Existence results for nonlinear elliptic equations with degenerate coercivity

Abstract

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Existence and regularity results for some parabolic equations with degenerate coercivity

“…The proof is essentially based on the approximate problems (P n ) with some nondegenerate coercivity and a priori estimates on the weak solutions of these problems. Similar problem to elliptic equations has already been studied in [13] (see also [1,2,9,10,18,19]). Recently, Porzio and Pozio in [24] have discussed the case of f ≡ 0, u(x, 0) = u 0 = 0.…”

“…Thus Theorem 1.5 shows that the regularity of solutions to parabolic equations with degenerate coercivity is essentially different from that of parabolic equations with coercivity, since the solutions belong to some Sobolev space for the latter so long as m > 1 (see [11,15] , but 1 for elliptic equations (see [1,2,9,10,13,18,19]). This is due to two different types of partial differential equations.…”

Uniqueness of solutions for some elliptic equations with a quadratic gradient term

“…This hypothesis, that the truncations of the solution are in the "energy space" H 1 0 (Ω), is quite natural when dealing with elliptic problems having a non regular datum (see [10,23]) or either a noncoercive principal term (see [2,19,20,30]). In our setting truncations are also used in a essential way in order to prove our main tool (see Prop.…”

“…2.3 below). (2) Observe that every term in (2.3) is well-defined because of condition (2.2). It is worthwhile to point out that it is natural to require some additional integrability to f in order to assure the existence of solution…”

“…[17,18]). On the other hand, (nonsingular) elliptic equations having a principal term with degenerate coercivity have been studied in the nineties (see [2,19] and references therein). In some sense, these two features are combined in [16], where the authors look for minima of noncoercive functionals whose corresponding Euler-Lagrange equations have both a noncoercive principal term and a quadratic gradient term.…”