By Badiale M., Benci V., Rolando S.
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Additional resources for A nonlinear elliptic equation with singular potential and applications to nonlinear field equations
To modify the model in Example 4 of § 2 slightly, let U denote a linear space in which a certain nonempty convex cone Q has been singled out as the "nonpositive orthant" for a partial ordering, and consider the problem (P) minimize / where C e X is convex, f0:C -» R is convex, and where O:C —>• U is convex in the sense that We suppose U is paired with a space Y in such a way that Q is closed. 2) the Lagrangian where Q* is the polar of Q. Thus, similar to Example 1' above, the dual consists of maximizing If for instance the set C is compact and the functions /0 and are continuous, it is simple to prove that the (convex) optimal value function q> is lower-semicontinuous and proper, implying the optimal values in (P) and (D) coincide.
The following conditions on an element y of Y are equivalent: (a) y solves (D), and sup(D) = inf (P); (b) -yed(p(0); (c) i n f K ( x , y ) = i n f / ( x ) . xeX xeX Proof. Recalling from Theorem 7 that — g(y) = >*( —y) and sup(D) = —
is convex), we have the equivalence of: (a) inf (P) = sup (D), and there exists at least one y solving (D); (b) lim inf u ,_ M (p'(0; u) is finite for at least one ueU.
23) dom y = dom h* - A* dom k*. Here A is a densely defined linear operator whose graph is closed, and A* is the adjoint operator (see Example 11). Furthermore, h and k are proper convex functions, so the convexity needed in Theorem 18 is present. , there exists x e (dom h) n (dom A) such that k is bounded above in a neighborhood of Ax. Alternatively, if U and V are Banach spaces in their "compatible" topologies, and if h and k are closed, the condition suffices for the conclusions of Theorem 17.
A nonlinear elliptic equation with singular potential and applications to nonlinear field equations by Badiale M., Benci V., Rolando S.