1. bookVolume 29 (2021): Edizione 3 (November 2021)
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Rivista
eISSN
1844-0835
Prima pubblicazione
17 May 2013
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1 volta all'anno
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Inglese
access type Accesso libero

Non-autonomous weighted elliptic equations with double exponential growth

Pubblicato online: 23 Nov 2021
Volume & Edizione: Volume 29 (2021) - Edizione 3 (November 2021)
Pagine: 33 - 66
Ricevuto: 01 Apr 2021
Accettato: 15 May 2021
Dettagli della rivista
License
Formato
Rivista
eISSN
1844-0835
Prima pubblicazione
17 May 2013
Frequenza di pubblicazione
1 volta all'anno
Lingue
Inglese
Abstract

We consider the existence of solutions of the following weighted problem: {L:=-div(ρ(x)|u|N-2u)+ξ(x)|u|N-2u=f(x,u)inBu>0inBu=0onB, \left\{ {\matrix{{L: = - div\left( {\rho \left( x \right){{\left| {\nabla u} \right|}^{N - 2}}\nabla u} \right) + \xi \left( x \right){{\left| u \right|}^{N - 2}}} \hfill & {u = f\left( {x,u} \right)} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u > 0} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u = 0} \hfill & {on} \hfill & {\partial B,} \hfill \cr } } \right. where B is the unit ball of ℝN, N #62; 2, ρ(x)=(loge|x|)N-1 \rho \left( x \right) = {\left( {\log {e \over {\left| x \right|}}} \right)^{N - 1}} the singular logarithm weight with the limiting exponent N − 1 in the Trudinger-Moser embedding, and ξ(x) is a positif continuous potential. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of double exponential type. We prove the existence of positive solution by using Mountain Pass theorem. In the critical case, the function of Euler Lagrange does not fulfil the requirements of Palais-Smale conditions at all levels. We dodge this problem by using adapted test functions to identify this level of compactness.

Keywords

MSC 2010

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