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Abstract: In this paper, we consider the existence problem to some type of the Lidstone boundary value problem at resonance with discontinuity of the Carathéodory type. We prove that a solution exists if the conditions similar to the Landesman–Lazer ones are satisfied. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the semilinear wave equation with boundary damping and source terms. This work is devoted to prove a finite time blow-up result under suitable condition on the initial data and positive initial energy. [Copyright &y& Elsevier]

Abstract: In this paper, we prove existence of small amplitude quasi-periodic solutions for a non-autonomous, quasi-periodically forced nonlinear wave equations with periodic spatial boundary conditions via KAM theory. [Copyright &y& Elsevier]

Abstract: In this paper we present two different results in the context of nonlinear analysis. The first one is essentially a nonlinear technique that, in view of its strong generality, may be useful in different practical problems. The second result, more technical, but also connected to the first one, is an extension of the well known Pietsch Domination Theorem. The last decade witnessed the birth of different families of Pietsch Domination-type results and some attempts of unification. Our result, that we call “full general Pietsch Domination Theorem” is potentially a definitive Pietsch Domination Theorem which unifies the previous versions and delimits what can be proved in this line. The connections to the recent notion of weighted summability are traced. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to prove the existence of a unique classical solution to the quasilinear elliptic equation , where at and where on the boundary ∂Ω. We prove that if the functions a, f, v, g satisfy certain conditions, then a unique classical solution exists. Applications include stationary heat/diffusion problems with convection and with a source/sink, where the value of the solution is known at a spatial location , and where is known on the boundary. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a discrete version of the following semilinear heat equation with absorption with , which is said to be the ω-heat equation with absorption on a network. Using the discrete Laplacian operator on a weighted graph, we define the ω-heat equations with absorption on networks and give their physical interpretations. The main concern is to investigate the large time behaviors of nontrivial solutions of the equations whose initial data are nonnegative and the boundary data vanish. It is proved that the asymptotic behaviors of the solutions as t tends to +∞ strongly depend on the sign of . [Copyright &y& Elsevier]

Abstract: In 1988, A. Bahri and P.L. Lions [A. Bahri, P.L. Lions, Morse-index of some min–max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988) 1027–1037] studied the following elliptic problem: where Ω is a bounded smooth domain of , and is not assumed to be odd in u. They proved the existence of infinitely many solutions under an appropriate growth restriction on f. In the present paper, we improve this result by showing that under the same growth assumption on f the problem admits in fact infinitely many sign-changing solutions. In addition we derive an estimate on the number of their nodal domains. We also deal with the corresponding fourth order equation with both Dirichlet and Navier boundary conditions, as well as with strongly coupled elliptic systems. [Copyright &y& Elsevier]

Abstract: In this paper we develop a unifying method to prove the existence and uniqueness of weak solutions for the initial–boundary value problem of a non-uniformly parabolic equation. Some well-known parabolic equations are the special cases of this equation. [Copyright &y& Elsevier]

Abstract: In this paper, we show the short time existence of the smooth solution for the porous medium equations in a smooth bounded domain: with zero boundary condition. On the fixed boundary , the flux is nonzero while the gradient of pressure, , is zero. As a consequence the parabolic equation above becomes degenerate. The proof is based on a definition of weighted space corresponding to the given degeneracy and the Schauder estimates in its linearized equation in for . [Copyright &y& Elsevier]