3 Existence of nodal solution

Least energy nodal solutions for elliptic equations with indefinite nonlinearity

Vladimir Bobkov

Institute of Mathematics, Ufa Science Center of RAS, Chernyshevsky str. 112, Ufa 450008, Russia Institut für Mathematik, Universität Rostock, Ulmenstraße 69, Haus 3, Rostock 18051, Germany

Received 20 July 2014, appeared 23 November 2014 Communicated by Patrizia Pucci

Abstract. We prove the existence of a nodal solution with two nodal domains for the Dirichlet problem with indefinite nonlinearity

−∆pu=λ|^u|^p−2u+f(x)|^u|^γ−2u

in a bounded domain Ω ⊂Rⁿ, providedλ ∈ (−∞,λ^∗₁), whereλ^∗₁ is a critical spectral value. The obtained solution has the least energy among all nodal solutions on the interval (−^{∞, min}{^λ1^∗,λ₂}), where λ₂is the second Dirichlet eigenvalue of −^∆p inΩ.

Moreover, the obtained solution forms a branch with continuous energy on(−∞,λ^∗₁). Keywords: nodal solutions, indefinite nonlinearity,p-Laplacian, fibering method, criti- cal spectral parameter.

2010 Mathematics Subject Classification: 35A01, 35A15, 35B05, 35B30, 35J92.

1 Introduction and main results

Let Ω ⊂ R^N be a bounded domain with the smooth boundary ∂Ω, N ≥ 1. We consider the Dirichlet boundary value problem

(−∆pu= λ|^u|^p−2u+ f(x)|^u|^{γ−2u, x} ∈Ω,

u|_∂Ω=0, (D⁾

where∆pu:=div(|∇^u|^p−2∇^u)is the p-Laplacian,λ,p,γ∈^Rand 1< p<γ< p^∗, where p^∗ =

( _pN

N−^p if p< N,

+_∞ if p≥ ^{N. (1.1)}

The function f ∈ ^L∞(_Ω)is assumed to be sign-changing and therefore the nonlinearity of (D⁾ is calledindefinite. Hereinafter we denote

Ω⁺ :={^x ∈Ω: f(x)>0}^.

BEmail: bobkovve@gmail.com

The questions of existence, nonexistence and multiplicity ofpositivesolutions to the prob- lems of type (D) have been comprehensively studied under various assumptions on differen- tial operator, spatial domain, coefficients and structure of nonlinearity, see, e.g., [5,11,14,15].

In particular, in [14] the explicit critical valueλ^∗ was introduced, such that (D) admits at least one positive solution for anyλ < λ^∗ and no positive solutions for λ > λ^∗. In spite of plenty of references, the multiplicity of solutions on a local interval (λ₁,λ₁+ε) was proved in [11]

using the fibering method, and this result was extended in [15] to the interval (λ₁,λ₁^∗) (see Figure1.1), where

λ^∗₁ := inf

u∈^W₀^1,p

(R

Ω|∇^u|^pdx R

Ω|^u|^{pdx :}

Z

Ω f(x)|^u|^γdx≥⁰ )

. (1.2)

Note that λ^∗₁ < +_∞ if ν(_Ω⁺) > 0, where ν is the n-th Lebesgue measure, and under the assumption R

Ω f(x)|^ϕ1|^γdx < 0 one can guarantee that λ^∗₁ > λ₁, whereas λ₁^∗ = λ₁ in the opposite case. Here by(λ₁,ϕ₁)we denote the first eigenpair of the operator −∆p on Ωwith zero Dirichlet boundary conditions [3].

At the same time, in the last few decades the questions of existence, multiplicity and qualitative properties ofnodal(sign-changing) solutions to the wide class of elliptic equations have attracted a lot of attention, cf. [1, 4, 6, 8, 9, 17] and survey [19] for historical overview and references. Nevertheless, to our best knowledge, there are only few articles concerning the existence of nodal solutions for the problems of type (D). We can mention [1,9,17], where some of existence and multiplicity results have been proved using different topological and variational arguments. Note that these works deal mainly with the Laplace operator (p =2).

Moreover, to the best of our knowledge, the questions of the qualitative properties of nodal solutions to (D) such as the precise number of nodal domains, property of the least energy among all nodal solutions, formation of branches, etc., have not been concerned.

λ₁ λ₂ λ^∗₁ λ^∗

E_λ

positive nodal

Figure 1.1: Branches of solutions w.r.t the energyE_λ;ν(_Ω⁺)>0, λ₂<λ^∗₁.

In the present article we apply the constructive minimization technique of the Nehari manifolds with the fibering approach (see, e.g., [4,8]) for the problem (D), which allows us to prove the existence of a nodal solution with two nodal domains for anyλ∈(−∞,λ^∗₁)and the least energy among all nodal solutions on(−^{∞, min}{^λ∗1,λ₂})(see Figure1.1). Here byλ₂ we denote the second eigenvalue of zero Dirichlet−∆p inΩ(see (1.4)).

A similar approach has been used in [6] to obtain the sign-changing solutions with positive

energy for the elliptic equations with convex-concave nonlinearity (−∆u=λ|^u|^q−2u+|^u|^{γ−2u, x} ∈Ω,

u=0, x ∈^{∂Ω, (1.3)}

where 1 < q < 2< γ < 2^∗. The method of proof carries over to the corresponding problem with the p-Laplacian.

Finally, we note that the disadvantage of the Nehari manifolds method consists in the fact that it cannot be used in proving the existence of nodal solutions with negative energy for (D⁾ and (1.3), however the existence of such solutions is known [17].

Before introducing our main results let us recall some common notations.

By aweak solutionof (D) we mean a critical pointu∈^W0^1,p(_Ω)of the energy functional E_λ(u):= ¹

pH_λ(u)− ¹ γF(u), where

H_λ(u):=

Z

Ω|∇^u|^pdx−^λ

Z

Ω|^u|^{pdx, F}(u):=

Z

Ωf(x)|^u|^γdx.

As usual, byW₀^1,p :=W₀^1,p(_Ω)we denote the standard Sobolev space equipped with the norm k^uk=

_Z

Ω|∇^u|^pdx _1/p

.

By a weak nodal solution of (D) we mean a critical point u ∈ ^W₀^{1,p of E}λ such that u^± 6≡ ⁰ a.e. in Ω, where u⁺ and u⁻ are the positive and negative parts of u, respectively. Note that u^± ∈ ^W₀^1,p(_Ω) (see, e.g., [16, Corollary A.5, p. 54]). Moreover, using the classical bootstrap arguments (see, e.g., [10, Lemma 3.2, p. 114]) it is not hard to show that under assumptions (1.1) and f ∈ ^L∞(_Ω)each weak solution of (D) belongs to L^∞(_Ω), and therefore toC^1,α(_Ω), by [18]. By a nodal domain of a function u ∈ ^C(_Ω) we denote any maximal connected open subset of{^x∈Ω:u(x)6=0}^.

From the definition of a weak solution it follows that any weak solution u ∈ ^W₀^{1,p of (}D⁾ satisfies

Qλ(u):= h^DEλ(u),ui= Hλ(u)−^F(u) =0,

and therefore any nontrivial solution belongs to the so-calledNehari manifold Nλ :={^u∈^W₀^1,p\ {⁰}^:Qλ(u) =₀}^.

Clearly, each nodal solution of (D) belongs to thenodal Nehari set Mλ := {^u ∈^W₀^1,p:u±∈ Nλ}^.

The Nehari manifolds method [4,8] enables one to find a nodal solution of (D) as a min- imum point of the energy functional E_λ on Mλ. However, due to the fact that E_λ possesses critical points both with positive and nonpositive energy (see Lemma2.2below), a minimiza- tion sequence for E_λ over Mλ will converge, in general, to a positive solution. To overcome this difficulty, we distinguish critical points with the different signs of energy and seek for a nodal solution of (D) as a minimum point ofE_λ on the following subset ofMλ:

Nλ¹:={^u∈^W₀^{1,p :u±} ∈ Nλ, E_λ(u^±)>0}^. Our main result is the following.

Theorem 1.1. Assume that(1.1)is satisfied andλ<λ₁^∗.

1. Ifν(_Ω⁺)>0, then there exists a weak nodal solution u_λ ∈ Nλ¹of the problem(D⁾with precisely two nodal domains. Moreover, u_λ has the least energy among all weak nodal solutions of (D^)on (−∞, min{^λ₁^∗,λ2}), i.e.,

−^∞<E_λ(u_λ)≤^Eλ(w_λ) for any weak nodal solution wλ of (D⁾on this interval.

2. If ν(_Ω⁺) = 0, then there are no weak nodal solutions of the problem (D^{) for any λ} ∈ (−∞, min{^λ₁^∗,λ2}).

In the proof it will be convenient to use the following variational characterization of the second eigenvalueλ₂ of the zero Dirichlet−^∆pinΩ(see [12], p. 195):

λ₂:= inf

A∈F2

sup

u∈A Z

Ω|∇^u|^{pdx, (1.4)}

where

F2 :=ⁿA ⊂ S ^:∃^h∈^C(S¹,A):h is oddo

, S ^:=

u∈^W0^1,p : Z

Ω|^u|^pdx=1

(1.5) andS¹ represents the unit sphere inR². By ϕ2 ∈ ^W₀^1,p we denote the corresponding second eigenfunction and note thatϕ^±₂ 6=0.

The second result concerns the formation of branches by the nodal solutions to (D^{). We} say that the family{^uλ}of critical points ofE_λformsa continuous branchon(a,b)(with respect to levels ofE_λ) if the map

E_(·)(u_(·)): (a,b)−→^R is a continuous function.

Theorem 1.2. Assume that(1.1) is satisfied andν(_Ω⁺)> 0. Then the set of nodal solutions u_λ for (D), given by Theorem1.1, forms a continuous branch on(−∞,λ^∗₁).

We note that the Nehari manifolds method leads to the similar results as above for more general class of problems of type (D^):

(−∆pu= λg(x)|^u|^p−2u+ f(x)|^u|^{γ−2u, x} ∈Ω, u|∂Ω=0,

whereg(x)∈ ^L∞(_Ω)also changes the sign. Nevertheless, we sacrifice this case for simplicity of exposition.

The paper is organized as follows. In Section 2, we give some auxiliary results concerning the properties of E_λ. Section 3 contains the proof of the existence of a weak nodal solution of (D). In Section 4, we show that the obtained solutions have precisely two nodal domains and the the least energy property. Moreover, in Section 4 we show the nonexistence result. In Section 5, we prove that the set of such nodal solutions forms a continuous branch.

2 Auxiliary results

First we show the following lemma.

Lemma 2.1. Assume that (1.1) is satisfied and Q_λ(u) = 0 for some u ∈ ^W₀^1,p. Then the following equivalences hold:

1. H_λ(u)>0⇐⇒ ^F(u)>0⇐⇒^Eλ(u)>0;

2. H_λ(u) =0⇐⇒ ^F(u) =0⇐⇒^Eλ(u) =0;

3. H_λ(u)<0⇐⇒ ^F(u)<0⇐⇒^Eλ(u)<0.

Proof. Letu∈^W₀^{1,p andQ}λ(u) =_{0. Then}

E_λ(u) = ^γ−^p

γp H_λ(u). (2.1)

Since 1< p<γ, all the statements of the lemma are satisfied.

Using this lemma it is easy to see that ifλ < λ₁, then E_λ(u) > 0 for any nontrivial weak solutionu∈^W0^1,p, whereas forλ≥ ^λ1the energyE_λ(u)may be either positive, or nonpositive.

Let us consider the fibered version of the functionalE_λ, given by Eλ(tu) = ^t

p

p Hλ(u)− ^tγ

γF(u), t >0.

The next lemma describes the structure of critical points of E_λ(tu)w.r.t.t>0.

Lemma 2.2. Assume that(1.1)is satisfied and u∈^W₀^1,p\ {⁰}^.

1. If H_λ(u),F(u)>0, then there exists only one positive critical point t(u)of E_λ(tu)w.r.t. t>0, which is a global maximum point, and E_λ(t(u)u)>0, Q_λ(t(u)u) =0.

2. If H_λ(u),F(u)<0, then there exists only one positive critical point t(u)of E_λ(tu)w.r.t. t>0, which is a global minimum point, and E_λ(t(u)u)<0, Q_λ(t(u)u) =0.

3. If H_λ(u)·^F(u)≤^0and(H_λ(u)_,F(u))6= (_{0, 0}), then E_λ(tu)has no positive critical points.

Proof. To obtain critical points ofE_λ(tu)w.r.t. t>0, let us find roots of

∂

∂tE_λ(tu) =t^p−1H_λ(u)−^tγ−1F(u) =t^p−1 H_λ(u)−^tγ−pF(u) =0.

Hence, if H_λ(u)·^F(u) ≤ ^{0 and} (H_λ(u),F(u)) 6= (0, 0), then E_λ(tu) has no positive critical points, and if H_λ(u)·^F(u)>0, then there exists exactly one positive critical point, given by

t(_u) =

H_λ(u) F(u)

_γ−¹_p

>_{0. (2.2)}

Assume first that H_λ(u),F(u)>0. Note that

∂

∂tE_λ(tu) = ¹

tQ_λ(tu), ∂²

∂t²E_λ(tu) = ¹

t²((p−¹)H_λ(tu)−(γ−¹)F(tu)).

Hence, ift(u) > 0 is a critical point of E_λ(tu), then Q_λ(t(u)u) = 0 and Lemma 2.1 implies thatEλ(t(u)u)>0. Moreover,

∂²

∂t²E_λ(tu) t=t(u)

=−(γ−^p)

t²(u) ^Hλ(t(u)u)<0.

Therefore, due to the fact that there is at most one critical point of E_λ(tu) w.r.t. t > 0, we conclude thatt(u)is a point of global maximum ofE_λ(tu).

The case H_λ(u),F(u)<0 of statement 2 may be handled in much the same way.

In the next result we provide the criterion for nonemptiness ofN_λ^1. Lemma 2.3. The following statements hold:

1. N_λ¹ 6=_∅for allλ∈R, wheneverν(_Ω⁺)>0;

2. N_λ¹ =_∅for allλ∈R, wheneverν(_Ω⁺) =0.

Proof. 1. Let λ ∈ ^{R and ν}(_Ω⁺) > 0. Then we are able to choose two open balls B₁,B₂ ⊂^Ω sufficiently small such thatB₁∩^B2 =_∅,λ₁(B₁),λ₁(B₂)>λandν(B₁∩Ω⁺),ν(B₂∩Ω⁺)>0.

Consider now the characteristic functionχ(B₁∩Ω⁺)of the setB₁∩Ω⁺. Sinceχ(B₁∩Ω⁺)

∈ ^L∞(_Ω),χ(B₁∩^Ω+)≥^{0 and suppχ}(B₁∩^Ω+)⊆^B1, the standard approximation arguments (see, e.g., [13, Lemma 7.2, p. 148]) imply the existence of uε ∈ ^C∞₀ (_Ω), uε ≥ 0, such that uε →^χ(B₁∩Ω⁺)in L^γ(_Ω)asε →0, and therefore

Z

Ω f|^uε|^γdx →

Z

Ω f|^χ(B₁∩^Ω+)|^γdx≡

Z

Ω fχ(B₁∩^Ω+)dx=

Z

B1∩Ω⁺ f dx>0,

i.e., F(u_ε) > 0 for sufficiently small ε > 0. The similar argumentation yields the existence of vε ∈ ^C₀^∞(_Ω), such that F(vε) > 0 for sufficiently small ε > 0. Moreover, due to the assumptionsB₁∩^B2 = _∅ andλ₁(B₁),λ₁(B₂)> λ, we can take ε > 0 small enough to satisfy suppu_ε ∩^suppvε = _∅ and H_λ(u_ε), H_λ(v_ε) > 0. Hence, Lemma 2.2 implies the existence of t(uε),t(vε)>0 such that

E_λ(t(uε)uε)>0, Q_λ(t(uε)uε) =0, E_λ(t(vε)vε)>0, Q_λ(t(vε)vε) =0.

Thus,t(uε)uε−^t(vε)vε ∈ N_λ^1.

2. Let nowν(_Ω⁺) =0. Then for anyu ∈^W₀^1,p\ {⁰}^{we haveF}(u)≤0, which is impossible for functions fromN_λ¹ in view of Lemma2.1.

Lemma 2.4. Assume that(1.1)is satisfied,λ<λ^∗₁and u ∈ N_λ^{1. Then} 1. E_λ(u^±)→+_∞ask^u±k →+_{∞, i.e., Eλ} is coercive onN_λ^1;

2. k^u±k> c₁ >0and E_λ(u^±)> c₂ >0, where the constants c₁,c₂do not depend on u.

Proof. 1. Letu ∈ N_λ¹. From Lemma 2.1 it follows that F(u^±)> 0. Hence, u^± are admissible functions for the minimization problem (1.2), and

λ₁^∗≤ R

Ω|∇^u±|^pdx R

Ω|^u±|^{pdx . (2.3)}

Using this fact and (2.1) we get E_λ(u^±) = ^γ−^p

γp H_λ(u^±)≥ ^λ1∗−^λ λ^∗₁

γ−^p γp

Z

Ω|∇^u±|^{pdx, (2.4)} ifλ≥^{0, and}

E_λ(u^±) = ^γ−^p

γp H_λ(u^±)≥ ^γ−^p γp

Z

Ω|∇^u±|^{pdx (2.5)}

forλ< 0. Therefore, by the assumptionλ<λ^∗₁, we conclude thatE_λ(u^±)→+_∞ask^u±k → +_∞.

2. Using (2.3) and the Sobolev embedding theorem we have the following chain for the caseλ≥^0:

λ^∗₁−^λ

λ^∗₁ k^u±k^p ≤ ^Hλ(u^±) =F(u^±)≤ ^Cγk^u±k^γ. Sinceγ> pandλ<λ^∗₁ we get

k^u±k ≥

λ^∗₁−^λ Cγλ^∗₁

_γ−¹_p

= c₁ >0. (2.6)

Combining this estimation with (2.4) we get the desired result. The caseλ<0 can be handled in the same way using the estimation (2.5).

3 Existence of nodal solution

In this section we prove the existence of a nodal solution for the problem (D). As noted above, we seek for a solution of (D) as a minimizer of the problem

(E_λ(w)→^inf,

w∈ Nλ¹. (3.1)

Lemma 3.1. Assume that(1.1) is satisfied,ν(_Ω⁺) > 0andλ < λ^∗₁. Then there exists a minimizer u∈^W₀^{1,pof (3.1)and u}∈ N_λ^1.

Proof. Sinceν(_Ω⁺) > 0, Lemma2.3 implies that N_λ¹ 6= _∅ for any λ ∈ Rand therefore there exists a minimizing sequence u_k ∈ Nλ¹,k∈^Nfor (3.1). Let us denote

c_λ :=inf{^Eλ(w):w∈ Nλ¹}^.

We havec_λ ∈(0,+_∞), sinceE_λ >c₂ >0 onN_λ^{1by Lemma}2.4. Hence, using the coercivity of E_λ onN_λ¹, given by Lemma2.4, we conclude that u^±_k are bounded inW₀^1,p. Hence, there exist u,v,w∈^W₀^1,p such that, up to subsequence,

u_k →^{u, u+}_k →^{v, u−}_k →^w, weakly inW₀^1,p and strongly inL^p∗(_Ω).

Let us introduce the map h: L^γ → ^{Lγ by h}(u) = u⁺. From [8, Lemma 2.3, p. 1046] it follows that h∈^C(L^γ,L^γ)_{. Hence,}u⁺=v≥^{0 andu−}=w≤^{0 in}Ω. Moreover,u^±6≡^{0 in} Ω.

Indeed, using Lemma2.4and Nehari constraintsQ_λ(u^±_k ) =0 we get 0< c₀ < lim

k→+_∞ Z

Ω|∇^u±_k |^pdx

= lim

k→+_∞

λ

Z

Ω|^u±_k |^pdx+

Z

Ω f|^u±_k |^γdx

=λ Z

Ω|^u±|^pdx+

Z

Ωf|^u±|^γdx.

Now we show thatu^±_k →^u±strongly inW₀^1,p. For this end note thatF(u^±)>0. Indeed, since F(u^±_k ) > 0, u^± are admissible functions for (1.2). Combining this fact with the weak lower semi-continuity of the norm inW₀^1,p, we get

0<(λ^∗₁−^λ)

Z

Ω|^u±|^pdx≤

Z

Ω|∇^u±|^pdx−^λ

Z

Ω|^u±|^pdx

≤^{lim inf}

k→+_∞

_Z

Ω|∇^u±_k |^pdx−^λ

Z

Ω|^u±_k |^pdx

=

Z

Ω f|^u±|^γdx.

From here it follows also that H_λ(u^±)>0.

Suppose now, by contradiction to the strong convergence inW₀^1,p, without loss of general- ity, thatk^u+k<lim inf_k→+_∞k^u+_k k^{. Since F}(u⁺)>0 and H_λ(u⁺)> 0, Lemma2.2 implies the existence of exactly one critical pointt(u⁺)>_{0 of}E_λ(tu⁺)_w.r.t.t >0, such that

E_λ(t(u⁺)u⁺)>0, Q_λ(t(u⁺)u⁺) =0.

By the same reason there existst(u⁻)>0, possibly equals to 1, such that E_λ(t(u⁻)u⁻)>0, Q_λ(t(u⁻)u⁻) =0.

Therefore,t(u⁺)u⁺+t(u⁻)u⁻∈ N_λ¹, and sinceu_k ∈ N_λ^{1, we get} E_λ(t(u⁺)u⁺+t(u⁻)u⁻)<lim inf

k→+_∞ E_λ(t(u⁺)u⁺_k ) +E_λ(t(u⁻)u⁺_k )

≤^{lim inf}

k→+_∞ E_λ(u⁺_k ) +E_λ(u⁺_k )=inf{^Eλ(w):w∈ Nλ¹}=c_λ. Thus, we get a contradiction. Consequently,u^±_k →^u±strongly inW₀^1,pandu∈ Nλ¹.

Now we adapt the proof of [4, Proposition 3.1, p. 8] to show that the minimizeru∈ N_λ^{1 of} (3.1) is, in fact, a solution of (D^).

Lemma 3.2. Assume that(1.1) is satisfied. If u ∈ N_λ¹ is a solution of (3.1), then DE_λ(u) = 0 in W^−1,p0(_Ω), i.e., u is a critical point of E_λ in W₀^1,p.

Proof. Letu∈ Nλ¹ is a solution of (3.1), i.e.,

Eλ(u) =cλ :=inf{^Eλ(w):w∈ Nλ¹}>0.

By Lemma2.2,t(u^±) =1 are the global maximum points of E_λ(tu^±)w.r.t.t >0 and hence E_λ(ru⁺+su⁻) =E_λ(ru⁺) +E_λ(su⁻)< E_λ(u⁺) +E_λ(u⁻) =E_λ(u) _(3.2)

for all (r,s)∈^R2+\ {(1, 1)}. Moreover, due to the fact thatE_λ(u^±)>0, we are able to choose κ >0 small enough, such that

t∈[1min−^κ,1+κ]E_λ(tu^±)>0. (3.3) Consider now the function

g: A:= (₁−^{κ, 1}+κ)² ⊂R²→^W₀^{1,p, g}(r,s) =ru⁺+su⁻. Hence, from (3.3) and (3.2) it follows that

0<_{c0 :}= _max

(r,s)∈^∂AEλ(_g(_r,s))<_cλ.

Assume now, by contradiction, thatDE_λ(u)6=0. Hence, using the continuity ofDE_λ we con- clude, that there exist some constantsα,δ > 0, such thatk^DEλ(v)k ≥ ^{αfor allv} ∈ ^U3δ(u) := {^w∈^W0^1,p :k^u−^wk<3δ}^.

Let us take some ε < min_c

λ−^c0

2 ,^αδ₈ and denote S_δ := U_2δ(u). Then the deformation lemma (see [20, Lemma 2.3, Parts (i), (v), (vi), p. 38]) implies the existence of homotopy η∈ ^C([0, 1]×^W0^1,p,W₀^1,p), such that

1) η(t,v) =vfor allt ∈[0, 1], if E_λ(v)<c_λ−^2ε, 2) E_λ(η(t,v))≤ ^Eλ(v)for allv ∈^W₀^1,pandt∈ [0, 1];

3) E_λ(η(t,v))<c_λ for all v∈ {^w∈^Sδ :E_λ(w)≤^c}^andt ∈(0, 1]. From 3)it follows that

{(r,s)∈^A:max^g(r,s)∈^Sδ}E_λ(η(t,g(r,s)))<c_λ, ∀^t ∈(_{0, 1}]_{. (3.4)} On the other hand, 2)and (3.2) imply that for allt ∈[0, 1]

{(r,s)∈^A:max^g(r,s)6∈^Sδ}E_λ(η(t,g(r,s)))≤ ^max

{(r,s)∈^A:g(r,s)6∈^Sδ}E_λ(g(r,s))< c_λ. (3.5) Furthermore, from 1)it follows thatη(t,g(r,s)) =g(r,s)for(r,s)∈ ^{∂Aand allt}∈[0, 1], since c₀ <c_λ−^2ε.

Now, due to the continuity ofηandE_λ, (3.3) implies the existence oft0 ∈(0, 1], such that E_λ(η^±(t,g(r,s)))>0 for allt∈ [0,t₀]and(r,s)∈ ^A.

Let us denote for simplicity

h(r,s):= η(t0,g(r,s)), and consider the maps

ψ₁: A→R², ψ₁(r,s):= Q_λ(h⁺(r,s)),Q_λ(h⁻(r,s)), ψ₂: A→^{R2, ψ}2(r,s):= Q_λ(ru⁺),Q_λ(su⁻).

Note that ψ₁(r,s) = (0, 0)if and only if h^±(r,s) ∈ Nλ. On the one hand, deg(ψ₂, 0,A) = 1, since there exists only one point (r,s) = (_{1, 1})∈ ^{A such that Q}λ(ru⁺)_, Q_λ(su⁻) =_{0 and the} Jacobian determinant

detJ_ψ2(1,1) = ^∂Q(ru⁺)

∂r r=1

· ^∂Q(su⁻)

∂s s=1

>_0.

On the other hand, sinceh(r,s) =g(r,s)for all(r,s)∈ ^{∂A, we get} ψ₁(r,s)≡^ψ2(r,s)_, (r,s)∈ ^∂A.

Consequently, using the homotopy invariance property of the degree (see [2, Theorem 3, (iv), p. 190 and Remark 7, (a), p. 192]), we get deg(ψ₁, 0,A) = deg(ψ₂, 0,A) = 1. Hence, there exists (r₀,s₀) ∈ ^{A such that Q}λ(h^±(r₀,s₀)) = 0. Furthermore, from the fact that E_λ(η^±(t0,g(r0,s0)))>0, we conclude thath(r0,s0)∈ N_λ^1.

Finally, from (3.4) and (3.5) we obtain

E_λ(h(r₀,s₀))<c_λ =inf{^Eλ(w): w∈ Nλ¹}^,

but it is a contradiction. Thus,DE_λ(u) =0 inW^−1,p0, i.e., u ∈ Nλ¹ is a critical point ofE_λ on W₀^1,p.

4 Least energy and number of nodal domains

Let us consider other subsets of the nodal Nehari setMλ:

Nλ² ={^u∈^W₀^{1,p :u±} ∈ Nλ, E_λ(u⁺)·^Eλ(u⁻)≤⁰}^, Nλ³ ={^u∈^W₀^{1,p :u±} ∈ Nλ, E_λ(u^±)<0}^.

It is easy to see thatMλ =N_λ¹∪ N_λ²∪ N_λ^3.

Lemma 4.1. Nλ²=_∅for allλ<λ₁^∗andNλ³ =_∅for allλ<λ₂.

Proof. 1. First we show thatN_λ²=_∅forλ<λ^∗₁. Assume, contrary to our claim, that for some λ < λ^∗₁ there exists w ∈ Nλ². Suppose first that E_λ(w⁺)·^Eλ(w⁻) < 0. From Lemma 2.1 it follows thatF(w⁺)·^F(w⁻)<0. Therefore, there existst>0, such that

F(tw⁺+w⁻) =t^γF(w⁺) +F(w⁻) =0.

This implies that tw⁺+w⁻ is an admissible function for minimization problem (1.2), which yields a contradiction, sinceλ<λ^∗₁.

Suppose now, without loss of generality, that E_λ(w⁺) = 0. Lemma 2.1 implies that F(w⁺) =0, and consequentlyw⁺is also an admissible function for (1.2), a contradiction.

2. Let us show thatN_λ³ =_∅forλ<λ₂. For this end we consider the critical point µ₂ := _inf

u∈^W0^1,p(_Ω) u_±6≡0

max

R

|∇^u+|^pdx R |^u+|^{pdx ,}

R |∇^u−|^pdx R |^u−|^pdx

. (4.1)

Proposition 4.2. µ₂=λ₂.

Proof. Note first that µ₂ ≤ ^λ2. Indeed, using the second eigenfunction ϕ₂ ∈ ^W₀^{1,p, which} corresponds toλ₂, as an admissible function for (4.1), we get

µ₂≤^max R

|∇^ϕ+₂|^pdx R |^ϕ₂⁺|^{pdx ,}

R |∇^ϕ−₂|^pdx R |^ϕ−₂|^pdx

=

R |∇^ϕ±₂|^pdx R |^ϕ±₂|^pdx =λ₂.

Now we show thatλ₂ ≤ ^µ2. Arguing as in the proof of [7, Proposition 4.2, p. 8] it is not hard to obtain a nonzero minimizer ψ2 ∈ ^W₀^1,p of (4.1), such that ψ₂^± 6= 0 and, due to the homogeneity of (4.1),R

Ω|^ψ2|^pdx=1. Consider the set A^:=

u∈^W₀^{1,p :u}=sψ₂⁺+tψ⁻₂, wheres,t ∈R, such that ^Z

Ω|^sψ+2 +tψ₂⁻|^pdx =1

. By construction,A ⊂ S^{, where}S is defined in (1.5). Moreover, taking

h(x,y)_:=|^x|^{2p−1x ψ}

+ 2

R |^ψ+2|^pdx1p

+|^y|^{2p−1y ψ−2} R |^ψ2⁻|^pdx1p

,

we conclude thath :S¹→ Ais continuous and odd, and consequentlyA ∈ F2. Therefore, λ₂ ≤^sup

u∈A Z

Ω|∇^u|^pdx= sup

s,t∈R:

R

Ω|^sψ+2+tψ₂⁻|^pdx=1

|^s|^p

Z

Ω|∇^ψ+2|^pdx+|^t|^p

Z

Ω|∇^ψ2⁻|^pdx

≤^µ2 sup

s,t∈R:

R

Ω|^sψ2⁺+tψ⁻₂|^pdx=1

|^s|^p

Z

Ω|^ψ+2|^pdx+|^t|^p

Z

Ω|^ψ−2|^pdx

=µ₂.

Hence,µ₂ =λ₂.

To finish the proof of Lemma 4.1 suppose a contradiction, i.e., there exists w ∈ N_λ^{3 for} someλ<λ₂. Lemma2.1implies that H_λ(w^±)<0, and therefore

R |∇^w±|^pdx

R |^w±|^pdx <λ<λ₂ =µ₂≤^max R

|∇^w+|^pdx R |^w+|^{pdx ,}

R |∇^w−|^pdx R |^w−|^pdx

, which is impossible.

The property of the least energy is given in the following lemma.

Lemma 4.3. Assume that(1.1)is satisfied,ν(_Ω⁺)>0and u_λ ∈ N_λ¹is a nodal solution of (D^)given by Lemma3.2. Then uλhas the least energy among all nodal solutions of (D^)on(−∞, min{^λ₁^∗,λ2}), i.e.,

−∞<E_λ(u_λ)≤^Eλ(w_λ), for any nodal solution w_λof (D⁾on this interval.

Proof. Lemma4.1 implies that N_λ^2,N_λ³ = _∅ forλ < min{^λ∗₁^,λ2}. Therefore, Mλ = N_λ¹ 6= _∅ for suchλ, due to Lemma 2.3, and thus any nodal solution of (D) belongs to Nλ¹. Sinceu_λ is obtained by minimization ofE_λ overN_λ¹, we get the desired result.

In the next lemma we prove nonexistence result for (D^).

Lemma 4.4. If ν(_Ω⁺) = 0, then there are no weak nodal solutions of the problem (D^{) for any λ} ∈ (−∞, min{^λ∗₁^,λ2}).

Proof. From the proof of Lemma4.3 it follows thatMλ = N_λ¹. However, Lemma2.3 implies that Nλ¹ = _∅ for any λ ∈ R, whenever ν(_Ω⁺) = 0. Thus, Mλ = ∅, which implies the nonexistence of weak nodal solutions for (D^{) on}(−∞, min{^λ∗₁^,λ2})_.

The next result gives the information about the precise number of nodal domains for solutions of (3.1).

Lemma 4.5. Assume that (1.1) is satisfied and λ < λ^∗₁. Then any solution u ∈ N_λ^{1 of (3.1) has} precisely two nodal domains.

Proof. Let u ∈ N_λ¹ be a solution of (3.1) and consequently a solution of (D). Recall that any solution of (D) is, in fact, of classC^1,α(_Ω),α∈(0, 1)(see Section 1). Suppose, by contradiction, that there exist three nodal domainsD_i, i= 1..3, and, without loss of generality, u> _{0 in} D₁ andD₃. We denoteu=u₁+u₂+u₃, where

u_i(x) =

(u(x) if x∈ ^Di,

0 if x∈Ω\^Di, i=1..3.

Hence,u_i ∈ ^C1,α(_Ω)andu₁,u₃> 0,u₂ <0 in their supports. Moreover, testing (D^{) byu}i one can getQ_λ(u_i) =0 for alli=1..3.

Assume first that E_λ(u_i)>0, i=1..3. However,u₁+u₂ ∈ N_λ^1andEλ(u₁+u₂)<E_λ(u) = c_λ. Hence, we get a contradiction.

Suppose now, without loss of generality, that E_λ(u₁) ≤^{0. Since E}λ(u₂) > 0, we conclude thatu₁+u₂∈ N_λ², which contradicts Lemma4.1.

5 Continuous branch

Let u_λ ∈ N_λ¹ be a nodal solution of the problem (D) given by a minimizer of (3.1). First we show that for any λ ∈ (−∞,λ^∗₁)and for any sequence ∆λ → 0, the corresponding sequence ofsolutions u_λ+_∆λ ∈ N_λ¹+_∆λ converges strongly inW₀^1,p, up to subsequence, to some u0 ∈ N_λ^1. It is not hard to see that for the sequenceu_λ+_∆λ we have

E_λ+_∆λ(u_λ+_∆λ)→^{c and DE}λ+_∆λ(u_λ+_∆λ) =0.

Moreover, using Lemma 2.4 it is not hard to show that there exist constants K₁,K₂, such that for all sufficiently small ∆λ it holds 0 < K₁ < k^uλ+_∆λk < K₂ < ∞. Hence, Sobolev’s embedding theorem and the Eberlein–Smulian theorem imply the existence ofu₀ ∈^W₀^1,psuch that, up to subsequence, u_λ+_∆λ → ^u0 strongly in L^p∗(_Ω) and u_λ+_∆λ * u₀ weakly in W₀^1,p. SinceDE_λ+_∆λ(u_λ+_∆λ) =0 for any∆λsmall enough, we have

h^DEλ+_∆λ(u_λ+_∆λ),u₀−^uλ+_∆λi=

Z

Ω|∇^uλ+_∆λ|^p−2∇^uλ+_∆λ∇(u₀−^uλ+_∆λ)dx

−(λ+_∆λ)

Z

Ω|^uλ+_∆λ|^p−2uλ+_∆λ(u0−^uλ+_∆λ)dx

−

Z

Ω f|^uλ+_∆λ|^γ−2uλ+_∆λ(u₀−^uλ+_∆λ)dx =0.

From here, using the strong convergenceu_λ+_∆λ →^u0 in L^p∗(_Ω), we obtain Z

Ω|∇^uλ+_∆λ|^p−2∇^uλ+_∆λ∇^u0dx−

Z

Ω|∇^uλ+_∆λ|^pdx →^0, which implies thatu_k →^u0 strongly inW₀^1,p. Moreover, since

Eλ(_u0) = _lim

∆λ→⁰Eλ+_∆λ(_uλ+∆λ)> _c2>_{0, Qλ}(_u0) = _lim

∆λ→⁰Qλ+_∆λ(_uλ+∆λ) =_0,