Nonlinear elliptic equations with subcoercive operators

4 Jul 2021, 17:45


Eugene Kalita


Let $X, Y$ be separable reflexive Banach spaces, $X\subset Y$ densely.
Let $A: X\to X^*$ be monotone operator (dissipative, in linear case),
coercive in the norm of $Y$, that is $(Au,u)/\|u\|_Y \to \infty$ as $\|u\|_Y \to \infty, \ u\in X$.

We introduce a notion of solution of equation $Au=f$ for $u\in Y$, $f\in Y^*$ in such the situation.
In case $\exists\,V\subset X$ dense (and so $V\subset Y$ dense) with $Av\in Y^*$ for $v\in V$,
our solution coincides with the solution in sense of monotonic extension of operator $A$.

We treat elliptic equations of nonstrictly divergent form
${\rm{div}}^t A(x,D^su)=f(x)$, $s\ne t$, $x\in R^n$,
under degenerate Cordes-type condition
and achieve existence and uniqueness results
in wider situation than already known.

Primary author

Eugene Kalita

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