Existence of Large Solutions to Semilinear Elliptic Equations with Multiple Terms
AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OH SCHOOL OF ENGINEERING AND MANAGEMENT
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We consider the semilinear elliptic equation Delta u pxusup alpha qxusup beta on a domain Omega reflex subset contained in real numbersup n, n or 3, where p and q are nonnegative continuous functions with the property that each of their zeroes is contained in a bounded domain Omegasub p or Omegasub q, respectively in Omega such that p is positive on the boundary of Omegasub p and q is positive on the boundary of Omegasub q. For Omega bounded, we show that there exists a nonnegative solution u such that ux towards infinity as x towards the derivative of Omega is 0 is alpha is or to beta, and beta is 1, and that such a solution does not exist is 0 is alpha is or beta is or to 1. For Omega real numbersup n, we established conditions on p and q to guarantee the existence of a nonnegative solution u satisfying ux towards infinity as the absolute value of x approaches infinity for 0 alpha is or beta, and beta 1, and for 0 alpha is or beta is or 1. For Omegareal numbersup n and 0 , alpha and beta and 1, we also establish conditions on p and q for the existence and nonexistence of a solution of u where u is bounded on Real numbersup n.
- Numerical Mathematics
- Theoretical Mathematics