On the classification of solutions of the Lane–Emden equation on unbounded domains of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mi>N</mml:mi></mml:msup></mml:math>

Farina, Alberto

doi:10.1016/j.matpur.2007.03.001

Cited by 237 publications

(72 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The proof of the case , bears resemblance to an argument found in [5, 6, 8]. Now, we prove the case .…”

Section: Auxiliary Resultssupporting

confidence: 76%

“…The proof of the case , bears resemblance to an argument found in [5, 6, 8]. For more details, please see the proof of proposition 4 in [6] for the case , the proof of Lemma 4.2 in [5] for the case and the proof of Proposition 1.2 in [8] for the case .…”

Section: Auxiliary Resultsmentioning

confidence: 56%

“…For more details, please see the proof of proposition 4 in [6] for the case , the proof of Lemma 4.2 in [5] for the case and the proof of Proposition 1.2 in [8] for the case . For this reason, we omit the details.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…To simplify the proof, we will concentrate on the case which is the most delicate case; even we believe that the results should hold true for , for more details, see for example [5, 6, 8, 23]. Let .…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…We start with the second-order Lane-Emden equation Farina [6] proved that nontrivial finite Morse index solutions of (1.2) exist if and only if and , or and , where is the so-called Joseph-Lundgren exponent. Also, in [13] several Liouville-type theorems are presented for stable solutions, where is a general convex, nondecreasing function.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Classification of stable solutions for non-homogeneous higher-order elliptic PDEs

2017

View full text Add to dashboard Cite

Under some assumptions on the nonlinearity f, we will study the nonexistence of nontrivial stable solutions or solutions which are stable outside a compact set of for the following semilinear higher-order problem: with . The main methods used are the integral estimates and the Pohozaev identity. Many classes of nonlinearity will be considered; even the sign-changing nonlinearity, which has an adequate subcritical growth at zero as for example , where , , , . More precisely, we shall revise the nonexistence theorem of Berestycki and Lions (Arch. Ration. Mech. Anal. 82:313-345, 1983) in the class of smooth finite Morse index solutions as the well known work of Bahri and Lions (Commun. Pure Appl. Math. 45:1205-1215, 1992). Also, the case when is a nonnegative function will be studied under a large subcritical growth assumption at zero, for example or , and . Extensions to solutions which are merely stable are discussed in the case of supercritical growth with .

show abstract

“…The proof of the case , bears resemblance to an argument found in [5, 6, 8]. Now, we prove the case .…”

Section: Auxiliary Resultssupporting

confidence: 76%

Section: Auxiliary Resultsmentioning

confidence: 56%

Section: Auxiliary Resultsmentioning

confidence: 99%

Section: Auxiliary Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Classification of stable solutions for non-homogeneous higher-order elliptic PDEs

2017

View full text Add to dashboard Cite

show abstract

Uniform boundedness for finite Morse index solutions to supercritical semilinear elliptic equations

Figalli

Zhang

2023

Comm Pure Appl Math

View full text Add to dashboard Cite

We consider finite Morse index solutions to semilinear elliptic questions, and we investigate their smoothness. It is well-known that: -For 𝑛 = 2, there exist Morse index 1 solutions whose 𝐿 ∞ norm goes to infinity.-For 𝑛 ≥ 3, uniform boundedness holds in the subcritical case for power-type nonlinearities, while for critical nonlinearities the boundedness of the Morse index does not prevent blow-up in 𝐿 ∞ .In this paper, we investigate the case of general supercritical nonlinearities inside convex domains, and we prove an interior a priori 𝐿 ∞ bound for finite Morse index solution in the sharp dimensional range 3 ≤ 𝑛 ≤ 9. As a corollary, we obtain uniform bounds for finite Morse index solutions to the Gelfand problem constructed via the continuity method.

show abstract

Low dimensional instability for quasilinear problems of weighted exponential nonlinearity

2018

Mathematische Nachrichten

View full text Add to dashboard Cite

We prove a sharp Liouville type theorem for stable Wloc1,p solutions of the equation −normalΔpu=ffalse(xfalse)euon the entire Euclidean space RN, where p>2 and f is a continuous and nonnegative function in double-struckRN∖false{0false} such that ffalse(xfalse)≥a|x|q as false|xfalse|≥R0>0, where q>−p and a>0. Our theorem holds true for 0false2≤N<0falsep2+3p+4qp−1 and is sharp in the case ffalse(xfalse)=a|x|q.

show abstract

On the classification of solutions of the Lane–Emden equation on unbounded domains of RN

Cited by 237 publications

References 32 publications

Classification of stable solutions for non-homogeneous higher-order elliptic PDEs

Classification of stable solutions for non-homogeneous higher-order elliptic PDEs

Uniform boundedness for finite Morse index solutions to supercritical semilinear elliptic equations

Low dimensional instability for quasilinear problems of weighted exponential nonlinearity

Contact Info

Product

Resources

About