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This paper establishes the existence of infinitely many solutions for nonlinear problems without any symmetry, achieving three major advances. First, in the setting of semilinear elliptic PDEs, we introduce a refined variational truncation…

We establish uniqueness results for quasilinear elliptic problems through the criterion recently provided in \cite{DFMST}. We apply it to generalized $p$-Laplacian subhomogeneous problems that may admit multiple nontrivial nonnegative…

In this paper, we consider the Minkowski problem associated with the solution to the anisotropic $p$-Laplacian (or Finsler $p$-Laplacian) equation, namely, the Minkowski problem of anisotropic $p$-torsional rigidity. The sufficient and…

In this paper we develop a new theory for the existence, localization and multiplicity of positive solutions for a class of non-variational,quasilinear, elliptic systems. In order to do this, we provide a fairly general abstract framework…

In this paper we prove the existence of a nontrivial non-negative radial solution for a quasilinear elliptic problem. Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space.…

In this paper, we investigate Liouville theorems for solutions to the anisotropic $p$-Laplace equation $$-\Delta_p^H u=-\operatorname{div}(a(\nabla u))=f(u),\quad\text{in }\mathbb{R}^n,$$ where the semilinear term $f$ may be positive,…

We consider perturbations of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator plus an indefinite potential. We consider both sublinear and superlinear perturbations and we determine how the set of positive…

Let us consider the quasilinear problem \[ (P_\varepsilon) \ \ \left\{ \begin{array}{ll} - \varepsilon^p \Delta _{p}u + u^{p-1} = f(u) & \hbox{in} \ \Omega \newline u>0 & \hbox{in} \ \Omega \newline u=0 & \hbox{on} \ \partial \Omega…

In this work we analyze the existence of solutions to the nonlinear elliptic system: \begin{equation*} \left\{ \begin{array}{rcll} -\Delta u & = & v^q+\a g & \text{in }\Omega , \\ -\Delta v& = &|\nabla u|^{p}+\l f &\text{in }\Omega , \\…

This paper studies a non-singular coupling scheme for solving the acoustic and elastic wave scattering problems and its extension to the problems of Laplace and Lam\'e equations and the problem with a compactly supported inhomogeneity is…

In this paper, we prove a multiplicity result of solutions for the following stationary Schr\"odinger-Poisson-Slater equations \begin{equation}\label{eq-abstract} -\Delta u - \lambda u + (\left | x \right |^{-1}\ast \left | u \right |^2) u…

Let $1<p<+\infty$ and let $\Omega\subset\mathbb R^N$ be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the…

In this note, we consider a Robin-type traction problem for a linearly elastic body occupying an infinite periodically perforated domain. After proving the uniqueness of the solution we use periodic elastic layer potentials to show that the…

In this paper we examine the existence of multiple solutions of parametric fractional equations involving the square root of the Laplacian $A_{1/2}$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ ($n\geq 2$) and with Dirichlet…

The relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold itself is studied. The technique is based on Ljusternik-Schnirelmann category and Morse theory.

We investigate the existence and concentration of normalized solutions for a $p$-Laplacian problem with logarithmic nonlinearity of type \[ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^p\Delta_p u+V(x)|u|^{p-2}u=\lambda…

We are concerned with the existence and asymptotic behavior of multiple radial sign-changing solutions with the nodal characterization for a Kirchhoff-type problem involving the nonlinearity $|u|^{p-2}u(2<p<4)$ in $\mathbb{R}^3$. By…

We consider a nonlinear Dirichlet problem driven by the $(p,q)$-Laplacian with $1<q<p$. The reaction is parametric and exhibits the competing effects of a singular term and of concave and convex nonlinearities. We are looking for positive…

In this paper we consider an optimal control problem for the coupled system of a nonlinear monotone Dirichlet problem with anisotropic p-Laplacian and matrix-valued nonsmooth controls in its coefficients and a nonlinear equation of…

Let $p,q$ be functions on $\mathbb{R}^{N}$ satisfying $1\ll q\ll p\ll N$, we consider $p(x)$-Laplacian problems of the form \[ \left\{ \begin{array} [c]{l}% -\Delta_{p(x)}u+V(x)\vert u\vert ^{p(x)-2}u=\lambda\vert u\vert…