Entire radial bounded solutions for Leray-Lions equations of (p, q)-type

1 Introduction

This article aims to investigate the existence of weak bounded solutions for the generalized quasilinear Schrödinger equation:

where 1 < q ≤ p , N ≥ 2 , A : R N × R × R N → R is C 1 -Carathéodory function with partial derivatives

while g : R N × R → R is a given Carathéodory function. The Leray-Lions operators we can handle include the case of the so-called double-phase operator A ( x , t , ξ ) = A 1 ( x ) ∣ ξ ∣ p + A 2 ( x ) ∣ ξ ∣ q , which, in particular, covers the fundamental example:

where Δ Q u = div ( ∣ ∇ u ∣ Q − 2 ∇ u ) for any Q > 1 . Such an operator permits to consider more general equations than the classical Schrödinger one, and it is useful to treat some phenomena in biophysics [18], plasma physics [30], chemical reaction design [16], quantum field theory [2,25], and nonlinear optics and fluid mechanics [2]. Thus, it is not surprising that it has been widely investigated in the last decade, both in bounded and in unbounded domains, for instance, see [4,26,28] and the recent overview on this topic given in [24].

We notice that equation (1.1) has a variational structure since its solutions coincide, at least formally, with critical points of the functional

with G ( x , t ) = ∫ 0 t g ( x , s ) d s , whose formal Euler-Lagrange equation is precisely (1.1).

Such a functional has two basic difficulties, which arise independently on G . First, the problem being settled in R N , there is a natural lack of compactness; hence, standard variational tools cannot work directly; second, in general, J is Gâteaux differentiable only along directions in the space W 1 , p ( R N ) ∩ W 1 , q ( R N ) ∩ L ∞ ( R N ) .

Let us point out that even if p = q , G = 0 , A ( x , t , ξ ) = 1 p A 1 ( x , t ) ∣ ξ ∣ p and we look for solutions verifying homogeneous Dirichlet condition in a bounded domain Ω , the corresponding action functional

is not well defined on W 0 1 , p ( Ω ) if A 1 ( x , t ) is unbounded with respect to t . Moreover, even if A 1 ( x , t ) is strictly positive and bounded with respect to t but ∂ A 1 ∂ t ( x , t ) ≢ 0 , then J ¯ is well defined on W 0 1 , p ( Ω ) , but it is Gâteaux differentiable only along special directions in the space W 0 1 , p ( Ω ) .

For this reason, several abstract approaches have been introduced, see [1,8,9]. In particular, in the last two papers, the authors have introduced an approach in which the natural space is the intersection of L ∞ ( R N ) with the reference Sobolev one and the classical Palais-Smale condition is replaced by a weak Cerami-type one, see Definition 2.1. Such an approach has been fruitfully used in several situations, such as [6,12,13] (for equations in bounded domains), and [10,14,15, 20–22,29] (for equations in the entire space).

In particular, in [20], the authors consider a quasilinear elliptic equation driven by a general operator of Leray-Lions type when the terms ∣ u ∣ p − 2 u and ∣ u ∣ q − 2 u are multiplied by two suitable unbounded potentials V ( x ) and W ( x ) . By using an approximation argument over bounded domains, they prove the existence of two bounded solutions, one negative and one positive (see also [22] if p = q ). An approximation method has been used also in [29], where V and W are as in [20], but the main term has the simplest form A ( x , t , ξ ) = 1 p A 1 ( x , t ) ∣ ξ ∣ p + 1 q A 2 ( x , t ) ∣ ξ ∣ q and g ( x , t ) has a super- p -linear growth.

Differently from [20,29], in this article, the coefficients V and W are constant, and we look for solutions of equation (1.1) by working directly in the whole Euclidean space and assuming that g is sub- p -linear at infinity.

Proving the validity of a weak Cerami-type condition is the core of our study, since the non compactness of R N is the main obstruction. For this reason, we assume that the problem is radially symmetric and we restrict our investigation to the subspace of radial functions, which turns out to be a natural constraint, in the spirit of the celebrated Palais’ result [27]. However, the setting being so general, the required convergence is not easy at all even in this framework. Indeed, we need a convergence result in the spirit of the one proved by Boccardo, Murat and Puel in [3, Lemma 5] (Lemma 3.9), but the novelty of our result is that it holds when we deal with Leray-Lions operators having (p, q)-growth. In particular, the q -growth introduces several difficulties since no compact embedding is possible when working with entire functions and low exponents, as q is. However, with the aid of a suitable a priori L ∞ -estimate, see Lemma 3.6, we are able to establish the desired result.

This article is organized as follows. In Section 2, we introduce the abstract framework and the general assumptions we make for our problem and we state our main theorem, which establishes the existence of two signed solutions (Theorem 2.14). In Section 3, we make precise the variational approach and we give some preliminary properties of the problem, together with the Boccardo-Murat-Puel-type result. In Section 4, we prove the weak Cerami-Palais-Smale condition for the functional J , and finally, we establish the existence of two radial weak bounded solutions for (1.1), one being positive and the other one being negative.

2 Abstract setting and statement of the main result

In this section, we assume that

• ( X , ‖ ⋅ ‖ X ) is a Banach space with dual ( X ′ , ‖ ⋅ ‖ X ′ ) ;

• ( S , ‖ ⋅ ‖ S ) is a Banach space such that X ↪ S continuously, i.e., X ⊂ S and a constant σ 0 > 0 exists such that

  ‖ u ‖ S ≤ σ 0 ‖ u ‖ X for all  u ∈ X ;

• J : D ⊂ S → R and J ∈ C 1 ( X , R ) with X ⊂ D .

If β ∈ R , we say that a sequence ( u n ) n ⊂ X is a Cerami-Palais-Smale sequence at level β , briefly ( CPS ) β -sequence, if

Moreover, β is a Cerami-Palais-Smale level, briefly (CPS)-level, if there exists a ( CPS ) β -sequence. We say that J satisfies the Cerami-Palais-Smale condition in X at level β if every ( CPS ) β -sequence converges in X (up to subsequences). However, considering the general setting of our problem, it may happen that a ( CPS ) β -sequence may be unbounded in ‖ ⋅ ‖ X but being convergent with respect to ‖ ⋅ ‖ S . In light of this, we need to weaken the classical Cerami-Palais-Smale condition in an appropriate way according to ideas developed in previous papers [7–9].

Let us point out that, due to the convergence only in S , the ( wCPS ) β condition implies that the set of critical points of J at the β level is compact with respect to ‖ ⋅ ‖ S ; this is enough to prove a Deformation Lemma and some abstract theorems about critical points [9]. In particular, the following generalized version of the Weierstrass theorem holds true (see the Minimum Principle given in [9, Theorem 1.6]).

As it is clear, norms play a crucial role in our setting. For this reason, now we need to make precise the notation we will use throughout this article.

Here and in the following, let N = { 1 , 2 , … } be the set of strictly positive integers, denote by x ⋅ y the inner product in R N and by ∣ ⋅ ∣ the standard norm on any Euclidean space as the dimension of the considered vector is clear and no ambiguity arises. Furthermore, we denote by:

• B R ( x ) = { y ∈ R N : ∣ y − x ∣ < R } the open ball in R N with center in x ∈ R N and radius R > 0 ;

• B R c = R N \ B R ( 0 ) the complement of the open ball B R ( 0 ) in R N ;

• meas ( Ω ) the usual Lebesgue measure of a measurable set Ω in R N ;

• L ℓ ( R N ) the Lebesgue space with norm ∣ u ∣ ℓ = ∫ R N ∣ u ∣ ℓ d x 1 ⁄ ℓ if 1 ≤ ℓ < + ∞ ;

• L ∞ ( R N ) the space of Lebesgue-measurable and essentially bounded functions u : R N → R with norm ∣ u ∣ ∞ = ess sup R N ∣ u ∣ ;

• W 1 , l ( R N ) the classical Sobolev space with norm ‖ u ‖ l = ( ∣ ∇ u ∣ l l + ∣ u ∣ l l ) 1 l if 1 ≤ l < + ∞ ;

• W r 1 , l ( R N ) = { u ∈ W 1 , l ( R N ) : u ( x ) = u ( ∣ x ∣ ) a.e. x ∈ R N } the subspace of radial functions of W 1 , l ( R N ) equipped with the same norm ‖ ⋅ ‖ l ;

• W = W 1 , p ( R N ) ∩ W 1 , q ( R N ) with norm ‖ ⋅ ‖ W = ‖ ⋅ ‖ p + ‖ ⋅ ‖ q .

From the Sobolev embedding theorems, for any ℓ ∈ [ l , l * ] with l * = l N N − l if N > l , or any ℓ ∈ [ l , + ∞ [ if l = N , the Sobolev space W 1 , l ( R N ) is continuously embedded in L ℓ ( R N ) , i.e., there exists a constant σ ℓ > 0 such that

[5, Corollaries 9.10 and 9.11]. Clearly, it is σ l = 1 . On the other hand, if l > N , then W 1 , l ( R N ) is continuously imbedded in L ∞ ( R N ) [5, Theorem 9.12].

Finally, we define

From now on, we assume 1 < q ≤ p ≤ N as, otherwise X = W 1 , p ( R N ) ∩ W 1 , q ( R N ) and the proofs can be simplified.

From Remark 2.3, if ( u n ) n ⊂ L q ( R N ) ∩ L ∞ ( R N ) and u ∈ L q ( R N ) ∩ L ∞ ( R N ) are such that u n → u in L q ( R N ) ∩ L ∞ ( R N ) , then u n → u also in L ℓ ( R N ) for any ℓ ≥ q .

This result can be weakened as follows.

From now on, we consider A : R N × R × R N → R and g : R N × R → R such that:

1. A is a C 1 -Carathéodory function, i.e., A ( ⋅ , t , ξ ) is measurable for all ( t , ξ ) ∈ R × R N , and A ( x , ⋅ , ⋅ ) is C 1 for a.e. x ∈ R N ;

2. positive continuous functions φ i , Φ i , ψ i , Ψ i : R → R , i ∈ { 0 , 1 , 2 } , exist such that

   ∣ A ( x , t , ξ ) ∣ ≤ φ 0 ( t ) ∣ t ∣ p + Φ 0 ( t ) ∣ ξ ∣ p + ψ 0 ( t ) ∣ t ∣ q + Ψ 0 ( t ) ∣ ξ ∣ q , ∣ A t ( x , t , ξ ) ∣ ≤ φ 1 ( t ) ∣ t ∣ p − 1 + Φ 1 ( t ) ∣ ξ ∣ p + ψ 1 ( t ) ∣ t ∣ q − 1 + Ψ 1 ( t ) ∣ ξ ∣ q , ∣ a ( x , t , ξ ) ∣ ≤ φ 2 ( t ) ∣ t ∣ p − 1 + Φ 2 ( t ) ∣ ξ ∣ p − 1 + ψ 2 ( t ) ∣ t ∣ q − 1 + Ψ 2 ( t ) ∣ ξ ∣ q − 1

   for a.e. x ∈ R N and for all ( t , ξ ) ∈ R × R N ;

3. a constant α 0 > 0 exists such that

   A ( x , t , ξ ) ≥ α 0 ( ∣ ξ ∣ p + ∣ ξ ∣ q ) for a.e.  x ∈ R N  and for all  ( t , ξ ) ∈ R × R N ;

4. a constant η 0 > 0 exists such that

   A ( x , t , ξ ) ≤ η 0 a ( x , t , ξ ) ⋅ ξ for a.e.  x ∈ R N  and for all  ( t , ξ ) ∈ R × R N ;

5. a constant α 1 > 0 exists such that

   a ( x , t , ξ ) ⋅ ξ + A t ( x , t , ξ ) t ≥ α 1 a ( x , t , ξ ) ⋅ ξ for a.e.  x ∈ R N  and for all  ( t , ξ ) ∈ R × R N ;

6. constants μ > p and α 2 > 0 exist such that

   μ A ( x , t , ξ ) − a ( x , t , ξ ) ⋅ ξ − A t ( x , t , ξ ) t ≥ α 2 A ( x , t , ξ ) for a.e.  x ∈ R N  and for all  ( t , ξ ) ∈ R × R N ;

7. for all ξ , ξ * ∈ R N , ξ ≠ ξ * , for a.e. x ∈ R N and for all t ∈ R , we have

   [ a ( x , t , ξ ) − a ( x , t , ξ * ) ] ⋅ [ ξ − ξ * ] > 0 ;

8. A ( x , t , ξ ) = A ( ∣ x ∣ , t , ξ ) for a.e. x ∈ R N , and for all ( t , ξ ) ∈ R × R N ;

9. some real constants l 1 , l 2 , η 1 , η 2 exist such that

   lim t → 0 ψ 1 ( t ) ∣ t ∣ η 1 = l 1 , lim t → 0 ψ 2 ( t ) ∣ t ∣ η 2 = l 2 ,

   with ψ 1 , ψ 2 as in ( h 1 ) and

   (2.1) η 1 > q N − 1 η 2 > q − 1 N − 1 ;

1. g ( x , t ) is a Carathéodory function;

2. real numbers r ∈ ( 1 , p ) and s ∈ ( 1 , q ) and functions η ∈ L p p − r ( R N ) ∩ L ∞ ( R N ) and ζ ∈ L q q − s ( R N ) ∩ L ∞ ( R N ) exist such that

   0 ≤ g ( x , t ) t ≤ η ( x ) ∣ t ∣ r + ζ ( x ) ∣ t ∣ s for a.e.  x ∈ R N ,  for all  t ∈ R ,

3. g ( x , t ) = g ( ∣ x ∣ , t ) for a.e. x ∈ R N , for all t ∈ R :

4. we have that

   lim t → 0 + g ( x , t ) t q − 1 = + ∞ uniformly with respect to  x ∈ R N .

We point out some direct consequences of the previous hypotheses.

Now, we are ready to state our main result.

3 Variational approach and preliminary tools

First of all, we want to make precise the variational approach to the problem.

Let us start noticing the following facts.

Let us point out that assumptions ( h 0 ) – ( h 1 ) and the definition of the Banach space X imply that A ( ⋅ , u , ∇ u ) ∈ L 1 ( R N ) for any u ∈ X .

Therefore, by Proposition 3.1, the functional

is well defined for all u ∈ X . Moreover, taking v ∈ X , the Gâteaux differential of the functional J in u along the direction v is given by

The following regularity result holds (for the proof, see [20, Proposition 2.14]).

Thanks to Proposition 3.2, the search for solutions of equation (1.1) reduces to the search for critical points of the functional J ; thus, our aim is to prove that J satisfies the hypotheses of Proposition 2.2.

Let us start with the following simple facts.