Positive Solutions For Nonlinear Problems Research Articles

Positive solutions of generalized nonlinear logistic equations via sub-super solutions

Let Ω be a smooth bounded domain in RN, N≥1, let m,n be two nonnegative functions defined in Ω, and let ϕ:RN→RN be a continuous and strictly monotone mapping. We consider the existence and nonexistence of positive solutions for nonlinear problems involving the ϕ-Laplacian, of the form{−divϕ(∇u)=λm(x)f(u)−n(x)g(u)in Ω,u=0on ∂Ω, where λ>0 is a real parameter, and f,g:[0,∞)→[0,∞) are continuous functions modeled by f(t)=tq and g(t)=tr with 0<q<r. The subdiffusive, equidiffusive and superdiffusive cases are all treated, and some qualitative properties of the solutions are also given. As a byproduct, we obtain results in the case of a (roughly speaking) sublinear nonlinearity f and n≡0. We point out that some of our results are new even for the usual p-Laplacian. Our main tool is the sub-supersolutions method combined with some estimates on related nonlinear problems.

Positive solutions for nonlinear problems involving the one-dimensional ϕ-Laplacian

Let Ω:=(a,b)⊂R, m∈L1(Ω) and λ>0 be a real parameter. Let L be the differential operator given by Lu:=−ϕ(u′)′+r(x)ϕ(u), where ϕ:R→R is an odd increasing homeomorphism and 0≤r∈L1(Ω). We study the existence of positive solutions for problems of the form{Lu=λm(x)f(u)in Ω,u=0on ∂Ω, where f:[0,∞)→[0,∞) is a continuous function which is, roughly speaking, sublinear with respect to ϕ. Our approach combines the sub and supersolution method with some estimates on related nonlinear problems. We point out that our results are new even in the cases r≡0 and/or m≥0.

Nonlinear Liouville Problems in a Quarter Plane

We answer affirmatively the open problem proposed by Cabr\'e and Tan in their paper Positive solutions of nonlinear problems involving the square root of the Laplacian (see Adv. Math. {\bf 224} (2010), no. 5, 2052-2093).

Existence and global asymptotic behavior of positive solutions for nonlinear problems on the half-line

In this paper, we aim at studying the existence, uniqueness and the exact asymptotic behavior of positive solutions to the following boundary value problem{1A(Au′)′+a(t)uσ=0,t∈(0,∞),limt→0+u(t)=0,limt→∞u(t)ρ(t)=0, where σ<1, A is a continuous function on [0,∞), positive and differentiable on (0,∞) such that 1A is integrable on [0,1] and ∫0∞1A(t)dt=∞. Here ρ(t)=∫0t1A(s)ds, for t⩾0 and a is a nonnegative continuous function that is required to satisfy some assumptions related to the Karamata classes of regularly varying functions. Our arguments are based on monotonicity methods.

Some Remarks on Positive Solutions of Nonlinear Problems at Resonance

The proof of a result of J.J. Nieto [3] appeared in Acta Math. Hung. (1992) concerning the positive solutions of nonlinear problems at resonance is corrected and improved.

Positive solutions of nonlinear problems involving the square root of the Laplacian

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas–Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas–Ni–Nirenberg type.

Positive solutions for nonlinear hemivariational inequalities

In this paper we study the existence of positive solutions for nonlinear problems driven by the p-Laplacian or more generally, by multivalued p-Laplacian-like operators. Both problems have a nonsmooth locally Lipschitz potential (hemivariational inequalities). Using variational methods based on the nonsmooth critical point theory, we prove two existence results with the p-Laplacian and multivalued p-Laplacian-like operators.

Positive solutions for non-homogeneous semilinear elliptic equations with data that changes sign

Let Ω ⊂ RN be a bounded domain. We consider the nonlinear problem and prove that the existence of positive solutions of the above nonlinear problem is closely related to the existence of non-negative solutions of the following linear problem: .In particular, if p &gt; (N + 2)/(N − 2), then the existence of positive solutions of nonlinear problem is equivalent to the existence of non-negative solutions of the linear problem (for more details, we refer to theorems 1.2 and 1.3 in § 1 of this paper).

Uniqueness and nondegeneracy for some nonlinear elliptic problems in a ball

In this paper we study the uniqueness and nondegeneracy of positive solutions of nonlinear problems of the type Δ p u+ f( r, u)=0 in the unit ball B, u=0 on ∂B. Here Δ p denotes the p Laplace operator Δ p =div(|∇ u| p−2 ∇ u), p>1. The main ideas rely on the Maximum Principle and an implicit function theorem that we derive in a suitable weighted space. This space is essential to deal with the case p≠2.

Positive solutions of nonlinear problems at resonance

where L : D(L) C E -* F is a linear operator , N : E -~ F is a nonlinear operator , and E and F are Banach spaces. If L is noninvertible we say tha t (1) is a problem at resonance. In tha t case, one can use the Lyapunov-Schmidt me thod [15] or some of its extensions, such as the alternative method [1, 2, 4, 5], to give existence results for the opera tor equat ion (1). On the other hand, in numerous cases it is of interest to determine the existence of positive solutions. For instance, in the s tudy of a model of an infectious disease [7] or in the analysis of a nuclear reactor [14], By a positive solution of (1) we mean a solution which belongs to a given cone C of E . In applications, one usually takes E as a subspace of L2(A), A a domain of R n, and C as the cone of the positive functions, tha t is, C = {u ~ E : u => 0 a.e. in A}. In Section 2, we present an existence result in a cone (Theorem 2) for nonlinear problems at resonance of type (1). Theorem 2 generalizes the result in [10] since we do not require N ( C ) to be bounded. In Section 3, we use the me thod of upper and lower solutions to show the existence of positive periodic solutions for a second order nonlinear differential equat ion (Theorem 6). Taking into account the symmetries of the equation, we give sufficient conditions for the problem to have a negative periodic solution (Theorem 7). These two theorems improve some of the results of Nieto and Rao in [11].