Journal
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
Volume 144, Issue 2, Pages 575-586Publisher
AMER MATHEMATICAL SOC
DOI: 10.1090/proc12724
Keywords
Indefinite variational problem; critical growth in the gradient; superlinear term with slow growth; Cerami condition
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Funding
- FONDECYT project [11121567]
- project NONLOCAL - French National Research Agency (ANR) [ANR-14-CE25-0013]
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We consider the problem (P) -Delta u = c(x)u+mu vertical bar Delta u vertical bar(2) + f(x), u is an element of H-0(1)(Omega) boolean AND L-infinity(Omega), where Omega is a bounded domain of R-N, N >= 3, mu > 0 and c, f is an element of L-q(Omega) for some q > N/2 with f not greater than or equal to 0. Here c is allowed to change sign and we assume that c(+) not equivalent to 0. We show that when c(+) and mu f are suitably small this problem has at least two positive solutions. This result contrasts with the case c <= 0, where uniqueness holds. To show this multiplicity result we first transform (P) into a semilinear problem having a variational structure. Then we are led to the search of two critical points for a functional whose superquadratic part is indefinite in sign and has a so-called slow growth at infinity. The key point is to show that the Palais-Smale condition holds.
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