The aim of this work is to provide uniform
L
∞
-estimates for the solutions of a quite general class of (
p
,
q
)-quasilinear elliptic systems depending on two parameters
α
and
δ
.
We establish an Amann-Zehnder-type result for resonance systems of quasilinear elliptic equations with homogeneous Dirichlet boundary conditions, involving nonlinearities growing asymptotically
...-linear at infinity. The proof relies on a cohomological linking in a product Banach space where the properties of cones of the sublevels are missing, differently from the single quasilinear equation. We also perform critical group computations of the energy functional at the origin, in spite of the lack of its
regularity, to exclude that the found mini-max solution is trivial. Finally, we furnish a local condition which guarantees that the found solution is not semi-trivial.
Amann–Zehnder type results for p-Laplace problems Cingolani, Silvia; Degiovanni, Marco; Vannella, Giuseppina
Annali di matematica pura ed applicata,
04/2018, Letnik:
197, Številka:
2
Journal Article
Recenzirano
Odprti dostop
The existence of a nontrivial solution is proved for a class of quasilinear elliptic equations involving, as principal part, either the
p
-Laplace operator or the operator related to the
p
-area ...functional, and a nonlinearity with
p
-linear growth at infinity. To this aim, Morse theory techniques are combined with critical groups estimates.
In this paper we consider the quasilinear critical problem(Pλ){−Δpu=λuq−1+up⁎−1inΩu>0inΩu=0on∂Ω where Ω is a regular bounded domain in RN, N≥p2, 1<p<2, p≤q<p⁎, p⁎=Np/(N−p), λ>0 is a parameter. In ...spite of the lack of C2 regularity of the energy functional associated to (Pλ), we employ new Morse techniques to derive a multiplicity result of solutions. We show that there exists λ⁎>0 such that, for each λ∈(0,λ⁎), either (Pλ) has P1(Ω) distinct solutions or there exists a sequence of quasilinear problems approximating (Pλ), each of them having at least P1(Ω) distinct solutions. These results complete those obtained in 23 for the case p≥2.
In this work, we study a class of Euler functionals defined in Banach spaces, associated with quasilinear elliptic problems involving p-Laplace operator (p > 2). First we obtain perturbation results ...in the spirit of the remarkable paper by Marino and Prodi (Boll. U.M.I. (4) 11(Suppl. fasc. 3): 1–32, 1975), using the new definition of nondegeneracy given in (Ann. Inst. H. Poincaré: Analyse Non Linéaire. 2:271–292, 2003). We also extend Morse index estimates for minimax critical points, introduced by Lazer and Solimini (Nonlinear Anal. T.M.A. 12:761–775, 1988) in the Hilbert case, to our Banach setting.
Let us consider the quasilinear problem
(
P
ε
)
{
−
ε
p
Δ
p
u
+
u
p
−
1
=
f
(
u
)
in
Ω
,
u
>
0
in
Ω
,
u
=
0
on
∂
Ω
where
Ω is a bounded domain in
R
N
with smooth boundary,
N
>
p
,
2
⩽
p
<
p
∗
,
p
∗
=
...N
p
/
(
N
−
p
)
,
ε
>
0
is a parameter. We prove that there exists
ε
∗
>
0
such that, for any
ε
∈
0
,
ε
∗
,
(
P
ε
)
has at least
2
P
1
(
Ω
)
−
1
solutions, possibly counted with their multiplicities, where
P
t
(
Ω
)
is the Poincaré polynomial of
Ω. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on
Ω, approximating
(
P
ε
)
.
We consider a Neumann problem of the type -εΔu+F′(u(x))=0 in an open bounded subset Ω of Rn, where F is a real function which has exactly k maximum points.Using Morse theory we find that, for ε ...suitably small, there are at least 2k nontrivial solutions of the problem and we give some qualitative information about them.
We deal with the existence of solutions for the quasilinear problem(Pλ){−Δpu=λuq−1+up∗−1inΩ,u>0inΩ,u=0on∂Ω, where Ω is a bounded domain in RN with smooth boundary, N⩾p2, 1<p⩽q<p∗, p∗=Np/(N−p), λ>0 is ...a parameter. Using Morse techniques in a Banach setting, we prove that there exists λ∗>0 such that, for any λ∈(0,λ∗), (Pλ) has at least P1(Ω) solutions, possibly counted with their multiplicities, where Pt(Ω) is the Poincaré polynomial of Ω. Moreover for p⩾2 we prove that, for each λ∈(0,λ∗), there exists a sequence of quasilinear problems, approximating (Pλ), each of them having at least P1(Ω) distinct positive solutions.
On s'interesse à l'existence de solutions pour l'équation quasi-linéaire(Pλ){−Δpu=λuq−1+up∗−1inΩ,u>0inΩ,u=0on∂Ω, où Ω est un domaine de RN avec frontière régulière, N⩾p2, 1<p⩽q<p∗, p∗=Np/(N−p), λ>0 est un paramètre. Par des techniques de la théorie de Morse dans le cadre des espaces de Banach, un démontre l'existence de λ∗>0 tel que, pour tout λ∈(0,λ∗), (Pλ) possède au moins P1(Ω) solutions, considerées avec leur multiplicité, où P1(Ω) est le polynôme de Poincaré de Ω. En outre, pour p⩾2, on démontre que, pour tout λ∈(0,λ∗), il existe une suite de problèmes quasi-linéaires qui approchent (Pλ), chacun desquels a au moins P1(Ω) solutions positives différentes.
We consider a class of quasilinear elliptic equations whose principal part includes the $p$-area (for $1 < p < \infty$) and the $p$-Laplace (for $1 < p≤ 2)$ operator. For the critical points of the ...associated functional, we provide estimates of the corresponding critical polynomial.
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