Multiple solutions for semilinear elliptic equations with Neumann boundary condition and jumping nonlinearities

We obtain nonconstant solutions of semilinear elliptic Neumann boundary value problems with jumping nonlinearities when the asymptotic limits of the nonlinearity fall in the type (Il), l>2 and (IIl), l?1 regions formed by the curves of the Fucik spectrum. Furthermore, we have at least two nonconstant solutions in every order interval under resonance case. In this paper, we apply the sub-sup solution method, Fucik spectrum, mountain pass theorem in order intervals, degree theory and Morse theory to get the conclusions.     

Existence of positive solutions to semilinear elliptic problems with nonlinear boundary condition

In this paper, a semilinear elliptic equation with a nonlinear boundary condition and a perturbation in the reaction term is studied. The existence of a positive solution and another non-zero solution to the problem is proved when \(|\lambda |\) is small enough without any specific assumptions on the perturbation term. Moreover, it is shown that the non-zero solution becomes a positive one for small \(\lambda >0\) under suitable assumptions on the perturbation term.     

Multiple positive solutions for a class of semilinear elliptic systems with nonlinear boundary condition

Two positive solutions are obtained for the nonlinear homogeneous system with nonlinear homogeneous boundary condition via the Nehari manifold approach.     

Positive solutions for semilinear elliptic equations with singular forcing terms

We consider the existence of solutions to the semilinear elliptic problem

(∗)_κ     

Some semilinear elliptic equations with nonlinear boundary conditions: Radial solutions and simple symmetry-breaking

The subject of nonnegative solutions for semilinear equationshas received considerable attention in recent years. A significantomission is the consideration of nonlinear boundary conditions,and their impact upon the structure of solutions. In this papersome such problems defined on spherically symmetric domainsare presented. The author considers the existence of radiallysymmetric solutions as a function of domain size, and also showsthat infinitesimal symmetry-breaking bifurcations may occurin the simplest eigenmode. This is precluded in the case ofhomogeneous Neumann or Dirichlet problems for similar sourceterms. Given the importance of robust simple symmetry-breaking(in cell division for example), this result suggests attentionshould be further focused upon modelling nonlinear boundaryconditions.     

Positive solutions of linear elliptic equations with critical growth in the Neumann boundary condition

We study the existence of positive solutions of a linear elliptic equation with critical Sobolev exponent in a nonlinear Neumann boundary condition. We prove a result which is similar to a classical result of Brezis and Nirenberg who considered a corresponding problem with nonlinearity in the equation. Our proof of the fact that the dimension three is critical uses a new Pohoaev-type identity.AMS Subject Classification: Primary: 35J65; Secondary: 35B33.     

Positive solutions for nonlinear singular superlinear elliptic equations

Positivity - We consider a nonlinear nonparametric elliptic Dirichlet problem driven by the p-Laplacian and reaction containing a singular term and a $$(p-1)$$ -superlinear perturbation. Using...     

Exact boundary behavior of large solutions to semilinear elliptic equations with a nonlinear gradient term

This paper is concerned with exact boundary behavior of large solutions to semilinear elliptic equations △u=b(x)f(u)(C0+|▽u|q),x∈Ω,where Ω is a bounded domain with a smooth boundary in RN,C0≥0,q E [0,2),b∈Clocα(Ω) is positive in but may be vanishing or appropriately singular on the boundary,f∈C[0,∞),f(0)=0,and f is strictly increasing on [0,∞)(or f∈C(R),f(s)> 0,■s∈R,f is strictly increasing on R).We show unified boundary behavior of such solutions to the problem under a new structure condition on f.     

Positive entire stable solutions of inhomogeneous semilinear elliptic equations

For n≥3 and p>1, the elliptic equation Δu+K(x)u^p+μf(x)=0 in possesses a continuum of positive entire solutions, provided that (i) locally Hölder continuous functions K and f vanish rapidly, for instance, K(x),f(x)=O(|x|^l) near ∞ for some l<−2 and (ii) μ≥0 is sufficiently small. Especially, in the radial case with K(x)=k(|x|) and f(x)=g(|x|) for some appropriate functions k,g on [0,∞), there exist two intervals I_μ,1, I_μ,2 such that for each α∈I_μ,1 the equation has a positive entire solution u_α with u_α(0)=α which converges to l∈I_μ,2 at ∞, and u_α1<u_α2 for any α₁<α₂ in I_μ,1. Moreover, the map α to l is one-to-one and onto from I_μ,1 to I_μ,2. If K≥0, each solution regarded as a steady state for the corresponding parabolic equation is stable in the uniform norm; moreover, in the radial case the solutions are also weakly asymptotically stable in the weighted uniform norm with weight function |x|ⁿ⁻².     

Large solutions of semilinear elliptic equations under the Keller-Osserman condition

We consider the equation Δu=p(x)f(u) where p is a nonnegative nontrivial continuous function and f is continuous and nondecreasing on [0,∞), satisfies f(0)=0, f(s)>0 for s>0 and the Keller-Osserman condition where . We establish conditions on the function p that are necessary and sufficient for the existence of positive solutions, bounded and unbounded, of the given equation.