We investigate the existence and stability of solutions for higher-order two-point boundary value problems in case the differential operator is not necessarily positive definite, i.e. with superlinear nonlinearities. We write an abstract realization of the Dirichlet problem and provide abstract existence and stability results which are further applied to concrete problems.

In this paper we study a nonlinear Dirichlet elliptic differential equation driven by the p-Laplacian and with a nonsmooth potential. The hypotheses on the nonsmooth potential allow resonance with respect to the principal eigenvalue λ₁ > 0 of $(-\Delta ₚ,W{₀}^{1,p}\left(Z\right))$. We prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal.

We show, by variational methods, that there exists a set $\mathcal{A}$ open and dense in $a\in L^{\mathrm{\infty }}\left({\mathbb{R}}^N\right):a\ge 0$ such that if $a\in \mathcal{A}$ then the problem $-\mathrm{△}u+u=a\left(x\right){\left|u\right|}^{p-1}u,u\in H^1\left({\mathcal{R}}^N\right)$, with $p$ subcritical (or more general nonlinearities), admits infinitely many solutions.

We investigate the existence of solutions for the Dirichlet problem including the generalized balance of a membrane equation. We present a duality theory and variational principle for this problem. As one of the consequences of the duality we obtain some numerical results which give a measure of a duality gap between the primal and dual functional for approximate solutions.

Following Morrey [14] we associate to any measurable symmetric $2\times 2$ matrix valued function $A\left(x\right)$ such that $$\frac{{\left|\xi \right|}^2}{K}\le \left(A\left(x\right)\xi ,\xi \right)\le K{\left|\xi \right|}^2\text{a.e.}x\in \mathrm{\Omega },\mathrm{\forall }\xi \in {\mathbb{R}}^2,$$$\mathrm{\Omega }\in {\mathbb{R}}^2...$

By deriving a variant of interpolation inequality, we obtain a sharp criterion for global existence and blow-up of solutions to the inhomogeneous nonlinear Schrödinger equation with harmonic potential $$\mathrm{i}{\varphi }_t=-▵\varphi +{\left|x\right|}^2\varphi -{\left|x\right|}^b{\left|\varphi \right|}^{p-2}\varphi .$$ We also prove the existence of unstable standing-wave solutions via blow-up under certain conditions on the unbounded inhomogeneity and the power of nonlinearity, as well as the frequency of the wave.

Some general multiplicity results for critical points of parameterized functionals on reflexive Banach spaces are established. In particular, one of them improves some aspects of a recent result by B. Ricceri. Applications to boundary value problems are also given.

Recent papers have studied the existence of phase transition solutions for Allen–Cahn type equations. These solutions are either single or multi-transition spatial heteroclinics or homoclinics between simpler equilibrium states. A sufficient condition for the construction of the multitransition solutions is that there are gaps in the ordered set of single transition solutions. In this paper we explore the necessity of these gap conditions.