Some Aspects of Semilinear Elliptic Equations on ℝn

In this expository paper, I intend to give a partial survey of semilinear elliptic equations on ℝ n.

[1]  J. Moser On a Nonlinear Problem in Differential Geometry , 1973 .

[2]  Ding Weiyue,et al.  On a conformally invariant elliptic equation onRn , 1986 .

[3]  Wei-Ming Ni,et al.  On the elliptic equation $\Delta u+Ku^{(n+2)/(n-2)}=0$ and related topics , 1985 .

[4]  H. Wittich Ganze Lösungen der Differentialgleichung Δu=eu , 1943 .

[5]  Shoji Yotsutani,et al.  Semilinear elliptic equations of Matukuma-type and related topics , 1988 .

[6]  Pierre-Louis Lions,et al.  The concentration-compactness principle in the Calculus of Variations , 1985 .

[7]  W. Ni,et al.  On the existence of positive entire solutions of a semilinear elliptic equation , 1986 .

[8]  Avner Friedman,et al.  Bounded entire solutions of elliptic equations. , 1973 .

[9]  L. Nirenberg,et al.  Partial Differential Equations Invariant under Conformal or Projective Transformations , 1974 .

[10]  E. Calabi An extension of E. Hopf’s maximum principle with an application to Riemannian geometry , 1958 .

[11]  On the least growth of harmonic functions and the boundary behavior of riemann mappings , 1985 .

[12]  R. McOwen On the equation Δu + Ke2u = f and prescribed negative curvature in R2 , 1984 .

[13]  J. Batt,et al.  Stationary spherically symmetric models in stellar dynamics , 1986 .

[14]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, I existence of a ground state , 1983 .

[15]  M. Berger On the existence and structure of stationary states for a nonlinear Klein-Gordan equation , 1972 .

[16]  C. Swanson,et al.  Existence theorems for generalized Klein-Gordon equations , 1983 .

[17]  F. Lin On the Elliptic Equation D i [ a ij (x)D j U ] - k(x)U + K(x)U p = 0 , 1985 .

[18]  O. Oleinik ON THE EQUATION Δu + k(x)eu = 0 , 1978 .

[19]  H. Walker,et al.  The null spaces of elliptic partial differential operators in Rn , 1973 .

[20]  M. Kwong,et al.  Uniqueness of non-negative solutions of a class of semi-linear elliptic equations , 1988 .

[21]  R. McOwen,et al.  Conformal deformations of complete manifolds with negative curvature , 1985 .

[22]  W. Ni,et al.  A counterexample to the nodal domain conjecture and a related semilinear equation , 1988 .

[23]  Robert Osserman,et al.  On the inequality $\Delta u\geq f(u)$. , 1957 .

[24]  G. Stampacchia,et al.  Regular points for elliptic equations with discontinuous coefficients , 1963 .

[25]  J. Kazdan,et al.  Prescribing the Curvature of a Riemannian Manifold , 1985 .

[26]  Lambertus A. Peletier,et al.  Uniqueness of positive solutions of semilinear equations in ℝn , 1983 .

[27]  A. Eddington The Dynamics of a Globular Stellar System. Second Paper , 1915 .

[28]  C. V. Coffman Uniqueness of the ground state solution for Δu−u+u3=0 and a variational characterization of other solutions , 1972 .

[29]  Walter A. Strauss,et al.  Existence of solitary waves in higher dimensions , 1977 .

[30]  Shoji Yotsutani,et al.  On Matukuma's equation and related topics , 1986 .

[31]  P. Rabinowitz Some aspects of nonlinear eigenvalue problems , 1973 .

[32]  P. Lions On the Existence of Positive Solutions of Semilinear Elliptic Equations , 1982 .

[33]  James Serrin,et al.  Uniqueness of positive radial solutions of Δu+f(u)=0 in ℝn , 1987 .

[34]  R. Schoen The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation , 1988 .

[35]  D. Sattinger Conformal Metrics in R^2 with Prescribed Curvature , 1972 .

[36]  Manabu Naito A note on bounded positive entire solutions of semilinear elliptic equations , 1984 .

[37]  L. Ahlfors An extension of Schwarz’s lemma , 1938 .

[38]  L. Ahlfors,et al.  RIEMANN'S MAPPING THEOREM FOR VARIABLE METRICS* , 1960 .

[39]  W. Ni On the elliptic equation Δu+K(x)e2u=0 and conformal metrics with prescribed Gaussian curvatures , 1982 .

[40]  Nichiro Kawano,et al.  On bounded entire solutions of semilinear elliptic equations , 1984 .

[41]  Z. Nehari,et al.  ON A NONLINEAR DIFFERENTIAL EQUATION ARISING IN NUCLEAR PHYSICS , 1963 .

[42]  C. Kenig,et al.  On the elliptic equation $Lu - k + K \exp [2u] = 0$ , 1985 .

[43]  B. Gidas,et al.  Symmetry and related properties via the maximum principle , 1979 .

[44]  J. Serrin,et al.  Nonexistence theorems for singular solutions of quasilinear partial differential equations , 1986 .

[45]  R. Schoen Conformal deformation of a Riemannian metric to constant scalar curvature , 1984 .

[46]  C. Kenig,et al.  An exterior Dirichlet problem with applications to some nonlinear equations arising in geometry , 1984 .

[47]  C. Stuart Bifurcation for Dirichlet problems without eigenvalues , 1982 .

[48]  R. McOwen The behavior of the laplacian on weighted sobolev spaces , 1979 .