Solutions For Semilinear Dirichlet Problems Research Articles

Ground state sign-changing solutions for semilinear Dirichlet problems

In the present paper, we consider the existence of ground state sign-changing solutions for the semilinear Dirichlet problem \t\t\t0.1{−△u+λu=f(x,u),x∈Ω;u=0,x∈∂Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\left \\{ \\textstyle\\begin{array}{l@{\\quad}l} -\\triangle u+\\lambda u=f(x, u), & \\hbox{$x\\in\\Omega$;} \\\\ u=0, & \\hbox{$x\\in\\partial\\Omega$,} \\end{array}\\displaystyle \\right . $$\\end{document} where Omegasubsetmathbb{R}^{N} is a bounded domain with a smooth boundary ∂Ω, lambda>-lambda_{1} is a constant, lambda_{1} is the first eigenvalue of (-triangle, H_{0}^{1}(Omega)), and fin C(Omegatimesmathbb{R}, mathbb{R}). Under some standard growth assumptions on f and a weak version of Nehari type monotonicity condition that the function tmapsto f(x, t)/|t| is non-decreasing on (-infty, 0)cup(0, infty) for every xinOmega, we prove that (0.1) possesses one ground state sign-changing solution, which has precisely two nodal domains. Our results improve and generalize some existing ones.

Asymptotic behavior of positive large solutions of semilinear Dirichlet problems

Let be a smooth bounded domain in R n ; n 2: This paper deals with the existence and the asymptotic behavior of positive solutions of the following problems u = a(x)u ; > 1 and u = a(x)e u ; with the boundary conditionuj@ = +1: The weight functiona(x) is positive inC loc () ; 0 < < 1, and satises an appropriate assumption related to Karamata regular variation theory. Our arguments are based on the sub-supersolution method.

A note on existence and stability of solutions for semilinear Dirichlet problems

We provide existence and stability results for a fourth-order semilinear Dirichlet problem in the case when both the coefficients of the differential operator and the nonlinear term depend on the numerical parameter. We use a dual variational method.

Multiple positive solutions for semilinear Dirichlet problems with sign-changing weight function in infinite strip domains

In this paper, existence and multiplicity results to the following Dirichlet problem { − Δ u + u = λ f ( x ) | u | q − 1 + h ( x ) | u | p − 1 , in Ω u > 0 , in Ω u = 0 , on ∂ Ω are established, where Ω = Ω ′ × R , Ω ′ ⊂ R N − 1 is bounded smooth domain and N ≥ 2 . Here 1 < q < 2 < p < 2 ∗ ( 2 ∗ = 2 N N − 2 if N ≥ 3 , 2 ∗ = ∞ if N = 2 ) , λ is a positive real parameter, the function f , among other conditions, can possibly change sign in Ω , and the function h satisfies suitable conditions. The study is based on the comparison of energy levels on Nehari manifold.

Corrections to ``Existence and stability of solutions for semilinear Dirichlet problems'' (Ann. Polon. Math. 88 (2006), 127–139)

Corrections to ``Existence and stability of solutions for semilinear Dirichlet problems'' (Ann. Polon. Math. 88 (2006), 127–139)

Existence and stability of solutions for semilinear Dirichlet problems

Existence and stability of solutions for semilinear Dirichlet problems

Multiple Solutions of Semilinear Dirichlet Problems on a Domain withS1-Group Actions

Let Ω be a bounded domain with a smooth boundary such that there exists anS1group of isometric linear transformations on Ω. In the present paper, we consider the multiple existence of solutions of the problem[formula]whereg∈C1(R) andh∈L2(Ω).

Solution curves for semilinear equations on a ball

We show that the set of positive solutions of semilinear Dirichlet problem on a ball of radius R in RI Au+ Af(u) = O for lxl < R, u = O on lxl = R consists of smooth curves. Our results can be applied to compute the direction of bifurcation. We also give an easy proof of a uniqueness theorem due to Smoller and Wasserman (1984).

Positive solutions of semilinear Dirichlet problems

We consider the existence problem of positive solutions of the semilinear Dirichlet problem with a different class of nonlinearities which does not depend on the behaviour of the function at zero and at infinity.

On the existence of sign changing solutions for semilinear Dirichlet problems

On the existence of sign changing solutions for semilinear Dirichlet problems

On the existence of sign changing solutions for semilinear Dirichlet problems

where Ω ⊂ R is a bounded domain with Lipschitz boundary and f : R → R is of class C with f(0) = 0. Thus u0 ≡ 0 is a trivial solution of (D) and we are interested in finding and studying nontrivial solutions. One way of obtaining these is to compare the behavior of f near the origin and near infinity. We shall always assume that f grows subcritically at infinity so that variational methods can be applied and the associated functional satisfies the Palais–Smale condition. Suppose f ′(0) < λ1 where 0 < λ1 < λ2 ≤ λ3 ≤ . . . are the eigenvalues (counted with multiplicities) of −∆ on Ω with homogeneous Dirichlet boundary conditions. If f grows superlinearly at infinity then the mountain pass theorem of Ambrosetti and Rabinowitz [AR], [R] together with the maximum principle guarantees the existence of a positive solution u+ and a negative solution u− of (D). Using linking or Morse type arguments Wang [Wa] obtained a third nontrivial solution u1. In this paper we shall refine Wang’s result and obtain more information on u1 and on other solutions whose existence is proved via Morse theory. Let us illustrate this with the following two theorems. More general results will be stated and proved later.

Positive solutions for semilinear dirichlet problems in an annulus

Positive solutions for semilinear dirichlet problems in an annulus

Solutions of semilinear elliptic problems in shrinking annuli

The positive, radially symmetric solutions of semilinear Dirichlet problems in annuli is studied, the inner radius of which is sufficiently small. By means of a phase plane analysis their asymptotic behaviour is computed as the inner radius shrinks. It is of particular interest in the cases where the Dirichlet problem for the sphere has no non-trivial solution. This work extends results of Bandle and Peletier [ Math. Ann. 280 (1988), 1–19].

Corner Behavior of Solutions of Semilinear Dirichlet Problems

In recent years there has been considerable attention paid to the behavior of solutions of elliptic boundary value problems in domains with piecewise smooth boundary. In two dimensions the study concerns the behavior of a solution near a corner, and in three (or more) dimensions two cases have been given considerable attention: a conical vertex on the boundary, or an edge.The solution of such a problem may be singular at the nonsmooth boundary points. The standard example in two dimensions is a solution in polar coordinates of the Dirichlet problem near a corner of interior angle πα;u = r1/α sin θ/α is a function which is harmonic in the sector 0 &lt; θ &lt; πα, has zero boundary values near the corner, and yet at the origin has unbounded derivatives of order &gt; 1/α unless 1/α is an integer.