On a priori bounds for positive solutions of subcritical elliptic problems

Abstract:

We provide a priori bounds for positive solutions of \(p\)-Laplacian equations

$$-\Delta_p u =f(u)\quad \text{in}\ \Omega, \qquad u= 0\quad \text{on}\ \partial\Omega,$$

in a bounded domain, with a subcritical nonlinearity of the form

$$f(u)=u^{p^*-1}/[\ln (e+u)]^\alpha,$$

with \(p^*= Np/(N-p)\) the critical Sobolev exponent and \(\alpha >p/(N-p)\); see [1,2].

We extend our results to Hamiltonian elliptic systems

$$-\Delta u=f(v), \ -\Delta v=g(u)\ \text{in}\ \Omega, \ u=v=0\ \text{on}\ \partial\Omega,$$

where

$$f(v)=v^{p-1}/[\ln (e+v)]^\alpha,\quad g(u) =u^{q-1}/[\ln (e+u)]^\beta,$$

with \(\alpha, \beta> 2/(N-2)\), and \((p,q)\) on the critical hyperbola \(\frac{1}{p} +\frac{1}{q} = \frac{N-2}{N}\); see [3].

References:

[1] A. Castro and R. Pardo: A priori bounds for positive solutions of subcritical elliptic equations. Rev. Mat. Complut. 28 (2015), 715–731.

[2] L. Damascelli and R. Pardo: A priori estimates for some elliptic equations involving the p-Laplacian. Nonlinear Anal. 41 (2018), 475 – 496.

[3] N. Mavinga and R. Pardo: A priori bounds and existence of positive solutions for subcritical semilinear elliptic systems. J. Math. Anal. Appl. 449 (2017), 1172–1188.