Dirichlet problem in one-dimensional billiard space with velocity dependent right-hand side

We present some existence and multiplicity results for positive solutions to the Dirichlet problem associated with

This paper deals with existence and regularity of positive solutions of sublinear equations of the form $-\Delta u + b(x)u =\lambda f(u)$ in $\Omega$ where either $\Omega\in R^N$ is a bounded smooth domain in which case we consider the Dirichlet problem or $\Omega =R^N$, where we look for positive solutions, $b$ is not necessarily coercive or continuous and $f$ is a real function with sublinear growth which may have certain discontinuities. We explore the method of lower and upper solutions associated with some subdifferential calculus.

Symmetry of components and Liouville theorems for noncooperative elliptic systems on the half-space

We study the periodic boundary value problem associated with the second order nonlinear equation \begin{equation*} u'' + ( \lambda a^{+}(t) - \mu a^{-}(t) ) g(u) = 0, \end{equation*} where $g(u)$ has superlinear growth at zero and sublinear growth at infinity. For $\lambda, \mu$ positive and large, we prove the existence of $3^{m}-1$ positive $T$-periodic solutions when the weight function $a(t)$ has $m$ positive humps separated by $m$ negative ones (in a $T$-periodicity interval). As a byproduct of our approach we also provide abundance of positive subharmonic solutions and symbolic dynamics. The proof is based on coincidence degree theory for locally compact operators on open unbounded sets and also applies to Neumann and Dirichlet boundary conditions. Finally, we deal with radially symmetric positive solutions for the Neumann and the Dirichlet problems associated with elliptic PDEs.

New existence and multiplicity of solutions for some Dirichlet problems with impulsive effects

Semilinear elliptic problems near resonance with a nonprincipal eigenvalue

We deal with a Dirichlet problem driven by the degenerate fractional p-Laplacian and involving a nonlinear reaction which satisfies, among other hypotheses, a (p-1)-linear growth at infinity with non-resonance above the first eigenvalue. The energy functional governing the problem is thus noncoercive. We focus on the behavior of the reaction near the origin, assuming that it has a (p-1)-sublinear growth at zero, vanishes at three points, and satisfies a reverse Ambrosetti–Rabinowitz condition. Under such assumptions, by means of critical point theory and Morse theory, and using suitably truncated reactions, we show the existence of five nontrivial solutions: two positive, two negative, and one nodal.

In this work we obtain an existence result for a class of singular quasilinear elliptic Dirichlet problems on a smooth bounded domain containing the origin. By using a Caffarelli-Kohn-Nirenberg type inequality, a critical point result for differentiable functionals is exploited, in order to prove the existence of a precise open interval of positive eigenvalues for which the treated problem admits at least one nontrivial weak solution. In the case of terms with a sublinear growth near the origin, we deduce the existence of solutions for small positive values of the parameter. Moreover, the corresponding solutions have smaller and smaller energies as the parameter goes to zero.

Abstract. A numerical method for solving the Dirichlet problem for the wave equation in the two-dimensional space is proposed. The problem is analyzed for ill-posedness and a regularization algorithm is constructed. The first stage in the regularization process consists in the Fourier series expansion with respect to one of the variables and passing to a finite sequence of Dirichlet problems for the wave equation in the one-dimensional space. Each of the obtained Dirichlet problems for the wave equation in the one-dimensional space is reduced to the inverse problem with respect to a certain direct (well-posed) problem. The degree of ill-posedness of the inverse problem is analyzed based on the character of decreasing of the singular values of the operator A. The numerical solution of the inverse problem is reduced to minimizing the objective functional . The results of numerical calculations are presented.

Blow-up time estimates and simultaneous blow-up of solutions in nonlinear diffusion problems

The purpose of the article is to investigate Dirichlet boundary-value problems of the fractional p-Laplacian equation with impulsive effects. By using the Nehari manifold method, mountain pass theorem and three critical points theorem, some new results are achieved under more general growth conditions. In addition, this paper weakens the commonly used p-suplinear and p-sublinear growth conditions.

Multiple solutions of a Dirichlet problem in one-dimensional billiard space