Radial Solutions Of Semilinear Equations Research Articles

Regularity of the extremal solution for a biharmonic problem with general nonlinearity

We consider the class of radial solutions of semilinear equations $\Delta^2 u=\lambda f(u)$ in the unit ball of $\mathbb R^N$. It is the class of stable solutions which includes minimal solutions and extremal solution. We establish the regularity of this extremal solution for $N\leq 9$. Our regularity results do not depend on the specific nonlinearity $f$.

Computation of radial solutions of semilinear equations

We express radial solutions of semilinear elliptic equations on R n as convergent power series in r, and then use Pade approximants to compute both ground state solutions, and solutions to Dirichlet problem. Using a similar approach we have discovered existence of singular solutions for a class of subcritical problems. We prove convergence of the power series by modifying the classical method of majorants.

Regularity of radial minimizers and extremal solutions of semilinear elliptic equations

We consider a special class of radial solutions of semilinear equations − Δ u = g ( u ) in the unit ball of R n . It is the class of semi-stable solutions, which includes local minimizers, minimal solutions, and extremal solutions. We establish sharp pointwise, L q , and W k , q estimates for semi-stable radial solutions. Our regularity results do not depend on the specific nonlinearity g. Among other results, we prove that every semi-stable radial weak solution u ∈ H 0 1 is bounded if n ⩽ 9 (for every g), and belongs to H 3 = W 3 , 2 in all dimensions n (for every g increasing and convex). The optimal regularity results are strongly related to an explicit exponent which is larger than the critical Sobolev exponent.