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Entire solutions with exponential growth for an elliptic system modelling phase-transition QDD Terracini, S. Uniform Holder bounds for strongly competing systems involving the square root of the laplacian QDD Hartshorne, R. Wide oscillations finite time blow up for solutions to nonlinear fourth order differential equations QDD Gazzola, F. On the moments of solutions to linear parabolic equations involving the biharmonic operator QDD Pagani, C. Isola, T. Sauvageot J. In vitro tissue growth: a multiscale computational model of the dynamically evolving biophysical environment QDD Dulio, P.

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The inequality 4. In fact, the Linking Theorem provides two distinct critical levels cf. Remark 5. Utilizing g1 , g2 00 and g3 00 , it is easy to see that J is still weakly lower semicontinuous and coercive; thus, by a classical argument cf. It is clear that the growth condition will imply that the functional J will still satisfy Lemma 4. Proof of Theorem 5. J is an even functional, and thus we may apply Theorem 8 of [6] to get our result.

In fact, in such a case, denoting by Kcm the set of critical points at level cm 6 An ODE analog In this section we consider an analogous nonlocal semilinear problem for a second order ordinary differential operator with Cauchy conditions whose manifest asymmetry can be resolved by composition with a reflection operator which results in a variational structure amenable to the very same dual variational methods we have used for N ST.

Moreover, in this simplified setting some additional remarks concerning the role of the reflection operator with respect to existence of nontrivial solutions follow easily. One can easily prove cf. However, the problem SO can be transformed into an initial value problem for a Hamiltonian system for which there is an isolated equilibrium at the origin in the phase plane, and hence one obtains uniqueness of the trivial solution for the unreflected problem SO cf.

Therefore, one can say that the presence of the nonlocal effect in N SO , as represented by R, not only yields a variational structure but allows for nontrivial solutions as well. For the problem N ST , similar uniqueness considerations may well hold, although the lack of regularity in the nonlinearity f does not allow one to apply known results such as those of [13] and [21] to conclude that the unreflected problem has the unique trivial solution.

References [1] A. Ambrosetti and G. Agmon, L. Nirenberg, and M.

The dual variational method in nonlocal semilinear Trii

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Equations, to appear. Manes and A.