Hossein Tehrani

Existence of positive solutions for a class of indefinite elliptic problems in \({\mathbb R}^N\)

Abstract. We consider the question of existence of positive solutions for a class of elliptic problems in all of

Positive solutions to logistic type equations with harvesting

{mathbb R}^N

Positive Solutions to Semilinear Elliptic Equations with Logistic Type Nonlinearities and Constant Yield Harvesting in ℝ N

having a noncoercive linear part

A note on asymptotically linear elliptic problems in RN

- \Delta u - \lambda h(x) u

Non-zero solutions for a Schrödinger equation with indefinite linear and nonlinear terms

and a sign-changing nonlinearity of the form

A multiplicity result for the jumping nonlinearity problem

a(x) g(u)

Simple existence proofs for one-dimensional semi-positone problems

.

1,2(ℝ N ) versus C(ℝ N ) local minimizer on manifolds and multiple solutions for zero-mass equations in ℝ N

Abstract We use comparison principles, variational arguments and a truncation method to obtain positive solutions to logistic type equations with harvesting both in R N and in a bounded domain Ω ⊂ R N , with N ⩾ 3 , when the carrying capacity of the environment is not constant. By relaxing the growth assumption on the coefficients of the differential equation we derive a new equation which is easily solved. The solution of this new equation is then used to produce a positive solution of our original problem.

Diffusive logistic equations with harvesting and heterogeneity under strong growth rate

We consider a class of semilinear elliptic equations − Δ u = f(x, u) in all of ℝ N with nonlinearities of the form where λ, μ are positive parameters, a(x), h(x) are positive functions, and g(u) is a super-linearly increasing function in a more general fashion than the classical logistic term u 2. From a practical point of view, these problems can provide models for fishery or hunting management (cf. [8]) where μ h(x) denotes a harvesting term and, as such, one is interested in situations allowing the existence of positive solutions. From a mathematical point of view, these elliptic problems belong to the class of so-called semi-positone problems (cf. [2]) because the nonlinearity f(x, u) satisfies f(x, 0) < 0. Under suitable assumptions on a(x), h(x), we use variational methods to show that, for each λ > λ1 (a) (where λ1 (a) denotes the principal eigenvalue of − Δ u = λ a(x)u ∈ D 1,2(ℝ N )), there exists a positive solution decaying at infinity like O(|x|−(N−2)), provided that 0 < μ < (λ).

On the Fučik spectrum of the wave operator and an asymptotically linear problem

for some V (x) ∈ C(R,R) and v∞ ∈ R. In case this equation is considered in a bounded domain ⊂ R (with, say, Dirichlet boundary condition) there is a large literature on existence and multiplicity results, with the case of resonance being of particular interest (see [1,4,5,7,9,12,15]). We recall that the problem is said to be at resonance if −λ ∈ σ(S), where σ(S) denotes the spectrum of S, the “asymptotic linearization” of the problem. In other words, S :D(S) ⊂ L2( )→ L2( ) is the operator given by Su(x)=−∆u(x)− V (x)u(x), D(S)=H 1 0 ( )∩H 2( ). (2)