Semilinear elliptic equations are ubiquitous in natural sciences. They give rise to a variety of important phenomena in quantum mechanics, nonlinear optics, astrophysics, etc because they have rich multiple solutions. But the nontrivial solutions of semilinear equations are hard to be solved for the lack of stabilities, such as Lane-Emden equation, Henon equation and Chandrasekhar equation. In ...

we prove the existence of steady 2-dimensional flows, containing a bounded vortex, and approaching a uniform flow at infinity. the data prescribed is the rearrangement class of the vorticity field. the corresponding stream function satisfies a semilinear elliptic partial differential equation. the result is proved by maximizing the kinetic energy over all flows whose vorticity fields are rearra...

We prove the existence of steady 2-dimensional flows, containing a bounded vortex, and approaching a uniform flow at infinity. The data prescribed is the rearrangement class of the vorticity field. The corresponding stream function satisfies a semilinear elliptic partial differential equation. The result is proved by maximizing the kinetic energy over all flows whose vorticity fields are rearra...

In this paper, we study the multiplicity of nontrivial nonnegative solutions for a semilinear elliptic equation involving nonlinear boundary condition and sign-changing potential. With the help of the Nehari manifold, we prove that the semilinear elliptic equation: −∆u+ u = λf(x)|u|q−2u in Ω, ∂u ∂ν = g(x)|u|p−2u on ∂Ω, has at least two nontrivial nonnegative solutions for λ is sufficiently small.

Let Ω ⊂ RN be the upper half strip with a hole. In this paper, we show there exists a positive higher energy solution of semilinear elliptic equations in Ω and describe the dynamic systems of solutions of equation (1) in various Ω. We also show there exist at least two positive solutions of perturbed semilinear elliptic equations in Ω.

We study the large time behavior of solutions for the semilinear parabolic equation ∆u+V up−ut = 0. Under a general and natural condition on V = V (x) and the initial value u0, we show that global positive solutions of the parabolic equation converge pointwise to positive solutions of the corresponding elliptic equation. As a corollary of this, we recapture the global existence results on semil...

We consider a semilinear elliptic equation ∆u+ λf(u) = 0, x ∈ Ω, ∂u

Some oscillation criteria for the second order semilinear elliptic differential equation

Some oscillation criteria for the second order semilinear elliptic differential equation

In this note we establish multiple solutions for a semilinear elliptic equation with superlinear nonlinearility without assuming any symmetry.