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In this paper, we study the existence of two positive solutions of superlinear elliptic equations without assuming the conditions which have been used in the literature to deduce either the P.S. condition or a priori bounds of positive solutions.
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Small asymmetries between EP state 1 (EP1) and EP state 2 (EP2) have already been reported, but were deduced either from asymmetries in the residence time statistics31 or from the occurrence of dynamical bistability27.
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We deduce that either both α and β are transcendental functions or both α and β are polynomials.
Since bookmaker 1 gets a positive payoff, we deduce that either (pi ^H_1 = pi ^H_{min }) or (pi ^A_1 = pi ^A_{min }) or both.
Further, as in Step 2, we deduce that either fulfils (2.21) or has the fourth zero, is positive on with just one local maximum on, and.
end{cases} Applying the Liouville theorem (see Theorem 2.1), we deduce that either w_{i}(x equiv0,quad x in{R^{n}} quadmbox{or}quad w_{i} (x equiv c_{i}x_{n}^{frac{alpha}{2}},quad forall x in{R^{n}_, i=1,2, for some positive constants (c_{i}>0), then we could write (w_{i} = c_{i}x_{n}^{frac{alpha}{2}}), (c_{i}geq0).
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