Solutions for a quasilinear Schrödinger equation: a dual approach

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Abstract

We consider quasilinear stationary Schrödinger equations of the form (1)Δu−Δ(u2)u=g(x,u),x∈RN.Introducing a change of unknown, we transform the search of solutions u(x) of (1) into the search of solutions v(x) of the semilinear equation (2)Δv=11+2f2(v)g(x,f(v)),x∈RN,where f is suitably chosen. If v is a classical solution of (2) then u=f(v) is a classical solution of (1). Variational methods are then used to obtain various existence results.

Introduction

In this paper we deal with equations of the formΔu−Δ(u2)u=g(x,u),u∈H1(RN).These equations model several physical phenomena but until recently little had been done to prove rigorously the existence of solutions.

A major difficulty associated with (1.1) is the following; one may seek to obtain solutions by looking for critical points of the associated “natural” functional, J:H1(RN)→R given byJ(u)=12RN|u|2dx+RN|u|2u2dx−RNG(x,u)dx,where G(x,s)=0sg(x,t)dt. However except when N=1 this functional is not defined on all H1(RN).

The first existence results for equations of the form of (1.1) are, up to our knowledge, due to [12], [7]; papers to which we refer for a presentation of the physical motivations of studying (1.1). In [12], [7], however, the main existence results are obtained, through a constrained minimization argument, only up to an unknown Lagrange multiplier.

Subsequently a general existence result for (1.1) was derived in [8]. To overcome the undefiniteness of J the idea in [8] is to introduce a change of variable and to rewrite the functional J with this new variable. Then critical points are search in an associated Orlicz space (see [8] for details).

The aim of the present paper is to give a simple and shorter proof of the results of [8], which do not use Orlicz spaces, but rather is developed in the usual H1(RN) space. The fact that we work in H1(RN) also permit to cover a different class of nonlinearities. In particular we give full treatment of the autonomous case and for nonautonomous problems we do not assume that,s→g(x,s)s:]0,∞[→Risnondecreasingins.Following the strategy developed in [4] on a related problem, we also make use of a change of unknown v=f−1(u) and define an associated equation that we shall call dual. If v∈H1(RN) is classical solution ofΔv=11+2f2(v)g(x,f(v)),u=f(v) is a classical solution of (1.1).

Equations of form (1.2) are of semilinear elliptic type and one can try to solve them by a variational approach. In particular we shall see that, under very general conditions on g, the “natural” functional associated to (1.2), I:H1(RN)→R given byI(v)=12RN|v|2dx−RNG(x,f(v))dxis well defined and of class C1 on H1(RN).

The dual approach is introduced in Section 2. In Section 3, we deal with autonomous problems, when (1.1) is of the formΔu−Δ(u2)u=g(u),u∈H1(RN).Autonomous problems seems to play an important role in physical phenomena (see [3] for example) and we obtain here an existence result under assumptions we believe to be nearly optimal. We assume that the nonlinear term g satisfies:

  • (g0)

    g(s) is locally Hölder continuous on [0,∞[.

  • (g1)

    −∞<liminfs→0g(s)/s⩽limsups→0g(s)/s=−ν<0 for N⩾3,

    lims→0g(s)/s=−ν∈(−∞,0) for N=1,2.

  • (g2)

    When N⩾3,lims→∞|g(s)|/s(3N+2)/(N−2)=0.

    When N=2, for any α>0 there exists Cα>0 such that|g(s)|⩽Cαeαs2foralls⩾0.

  • (g3)

    When N⩾2, there exists ξ0>0 such that G(ξ0)>0,

    When N=1, there exists ξ0>0 such thatG(ξ)<0forallξ∈]0,ξ0[,G(ξ0)=0andg(ξ0)>0.

Remark 1.1

An easy calculation shows that (g0)–(g3) are satisfied in the model case g(s)=|s|2sνs.

Theorem 1.2

Assume that (g0)–(g3) hold. Then (1.3) admits a solution u0∈H1(RN) having the following properties:

  • (i)

    u0>0 on RN.

  • (ii)

    u0 is spherically symmetric: u0(x)=u0(r) with r=|x| and u0 decreases with respect to r.

  • (iii)

    u0∈C2(RN).

  • (iv)

    u0 together with its derivatives up to order 2 have exponential decay at infinity|Dαu0(x)|⩽Ce−δ|x|,x∈RNfor some C,δ>0 and for |α|⩽2.

We prove Theorem 1.2 searching for a critical point of the functional I, which is here autonomous. As we shall see the existence of a critical point follows almost directly, from classical results on scalar field equations due to Berestycki–Lions [2] when N=1 or N⩾3 and Berestycki–Gallouët–Kavian [1] when N=2.

In Section 4 we assume that (1.1) is of the form,Δu−Δ(u2)u+V(x)u=h(u).We require V∈C(RN,R) and h∈C(R+,R), to be Hölder continuous and to satisfy

  • (V0)

    There exists V0>0 such that V(x)⩾V0>0 on RN.

  • (V1)

    lim|x|→∞V(x)=V(∞) and V(x)⩽V(∞) on RN.

  • (h0)

    lims→0h(s)/s=0.

  • (h1)

    There exists p<∞ if N=1,2 and p<(3N+2)/(N−2) if N⩾3 such that |h(s)|⩽C(1+|s|p), ∀s∈R, for a C>0.

  • (h2)

    There exists μ⩾4 such that, ∀s>0,0<μH(s)⩽h(s)swithH(s)=0sh(t)dt.

Our main result is the following:

Theorem 1.3

Assume that (V0)–(V1) and (h0)–(h1) hold. Then (1.4) has a positive nontrivial solution if one of the following conditions hold:

  • (1)

    (h2) hold with a μ>4.

  • (2)

    (h2) hold with μ=4 with p⩽5 if N=3 and p<(3N+4)/N if N⩾4 in (h1).

The proof of Theorem 1.3 also relies on the study of the functional I. We first show that I possess a mountain pass geometry and denote by c>0 the mountain pass level (see Lemma 4.2). To find a critical point the main difficulties to overcome are the possible unboundedness of the Palais–Smale (or Cerami) sequences and a lack of compactness since (1.4) is set on all RN.

For the second difficulty we use some recent results presented in [9], [10] which imply that, under conditions (V0)–(V1), the mountain pass level c>0 is below (if VV(∞)) the first level of possible loss of compactness (see Theorem 3.4 and Lemma 4.3).

For the first difficulty we distinguish the cases μ>4 and μ=4 in (h2). In the case μ>4, it is direct to prove that all Cerami sequences of I are bounded. To show it in the case μ=4 is more involved and for this we make use of an idea introduced in [8].

Notation. Throughout the article the letter C will denote various positive constants whose exact value may change from line to line but are not essential to the analysis of the problem. Also if we take a subsequence of a sequence {vn} we shall denote it again {vn}.

Section snippets

The dual formulation

We start with some preliminary results. Let f be defined byf′(t)=11+2f2(t)andf(0)=0on [0,+∞[ and by f(t)=−f(−t) on ]−∞,0].

Lemma 2.1

(1) f is uniquely defined, C and invertible.

(2) |f′(t)|⩽1, for all t∈R.

(3) f(t)t→1 as t→0.

(4) f(t)t→21/4 as t→+∞.

Proof

Points (1)–(3) are immediate. To see (4) we integrate0tf′(s)1+2f2(s)ds=t.Using the changes of variables x=f(s) and x=12Sh(y) we obtain that122[sinh−1(2f(t))]+142sinh2[Sh−1(2f(t))]=t.Thus, sinh2[Sh−1(2f(t))]∼42t in the sense that, as t→+∞,sinh2[sinh−1(2f(t))]42t

Autonomous cases

In this section (1.1) is of the formΔu−Δ(u2)u=g(u),u∈H.with the nonlinearity g satisfying (g0)–(g3). Because we look for positive solutions we may assume without restriction that g(s)=0,∀s⩽0. Following our dual approach we shall obtain the existence of solutions for (3.1) studying the associated dual equationΔv=11+2f2(v)g(f(v)),v∈H.In this aim, we now recall some classical results due to Berestycki–Lions [2] and Berestycki–Gallouët–Kavian [1] on equations of the formΔv=k(v),v∈H.These authors

Nonautonomous cases

In this section we assume that (1.1) is of the formΔu−Δ(u2)u+V(x)u=h(u),u∈H.with the potential V(x) satisfying (V0)–(V1) and the nonlinearity h(s), (h0)–(h2). Here again we use our dual approach and first look to critical points of I:H→R given byI(v)=12RN|v|2+V(x)f2(v)dx−RNH(f(v))dx.Namely for solutions vH ofΔv=11+2f2(v)[−V(x)f(v)+h(f(v))].From Section 3 we readily deduce that I is well defined and of class C1 under conditions (V0)–(V1) and (h0)–(h1). Let us show that I has a mountain

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