Open Access. ©2021 S. El Manouni et al., published by De Gruyter. This work is licensed under the Creative Commons Attri-
bution alone 4.0 License.
Adv. Nonlinear Anal. 2022; 11: 304–320
Research Article
Said El Manouni, Greta Marino, and Patrick Winkert*
Existence results for double phase problems
depending on Robin and Steklov eigenvalues
for the p-Laplacian
https://doi.org/10.1515/anona-2020-0193
Received February 18, 2021; accepted May 20, 2021.
Abstract: In this paper we study double phase problems with nonlinear boundary condition and gradient
dependence. Under quite general assumptions we prove existence results for such problems where the per-
turbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools,
truncation techniques and comparison methods. The obtained solutions depend on the first eigenvalues of
the Robin and Steklov eigenvalue problems for the p-Laplacian.
Keywords: Convection term, double phase operator, multiplicity results, nonlinear boundary condition,
Robin eigenvalue problem, Steklov eigenvalue problem
MSC: 35J15, 35J62, 35J92, 35P30
1Introduction
Let Ω⊂RN,N>1, be a bounded domain with Lipschitz boundary ∂Ω. We consider the following double
phase problem with nonlinear boundary condition and convection term given by
−div |∇u|p−2∇u+µ(x)|∇u|q−2∇u=h1(x,u,∇u)in Ω,
|∇u|p−2∇u+µ(x)|∇u|q−2∇u·ν=h2(x,u)on ∂Ω,
(1.1)
where ν(x)is the outer unit normal of Ωat the point x∈∂Ω,1<p<q<N,0≤µ(·)∈L1(Ω)and h1:Ω×R×
RN→Ras well as h2:∂Ω ×R→Rare Carathéodory functions which satisfy suitable structure conditions
and behaviors near the origin and at infinity, see Sections 3 and 4 for the precise assumptions.
The differential operator that appears in (1.1) is the so-called double phase operator which is defined by
−div |∇u|p−2∇u+µ(x)|∇u|q−2∇ufor u∈W1,H(Ω)(1.2)
with an appropriate Musielak-Orlicz Sobolev space W1,H(Ω), see its definition in Section 2. Note that when
infΩµ>0or µ≡0then the operator becomes the weighted (q,p)-Laplacian or the p-Laplacian, respectively.
Said El Manouni, Imam Mohammad Ibn Saud Islamic University (IMSIU), Faculty of Sciences, Department of Mathematics P. O.
Greta Marino, Technische Universität Chemnitz, Fakultät für Mathematik, Reichenhainer Straße 41, 09126 Chemnitz, Germany,
E-mail: greta.mar[email protected]u-chemnitz.de
*Corresponding Author: Patrick Winkert, Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623
Said El Manouni et al., Existence results for double phase problems |305
The energy functional J:W1,H(Ω)→Rrelated to the double phase operator (1.2) is given by
J(u) = Z
Ω|∇u|p+µ(x)|∇u|qdx,(1.3)
where the integrand has unbalanced growth. The main characteristic of the functional Jis the change of
ellipticity on the set where the weight function is zero, that is, on the set {x∈Ω:µ(x)=0}. Precisely, the
energy density of Jexhibits ellipticity in the gradient of order qon the points xwhere µ(x)is positive and of
order pon the points xwhere µ(x)vanishes.
The first who introduced and studied functionals whose integrands change their ellipticity according to
a point was Zhikov [37] (see also the monograph of Zhikov-Kozlov-Oleinik [38]) in order to provide models for
strongly anisotropic materials. Functionals stated in (1.3) have been intensively studied in the past decade
concerning regularity for isotropic and anisotropic functionals. We mention the papers of Baroni-Colombo-
Mingione [3–5], Baroni-Kuusi-Mingione [6], Byun-Oh [7], Colombo-Mingione [9, 10], Marcellini [21, 22], Ok
[25, 26], Ragusa-Tachikawa [33] and the references therein.
Inthispaperwearegoingtostudyproblem(1.1)concerningmultiplicityofsolutions.Inthefirstpartofthe
paper,seeSection 3,we provetheexistenceofanontrivialweaksolutionwhenthefunction h1dependsonthe
gradient of the solution.Hence,novariationaltoolslikecriticalpointtheoryareavailable.Wewillmakeuseof
the surjectivity result for pseudomonotone operators where in the proof the first eigenvalues of the Robin and
Steklov eigenvalue problems for the p-Laplacian play an important role. In the second part of the paper we
will skip the gradient dependence and prove the existence of two constant sign solutions, one is nonnegative
and the other one is nonpositive. Here, we need some stronger conditions on the nonlinearities, for example
superlinearity at ±∞. Again, the solutions depend on the first Robin and Steklov eigenvalues, respectively. We
will see that the Steklov eigenvalue problem is the more natural one for problems with nonlinear boundary
condition than the Robin eigenvalue problem.
There are only few works dealing with double phase operators along with a nonlinear boundary condi-
tion. Papageorgiou-Vetro-Vetro [29] studied the Robin problem
−div a(z)|∇u|p−2∇u)−∆qu+ξ(z)|u|p−2u=λf(z,u(z)) in Ω,
∂u
∂nθ
+β|u|p−2u= 0 on ∂Ω,
(1.4)
where 1<q<p<N,ξ∈L∞(Ω)is a positive potential, a(z)>0for a. a. z∈Ωand
∂u
∂nθ
= [a(z)|∇u|p−2+|∇u|q−2]∂u
∂n
with n(·)being the outward unit normal on ∂Ω. Under different assumptions it is shown that problem (1.4)
admits two nontrivial solutions uλ,ˆ
uλ∈W1,H(Ω)for small λ>0such that kuλk1,H→+∞and kˆ
uλk1,H→0
as λ→0+. In Papageorgiou-Rădulescu-Repovš [28] the authors proved the existence of multiple solutions in
the superlinear and the resonant case for the problem
−div a0(z)|∇u|p−2∇u)−∆qu+ξ(z)|u|p−2u=f(z,u(z)) in Ω,
∂u
∂nθ
+β|u|p−2u= 0 on ∂Ω,
where 1<q<p≤Nand with a positive Lipschitz function a0(·). Note that our assumptions and our treatment
differ from the ones in [28] and [29]. Also, we allow that the weight function could be zero at some points.
Recently, Gasiński-Winkert [17] considered the problem
−div |∇u|p−2∇u+µ(x)|∇u|q−2∇u=f(x,u)−|u|p−2u−µ(x)|u|q−2uin Ω,
|∇u|p−2∇u+µ(x)|∇u|q−2∇u·ν=g(x,u)on ∂Ω.
(1.5)
Based on the Nehari manifold method it is shown that problem (1.5) has at least three nontrivial solutions.
We point out that the proof for the constant sign solutions in [17] is based on a mountain-pass type argument
306 |Said El Manouni et al., Existence results for double phase problems
and so different from the treatment we used in this paper. Very recently, Farkas-Fiscella-Winkert [13] studied
singular Finsler double phase problems with nonlinear boundary condition and critical growth of the form
−div(A(u)) + up−1+µ(x)uq−1=up*−1+λuγ−1+g1(x,u)in Ω,
A(u)·ν=up*−1+g2(x,u)on ∂Ω,
u>0in Ω,
(1.6)
where
div(A(u)) := divFp−1(∇u)∇F(∇u) + µ(x)Fq−1(∇u)∇F(∇u)
is the so-called Finsler double phase operator and (RN,F)stands for a Minkowski space. The existence of one
weak solution of (1.6) is proven by applying variational tools and truncation techniques.
For existence results for double phase problems with homogeneous Dirichlet boundary condition we
refer to the papers of Colasuonno-Squassina [8] (eigenvalue problem for the double phase operator), Farkas-
Winkert [12] (Finsler double phase problems), Gasiński-Papageorgiou [14] (locally Lipschitz right-hand side),
Gasiński-Winkert [15, 16] (convection and superlinear problems), Liu-Dai [19] (Nehari manifold approach),
Marino-Winkert[23] (systemsofdoublephaseoperators),Perera-Squassina[31](Morsetheoreticalapproach),
Zeng-Bai-Gasiński-Winkert[35,36](multivaluedobstacleproblems)andthereferencestherein.Relatedworks
dealing with certain types of double phase problems can be found in the works of Bahrouni-Rădulescu-
Winkert [1] (Baouendi-Grushin operator), Barletta-Tornatore [2] (convection problems in Orlicz spaces), Liu-
Dai [20] (unbounded domains), Papageorgiou-Rădulescu-Repovš [27] (discontinuity property for the spec-
trum), Rădulescu [32] (overview about isotropic and anisotropic double phase problems) and Zeng-Bai-
Gasiński-Winkert [34] (convergence properties for double phase problems). Finally, we mention the nice
overview article of Mingione-Rădulescu [24] about recent developments for problems with nonstandard
growth and nonuniform ellipticity.
The paper is organized as follows. In Section 2 we recall the main properties of the double phase operator
including the properties of the Musielak-Orlicz Sobolev space W1,H(Ω). In Section 3 we prove the existence
of at least one solution of (1.1) when h1depends on the gradient of the solution, see Theorem 3.1. The proof
is based on the surjectivity result for pseudomonotone operators and on the properties of the eigenvalues
of the Robin and Steklov eigenvalue problems for the p-Laplacian. Finally, in the last section, we skip the
convection term and use variational tools in order to prove the existence of two constant sign solutions for
superlinear problems. We consider two different problems. The first problem is treated by properties of the
first Steklov eigenvalue and the second one by the first Robin eigenvalue, see Theorems 4.1 and 4.2.
2Preliminaries
In this section we recall some definitions and present the main tools which will be needed in the sequel.
For every 1≤r<∞we denote by Lr(Ω)and Lr(Ω;RN)the usual Lebesgue spaces equipped with the
norm k·krand for 1<r<∞we consider the corresponding Sobolev space W1,r(Ω)endowed with the norm
k·k1,r. It is known that W1,r(Ω),→Lˆ
r(Ω)is compact forˆ
r<r*and continuous forˆ
r=r*, where r*is the critical
exponent of rdefined by
r*=(Nr
N−rif r<N,
any `∈(r,∞)if r≥N.(2.1)
On the boundary ∂Ω of Ωwe consider the (N−1)-dimensional Hausdorff (surface) measure σand denote
by Lr(∂Ω)the boundary Lebesgue space with norm k·kr,∂Ω. From the definition of the trace mapping we know
that W1,r(Ω),→L˜
r(∂Ω)is compact for ˜
r<r*and continuous for ˜
r=r*, where r*is the critical exponent of r
Said El Manouni et al., Existence results for double phase problems |307
on the boundary given by
r*=((N−1)r
N−rif r<N,
any `∈(r,∞)if r≥N.(2.2)
For simplification we will avoid the notation of the trace operator throughout the paper.
In the entire paper we will assume that
1<p<q<Nand 0≤µ(·)∈L1(Ω).(2.3)
Note that the conditions in (2.3) are quite general. In all the other mentioned works for Neumann double
phase problems (see, for example, [13], [17], [28], [29]) the condition
Nq
N+q−1<p
is needed, which is equivalent to q<p*and so q<p*is also satisfied. We do not need this restriction in the
current paper.
Let H:Ω×[0,∞)→[0,∞)be the function defined by
H(x,t) = tp+µ(x)tq.
Based on this we can introduce the modular function given by
ρH(u) := Z
Ω
H(x,|u|)dx=Z
Ω|u|p+µ(x)|u|qdx.
Then, the Musielak-Orlicz space LH(Ω)is defined by
LH(Ω) = nuu:Ω→Ris measurable and ρH(u)<+∞o
equipped with the Luxemburg norm
kukH= inf nτ>0 : ρHu
τ≤1o.
From Colasuonno-Squassina [8, Proposition 2.14] we know that the space LH(Ω)is a reflexive Banach space.
Moreover, we need the seminormed space
Lq
µ(Ω) =
uu:Ω→Ris measurable and Z
Ω
µ(x)|u|qdx<+∞
,
which is endowed with the seminorm
kukq,µ=
Z
Ω
µ(x)|u|qdx
1
q
.
Analogously, we define Lq
µ(Ω;RN).
The Musielak-Orlicz Sobolev space W1,H(Ω)is defined by
W1,H(Ω) = nu∈LH(Ω) : |∇u| ∈ LH(Ω)o
equipped with the norm
kuk1,H=k∇ukH+kukH,
where k∇ukH=k |∇u| kH. As before, we know that W1,H(Ω)is a reflexive Banach space.
The following proposition states the main embedding results for the spaces LH(Ω)and W1,H(Ω). We
refer to Crespo-Blanco-Gasiński-Harjulehto-Winkert [11, Proposition 2.17].
308 |Said El Manouni et al., Existence results for double phase problems
Proposition 2.1. Let (2.3) be satisfied and let
p*:= Np
N−pand p*:= (N−1)p
N−p(2.4)
be the critical exponents to p, see (2.1) and (2.2) for r=p. Then the following embeddings hold:
(i) LH(Ω),→Lr(Ω)and W1,H(Ω),→W1,r(Ω)are continuous for all r∈[1,p];
(ii) W1,H(Ω),→Lr(Ω)is continuous for all r∈[1,p*];
(iii) W1,H(Ω),→Lr(Ω)is compact for all r∈[1,p*);
(iv) W1,H(Ω),→Lr(∂Ω)is continuous for all r∈[1,p*];
(v) W1,H(Ω),→Lr(∂Ω)is compact for all r∈[1,p*);
(vi) LH(Ω),→Lq
µ(Ω)is continuous.
We equip the space W1,H(Ω)with the equivalent norm
kuk0:= inf
λ>0 : Z
Ω|∇u|
λp
+µ(x)|∇u|
λq
+|u|
λp
+µ(x)|u|
λqdx≤1
.
For u∈W1,H(Ω)let
ˆ
ρH(u) = Z
Ω|∇u|p+µ(x)|∇u|qdx+Z
Ω|u|p+µ(x)|u|qdx.(2.5)
Based on the proof of Liu-Dai [19, Proposition 2.1] we have the following relations between the norm k·k0
and the modular function ˆ
ρH, see also Crespo-Blanco-Gasiński-Harjulehto-Winkert [11, Proposition 2.16].
Proposition 2.2. Let (2.3) be satisfied, let y∈W1,H(Ω)and let ˆ
ρHbe defined as in (2.5).
(i) If y=0, then kyk0=λif and only if ˆ
ρH(y
λ) = 1;
(ii) kyk0<1(resp. >1,= 1) if and only if ˆ
ρH(y)<1(resp. >1,= 1);
(iii) If kyk0<1, then kykq
0⩽ˆ
ρH(y)⩽kykp
0;
(iv) If kyk0>1, then kykp
0⩽ˆ
ρH(y)⩽kykq
0;
(v) kyk0→0if and only if ˆ
ρH(y)→0;
(vi) kyk0→+∞if and only if ˆ
ρH(y)→+∞.
Let us recall some definitions which we will need in the next sections.
Definition 2.3. Let (X,k·kX)be a reflexive Banach space, X*its dual space and denote by h·,·iits duality
pairing. Let A:X→X*, then Ais called
(i) to satisfy the (S+)-property if un*uin Xand lim supn→∞hAun,un−ui≤0imply un→uin X;
(ii) pseudomonotone if un*uin Xand lim supn→∞hAun,un−ui≤0imply Aun*Au and hAun,uni→
hAu,ui;
(iii) coercive if
lim
kukX→∞
hAu,ui
kukX
=∞.
Remark 2.4. The classical definition of pseudomonotonicity is the following one: From un*uin Xand
limsupn→∞hAun,un−ui≤0we have
liminf
n→∞ hAun,un−vi≥hAu,u−vifor all v∈X.
This definition is equivalent to the one in Definition 2.3(ii) when the operator is bounded. Since we are only
considering bounded operators, we will use the one in Definition 2.3(ii).
Said El Manouni et al., Existence results for double phase problems |309
The following surjectivity result for pseudomonotone operators will be used in Section 3. It can be found, for
example, in Papageorgiou-Winkert [30, Theorem 6.1.57].
Theorem 2.5. Let Xbe a real, reflexive Banach space, let A:X→X*be a pseudomonotone, bounded, and
coercive operator, and let b∈X*. Then, a solution to the equation Au =bexists.
Let A:W1,H(Ω)→W1,H(Ω)*be the nonlinear map defined by
hA(u),φiH=Z
Ω|∇u|p−2∇u+µ(x)|∇u|q−2∇u·∇φdx
+Z
Ω|u|p−2u+µ(x)|u|q−2uφdx
(2.6)
for all u,φ∈W1,H(Ω), where h·,·iHis the duality pairing between W1,H(Ω)and its dual space W1,H(Ω)*.
Theoperator A:W1,H(Ω)→W1,H(Ω)*hasthefollowingproperties,see Crespo-Blanco-Gasiński-Harjulehto-
Winkert [11, Proposition 3.5].
Proposition 2.6. Let (2.3) be satisfied. Then, the operator Adefined by (2.6) is bounded (that is, it maps
bounded sets into bounded sets), continuous, strictly monotone (hence maximal monotone) and it is of type
(S+).
For s∈R, we set s±= max{±s,0}and for u∈W1,H(Ω)we define u±(·) = u(·)±. We have
u±∈W1,H(Ω),|u|=u++u−,u=u+−u−.
For r>1we write r′=r
r−1.
Further, we denote by C1(Ω)+the positive cone
C1(Ω)+=nu∈C1(Ω) : u(x)≥0for all x∈Ωo
of the ordered Banach space C1(Ω). This cone has a nonempty interior given by
intC1(Ω)+=nu∈C1(Ω) : u(x)>0for all x∈Ωo.
Let us now recall some basic facts about the spectrum of the negative r-Laplacian with Robin and Steklov
boundary condition, respectively, for 1<r<∞. We refer to the paper of Lê [18]. The r-Laplacian eigenvalue
problem with Robin boundary condition is given by
−∆ru=λ|u|r−2uin Ω,
|∇u|r−2∇u·ν=−β|u|r−2uon ∂Ω,(2.7)
where β>0. We know that problem (2.7) has a smallest eigenvalue λR
1,r,β>0which is isolated, simple and it
can be variationally characterized by
λR
1,r,β= inf
u∈W1,r(Ω)\{0}RΩ|∇u|rdx+βR∂Ω |u|rdσ
RΩ|u|rdx.(2.8)
By uR
1,r,βwe denote the normalized (that is, kuR
1,r,βkr= 1) positive eigenfunction corresponding to λR
1,r,β. We
know that uR
1,r,β∈intC1(Ω)+.
Further, we recall the r-Laplacian eigenvalue problem with Steklov boundary condition which is given
by
−∆ru=−|u|r−2uin Ω,
|∇u|r−2∇u·ν=λ|u|r−2uon ∂Ω.(2.9)
310 |Said El Manouni et al., Existence results for double phase problems
As before, problem (2.9) has a smallest eigenvalue λS
1,r>0which is isolated, simple and which can be char-
acterized by
λS
1,r= inf
u∈W1,r(Ω)\{0}RΩ|∇u|rdx+RΩ|u|rdx
R∂Ω |u|rdσ.(2.10)
The first eigenfunction associated to the first eigenvalue λS
1,rwill be denoted by uS
1,rand we can assume it is
normalized, that is, kuS
1,rkr,∂Ω = 1. We have uS
1,r∈intC1(Ω)+.
3Existence results in case of convection
In this section we are interested in the existence of a solution of problem (1.1) depending on the first eigen-
values of the Robin and Steklov eigenvalue problems of the p-Laplacian. We choose
h1(x,s,ξ) = f(x,s,ξ)−|s|p−2s−µ(x)|s|q−2sfor a. a. x∈Ω,
h2(x,s) = g(x,s)−ζ|s|p−2sfor a. a. x∈∂Ω,
for all s∈Rand for all ξ∈RNwith ζ>0specified later and Carathéodory functions fand gcharacterized
in hypotheses (H1) below. Then (1.1) becomes
−div(|∇u|p−2∇u+µ(x)|∇u|q−2∇u) = f(x,u,∇u)−|u|p−2u−µ(x)|u|q−2uin Ω,
(|∇u|p−2∇u+µ(x)|∇u|q−2∇u)·ν=g(x,u)−ζ|u|p−2uon ∂Ω,(3.1)
where we assume the following hypotheses:
(H1) The mappings f:Ω×R×RN→Rand g:∂Ω ×R→Rare Carathéodory functions with f(x,0,0) =0for
a. a. x∈Ωsuch that the following conditions are satisfied:
(i) There exist α1∈L
r1
r1−1(Ω),α2∈L
r2
r2−1(∂Ω)and a1,a2,a3≥0such that
|f(x,s,ξ)|≤a1|ξ|pr1−1
r1+a2|s|r1−1+α1(x)for a. a. x∈Ω,
|g(x,s)|≤a3|s|r2−1+α2(x)for a. a. x∈∂Ω,
for all s∈Rand for all ξ∈RN, where 1<r1<p*and 1<r2<p*with the critical exponents p*and
p*stated in (2.4).
(ii) There exist w1∈L1(Ω),w2∈L1(∂Ω)and b1,b2,b3≥0such that
f(x,s,ξ)s≤b1|ξ|p+b2|s|p+w1(x)for a. a. x∈Ω,
g(x,s)s≤b3|s|p+ω2(x)for a. a. x∈∂Ω,
for all s∈Rand for all ξ∈RN.
A function u∈W1,H(Ω)is called a weak solution of problem (3.1) if
Z
Ω|∇u|p−2∇u+µ(x)|∇u|q−2∇u·∇φdx+Z
Ω|u|p−2u+µ(x)|u|q−2uφdx
=Z
Ω
f(x,u,∇u)φdx+Z
∂Ω
g(x,u)φdσ−ζZ
∂Ω
|u|p−2uφ dσ
(3.2)
is satisfied for all φ∈W1,H(Ω). It is clear that this definition is well-defined.
The main result in this section is the following one.
Said El Manouni et al., Existence results for double phase problems |311
Theorem 3.1. Let hypotheses (2.3) and (H1) be satisfied. Then, there exists a nontrivial weak solution ˆ
u∈
W1,H(Ω)∩L∞(Ω)of problem (3.1) provided one of the following assertions is satisfied:
(A) b1+b2λR
1,p,β−1<1and b2βλR
1,p,β−1+b3<ζ;
(B) max{b1,b2}+b3λS
1,p−1<1and ζ≥0.
Here λR
1,p,βis the first eigenvalue of the p-Laplacian with Robin boundary condition with β>0and λS
1,pstands
for the first eigenvalue of the p-Laplacian with Steklov boundary condition, see (2.7) and (2.9), respectively.
Proof. Let ˜
Nf:W1,H(Ω)⊂Lr1(Ω)→Lr′
1(Ω)and ˜
Ng:Lr2(∂Ω)→Lr′
2(∂Ω)be the Nemytskij operators corre-
sponding to the functions f:Ω×R×RN→Rand g:∂Ω ×R→R, respectively. Furthermore, we denote
by i*:Lr′
1(Ω)→W1,H(Ω)*the adjoint operator of the embedding i:W1,H(Ω)→Lr1(Ω)and j*:Lr′
2(∂Ω)→
W1,H(Ω)*stands for the adjoint operator of the embedding j:W1,H(Ω)→Lr2(∂Ω). Then we define
Nf:= i*◦˜
Nf:W1,H(Ω)→W1,H(Ω)*,
Ng:= j*◦˜
Ng◦j:W1,H(Ω)→W1,H(Ω)*,
which are both bounded and continuous operators due to hypothesis (H1)(i). Moreover, we define
Nζ:W1,H(Ω)→W1,H(Ω)*by
Nζ:= i*
ζ◦ζ|·|p−2·◦iζ,
where i*
ζ:Lp′(Ω)→W1,H(Ω)*is the adjoint operator of the embedding iζ:W1,H(Ω)→Lp(Ω).
Now we can define the operator A:W1,H(Ω)→W1,H(Ω)*given by
A(u) := A(u)−Nf(u)−Ng(u) + Nζ(u).
Taking the growth conditions in (H1)(i) into account, it is clear that A:W1,H(Ω)→W1,H(Ω)*maps bounded
sets into bounded sets. In order to show the pseudomonotonicity, let {un}n∈N⊂W1,H(Ω)be such that
un*uin W1,H(Ω)and limsup
n→∞ hAun,un−uiH≤0.(3.3)
From the compact embeddings W1,H(Ω),→Lˆ
r(Ω)for any ˆ
r<p*and W1,H(Ω),→L˜
r(∂Ω)for any ˜
r<p*, see
Proposition 2.1(iii) and (v), along with (3.3) we have
un→uin Lr1(Ω)and un→uin Lr2(∂Ω),Lp(∂Ω).
Applying the growth conditions in (H1)(i) along with Hölder’s inequality gives
Z
Ω
f(x,un,∇un)(un−u)dx
≤a1Z
Ω
|∇un|pr1−1
r1|un−u|dx+a2Z
Ω
|un|r1−1|un−u|dx+Z
Ω
|α1(x)| |un−u|dx
≤a1k∇unkpr1−1
r1
pkun−ukr1+a2kunkr1−1
r1kun−ukr1+kα1kr1
r1−1kun−ukr1−→ 0
and
Z
∂Ω
g(x,un)(un−u)dσ≤a3Z
∂Ω
|un|r2−1|un−u|dσ+Z
∂Ω
|α2(x)| |un−u|dσ
≤a3kunkr2−1
r2,∂Ωkun−ukr2,∂Ω +kα2kr2
r2−1,∂Ωkun−ukr2,∂Ω −→ 0.
312 |Said El Manouni et al., Existence results for double phase problems
Furthermore, again by Hölder’s inequality, we have
ζZ
∂Ω
|un|p−2un(un−u)dσ≤ζkunkp−1
p,∂Ωkun−ukp,∂Ω −→ 0.
Replacing uby unand φby un−uin the weak formulation in (3.2) and using the considerations above leads
to
limsup
n→∞ hA(un),un−uiH= limsup
n→∞ hA(un),un−uiH≤0.(3.4)
From Proposition 2.6 we know that Afulfills the (S+)-property. Therefore, from (3.3) and (3.4) we conclude
that
un→uin W1,H(Ω).
Since Ais continuous we have A(un)→A(u)in W1,H(Ω)*which shows that Ais pseudomonotone.
Let us now prove that A:W1,H(Ω)→W1,H(Ω)*is coercive. We distinguish between two cases.
Case I: Condition (A) is satisfied.
From the p-Laplace eigenvalue problem with Robin boundary condition, see (2.7) and (2.8) for r=p, we
know that
kukp
p≤λR
1,p,β−1k∇ukp
p+βkukp
p,∂Ωfor all u∈W1,p(Ω).(3.5)
Let u∈W1,H(Ω)be such that kuk0>1and note that W1,H(Ω)⊆W1,p(Ω). Then, from (H1)(ii), (3.5), (A) and
Proposition 2.2(iv) we obtain
hA(u),uiH=Z
Ω|∇u|p−2∇u+µ(x)|∇u|q−2∇u·∇udx+Z
Ω|u|p−2u+µ(x)|u|q−2uudx
−Z
Ω
f(x,u,∇u)udx−Z
∂Ω
g(x,u)udσ+ζZ
∂Ω
|u|pdσ
≥k∇ukp
p+k∇ukq
q,µ+kukp
p+kukq
q,µ−b1k∇ukp
p−b2kukp
p−kω1k1
−b3kukp
p,∂Ω −kω2k1,∂Ω +ζkukp
p,∂Ω
≥1−b1−b2λR
1,p,β−1k∇ukp
p+kukp
p+k∇ukq
q,µ+kukq
q,µ
+ζ−b2βλR
1,p,β−1−b3kukp
p,∂Ω −kω1k1−kω2k1,∂Ω
≥1−b1−b2λR
1,p,β−1k∇ukp
p+kukp
p+k∇ukq
q,µ+kukq
q,µ−kω1k1−kω2k1,∂Ω
=1−b1−b2λR
1,p,β−1ˆ
ρH(u)−kω1k1−kω2k1,∂Ω
≥1−b1−b2λR
1,p,β−1kukp
0−kω1k1−kω2k1,∂Ω .
This shows the coercivity of A.
Case II: Condition (B) is satisfied.
From the Steklov p-Laplace eigenvalue problem, see (2.9) and (2.10) for r=p, we have the inequality
kukp
p,∂Ω ≤λS
1,p−1k∇ukp
p+kukp
pfor all u∈W1,p(Ω).(3.6)
As before, let u∈W1,H(Ω)be such that kuk0>1and note again that W1,H(Ω)⊆W1,p(Ω). Applying (H1)(ii),
(3.6), (B) and Proposition 2.2(iv) one gets
hA(u),uiH=Z
Ω|∇u|p−2∇u+µ(x)|∇u|q−2∇u·∇udx+Z
Ω|u|p−2u+µ(x)|u|q−2uudx
Said El Manouni et al., Existence results for double phase problems |313
−Z
Ω
f(x,u,∇u)udx−Z
∂Ω
g(x,u)udσ+ζZ
∂Ω
|u|pdσ
≥k∇ukp
p+k∇ukq
q,µ+kukp
p+kukq
q,µ−b1k∇ukp
p−b2kukp
p−kω1k1
−b3kukp
p,∂Ω −kω2k1,∂Ω +ζkukp
p,∂Ω
≥1−max{b1,b2}−b3λS
1,p−1k∇ukp
p+kukp
p+k∇ukq
q,µ+kukq
q,µ
−kω1k1−kω2k1,∂Ω
≥1−max{b1,b2}−b3λS
1,p−1ˆ
ρH(u)−kω1k1−kω2k1,∂Ω
≥1−max{b1,b2}−b3λS
1,p−1kukp
0−kω1k1−kω2k1,∂Ω .
Hence, A:W1,H(Ω)→W1,H(Ω)*is again coercive.
We have shown that A:W1,H(Ω)→W1,H(Ω)*is a bounded, pseudomonotone and coercive operator.
From Theorem 2.5 we find an element ˆ
u∈W1,H(Ω)such that A(ˆ
u)=0with ˆ
u=0since f(x,0,0) =0
for a. a. x∈Ω. In view of the definition of A, we see that ˆ
uturns out to be a nontrivial weak solution of
problem (3.1). Similar to Theorem 3.1 of Gasiński-Winkert [17] we can show the boundedness of ˆ
u. The proof
is complete.
4Constant sign solutions for superlinear perturbations
In this section we are interested in constant sign solutions for problems of type (1.1) without convection term
but with superlinear nonlinearities. We are going to consider the cases of the dependence on Robin and
Steklov eigenvalues separately. We start with the Steklov case and set
h1(x,s,ξ) = −ϑ|s|p−2s−µ(x)|s|q−2s−f(x,s)for a. a. x∈Ω,
h2(x,s) = ζ|s|p−2s−g(x,s)for a. a. x∈∂Ω,
for all s∈R,ϑ,ζ>0to be specified and Carathéodory functions fand gwhich satisfy hypotheses (H2) below.
With this choice, (1.1) can be written as
−div(|∇u|p−2∇u+µ(x)|∇u|q−2∇u) = −ϑ|u|p−2u−µ(x)|u|q−2u−f(x,u)in Ω,
(|∇u|p−2∇u+µ(x)|∇u|q−2∇u)·ν=ζ|u|p−2u−g(x,u)on ∂Ω,(4.1)
where the following conditions are supposed:
(H2) The nonlinearities f:Ω×R→Rand g:∂Ω ×R→Rare assumed to be Carathéodory functions which
satisfy the subsequent hypotheses:
(i) fand gare bounded on bounded sets.
(ii) It holds
lim
s→±∞
f(x,s)
|s|q−2s= +∞uniformly for a. a. x∈Ω,
lim
s→±∞
g(x,s)
|s|q−2s= +∞uniformly for a. a. x∈∂Ω.
(iii) It holds
lim
s→0
f(x,s)
|s|q−2s= 0 uniformly for a. a. x∈Ω,
lim
s→0
g(x,s)
|s|p−2s= 0 uniformly for a. a. x∈∂Ω.
314 |Said El Manouni et al., Existence results for double phase problems
We say that u∈W1,H(Ω)is a weak solution of problem (4.1) if
Z
Ω|∇u|p−2∇u+µ(x)|∇u|q−2∇u·∇φdx+Z
Ωϑ|u|p−2u+µ(x)|u|q−2uφdx
=Z
Ω−f(x,u)φdx+Z
∂Ω ζ|u|p−2u−g(x,u)φdσ
is fulfilled for all φ∈W1,H(Ω).
The following theorem states the existence of constant sign solutions where the parameter ζdepends on
the first Steklov eigenvalue for the p-Laplacian, namely λS
1,p.
Theorem 4.1. Let hypotheses (2.3) and (H2) be satisfied. Furthermore, let ϑ∈(0,1] and let ζ>λS
1,pwith λS
1,p
being the first eigenvalue of the Steklov eigenvalue problem of the p-Laplacian stated in (2.9). Then, problem
(4.1) has at least two nontrivial weak solutions u0,v0∈W1,H(Ω)∩L∞(Ω)such that u0≥0and v0≤0.
Proof. From hypothesis (H2)(ii) we know that we can find constants M1,M2=M2(ζ)>1such that
f(x,s)s≥|s|qfor a.a. x∈Ωand all |s|≥M1,
g(x,s)s≥ζ|s|qfor a.a. x∈Ωand all |s|≥M2.(4.2)
We set M3= max(M1,M2)and take a constant function u≡ς∈[M3,+∞). Applying (4.2), p<qand M3>1
yields
0≥ −f(x,u)for a. a. x∈Ωand 0≥ζ up−1−g(x,u)for a. a. x∈∂Ω.(4.3)
Analogously, we can choose v≡−ςin order to get
0≤ −f(x,v)for a. a. x∈Ωand 0≤ζ|v|p−2v−g(x,v)for a. a. x∈∂Ω.
Now, we introduce the cut-off functions θ±:Ω×R→Rand θ±
ζ:∂Ω ×R→Rdefined by
θ+(x,s) =
0if s<0
−f(x,s)if 0≤s≤u
−f(x,u)if u<s
,
θ+
ζ(x,s) =
0if s<0
ζsp−1−g(x,s)if 0≤s≤u
ζ up−1−g(x,u)if u<s
,
θ−(x,s) =
−f(x,v)if s<v
−f(x,s)if v≤s≤0
0if 0<s
,
θ−
ζ(x,s) =
ζ|v|p−2v−g(x,v)if s<v
ζ|s|p−2s−g(x,s)if v≤s≤0
0if 0<s
,
(4.4)
which are Carathéodory functions. We set
Θ±(x,s) =
s
Z
0
θ±(x,t)dtand Θ±
ζ(x,s) =
s
Z
0
θ±
ζ(x,t)dt.
Now we consider the C1-functionals Γ±:W1,H(Ω)→Rdefined by
Γ±(u) = 1
pk∇ukp
p+1
qk∇ukq
q,µ+ϑ
pkukp
p+1
qkukq
q,µ−Z
Ω
Θ±(x,u)dx−Z
∂Ω
Θ±
ζ(x,u)dσ.
Said El Manouni et al., Existence results for double phase problems |315
Furthermore, we write F(x,s) = Rs
0f(x,t)dtand G(x,s) = Rs
0g(x,t)dt.
We first investigate the existence of the nonnegative solution. Due to the truncations in (4.4) it is clear that
the functional Γ+is coercive and also sequentially weakly lower semicontinuous. Hence, its global minimizer
u0∈W1,H(Ω)exists, that is
Γ+(u0) = inf hΓ+(u) : u∈W1,H(Ω)i.
From hypotheses (H2)(iii), for given ε1,ε2>0, there exist δ1=δ1(ε1),δ2=δ2(ε2)∈(0,u)such that
F(x,s)≤ε1
q|s|qfor a.a. x∈Ωand for all |s|≤δ1,
G(x,s)≤ε2
p|s|pfor a.a. x∈∂Ω and for all |s|≤δ2.(4.5)
We set δ:= min(δ1,δ2). Recall that uS
1,pis the first eigenfunction corresponding to the first eigenvalue λS
1,pof
the eigenvalue problem of the p-Laplacian with Steklov boundary condition, see (2.9). We may suppose that
it is normalized, that is, kuS
1,pkp,∂Ω = 1. Since uS
1,p∈intC1(Ω)+, we may choose t∈(0,1) small enough
such that tuS
1,p(x)∈[0,δ]for all x∈Ω. Because of (4.4), (4.5) and δ<uit follows that
Γ+tuS
1,p
=1
p
∇tuS
1,p
p
p+1
q
∇tuS
1,p
q
q,µ+ϑ
p
tuS
1,p
p
p+1
q
tuS
1,p,
q
q,µ
−Z
Ω
Θ+x,tuS
1,pdx−Z
∂Ω
Θ+
ζx,tuS
1,pdσ
≤tp
pλS
1,p+tq
q
∇uS
1,p
q
q,µ+tq
q
uS
1,p
q
q,µ+Z
Ω
Fx,tuS
1,pdx−ζtp
p
+Z
∂Ω
Gx,tuS
1,pdσ
≤tp
pλS
1,p+tq
q
∇uS
1,p
q
q,µ+tq
q
uS
1,p
q
q,µ+ε1tq
q
uS
1,p
q
q−ζtp
p+ε2tp
p
=tp λS
1,p−ζ+ε2
p!+tq
∇uS
1,p
q
q,µ+
uS
1,p
q
q,µ+ε1
uS
1,p
q
q
q
.
(4.6)
By assumption, we know that ζ>λS
1,p. So we may choose ε1,ε2>0such that
0<ε1<∞and 0<ε2<ζ−λS
1,p.
From this choice and since p<qwe obtain from (4.6)
Γ+tuS
1,p<0for all sufficiently small t>0.
Therefore, we know now that
Γ+(u0)<0 = Γ+(0).
Hence, u0=0.
Since u0is a global minimizer of Γ+we have (Γ+)′(u0) = 0, that is,
Z
Ω|∇u0|p−2∇u0+µ(x)|∇u0|q−2∇u0·∇φdx
+Z
Ωϑ|u0|p−2u0+µ(x)|u0|q−2u0φdx
=Z
Ω
θ+(x,u0)φdx+Z
∂Ω
θ+
ζ(x,u0)φdσ
(4.7)
316 |Said El Manouni et al., Existence results for double phase problems
for all φ∈W1,H(Ω). First we take φ=−u−
0∈W1,H(Ω)as test function in (4.7). We obtain
k∇u−
0kp
p+k∇u−
0kq
q,µ+ku−
0kp
p+ku−
0kq
q,µ= 0,
which yields u−
0= 0 and so u0≥0. Second we choose φ=(u0−u)+∈W1,H(Ω)as test function in (4.7) which
results in
Z
Ω|∇u0|p−2∇u0+µ(x)|∇u0|q−2∇u0·∇(u0−u)+dx
+Z
Ωϑup−1
0+µ(x)uq−1
0(u0−u)+dx
=Z
Ω
θ+(x,u0)(u0−u)+dx+Z
∂Ω
θ+
ζ(x,u0)(u0−u)+dσ
=Z
Ω
(−f(x,u))(u0−u)+dx+Z
∂Ω ζ up−1−g(x,u)(u0−u)+dσ
≤0,
(4.8)
by (4.3). First note that
Z
Ω|∇u0|p−2∇u0+µ(x)|∇u0|q−2∇u0·∇(u0−u)+dx
≥ϑZ
Ω|∇(u0−u)+|p+µ(x)|∇(u0−u)+|qdx.
(4.9)
Since u0>u>1on the set {u0>u}we have
Z
Ωϑup−1
0+µ(x)uq−1
0(u0−u)+dx
≥ϑZ
{u0>u}up−1
0+µ(x)uq−1
0(u0−u)dx
≥ϑZ
{u0>u}(u0−u)p−1+µ(x)(u0−u)q−1(u0−u)dx
=ϑZ
Ω((u0−u)+)p+µ(x)((u0−u)+)qdx.
(4.10)
Combining (4.8) with (4.9) as well as (4.10) and using Proposition 2.2(iii), (iv) implies that
ϑmin{k(u0−u)+kp
0,k(u0−u)+kq
0}≤ϑˆ
ρH((u0−u)+)≤0.
Hence, u0≤uandso u0∈[0,u].Bythedefinitionofthe truncationsin (4.4)wesee that u0∈W1,H(Ω)∩L∞(Ω)
turns out to be a weak solution of our original problem (4.1).
For the nonpositive solution we consider the functional Γ−:W1,H(Ω)→Rand show in the same way
that it has a global minimizer v0∈W1,H(Ω)which belongs to [v,0].
Let us study now the case when the solutions depend on the first Robin eigenvalue. We set
h1(x,s,ξ) = (ζ−ϑ)|s|p−2s−µ(x)|s|q−2s−f(x,s)for a. a. x∈Ω,
h2(x,s) = −β|s|p−2sfor a. a. x∈∂Ω,
for all s∈Rwith parameters ζ>ϑ>0to be specified, β>0is the same parameter as in the Robin eigenvalue
problem and fis a Carathéodory function. Then, problem (1.1) becomes
−div(|∇u|p−2∇u+µ(x)|∇u|q−2∇u) = (ζ−ϑ)|u|p−2u−µ(x)|u|q−2u−f(x,u)in Ω,
(|∇u|p−2∇u+µ(x)|∇u|q−2∇u)·ν=−β|u|p−2uon ∂Ω,(4.11)
Said El Manouni et al., Existence results for double phase problems |317
where fsatisfies the following assumptions:
(H3) The function f:Ω×R→Ris a Carathéodory function such that:
(i) fis bounded on bounded sets.
(ii) It holds
lim
s→±∞
f(x,s)
|s|q−2s= +∞uniformly for a. a. x∈Ω.
(iii) It holds
lim
s→0
f(x,s)
|s|p−2s= 0 uniformly for a. a. x∈Ω.
We have the following multiplicity result concerning problem (4.11).
Theorem 4.2. Let hypotheses (2.3) and (H3) be satisfied. Further, let ζ>λR
1,p,β+ϑwith ϑ>0and λR
1,p,βbeing
the first eigenvalue of the Robin eigenvalue problem of the p-Laplacian with β>0stated in (2.7). Then, problem
(4.11) has at least two nontrivial weak solutions u1,v1∈W1,H(Ω)∩L∞(Ω)such that u1≥0and v1≤0.
Proof. Taking hypothesis (H3)(ii) into account we find a constant M=M(ζ)>1such that
f(x,s)s≥ζ|s|qfor a.a. x∈Ωand all |s|≥M.(4.12)
As in the proof of Theorem 4.1, by (4.12), we can take constant functions u∈(M,+∞)and v≡−usuch that
0≥ζ up−1−f(x,u)for a. a. x∈Ωand 0≤ζ|v|p−2v−f(x,v)for a. a. x∈Ω,(4.13)
because p<qand M>1.
Then we define truncations ψ±
ζ:Ω×R→Rand ψ±
β:∂Ω ×R→Ras follows
ψ+
ζ(x,s) =
0if s<0
ζsp−1−f(x,s)if 0≤s≤u
ζ up−1−f(x,u)if u<s
,
ψ+
β(x,s) =
0if s<0
−βsp−1if 0≤s≤u
−βup−1if u<s
,
ψ−
ζ(x,s) =
ζ|v|p−2v−f(x,v)if s<v
ζ|s|p−2s−f(x,s)if v≤s≤0
0if 0<s
,
ψ−
β(x,s) =
−β|v|p−2vif s<v
−β|s|p−2sif v≤s≤0.
0if 0<s
(4.14)
We set
Ψ±
ζ(x,s) =
s
Z
0
ψ±
ζ(x,t)dtand Ψ±
β(x,s) =
s
Z
0
ψ±
β(x,t)dt
and introduce the C1-functionals Π±:W1,H(Ω)→Rgiven by
Π±(u) = 1
pk∇ukp
p+1
qk∇ukq
q,µ+ϑ
pkukp
p+1
qkukq
q,µ−Z
Ω
Ψ±
ζ(x,u)dx−Z
∂Ω
Ψ±
β(x,u)dσ.
318 |Said El Manouni et al., Existence results for double phase problems
As before, we define F(x,s) = Rs
0f(x,t)dt.
We start with the existence of a nonnegative solution. Because of (4.14) we know that the functional Π+
is coercive and also sequentially weakly lower semicontinuous. Therefore, we find an element u1∈W1,H(Ω)
such that
Π+(u1) = inf hΠ+(u) : u∈W1,H(Ω)i.
By hypothesis (H3)(iii), we find for every ε>0a number δ∈(0,u)such that
F(x,s)≤ε
p|s|pfor a.a. x∈Ωand for all |s|≤δ.(4.15)
We recall that uR
1,p,βis the first eigenfunction corresponding to the first eigenvalue λR
1,p,βof the eigenvalue
problem of the p-Laplacian with Robin boundary condition, see (2.7). Without any loss of generality we can
assume that uR
1,p,βis normalized (that is, kuR
1,p,βkp= 1) and because of uR
1,p,β∈intC1(Ω)+we choose
t∈(0,1) sufficiently small such that tuR
1,p,β(x)∈[0,δ]for all x∈Ω. Applying (4.14), (4.15), δ<uand ϑ>0
gives
Π+tuR
1,p,β
=1
p
∇tuR
1,p,β
p
p+1
q
∇tuR
1,p,β
q
q,µ+ϑ
p
tuR
1,p,β
p
p+1
q
tuR
1,p,β
q
q,µ
−Z
Ω
Ψ+
ζx,tuR
1,p,βdx−Z
∂Ω
Ψ+
βx,tuR
1,p,βdσ
≤tp
pλR
1,p,β−βtp
p
uR
1,p,β
p
p,∂Ω +tq
q
∇uR
1,p,β
q
q,µ+tpϑ
p+tq
q
uR
1,p,β
q
q,µ
−ζtp
p+Z
Ω
Fx,tuR
1,p,βdx+βtp
p
uR
1,p,β
p
p,∂Ω
≤tp
pλR
1,p,β+tq
q
∇uR
1,p,β
q
q,µ+tpϑ
p+tq
q
uR
1,p,β
q
q,µ−ζtp
p+εtp
p
≤tp λR
1,p,β+ϑ−ζ+ε
p!+tq
∇uR
1,p,β
q
q,µ+
uR
1,p,β
q
q,µ
q
.
(4.16)
Due to ζ>λR
1,p,β+ϑand p<qone has from (4.16) for ε∈(0,ζ−λR
1,p,β−ϑ)that
Π+tuR
1,p,β<0for all sufficiently small t>0.
Hence, Π+(u1)<0 = Π+(0)and so u1=0.
We have (Π+)′(u1) = 0, that is,
Z
Ω|∇u1|p−2∇u1+µ(x)|∇u1|q−2∇u1·∇φdx
+Z
Ωϑ|u1|p−2u1+µ(x)|u1|q−2u1φdx
=Z
Ω
ψ+
ζ(x,u1)φdx+Z
∂Ω
ψ+
β(x,u1)φdσ
(4.17)
for all φ∈W1,H(Ω). As done in the proof of Theorem 4.1 we take φ=−u−
1∈W1,H(Ω)and φ=(u1−u)+∈
W1,H(Ω)as test functions in (4.17) which gives us 0≤u1≤u, see (4.13). Hence, by the definition of the
truncations in (4.14) we see that u1∈W1,H(Ω)∩L∞(Ω)solves problem (4.11).
In the same way we can show the existence of a nontrivial nonpositive solution v1∈W1,H(Ω)∩L∞(Ω)
by treating the functional Π−:W1,H(Ω)→Rinstead of Π+:W1,H(Ω)→R.
Said El Manouni et al., Existence results for double phase problems |319
Remark 4.3. In this section we decided to consider two different problems since in the proof of Theorem 4.1 the
use of the first Robin eigenfunction would have provided a condition of the form
λR
1,p,β+ϑ<(β+ζ)
uR
1,p,β
p
p,∂Ω ,(4.18)
which depends also on the boundary norm of the eigenfunction uR
1,p,β. So the statement of Theorem 4.1 still holds
true when we replace the assumption ζ>λS
1,pby (4.18) where uR
1,p,βis the first normalized (that is, kuR
1,p,βkp= 1)
eigenfunction associated to the first eigenvalue λR
1,p,βof the Robin eigenvalue problem.
Acknowledgments: The authors wish to thank knowledgeable referees for their corrections and remarks.
Furthermore, the authors wish to thank Ángel Crespo-Blanco for valuable comments and improvements.
We acknowledge support by the German Research Foundation and the Open Access Publication Fund of TU
Berlin.
Conflict of Interest: The authors declare that they have no conflict of interest.
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