A study of nonlinear problems for the -Laplacian in via Ricceri’s principle

Manouni, Said El

doi:10.1016/j.na.2011.04.013

Cited by 2 publications

(6 citation statements)

References 8 publications

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“…In view of and in El Manouni, we get

\max ({}, \underset{‖ x ‖ \to \infty}{lim sup} \frac{J false(u false)}{normalΦ false(u false)}, \underset{x \to 0}{lim sup} \frac{J false(u false)}{normalΦ false(u false)}) = 0 .

Hence, all the assumptions of Proposition are satisfied (with x 0 =0). Then there exists ρ >0, with the following property: For each

λ \in false[a, b false] \subset false(θ, + \infty false)

, there exists δ 0 >0 such that, for μ ∈[0, δ 0 ], the problem has at least 2 nontrivial solutions

u_{1}, u_{2} \in D^{1, p} false(R^{N} false)

, whose norms are less than ρ , ie, ‖ u i ‖⩽ ρ , i =1,2.…”

Section: Proof Of Theorem 14mentioning

confidence: 77%

“…As

normalΩ = R^{N}

, we deduce that

D^{1, p} false(R^{N} false) ↪ L^{p^{*}} false(R^{N} false)

by the Sobolev inequality

‖ u ‖_{L^{p^{*}}} ⩽ C_{1} ‖ u ‖

, C 1 >0 a constant, and obtain the compactness by the concentration‐compactness principle for the critical growth in

R^{N}

. The assumptions ( f 1 ), ( f 2 ), and are from El Manouni in which he only studies the subcritical nonlinear problem and he cannot deal with the supercritical or even exponential growth problem via Ricceri principle under the assumptions ( f 1 ), ( f 2 ), and . Under the same assumptions ( f 1 ), ( f 2 ), and in El Manouni, I can deal with the supercritical even exponential growth problem and extend the result obtained by El Manouni to the supercritical growth (Theorem ) and exponential case (Theorem ) in this paper.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Drabek studies the problem with μ =0, ie,

- div (a (x) | \nabla u {false|}^{p - 2} \nabla u) = λ f (x, u) .

He shows that the existence and C 1, α regularity of the weak solution, when the function f ( x , u ) is subcritical growth with respect to u . Manouni obtains 2 nontrivial solutions of the problem with f and g satisfying subcritical growth for λ ∈( a , b ) and small μ via Ricceri principle. () Joseph and Lundgren study the Dirichlet problem −Δ u = λ e u in Ω and obtain infinitely many positive solutions for λ = λ ∗ .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The assumptions ( f 1 ), ( f 2 ), and are from El Manouni in which he only studies the subcritical nonlinear problem and he cannot deal with the supercritical or even exponential growth problem via Ricceri principle under the assumptions ( f 1 ), ( f 2 ), and . Under the same assumptions ( f 1 ), ( f 2 ), and in El Manouni, I can deal with the supercritical even exponential growth problem and extend the result obtained by El Manouni to the supercritical growth (Theorem ) and exponential case (Theorem ) in this paper. We overcome this difficulty by the truncated technique and Moser iteration .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…It is easy to see that the critical points of the functional Φ− λ J − μ Ψ are the weak solutions of the problem . The derivative of J is given by

⟨ J^{'} false(u false), v ⟩ = \int_{R^{N}} f false(x, u false) v d x

for every

u, v \in D^{1, p} false(R^{N} false)

and

J^{'}

is a compact map from

D^{1, p} false(R^{N} false)

{false(D^{1, p} false(R^{N} false) false)}^{*}

(see lemma 2.1 of El Manouni).…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

See 4 more Smart Citations

A study of nonlinear elliptic problems involving supercritical and exponential growth in RN

Zhao

Yan

2017

Math Methods in App Sciences

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In this paper, we consider the multiplicity of solutions of the p-Laplacian problems involving supercritical Sobolev growth and exponential growth in R N via Ricceri principle. By means of the truncation combining with Moser iteration, we can extend the result about the subcritical growth to the supercritical and exponential growth. KEYWORDS degenerate elliptic equations, supercritical Sobolev exponent, L ∞ estimate 7162

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“…In view of and in El Manouni, we get