We present some symmetrization results which we apply to the same abstract eigenvalue problem in order to show the existence of three different solutions which are invariant by Schwarz symmetrization. In particular, we introduce two different methods in order to prove the existence of multiple symmetric solutions. The first is based on the symmetric version of the Ekeland variational principle and the mountain pass theorem, while the latter consists of an application of a suitable symmetric version of the three critical points theorem due to Pucci and Serrin [17, 18], see Theorem 2.13 and its Corollary 2.14. Using the second method, we are able to improve some recent results of Arcoya and Carmona [1] and Bonnano and Candito [2]. The methods we present work also for different types of symmetrization, see Van Schaftingen [22].

In the present paper, we study the weak lower semicontinuity of the functional \begin{align*} \Phi_{\lambda, \gamma}(u): & =\frac 1 2 \int_{\mathbb R ^n\times\mathbb R ^n} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dx\,dy -\frac \lambda 2 \int_\Omega |u(x)|^2 dx \\ & -\frac\gamma 2 \Big(\int_\Omega |u(x)|^{2^*} dx\Big)^{2/2^*}, \end{align*} where $\Omega$ is an open bounded subset of $\mathbb R ^n$, $n >2s$, $s\in (0,1)$, with continuous boundary, $\lambda$ and $\gamma$ are real parameters and $2^*:=2n/(n-2s)$ is the fractional critical Sobolev exponent. As a consequence of this regularity result for $\Phi_{\lambda, \gamma}$, we prove the existence of a nontrivial weak solution for two different nonlocal critical equations driven by the fractional Laplace operator $(-\Delta)^{s}$ which, up to normalization factors, may be defined as $$ -(-\Delta)^s u(x):= \int_{\mathbb{R}^{n}}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy, \quad x\in \mathbb R ^n. $$ These two existence results were obtained using, respectively, the direct method in the calculus of variations and critical points theory.

In this paper, we consider a wave model with non-effective mass and dissipation terms and provide sharp descriptions of its representation of solutions. In particular, we conclude estimates for a corresponding energy and estimates of dispersive type.

We study the global Cauchy problem for a system of Schrödinger equations with two wave interaction of quadratic, cubic and quintic degrees. For sufficiently small data with exponential decay at infinity we prove the existence and uniqueness of global solutions which are analytic with respect to Galilei and/or pseudo-conformal generators for sufficiently small data with exponential decay at infinity. This paper is a sequel to our paper [22], where three wave interaction is studied. We also discuss the associated Lagrange structure.

We study monotonicity properties of positive solutions to quasilinear elliptic systems, involving $p$-Laplace equations, in half spaces.

We study the regularity for nonlinear parabolic systems of $p$-Laplacian type, in the singular case $\frac{{2m}}{{m + 2}}<p<2$. We show an optimal condition on given external forces for a local Höolder continuity. Actually, our main result recovers the classical one for linear equations. The proof is based on Campanato's direct approach with the intrinsic scaling to the evolutionary $p$-Laplace operator. The iteration scheme is performed similarly as in [18], however, with some technical care peculiar to the singular case.

We are interested in an anisotropic singular diffusion equation in the plane and in its regularization. We establish existence, uniqueness and the basic regularity of solutions to both equations. We construct explicit solutions showing the creation of facets, i.e., flat parts of graphs of solutions. Inspired by the formula for solutions, we rigorously prove that both equations create ruled surfaces out of convex initial data. We also notice that at each positive time, the solutions do not have strict (local) extrema either. We present results of numerical experiments suggesting that the two flows do not seem to differ much. Possible applications to the image reconstruction is pointed out, too.