Nonexistence of Solutions of a Semilinear Biharmonic Equation with Singular Potential
The nonexistence of a global solution of the semilinear elliptic equation Δ2u − (C/|x|4)u − |x|σ|u|q = 0 in the exterior of a ball is studied. A sufficient condition for the nonexistence of a global solution is established. The proof is based on the test function method.
Keywordssemilinear elliptic equation biharmonic operator global solution critical exponent test function method
Unable to display preview. Download preview PDF.
- 1.É. Mitidieri and S. I. Pokhozhaev, “A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities,” in TrudyMat. Inst. Steklov (Nauka, Moscow, 2001), Vol. 234, pp. 3–383 [Proc. Steklov Inst. Math. 234, 1–362].Google Scholar
- 3.É. Mitidieri and S. I. Pokhozhaev, “Nonexistence of Positive Solutions for Quasilinear Elliptic Problems on RN,” in Trudy Mat. Inst. Steklov, Vol. 227: Studies in the Theory of Differentiable Functions of Several Variables and Its Applications. Pt. 18 (Nauka, Moscow, 1999), pp. 192–222 [Proc. Steklov Inst. Math. 227, 186–216 (1998)].Google Scholar
- 8.J. Serrin, “Positive solutions of prescribedmean curvature problem,” in Calculus of Variations and Partial Differential Equations, Lect. Notes in Math. (Springer-Verlag, Berlin, 1998), Vol. 1340, pp. 248–255.Google Scholar
- 14.G. G. Laptev, “On the absence of solutions for a class of singular semilinear differential inequalities,” in Trudy Mat. Inst. Steklov, Vol. 232: Function Spaces, Harmonic Analysis, and Differential Equations (Nauka, Moscow, 2001), pp. 223–235 [Proc. Steklov Inst. Math. 232, 216–228 (2001)].Google Scholar
- 16.Yu. V. Volodin, “The critical exponents of semilinear boundary-value problems with biharmonic operator in the exterior of a ball with boundary conditions of first type” Uchen. Zap. Ross. Gos. Sots. Univ., No. 8, 208–215 (2009).Google Scholar