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Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains
, 2010
"... We examine the regularity of the extremal solution of the nonlinear eigenvalue problem ∆ 2 u = λf(u) on a general bounded domain Ω in R N, with the Navier boundary condition u = ∆u = 0 on ∂Ω. Here λ is a positive parameter and f is a nondecreasing nonlinearity with f(0) = 1. We give general pointw ..."
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Cited by 17 (6 self)
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We examine the regularity of the extremal solution of the nonlinear eigenvalue problem ∆ 2 u = λf(u) on a general bounded domain Ω in R N, with the Navier boundary condition u = ∆u = 0 on ∂Ω. Here λ is a positive parameter and f is a nondecreasing nonlinearity with f(0) = 1. We give general pointwise bounds and energy estimates which show that for any convex and superlinear nonlinearity f, the extremal solution u ∗ is smooth provided N ≤ 5. f(t)f • If in addition lim inf t→+∞ ′ ′ (t) (f ′) 2 (t)> 0, then u ∗ is regular for N ≤ 7. • On the other hand, if γ: = lim sup t→+∞ f(t)f ′ ′ (t) (f ′ ) 2 (t) < +∞, then the same holds for N < 8 γ. It follows that u ∗ is smooth if f(t) = et and N ≤ 8, or if f(t) = (1 + t) p and N < 8p p−1. We also show that if f(t) = (1 − t) −p, p> 1 and p = 3, then u ∗ is smooth for N ≤ 8p p+1. We note that while these are major improvements on what is known for general domains, they still fall short of the expected optimal results as recently established for Dirichlet problems on radial domains, e.g., u ∗ is smooth for N ≤ 12 when f(t) = et [11], and for N ≤ 8 when f(t) = (1 − t) −2 [9] (see also [22]).
Linear instability of entire solutions for a class of nonautonomous elliptic equations
 IN: PROCEEDINGS OF ROYAL SOCIETY EDINBURGH SECT. A
, 2008
"... We study the effect of the potential yα on the stability of entire solutions for elliptic equations on RN, N 2, with exponential or smoooth/singular polynomial nonlinearities. Instability properties are crucial in order to establish regularity of the extremal solution to some related Dirichlet no ..."
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Cited by 9 (4 self)
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We study the effect of the potential yα on the stability of entire solutions for elliptic equations on RN, N 2, with exponential or smoooth/singular polynomial nonlinearities. Instability properties are crucial in order to establish regularity of the extremal solution to some related Dirichlet nonlinear eigenvalue problem on bounded domains. As a byproduct of our results, we will improve the known results about the regularity of such solutions.
LOW DIMENSIONAL INSTABILITY FOR SEMILINEAR AND QUASILINEAR PROBLEMS IN RN
"... Abstract. Stability properties for solutions of −∆m(u) = f(u) in RN are investigated, where N ≥ 2 and m ≥ 2. The aim is to identify a critical dimension N # so that every nonconstant solution is linearly unstable whenever 2 ≤ N < N#. For positive, increasing and convex nonlinearities f(u), glob ..."
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Cited by 4 (0 self)
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Abstract. Stability properties for solutions of −∆m(u) = f(u) in RN are investigated, where N ≥ 2 and m ≥ 2. The aim is to identify a critical dimension N # so that every nonconstant solution is linearly unstable whenever 2 ≤ N < N#. For positive, increasing and convex nonlinearities f(u), global bounds on f f (f ′)2 allows us to find a dimension N which is optimal for exponential and power nonlinearities. In the radial setting we can deal more generally with C1−nonlinearities and the dimension N # we find is still optimal.
pMEMS equation on a ball
 Methods Appl. Anal
"... Abstract. We investigate qualitative properties of the MEMS equation involving the p−Laplace operator, 1 < p ≤ 2, on a ball B in RN, N ≥ 2. We establish uniqueness results for semistable solutions and stability (in a strict sense) of minimal solutions. In particular, along the minimal branch we ..."
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Cited by 2 (2 self)
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Abstract. We investigate qualitative properties of the MEMS equation involving the p−Laplace operator, 1 < p ≤ 2, on a ball B in RN, N ≥ 2. We establish uniqueness results for semistable solutions and stability (in a strict sense) of minimal solutions. In particular, along the minimal branch we show monotonicity of the first eigenvalue for the corresponding linearized operator and radial symmetry of the first eigenfunction. Key words. AMS subject classifications. 35B05, 35B65, 35J70
Regularity of the extremal solution for singular pLaplace equations
, 2014
"... Your article is protected by copyright and all rights are held exclusively by SpringerVerlag Berlin Heidelberg. This eoffprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to selfarchive your article, please use the accepted manuscript version fo ..."
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Your article is protected by copyright and all rights are held exclusively by SpringerVerlag Berlin Heidelberg. This eoffprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to selfarchive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. manuscripta math. 146, 519–529 (2015) © SpringerVerlag Berlin Heidelberg 2014 Daniele Castorina Regularity of the extremal solution for singular pLaplace equations