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Existence and regularity of minima for integral functionals noncoercive in the energy space. (English) Zbl 1015.49014

From the introduction: “We are interested in the existence and regularity of minima for functionals whose model is \[ J(v)= \int_\Omega {|\nabla v|^p\over(1+|v|)^{\alpha p}} dx- \int_\Omega f v dx,\qquad v\in W^{1,p}_0(\Omega), \] where \(\Omega\) is a bounded, open subset of \(\mathbb{R}^N\), \(\alpha> 0\), \(p> 1\), and \(f\) belongs to \(L^r(\Omega)\) for some \(r\geq 1\).
This functional, which is clearly well defined thanks to Sobolev embedding if \(r\geq (p^*)'\), is however non-coercive on \(W^{1,p}_0(\Omega)\): there exists a function \(f\), and a sequence \(\{u_n\}\) whose norm diverges in \(W^{1,p}_0(\Omega)\), such that \(J(u_n)\) tends to \(-\infty\).
Thus, even if \(J\) is lower-semicontinuous on \(W^{1,p}_0(\Omega)\) as a consequence of the De Giorgi theorem, the lack of coerciveness implies that \(J\) may not attain its minimum on \(W^{1,p}_0(\Omega)\) even in the case in which \(J\) is bounded from below.
The structure of the functional has however enough properties in order to prove that if \(f\) belongs to \(L^r(\Omega)\), with \(r\geq [p^*(1- \alpha)]'\), then \(J\) (suitably extended) is coercive on \(W^{1,q}_0(\Omega)\) for some \(q< p\) depending on \(\alpha\). Thus, \(J\) attains its minimum on this larger space. Our aim is to prove some regularity results for these minima, depending on the summability of \(f\).
More precisely, we prove that if \(f\) is regular enough, then any minimum is bounded, so that (as a consequence of the structure of the functional) it belongs to \(W^{1,p}_0(\Omega)\). If we “decrease” the summability of \(f\), then the minima are no longer bounded, but they still belong to the “energy space” \(W^{1,p}_0(\Omega)\). Finally, there is a range of summability for \(f\) such that the minima are neither bounded, nor in \(W^{1,p}_0(\Omega)\).
We also prove some results concerning the regularity of the minima if the datum \(f\) belongs to Marcinkiewicz spaces, and a result of existence of solutions for a nonlinear elliptic equation whose model is the Euler equation of the functional \(J\)”.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49N60 Regularity of solutions in optimal control
49J10 Existence theories for free problems in two or more independent variables
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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