Semilinear Boundary Value Problems

Martin Schechter
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Martin Schechter: University of California, Department of Mathematics

Chapter Chapter 3 in Linking Methods in Critical Point Theory, 1999, pp 55-72 from Springer

Abstract: Abstract Many elliptic semilinear problems can be described in the following way. Let Ω be a domain in R n and let A be a selfadjoint operator on L2(Ω). We assume that A ≥ λ0 > 0 and that 3.1.1 $$C_0^\infty (\Omega ) \subset D: = D({A^{1/2}}) \subset {H^{m,2}}(\Omega )$$ for some m > 0, where C 0 ∞ (Ω) denotes the set of test functions in Ω (i.e., infinitely differentiable functions with compact supports in Ω) and Hm,2(Ω) denotes the Sobolev space described in Appendix I to this chapter. If m is an integer, the norm in Hm,2(Ω) is given by 3.1.2 $${\left\| u \right\|_{m,2}}: = {\left( {\sum\limits_{\left| \mu \right|m} {{{\left\| {{D^u}u} \right\|}^2}} } \right)^{1/2}}.$$

Keywords: Compact Subset; Sobolev Inequality; Generic Derivative; Selfadjoint Operator; Critical Point Theory (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1007/978-1-4612-1596-7_3

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