Nonexistence of positive weak solutions of elliptic inequalities

Filippucci, Roberta

doi:10.1016/j.na.2008.12.018

Cited by 65 publications

(60 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular this result improves the case β = γ = 0 of Corollary 1.1 in [12], where the stronger condition θ < p − 1 was required.…”

Section: Corollary 1 Any Entire Nonnegative Solution U Of the Inequasupporting

confidence: 61%

“…The corresponding case with gradient term is studied in [12]. A first attempt to treat the subhomogeneous and homogeneous case for systems can be found in [3], where nonexistence of positive entire solutions of (5), with θ 1 = θ 2 = 0, is deduced in a more restrictive class of solutions, when 0 < p 1 q 1 (p − 1)(q − 1) and (p − 1)(q − 1) < p 1 q 1 ; the latter condition, as discussed in [4], is a natural definition of "superhomogeneity" for the system under consideration.…”

Section: Corollary 1 Any Entire Nonnegative Solution U Of the Inequamentioning

confidence: 99%

“…To show the validity of Step 2 we argue as in Theorem 3.1 in [12] where positive solutions are taken under consideration. By using Young inequality we obtain for every > 0…”

Section: Main Theoremmentioning

confidence: 99%

“…We prove that any nonnegative entire (weak) solution is necessarily constant. The main argument of our proofs is based on previous estimates, given in Filippucci (2009) [12] for elliptic inequalities. Actually, the main technique for proving the central estimate has been developed by Mitidieri and Pohozaev (2001) [23] and relies on the method of test functions.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Nonexistence of nonnegative solutions of elliptic systems of divergence type

Filippucci

2011

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

In this paper we deal with noncoercive elliptic systems of divergence type, that include both the p-Laplacian and the mean curvature operator and whose right-hand sides depend also on a gradient factor. We prove that any nonnegative entire (weak) solution is necessarily constant. The main argument of our proofs is based on previous estimates, given in Filippucci (2009) [12] for elliptic inequalities. Actually, the main technique for proving the central estimate has been developed by Mitidieri and Pohozaev (2001) [23] and relies on the method of test functions. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required.

show abstract

“…In particular this result improves the case β = γ = 0 of Corollary 1.1 in [12], where the stronger condition θ < p − 1 was required.…”

Section: Corollary 1 Any Entire Nonnegative Solution U Of the Inequasupporting

confidence: 61%

Section: Corollary 1 Any Entire Nonnegative Solution U Of the Inequamentioning

confidence: 99%

“…To show the validity of Step 2 we argue as in Theorem 3.1 in [12] where positive solutions are taken under consideration. By using Young inequality we obtain for every > 0…”

Section: Main Theoremmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Nonexistence of nonnegative solutions of elliptic systems of divergence type

Filippucci

2011

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

show abstract

“…(5 ) Under these conditions, the conclusion for Example 1 is that if q > p + r − 1 and either s + t p (6 ) or…”

Section: It Is Clear That (3) Is Satisfied With Moreovermentioning

confidence: 96%

Entire solutions of completely coercive quasilinear elliptic equations

Farina

Serrin

2011

Journal of Differential Equations

View full text Add to dashboard Cite

A famous theorem of Sergei Bernstein says that every entire solution u = u(x), x ∈ R 2 , of the minimal surface equation div Duis an affine function; no conditions being placed on the behavior of the solution u.Bernstein's Theorem continues to hold up to dimension n = 7 while it fails to be true in higher dimensions, in fact if x ∈ R n , with n 8, there exist entire non-affine minimal graphs (Bombieri, De Giorgi and Giusti). Our purpose is to consider an extensive family of quasilinear elliptic-type equations which has the following strong BernsteinLiouville property, that u ≡ 0 for any entire solution u, no conditions whatsoever being placed on the behavior of the solution (outside of appropriate regularity assumptions). In many cases, moreover, no conditions need be placed even on the dimension n. We also study the behavior of solutions when the parameters of the problem do not allow the Bernstein-Liouville property, and give a number of counterexamples showing that the results of the paper are in many cases best possible.

show abstract

On maximum principles for solutions to nonlinear elliptic ODEs and radial solutions to nonlinear elliptic PDE's via Opial‐type inequalities

Kałamajska

Kosiorek

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

We consider degenerated nonlinear PDE of elliptic type: −div(a(|x|)|∇w(x)|p−2∇w(x))+h|x|,w(x),∇w(x),x|x|=ϕ(w(x)), a.e. in the ball in ℝn. Using the argument based on Opial‐type inequalities, we investigate qualitative properties of their radial solutions, like, for example, the a priori estimates, maximum principles, monotonicity, as well as nonexistence of the nontrivial solutions.

show abstract

Nonexistence of positive weak solutions of elliptic inequalities

Cited by 65 publications

References 19 publications

Nonexistence of nonnegative solutions of elliptic systems of divergence type

Nonexistence of nonnegative solutions of elliptic systems of divergence type

Entire solutions of completely coercive quasilinear elliptic equations

On maximum principles for solutions to nonlinear elliptic ODEs and radial solutions to nonlinear elliptic PDE's via Opial‐type inequalities

Contact Info

Product

Resources

About