# ON THE MINIMAL ENERGY SOLUTION IN A QUASILINEAR ELLIPTIC EQUATION

Park, Sang-Don;Kang, Chul

• Published : 2003.01.01
• 44 4

#### Abstract

In this paper we seek a positive, radially symmetric and energy minimizing solution of an m-Laplacian equation, -div$($\mid${\nabla}u$\mid$^{m-2}$\mid${\nabla}u)\;=\;h(u)$. In the variational sense, the solutions are the critical points of the associated functional called the energy, $J(v)\;=\;\frac{1}{m}\;\int_{R^N}\;$\mid${\nabla}v$\mid$^m\;-\;\int_{R^N}\;H(v)dx,\;where\;H(v)\;=\;{\int_0}^v\;h(t)dt$. A positive, radially symmetric critical point of J can be obtained by solving the constrained minimization problem; minimize{$\int_{R^N}$\mid${\nabla}u$\mid$^mdx$\mid$\;\int_{R^N}\;H(u)d;=\;1$}. Moreover, the solution minimizes J(v).

#### Keywords

quasilinear elliptic;m-Laplacian;constrained minimization;variational equation;radially symmetric;Lagrange multiplier

#### References

1. Nonlinear Analysis, Theory, Methods and Applications v.12 no.11 Boundary regularity for solutions degenerate elliptic equations G. M. Liberman https://doi.org/10.1016/0362-546X(88)90053-3
2. Nonlinear partial differential equations and free boundary v.I J. I. Diaz
3. Arch. Mech. Anal. v.82 Nonlinear scalar equations, I. Existence of ground state H. Berestycki;P. L. Lions
4. Variational methods M. Struwe
5. Comm. Math. Phys. v.55 Existence of solitary waves in higher dimensins W. A. Strauss https://doi.org/10.1007/BF01626517
6. Proc. London Math. soc. v.30 Boundary value problems for ordinary differential equations in infinite intervals E. N. Dancer https://doi.org/10.1112/plms/s3-30.1.76
7. Comm. Math. Phys. v.58 no.2 Action minima among solutions to a class of Euclidean scalar field equations S. Coleman;V. Glazer;A. Martin https://doi.org/10.1007/BF01609421
8. Nonlinear Analysis, Theory and Methods and Applications v.13 Quasilinear equations involving critical Sovolev exponents M. Guedda;L. Veron