On the fourth-order Leray–Lions problem with indefinite weight and nonstandard growth conditions

We prove the existence of at least three weak solutions for the fourth-order problem with indefinite weight involving the Leray–Lions operator with nonstandard growth conditions. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano [Appl. Anal. 89 (2010) 1–10].

Fourth-order elliptic problems with critical nonlinearities by a sublinear perturbation

In this paper, we get the existence of two positive solutions for a fourth-order problem with Navier boundary condition. Our nonlinearity has a critical growth, and the method is a local minimum theorem obtained by Bonanno.

Onset of Convection in Two-Dimensional Porous Cavities with Open and Conducting Boundaries

The onset of thermal convection in two-dimensional porous cavities heated from below is studied theoretically. An open (constant-pressure) boundary is assumed, with zero perturbation temperature (thermally conducting). The resulting eigenvalue problem is a full fourth-order problem without degeneracies. Numerical results are presented for rectangular and elliptical cavities, with the circle as a special case. The analytical solution for an upright rectangle confirms the numerical results. Streamlines penetrating the open cavities are plotted, together with the isotherms for the associated closed thermal cells. Isobars forming pressure cells are depicted for the perturbation pressure. The critical Rayleigh number is calculated as a function of geometric parameters, including the tilt angle of the rectangle and ellipse. An improved physical scaling of the Darcy–Bénard problem is suggested. Its significance is indicated by the ratio of maximal vertical velocity to maximal temperature perturbation.

Existence of three weak solutions for fourth-order elastic beam equations on the whole space

UDC 517.9 Multiplicity results for a perturbed fourth-order problem on the real line with a perturbed nonlinear term depending on one real parameter is investigated. Our approach is based on variational methods and critical point theory which are obtained in [G. Bonanno, <em>A critical point theorem via the Ekeland variational principle</em>, Nonlinear Anal., <strong>75</strong>, 2992-3007 (2012)].

Linear theory for beams with intermediate piers

The full linear theory for hinged beams with intermediate piers is developed. The analysis starts with the variational setting and the study of the linear stationary problem. Well-posedness results are provided and the possible loss of regularity, due to the presence of the piers, is analyzed. A complete spectral theorem is then proved, explicitly determining the eigenvalues according to the position of the piers and exhibiting the fundamental modes of oscillation. A related second-order eigenvalue problem is also studied, showing that it may display nonsmooth eigenfunctions and that the fourth-order problem cannot be seen as the square of a second-order problem.

Positive solutions of beam equations under nonlocal boundary value conditions

In this article, we study the fourth-order problem with the first and second derivatives in nonlinearity under nonlocal boundary value conditions $$\begin{aligned}& \left \{ \textstyle\begin{array}{l}u^{(4)}(t)=h(t)f(t,u(t),u'(t),u''(t)),\quad t\in(0,1),\\ u(0)=u(1)=\beta_{1}[u],\qquad u''(0)+\beta_{2}[u]=0,\qquad u''(1)+\beta_{3}[u]=0, \end{array}\displaystyle \right . \end{aligned}$$ {u(4)(t)=h(t)f(t,u(t),u′(t),u″(t)),t∈(0,1),u(0)=u(1)=β1[u],u″(0)+β2[u]=0,u″(1)+β3[u]=0, where $f: [0,1]\times\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}_{-}\to \mathbb{R}_{+}$f:[0,1]×R+×R×R−→R+ is continuous, $h\in L^{1}(0,1)$h∈L1(0,1) and $\beta_{i}[u]$βi[u] is Stieltjes integral ($i=1,2,3$i=1,2,3). This equation describes the deflection of an elastic beam. Some inequality conditions on nonlinearity f are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on a special cone in $C^{2}[0,1]$C2[0,1]. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel.