Some regularity results in small Lebesgue spaces for the weak or the very weak solution of the linear equation $-\Delta u = f$ are given, improving all the previous results obtained in the usual Lebesgue spaces. Some of our results have been derived using borderline Sobolev embeddings related to the Grand Lebesgue spaces. So, we provide new Sobolev inclusions using the Generalized Gamma spaces and generalizing the Fusco-Lions-Sbordone results.

We consider, in this note, the critical Schrödinger-Poisson system \begin{equation}\label{SP0} \begin{cases} \Delta_g u+ \omega^2 u +\varphi u = u^{\frac{n+2}{n-2}}~,\\ \Delta_g \varphi +m_0^2 \varphi = 4\pi q^2 u^2~ \end{cases} \end{equation} on a closed Riemannian $n$-dimensional manifold $(M^n,g)$, for $n=4$. If the scalar curvature is negative somewhere, we prove that this system admits positive solutions for small phases $\omega$ and that $\omega=0$ is an unstable phase (see Definition 1.1. By contrast, small phases are always stable (see [32]) when $n=4$ and the scalar curvature is positive everywhere, and unstable phases never exist when $n\ge 5$ (see [29, 31]).

The aim of this paper is to study connected, unbounded sets of solutions of the non-cooperative elliptic system of equations with Neumann boundary conditions. The existence of such sets is obtained by proving the bifurcation from infinity. To this end we apply the degree for $G$-invariant strongly indefinite

The present paper concerns the study of distributional travelling waves in models ruled by the nonlinear Klein-Gordon equation $u_{tt}-c^{2}u_{xx} =\phi(u)$, where $c>0$ is a real number and $\phi$ is an entire function which takes real values on the real axis. For this purpose, we use a product of distributions that extends the meaning of $\phi(u)$ to certain distributions $u$ and that allows us to define a solution concept consistent with the classical solution concept. The phi-four equation and the sine-Gordon equation are examined as particular cases.

In this work, we prove a weak Harnack estimate for the weak supersolutions to the porous medium equation. The proof is based on a priori estimates for the supersolutions and measure theoretical arguments.

Of interest in this note is the following geometric equation, $\Delta^2 u + u^{-q} = 0$ in $\mathbf R^3$. It was found by Choi--Xu (J. Differential Equations, 246, 216–234) and McKenna–Reichel (Electron. J. Differential Equations, 37 (2003)) that the condition $q > 1$ is necessary and any positive radially symmetric solution grows at least linearly and at most quadratically at infinity for any $q > 1$. In addition, when $q > 3$ any positive radially symmetric solution is either exactly linear growth or exactly quadratic growth at infinity. Recently, Guerra (J. Differential Equations, {253}, 3147–3157) has shown that the equation always admits a unique positive radially symmetric solution of exactly given linear growth at infinity for any $q > 3$ which is also necessary. In this note, by using the phase-space analysis, we show the existence of infinitely many positive radially symmetric solutions of exactly given quadratic growth at infinity for any $q > 1$, hence completing the picture of positive radially symmetric solutions of the equation.

In this paper, we mainly study the following semilinear Dirichlet problem $ -\Delta u=q(x)f(u),\; u > 0,\;x\in \Omega ,$ $u_{|\partial \Omega }=0,$ where $ \Omega $ is an annulus in $\mathbb{R}^{n},\;\big( n\geq 2\big) .$ The function $f$ is nonnegative in $\mathcal{C}^{1}(0,\infty )$ and $q\in \mathcal{C}_{loc}^{\gamma }(\Omega ),\;(0 < \gamma < 1),$ is positive and satisfies some required hypotheses related to Karamata regular variation theory. We establish the existence of a positive classical solution to this problem. We also give a global boundary behavior of such solution.

In this paper, we study strongly coupled elliptic systems in non-variational form involving fractional Laplace operators. We prove Liouville type theorems and, by mean of the blow-up method, we establish a priori bounds of positive solutions for subcritical and superlinear nonlinearities in a coupled sense. By using those latter, we then derive the existence of positive solutions through topological methods.

In this work, we deal with the following wave equation with localized dissipation given by a memory term $$ u_{tt} -u_{xx} + \partial_x \Big\{ a(x)\int_{0}^{t} g(t-s)u_{x}(x,s)ds \Big\}=0. $$ We consider that this dissipation is indefinite due to sign changes of the coefficient $a$ or by sign changes of the memory kernel $g$. The exponential decay of solutions is proved when the average of coefficient $a$ is positive and the memory kernel $g$ is small.

In this paper, based on variational methods and critical point theory, we investigate the existence of nontrivial solutions for a class of impulsive Sturm-Liouville differential equations with nonlinear derivative dependence.