By Michel Chipot

ISBN-10: 0080557317

ISBN-13: 9780080557311

ISBN-10: 044453217X

ISBN-13: 9780444532176

A suite of self contained state-of-the paintings surveys. The authors have made an attempt to accomplish clarity for mathematicians and scientists from different fields, for this sequence of handbooks to be a brand new reference for examine, studying and instructing. * Written through famous specialists within the box * Self contained quantity in sequence overlaying essentially the most swift constructing subject matters in arithmetic * expert and punctiliously up to date for college students, lecturers and researchers

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For the case of p 1 we apply the Mountain-Pass Theorem to the functional I (u) = p p+1 | u| p+1 p − Ω F (u) Ω p u p+1 p+1 p C −o u p+1 p C which is a C 1 -functional on the space E=W 2, p+1 p Observe that since p < W 2, p+1 p (Ω) 1, p+1 p (Ω) ∩ W0 2 N−2 , (Ω). it follows p+1 p >1+ N−2 2 > N 2, and thus C(Ω), which implies that the second term of the functional I is defined if F is continuous, and so no growth restriction on F is necessary! 5. Multiplicity of solutions for elliptic systems In this section we discuss the multiplicity of solutions for elliptic systems of the form studied previously.

7) D EFINITION 2. 4) is strongly coupled if there are positive numbers β1 , β2 such that β1 + 2 − β1 α11 > 0, β1 + 2 − β2 α12 = 0, β2 + 2 − β1 α21 = 0, β2 + 2 − β2 α22 > 0. G. 1. 9) which requires that α11 > 1, α22 > 1 and α12 < α22 − 1 α11 α11 − 1 and α21 < α11 − 1 α22 . 2. 11) which requires that α12 α21 > 1 and α11 < α21 + 1 α12 α12 + 1 and α22 < α12 + 1 α21 . 3. We observe that the requirements that α11 , α22 > 1 and α12 α21 > 1 are known as super-linearity conditions. W EAKLY C OUPLED S YSTEM .

7] C. Bandle and M. Essen, On positive solutions of Emden equations in cones, Arch. Ration. Mech. Anal. 112 (1990), 319–338. [8] T. Bartsch and M. Clapp, Critical point theory for indefinite functionals with symmetries, J. Funct. Anal. 138 (1996), 107–136. [9] T. G. de Figueiredo, Infinitely many solutions of nonlinear elliptic systems, Progr. , vol. 35 (The Herbert Amann Anniversary Volume) (1999), 51–68; (1996), 107–136. [10] T. Bartsch and M. Willem, Infinitely many nonradial solutions of a Euclidean scalar field equation, J.