A Singular Elliptic System with Higher Order Terms of <i>p</i>-Laplacian Type

Cave, Linda Maria De

doi:10.1515/ans-2015-5052

Cited by 4 publications

(8 citation statements)

References 13 publications

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“…Going further in details, in and the variational counterpart of system has been studied for

p = 2

and for the general case

p > 1

respectively; precisely, in the author considers the following system

\begin{matrix} true \\ right \{\begin{matrix} right z > 0 in normalΩ, z \in W_{0}^{1, p} (Ω) : & right - Δ_{p} z = q z^{q - 1} u^{θ}, \\ right u > 0 in normalΩ, u \in W_{0}^{1, p} (Ω) : & right - Δ_{p} u = θ z^{q} u^{θ - 1}, \end{matrix} \end{matrix}

namely system with

a (x) \equiv q

b (x) \equiv θ

and

q, θ, p

positive real numbers such that

\begin{matrix} true \\ right \{\begin{matrix} right 0 < θ < 1, 0 < q < p^{*} - θ, \\ right 1 < p < normalN . \end{matrix} \end{matrix}

We underline that, on one hand, assumption covers a wider range for the parameters

q, θ, p

with respect to and but, on the other hand, the weights in the lower order terms of , namely

…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Differently from the case studied in , here we no longer have a variational structure and in order to prove the existence of a solution to , we can not proceed as in the proof of [, Theorem ].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Theorem Let X be a Banach space,

S : X \to X

be a continuous map such that

\bar{S false(C false)}

is compact for every

C \subset X

bounded and

scriptK

be a convex, closed and bounded subset of X that is invariant for S . Then S has at least a fixed point in

scriptK

. The lack of variational structure for the system prevents also to prove that the solution we find is different from zero as done in , where this follows by using the property that such a solution is obtained as limit of a sequence of non‐trivial critical point of a certain approximating functionals

J_{n}

. The idea here is to define the invariant set

scriptK

in such a way it is far from zero, so that the solution

(u, z)

to found by applying Theorem , namely found as a fixed point of some map S defined on

scriptK

, will not be identically zero.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Section is the core of the paper. At first, we treat the trivial case

a (x) \equiv A

and

b (x) \equiv B

, with

A, B

positive constants, showing that elementary computations lead to the existence of a solution to by taking advantage of the existence of a solution in the case

A = q

and

B = θ

(see ); the second part of Section is entirely devoted to the proof of Theorem under its general assumptions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Going further in details, in [4] and [11] the variational counterpart of system (1.1) has been studied for p = 2 and for the general case p > 1 respectively; precisely, in [11] the author considers the following system…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 4 more Smart Citations

Existence of solutions to a non‐variational singular elliptic system with unbounded weights

Cave

Oliva

Strani

2016

Mathematische Nachrichten

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In this paper we prove an existence result for the following singular elliptic system truerightrightz>00.28emin0.28emnormalΩ0.16em,0.28emz∈W01,p(Ω)0.16em:right−Δpz=a(x)zq−1uθ0.16em,rightu>00.28emin0.28emnormalΩ0.16em,0.28emu∈W01,p(Ω)0.16em:right−Δpu=b(x)zquθ−10.16em,where Ω is a bounded open set in errorN (N≥2), −normalΔp is the p‐laplacian operator, a(x) and b(x) are suitable Lebesgue functions and q>0, 0<θ<1, p>1 are positive parameters satisfying suitable assumptions.

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“…Going further in details, in and the variational counterpart of system has been studied for

p = 2

and for the general case

p > 1

respectively; precisely, in the author considers the following system

\begin{matrix} true \\ right \{\begin{matrix} right z > 0 in normalΩ, z \in W_{0}^{1, p} (Ω) : & right - Δ_{p} z = q z^{q - 1} u^{θ}, \\ right u > 0 in normalΩ, u \in W_{0}^{1, p} (Ω) : & right - Δ_{p} u = θ z^{q} u^{θ - 1}, \end{matrix} \end{matrix}

namely system with

a (x) \equiv q

b (x) \equiv θ

and

q, θ, p

positive real numbers such that

\begin{matrix} true \\ right \{\begin{matrix} right 0 < θ < 1, 0 < q < p^{*} - θ, \\ right 1 < p < normalN . \end{matrix} \end{matrix}

We underline that, on one hand, assumption covers a wider range for the parameters

q, θ, p

with respect to and but, on the other hand, the weights in the lower order terms of , namely

…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Theorem Let X be a Banach space,

S : X \to X

be a continuous map such that

\bar{S false(C false)}

is compact for every

C \subset X

bounded and

scriptK

be a convex, closed and bounded subset of X that is invariant for S . Then S has at least a fixed point in

scriptK

J_{n}

. The idea here is to define the invariant set

scriptK

in such a way it is far from zero, so that the solution

(u, z)

to found by applying Theorem , namely found as a fixed point of some map S defined on

scriptK

, will not be identically zero.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Section is the core of the paper. At first, we treat the trivial case

a (x) \equiv A

and

b (x) \equiv B

, with

A, B

positive constants, showing that elementary computations lead to the existence of a solution to by taking advantage of the existence of a solution in the case