In this paper we are concerned with the regularity of solutions to a nonlinear elliptic system of m equations in divergence form, satisfying p growth from below and q growth from above, with p &lt;= q; this case is known as p, q-growth conditions. Well known counterexamples, even in the simpler case p = q, show that solutions to systems may be singular; so, it is necessary to add suitable structure conditions on the system that force solutions to be regular. Here we obtain local boundedness of solutions under a componentwise coercivity condition. Our result is obtained by proving that each component u(alpha) of the solution u = (u(1),..., u(m)) satisfies an improved Caccioppoli's inequality and we get the boundedness of u(alpha) by applying De Giorgi's iteration method, provided the two exponents p and q are not too far apart. Let us remark that, in dimension n = 3 and when p = q, our result works for 3/2 &lt; p &lt;= 3, thus it complements the one of Bjorn whose technique allowed her to deal with p &lt;= 2 only. In the final section, we provide applications of our result.

### Local boundedness for solutions of a class of nonlinear elliptic systems

#### Abstract

In this paper we are concerned with the regularity of solutions to a nonlinear elliptic system of m equations in divergence form, satisfying p growth from below and q growth from above, with p <= q; this case is known as p, q-growth conditions. Well known counterexamples, even in the simpler case p = q, show that solutions to systems may be singular; so, it is necessary to add suitable structure conditions on the system that force solutions to be regular. Here we obtain local boundedness of solutions under a componentwise coercivity condition. Our result is obtained by proving that each component u(alpha) of the solution u = (u(1),..., u(m)) satisfies an improved Caccioppoli's inequality and we get the boundedness of u(alpha) by applying De Giorgi's iteration method, provided the two exponents p and q are not too far apart. Let us remark that, in dimension n = 3 and when p = q, our result works for 3/2 < p <= 3, thus it complements the one of Bjorn whose technique allowed her to deal with p <= 2 only. In the final section, we provide applications of our result.
##### Scheda breve Scheda completa Scheda completa (DC)
2022
Cupini, G; Leonetti, F; Mascolo, E
File in questo prodotto:
File
cupleomas_Meier_pubblicato.pdf

accesso aperto

Tipo: Versione (PDF) editoriale
Licenza: Licenza per Accesso Aperto. Creative Commons Attribuzione (CCBY)
Dimensione 378.4 kB
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/900764`
• ND
• 2
• 2