function g : R → R + such that

where ‖ u ‖ W 1 , N = ‖ ∇ u ‖ N N + ‖ u ‖ N N 1 N and ‖ ⋅ ‖ _N is the usual norm of the Lebesgue space L N ( R N ) . It is noteworthy that many authors have independently proved that the maximum growth of such a function g is of exponential type. In that context, we mention the works of Adimurthi-Yang [1] and Li-Ruf [2]. In recent years, the existence and multiplicity of solutions to elliptic equations involving the N-Laplace operator with subcritical and critical growth in the sense of Trudinger-Moser inequality have been extensively studied, motivated by their applicability in many fields of modern mathematics. For a detailed study, we refer to Beckner [3], Chang-Yang [4], Chen-Lu-Zhu [5], Lam-Lu [6], Zhang-Zhu [7] and the references therein.

It should be pointed out here that the Trudinger-Moser type inequalities and the Adams type inequalities have been widely studied by many authors across diverse domains such as Euclidean spaces, Heisenberg groups, Riemannian manifolds, and so on. In this context, we recommend that readers take a look at some works by Chen-Wang-Zhu [8], Cohn-Lu [9], do Ó-Lu-Ponciano [10], Duy-Phi [11], Jiang-Xu-Zhang-Zhu [12], Lam-Lu [13], [14], Li-Lu-Zhu [15], Wang [16], Xue-Zhang-Zhu [17] and the references cited therein.

In the last decade, great attention has been focused on the study of (p, q)-Laplace equations as well as double phase problems in the whole Euclidean space R N due to the broad applications in biophysics, plasma physics, solid state physics, and chemical reaction design, see, for example, the books of Aris [18], Fife [19] and Murray [20] as well as the papers of Myers-Beaghton-Vvedensky [21] and Wilhelmsson [22] and the references therein. On the other hand, concerning the Sobolev limiting case, that is, p < q and q = N, such types of problems are often comparatively less looked upon. This is one of the main motivations for the study in this article. More details on (p, N)-Laplace equations can be found in the papers of Carvalho-Figueiredo-Furtado-Medeiros [23], Chen-Fiscella-Pucci-Tang [24], Fiscella-Pucci [25], Mahanta-Mukherjee-Sarkar [26] and Mahanta-Winkert [27], as well as the references therein.

Moreover, we make a note that one of the hypotheses on the potential function V that appears in (V1) says that the corresponding first-order weighted Sobolev spaces make sense and are well-behaved; see, for instance, Lemma 2.2 and Lemma 2.3, respectively. To address the challenge posed by the lack of compactness, Bartsch-Wang [28] were the first to place assumptions on the potential function V. Further, as an application, they studied the existence and multiplicity of solutions for a superlinear Schrödinger type equation in R N . Afterwards, reducing the conditions on the potential and the nonlinearity, Tang [29] achieved some more general results. Later on, Chen-Lu-Zhu [5], [30], [31], [32] developed and introduced some more generalized form of assumptions on the potential function V, called degenerated and trapping types of potentials, to establish the Trudinger-Moser type inequalities as well as the consequences of the Adams type inequalities, and by employing such inequalities, they studied elliptic and subelliptic PDEs. In addition, we also mention here that Chen-Lu-Zhu [33] showed the existence of extremals for Trudinger-Moser inequalities in R 2 in the presence of trapping potential.

Nowadays, there is a great interest in the study of the time-dependent nonlinear logarithmic Schrödinger equation of the form

where Ψ : R N × [ 0 , + ∞ ) → C , N ≥ 2, E ∈ R , ɛ is a positive parameter and V is a continuous function satisfying certain hypotheses. It is worth noting that the standing wave solution of (NLS) is of the form Ψ(x, t) = exp(−iEt/ɛ)u(x), where u is a solution of the equation

From the point of view of the application, such equations are the main tools for studying quantum physics, quantum optics, effective quantum gravity, nuclear physics, transport and diffusion phenomena, theory of superfluidity and Bose–Einstein condensation. For more information in this direction, we refer to Białynicki-Birula-Mycielski [34], Carles-Gallagher [35], Cazenave [36], Cazenave-Lions [37], Zloshchastiev [38] and the references therein. In addition, in order to study (1.1), there have been several technical difficulties due to the presence of logarithmic nonlinearity. For example, let u be a smooth function satisfying

By direct computation, one has u ∈ H 1 ( R N ) but ∫ R N u 2 ⁡ log u 2 d x = − ∞ . So, the Euler–Lagrange functional associated to (1.1) is not finite and is no longer C ¹ on H 1 ( R N ) . As a result, we cannot directly use the classical critical point theory to study the behavior of solutions of (1.1). To overcome these difficulties, several approaches have been developed in the mathematical literature so far. We will discuss some of them below.

Initially, Cazenave [36] studied the following time-dependent logarithmic Schrödinger equation

by considering the N-function A and the function space W defined as

endowed with the Luxemburg norm  ‖ ⋅ ‖ W = ‖ ⋅ ‖ H 1 ( R N ) + ‖ ⋅ ‖ A , where

The author defined the associated functional L : W → R given by

and proved the existence of infinitely many critical points of L on the set u ∈ W : ∫ R N | u | 2 d x = 1 . As a result, he also provided a lot of information about the behavior of the solutions of equation (1.2).

Later, Squassina-Szulkin [39], [40] investigated the following logarithmic Schrödinger equation

where V , Q ∈ C ( R N , R ) are 1-periodic functions of the variables x ₁, x ₂, …, x _N satisfying the hypotheses

Employing the standard nonsmooth critical point theory of lower semicontinuous functionals, which was developed by Szulkin [41], the authors showed first the existence of positive ground state solutions by adopting the deformation lemma. Then, by using the genus theory, they proved the existence of infinitely many high-energy solutions, which are geometrically distinct under Z N -action. Moreover, several authors used nonsmooth variational techniques to study the logarithmic Schrödinger equations, such as Alves-Ambrosio [42], Alves-de Morais Filho [43], Alves-Ji [44], [45], d’Avenia-Montefusco-Squassina [46], Deng-He-Pan-Zhong [47], Ji-Szulkin [48], Li-Peng-Shuai [49] and Liu-Peng-Zou [50]. In contrast to this, Tanaka-Zhang [51] have also studied (1.3) by considering V, Q as spatially 1-periodic functions of class C ¹. The authors showed the existence of infinitely many multi-bump solutions for (1.3), which are distinct under Z N -action, by taking an approach using spatially 2L-periodic problems with L ≫ 1.

During the last decade, Wang-Zhang [52] introduced an advanced way of studying logarithmic equations, which is known as the power approximation method. First, they considered the following semiclassical scalar field equation with power-law nonlinearity

where p ∈ (2, 2*) with 2 * = 2 N N − 2 if N ≥ 3 and 2* = +∞ if N ≤ 2. The authors showed that when p ↘ 2, then the ground state solutions of (1.4) either blow up or vanish, and converge to the ground state solutions of the logarithmic-scalar field equation

In addition, they also proved that the same result holds for bound-state solutions. Later, the authors studied the concentration behavior of nodal solutions of (1.1) in [53] by employing the same idea discussed above.

On the other hand, concerning the penalization method and the Lusternik–Schnirelmann category theory, which are generally used to study the multiplicity of the positive solutions of nonlinear PDEs and their concentration phenomena, we recommend the readers to study the papers of Alves-Figueiredo [54], Ambrosio-Repovš [55], Thin [56] and Zhang-Sun-Liang-Thin [57], see also the references therein. The most important features and novelties of our problem are listed below:

1. The appearance of the (p, N)-Laplace operator in our problem is nonhomogeneous, and thus, the calculations are more complicated.

2. Due to the lack of compactness caused by the unboundedness of the domain, the Palais-Smale sequences do not have the compactness property.

3. The reaction combines the multiple effects generated by the logarithmic term and a term with critical growth with respect to the exponential nonlinearity, making our study more delicate and challenging.

4. The concentration phenomena create a bridge between the global maximum point of the solution and the global minimum of the potential function.

5. The proofs combine refined techniques, including variational and topological tools.

To the best of our knowledge, this is the first time in the literature, in which two penalized functions are used simultaneously, one corresponds to the logarithmic nonlinearity and the other one corresponds to the exponential growth. Motivated by all the cited works, especially by the papers of Alves-da Silva [58], Alves-Ji [44] and Squassina-Szulkin [39], we study the existence, multiplicity and concentration phenomena of solutions for problem ( P ε ).

Note that, by the change of variable x↦ɛx, we can see that ( P ε ) is equivalent to the problem

where

Now, we state the main results of this article.

The paper is organized as follows. In Section 2, we introduce the underlying function spaces, the main tools of the variational framework and some preliminary results. Section 3 is devoted to the study of the penalized problem by using the mountain pass geometry and some topological tools. In Section 4, the properties of the Nehari manifold associated with the penalized problem and the concentration behavior of the positive solutions for ( P ε ) are studied. Finally, in Section 5, we prove Theorem 1.4 by invoking the Lusternik–Schnirelmann category theory.

2 Some preliminary results

This section is devoted to some basic results on Sobolev spaces, Orlicz spaces and related lemmas that will be used to establish the main results of this article. To this end, for t ∈ (1, + ∞), L t ( R N ) denotes the standard Lebesgue space with the norm ‖ ⋅ ‖ _t . Further, if Ω ⊂ R N , then we define the norm of L ^t (Ω) by ‖ ⋅ ‖ L t ( Ω ) . For nonnegative measurable functions V : R N → R , the space L V ε t ( R N ) consists of all real-valued measurable functions such that V ( ε x ) | u | t ∈ L 1 ( R N ) and is equipped with the seminorm

which turns into a norm due to hypothesis (V1). The space L V ε t ( R N ) , ‖ ⋅ ‖ t , V ε is a separable and uniform convex Banach space (see Pucci-Xiang-Zhang [59]). Note that under the assumption (V1), the embedding L V ε t ( R N ) ↪ L t ( R N ) is continuous.

Next, we define

endowed with the norm

It is well-known that the space W 1 , t ( R N ) , ‖ ⋅ ‖ W 1 , t is a separable and uniformly convex Banach space. Note that C c ∞ ( R N ) is a dense subset of W 1 , t ( R N ) . Moreover, the critical Sobolev exponent of t is defined by t * = N t N − t if t < N and t* = +∞ otherwise. Further, we set

and endow it with the norm

Then, the space (X, ‖ ⋅‖ _X ) is a reflexive and separable Banach space.

The weighted Sobolev space W V ε 1 , t ( R N ) is defined by

equipped with the norm

The space W V ε 1 , t ( R N ) , ‖ ⋅ ‖ W V ε 1 , t is a separable and uniformly convex Banach space, see Proposition 2.1 in Bartolo-Candela-Salvatore [60], thanks to (V1). Moreover, C c ∞ ( R N ) is a dense subset of W V ε 1 , t ( R N ) , see Bartolo-Candela-Salvatore [60] and Chen-Chen [61]. From now on, our function space is given by

which is endowed with the norm

Because of assumptions (V1) and Proposition 2.1 in Bartolo-Candela-Salvatore [60], it is easy to see that ( X ε , ‖ ⋅ ‖ X ε ) is a reflexive and separable Banach space. In the entire paper, C, C ₁, C ₂, C ₃, … denote some fixed positive constants possibly different at different places. Moreover, for any Banach space (X, ‖ ⋅‖ _X ), we denote its continuous dual by X * , ‖ ⋅ ‖ X * and o _n (1) denotes the real sequence such that o _n (1) → 0 as n → ∞. By ⇀ we mean the weak convergence and → means the strong convergence, while u ^± = max {±u, 0} stand for the positive and negative part of a function u, respectively. Furthermore, B _r (x ₀) is an open ball centered at x 0 ∈ R N with radius r > 0 and B _r = B _r (0). Finally, for any set S ⊂ R N , we denote its Lebesgue measure by |S| and its complement by S ^c .

Now, we shall discuss some basic properties of Orlicz spaces.

Further, we say that a N-function F satisfies the Δ₂-condition, which is denoted by F ∈ Δ 2 , if there exists a constant c > 0 such that

The conjugate of the N-function F is denoted by F ̃ and defined as

Note that F ̃ is always an N-function and F ̃ ̃ = F , i.e., F and F ̃ are complementary to each other. Now, we define the Orlicz space associated with the N-function F by

endowed with the Luxemburg norm

Clearly, the space L F ( R N ) , ‖ ⋅ ‖ F is a Banach space. Consequently, the Young’s type and Hölder’s type inequalities in Orlicz spaces are given by

and

Moreover, if F , F ̃ ∈ Δ 2 , then the space L F ( R N ) is reflexive and separable. Again, the Δ₂-condition implies at once that

and

Now, we shall characterize an important property of the N-function. It states that if F is an N-function of class C ¹ and F ̃ be its conjugate such that the following condition holds

then F , F ̃ ∈ Δ 2 .

Due to some mathematical difficulties in ( S ε ), we cannot directly apply smooth variational techniques to study the problem ( S ε ). Indeed, if we set the energy functional I _ɛ associated with ( S ε ), which is defined on the space X _ɛ as

where

then the energy functional I _ɛ is not well-defined on X _ɛ , since it may happen that there exists u ∈ X _ɛ satisfying ∫ R N | u | N ⁡ log | u | N d x = − ∞ and hence, I _ɛ (u) = +∞. Inspired by the works of [39], [43], [44], we can avoid this difficulty by choosing

where H 1 is a nonnegative C ¹-function, which is also convex and H 2 is also a nonnegative C ¹-function satisfying some growth condition. Hence, one can easily obtain from (2.4) that

This technique guarantees that I _ɛ may be expressed as the combination of a C ¹-functional with a convex and lower semicontinuous functional. Therefore, the critical point theory for lower semicontinuous functionals, as established by Szulkin [41], can be used to examine solutions of ( P ε ).

Another feature and novelty of ( S ε ) is the fact that the corresponding energy functional defined in (2.4) is not a C ¹-functional and hence, we shall not be able to find the multiplicity of solutions of ( S ε ) by using smooth variational methods together with the Lusternik–Schnirelmann category theory. To overcome this difficulty, we shall work with a newly constructed Banach space, where the functional I _ɛ is C ¹. Inspired by the work of Shen-Squassina [62], we fix δ > 0 sufficiently small and define H 1 and H 2 by

and

so that

The functions H 1 and H 2 have the following properties, respectively.

• ( H 1 )

  1. For δ > 0 small enough, H 1 is convex, even and of class C 1 ( R , R ) .

  2. H 1 ( s ) ≥ 0 and H 1 ′ ( s ) s ≥ 0 for all s ∈ R .

  3. For any q > N, there exists constants C ₁, C ₂ > 0 such that

     | H 1 ′ ( s ) | ≤ C 1 | s | q − 1 + C 2   for all  s ∈ R .

  4. There exists constants C ₁, C ₂ > 0 such that

     | H 1 ( s ) | ≤ C 1 | s | N + C 2   for all  s ∈ R .

• ( H 2 )

  1. There hold H 2 ′ ( s ) ≥ 0 for all s > 0 and H 2 ′ ( s ) > 0 for all s > (N − 1)δ.

  2. H 2 ∈ C 2 ( R , R ) and for any q > N, there exists a constant C = C _q > 0 such that

     | H 2 ′ ( s ) | ≤ C | s | q − 1   and | H 2 ( s ) | ≤ C | s | q   for all  s ∈ R .

  3. The map s ↦ H 2 ′ ( s ) s N − 1 is nondecreasing for s > 0 and strictly increasing for s > (N − 1)δ.

  4. H 2 ′ is an odd function and there holds lim s → ∞ H 2 ′ ( s ) s N − 1 = ∞ .

The next two results can be found in Shen-Squassina [62].

We define now the spaces

endowed with the norms

Thanks to Lemma 2.6, the spaces (Y, ‖ ⋅‖ _Y ) and ( Y ε , ‖ ⋅ ‖ Y ε ) are reflexive and separable Banach spaces. Moreover, we have the continuous embeddings

Similarly, these continuous embeddings are also true, if we replace Y _ɛ by Y and X _ɛ by X, respectively. Next, we denote by S _r (or by S _r ) the best constant in the embedding from Y _ɛ (or Y) into some Lebesgue space L r ( R N ) .

Now, we recall the following version of Lions’ compactness lemma, see Alves-Figueiredo [[54], Proposition 4].

The next lemma can be found in Alves-da Silva [58].

From Yang [63], we have the following result.

The inequality in the following lemma is known as the Trudinger-Moser inequality, which was first studied by Adimurthi-Yang [1].