A note on the dependence of solutions on functional parameters for nonlinear sturm-liouville problems

We deal with the existence and the continuous dependence of solutions on functional parameters for boundary valued problems containing the Sturm-Liouville equation. We apply these result to prove the existence of at least one solution for a certain class of optimal control problems

Minimal positive solutions for systems of semilinear elliptic equations

The paper is devoted to a system of nonlinear PDEs containing gradient terms. Applying the approach based on Sattinger's iteration procedure we use sub and supersolutions methods to prove the existence of positive solutions with minimal growth. These results can be applied for both sublinear and superlinear problems

Positive evanescent solutions of singular elliptic problems in exterior domains

We investigate the existence of positive solutions for the following class of nonlinear elliptic problems $$\operatorname{div}(a(\|x\|)\nabla u(x))+f(x,u(x))-(u(x))^{-\alpha}\|\nabla u(x)\|^{\beta}+g(\|x\|)x\cdot\nabla u(x)=0,$$ where $x\in\mathbb{R}^{n}$ and $\|x\|>R,$ with the condition $\lim_{\|x\|\rightarrow\infty}u(x)=0$. We present the approach based on the subsolution and supersolution method for bounded subdomains and a certain convergence procedure. Our results cover both sublinear and superlinear cases of $f$. The speed of decaying of solutions will be also characterized more precisely

The existence of minimizers of the action functional without convexity assumption

We shall prove the existence of minimizers of the following functional $f(u)=\int_{0}^{T}L(x,u(x),u'(x))dx$ without convexity assumption. As a consequence of this result and the duality described in [A. Nowakowski, Metody wariacyjne dla nieliniowych problemów Dirichleta (Chapter 6), Wydawnictwo Naukowo Techniczne, Warszawa, 1994] we derive the existence of solutions for the Dirichlet problem for a certain differential inclusion being a generalization of the Euler-Lagrange equation of the functional $f$

A note on the dependence of solutions on functional parameters for nonlinear Sturm-Liouville problems

Tyt. z nagłówka.Bibliogr. s. 847-848.We deal with the existence and the continuous dependence of solutions on functional parameters for boundary valued problems containing the Sturm-Liouville equation. We apply these result to prove the existence of at least one solution for a certain class of optimal control problems.Dostępny również w formie drukowanej.KEYWORDS: positive solution, continuous dependence of solutions on functional parameters, Sturm-Liouville equation

Positive stationary solutions of convection-diffusion equations for superlinear sources

We investigate the existence and multiplicity of positive stationary solutions for a certain class of convection-diffusion equations in exterior domains. This problem leads to the following elliptic equation Δu(x) + f(x, u(x)) + g(x)x · ∇u(x) = 0, for x ∈ ΩR = {x ∈ Rn, ∥x∥ > R}, n > 2. The goal of this paper is to show that our problem possesses an uncountable number of nondecreasing sequences of minimal solutions with finite energy in a neighborhood of infinity. We also prove that each of these sequences generates another solution of the problem. The case when f(x, ·) may be negative at the origin, so-called semipositone problem, is also considered. Our results are based on a certain iteration schema in which we apply the sub and supersolution method developed by Noussair and Swanson. The approach allows us to consider superlinear problems with convection terms containing functional coefficient g without radial symmetry

Continous dependence of solutions of elliptic BVPs on parameters

Tyt. z nagłówka.Bibliografia s. 358.Dostępny również w formie drukowanej.ABSTRACT: The continuous dependence of solutions for a certain class of elliptic PDE on functional parameters is studied in this paper. The main result is as follow: the sequence [formula] of solutions of the Dirichlet problem discussed here (corresponding to parameters [formula] converges weakly to [formula] (corresponding to [formula]), in [formula], provided that [formula] tends to [formula] a.e. in Ω. Our investigation covers both sub and superlinear cases. We apply this result to some optimal control problems. KEYWORDS: continuous dependence on parameters, elliptic Dirichlet problems, optimal control problem

Positive solutions for a nonconvex elliptic Dirichlet problem with superlinear response

The existence of bounded solutions of the Dirichlet problem for a ceratin class of elliptic partial differential equations is discussed here. We use variational methods based on the subdifferential theory and the comparison principle for difergence form operators. We present duality and variational principles for this problem. As a consequences of the duality we obtain also the variational principle for minimizing sequences of $J$ which gives a measure of a duality gap between primal and dual functional for approximate solutions