In this paper, we are concerned with existence, qualitative properties, and uniform Gaussian estimates of the global fundamental solutions of a family of heat operators on Carnot groups. As a byproduct, we obtain existence and uniqueness theorems of Thychonov type for the Cauchy problem. Our effort here is also to provide simple and direct proofs relying on few basic tools such as invariant Harnack inequalities and maximum principles. In our study, we thoroughly exploit some structural properties of Carnot groups pointed out in the previous paper [4].

We consider a semilinear problem of the type $Lu=f(b,u),$ where $f(b,u)\simeq bu$ as $u\to 0$ and $f(b,u)\simeq b_\infty u$ as $\|u \| \to\infty$ assuming that there exist a finite number of eigenvalues of the linear operator $L$ between $b$ and $b_\infty$. Under suitable assumptions we prove the existence of four nontrivial solutions for $b$ close to an eigenvalue. We give an application to problems of oscillations of a forced beam.

Consider the nonlinear Sturm-Liouville eigenvalue problem \begin{align*} u''-Q(x)u & + \lambda(Mu+f(u))=0,\qquad x\in{\mathbb R }, \\ \lim\limits_{|x|\to\infty} u(x) & =\lim\limits_{|x|\to\infty} u'(x) =0, \end{align*} where the potential $Q$ is positive and coercive, the function $f(s)$ behaves like $s^p$, $p>1$, $M$ is a positive constant and $\lambda$ is a positive parameter. When the domain is a bounded interval, Rabinowitz global bifurcation theory applies to this problem, showing the existence of unbounded branches of nontrivial solutions. Even more, Rabinowitz proved that the branches bend back. This last fact has as a consequence a multiplicity result for solutions of a related nonlinear Schr\"odinger equation. In this paper we prove that this result holds true when the domain is ${\mathbb R }$. The main point of the article is the proof that the branches bend back, the place where the noncompactness of ${\mathbb R }$ poses a difficulty.

In this paper we characterize the existence of large solutions for a general class of sublinear elliptic problems of logistic type related to the porous media equation. Our main result shows that large solutions do exist if, and only if, the nonlinear diffusion is not too large. As a byproduct of the general theory developed by the authors in [3], those large solutions must be unstable with respect to the positive solutions of the parabolic counterpart of the elliptic model. This seems to be the first result of this nature available in the literature. Most precisely, as the diffusion becomes non-linear the metasolutions become unstable, so arising a classical steady-state gaining the stability lost by the metasolution. In particular, a dynamical bifurcation occurs from the linear diffusion case.

In this paper, we consider the problem $-\Delta u =|u| ^{2^*-2}u+\lambda u$ in $\Omega$, $u= 0$ on $\partial \Omega$, where $\Omega$ is an open regular bounded subset of $\mathbb R^N$ $(N\geq 3)$, $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent and $\lambda>0$. Our main result asserts that, if $N\geq 7$, the problem has infinitely many solutions and, from the point of view of the compactness arguments employed here, the restriction on the dimension $N$ cannot be weakened.