We study the following bifurcation problem in any bounded domain $\Omega$ in ${ℝ}^N$: $$\{\begin{array}{l}{A}_{p}u:=-\sum _{i,j=1}^N\frac{\partial }{\partial x_i}\left[{\left(\sum _{m,k=1}^N{a}_{mk}\left(x\right)\frac{\partial u}{\partial x_m}\frac{\partial u}{\partial x_k}\right)}^{\frac{p-2}{2}}{a}_{ij}\left(x\right)\frac{\partial u}{\partial x_j}\right]=\\ \lambda g\left(x\right){\left|u\right|}^{p-2}u+f\left(x,u,\lambda \right),\\ \\ u\in W_0^{1,p}\left(\Omega \right).\end{array}$$. We prove that the principal eigenvalue ${\lambda }_1$ of the eigenvalue problem $$\{\begin{array}{l}{A}_{p}u=\lambda g\left(x\right){\left|u\right|}^{p-2}u,\\ u\in W_0^{1,p}\left(\Omega \right),\end{array}$$ is a bifurcation point of the problem mentioned above.

Various eigenvalue and range results are given for perturbations of $m$-accretive and maximal monotone operators. The eigenvalue results improve and extend some recent results by Guan and Kartsatos, while the range theorem gives an affirmative answer to a recent problem of Kartsatos.

We shall investigate a size structured population dynamics with aging and birth functions having general forms. The growth rate we deal with depends not only on the size but also on time. We show the existence of a local solution and continuous dependence on the initial data, which shows the uniqueness of the solution as well.

We prove the solvability of the Dirichlet problem $$\{\begin{array}{l}-{\Delta }_{p}u=f\left(u\right)+h\text{in}\Omega ,\\ u=0\text{on}\partial \Omega \end{array}$$ for every given $h$, under a condition involving only the asymptotic behaviour of the potential $F$ of $f$ with respect to the first eigenvalue of the $p$-Laplacian ${\Delta }_p$. More general operators are also considered.

The theory of nonlinear weakly hyperbolic equations was developed during the last decade in an astonishing way. Today we have a good overview about assumptions which guarantee local well posedness in spaces of smooth functions $(C^{\infty }$, Gevrey). But the situation is completely unclear in the case of Sobolev spaces. Examples from the linear theory show that in opposite to the strictly hyperbolic case we have in general no solutions valued in Sobolev spaces. If so-called Levi conditions are satisfied, then the situation will be better. Using sharp Levi conditions of $C^{\infty }$-type leads to an interesting effect. The linear weakly hyperbolic Cauchy problem has a Sobolev solution if the data are sufficiently smooth. The loss of derivatives will be determined in essential by special lower order terms. In the present paper we show that it is even possible to prove the existence of Sobolev solutions in the quasilinear case although one has the finite loss of derivatives for the linear case. Some of the tools are a reduction process to problems with special asymptotical behaviour, a Gronwall type lemma for differential inequalities with a singular coefficient, energy estimates and a fixed point argument.

In this paper we consider a quasilinear parabolic equation in a bounded domain under periodic Dirichlet boundary conditions. Our main goal is to prove the existence of extremal solutions among all solutions lying in a sector formed by appropriately defined upper and lower solutions. The main tools used in the proof of our result are recently obtained abstract results on nonlinear evolution equations, comparison and truncation techniques and suitably constructed special testfunction.

The $n$th order eigenvalue problem: $$\begin{array}{l}{\Delta }^{n}x\left(t\right)={\left(-1\right)}^{n-k}\lambda f\left(t,x\left(t\right)\right),t\in \left[0,T\right],\\ x\left(0\right)=x\left(1\right)=\cdots =x\left(k-1\right)=x\left(T+k+1\right)=\cdots =x\left(T+n\right)=0,\end{array}$$ is considered, where $n\ge 2$ and $k\in \left\{1,2,\dots ,n-1\right\}$ are given. Eigenvalues $\lambda$ are determined for $f$ continuous and the case where the limits $f_0\left(t\right)=\underset{n\to 0^{+}}{lim}\frac{f\left(t,u\right)}{u}$ and $f_{\infty }\left(t\right)=\underset{n\to \infty }{lim}\frac{f\left(t,u\right)}{u}$ exist for all $t\in \left[0,T\right]$. Guo′s fixed point theorem is applied to operators defined on annular regions in a cone.

For the damped Boussinesq equation $u_{tt}-2b{u}_{txx}=-\alpha u_{xxxx}+u_{xx}+\beta {\left(u^2\right)}_{xx},x\in \left(0,\pi \right),t>0;\alpha ,b=const>0,\beta =const\in R^1$, the second initial-boundary value problem is considered with small initial data. Its classical solution is constructed in the form of a series in small parameter present in the initial conditions and the uniqueness of solutions is proved. The long-time asymptotics is obtained in the explicit form and the question of the blow up of the solution in a certain case is examined. The possibility of passing to the limit $b\to +0$ in the constructed solution is investigated.

We study the existence of nontrivial solutions to the following problem: $$\{\begin{array}{l}u\in W^{1,N}\left({ℝ}^N\right),u\ge 0\text{and}\\ -div\left({\left|\nabla u\right|}^{N-2}\nabla u\right)+a\left(x\right){\left|u\right|}^{N-2}u=f\left(x,u\right)\text{in}{ℝ}^N\left(N\ge 2\right),\end{array}$$ where $a$ is a continuous function which is coercive, i.e., $a\left(x\right)\to \infty \text{as}\left|x\right|\to \infty$ and the nonlinearity $f$ behaves like $exp\left(\alpha {\left|u\right|}^{N/\left(N-1\right)}\right)$ when $\left|u\right|\to \infty$.