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The main goal of this paper is to introduce a new fractional anisotropic Sobolev space with variable exponent where the basic qualitative properties (completeness, separability, reflexivity, ...) are established, including the continuous…

For $N\geq 4$, we let $\Omega$ be a bounded domain of $\mathbb{R}^N$ and $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u \in H^1_0\left(\Omega\right)$ to the equation…

This paper investigates the existence of positive normalized solutions to the Sobolev critical Schr\"{o}dinger equation: \begin{equation*} \left\{ \begin{aligned} &-\Delta u +\lambda u =|u|^{2^*-2}u \quad &\mbox{in}& \ \Omega,\\…

We prove the existence of a mountain-pass solution and the a priori bound property for the electrostatic Klein-Gordon-Maxwell equations in high dimension.

We study the existence and multiplicity of positive solutions with prescribed $L^2$-norm for the Sobolev critical Schr\"odinger equation on a bounded domain $\Omega\subset\mathbb{R}^N$, $N\ge3$: \[ -\Delta U = \lambda U + U^{2^{*}-1},\qquad…

The purpose of this paper is to establish a critical point theorem, which is an infinite-dimensional generalization of the classical generalized Mountain Pass Theorem of P. H. Rabinowitz \cite[Theorem 5.3]{Ra}. As application, we obtain the…

In this paper we prove some linking theorems and mountain pass type results for dynamical systems in terms of local semiflows on complete metric spaces. Our results provide an alternative approach to detect the existence of compact…

In this paper, we study a class of quasilinear elliptic equations which appears in nonlinear optics. By using the mountain pass theorem together with a technique of adding one dimension of space, we prove the existence of a non-trivial weak…

We study normalized solutions for the nonlinear Schrodinger (NLS) equation with potential and Sobolev critical nonlinearity. By establishing suitable assumptions on the potential, together with new techniques, we find a mountain-pass type…

In this paper, we study the following nonlocal problem in fractional Orlicz Sobolev spaces \begin{eqnarray*} (-\Delta_{\Phi})^{s}u+V(x)a(|u|)u=f(x,u),\quad x\in\mathbb{R}^N, \end{eqnarray*} where $(-\Delta_{\Phi})^{s}(s\in(0, 1))$ denotes…

In this paper we establish the existence and multiplicity of nontrivial solutions to the following problem \begin{align*} \begin{split} (-\Delta)^{\frac{1}{2}}u+u+(\ln|\cdot|*|u|^2)&=f(u)+\mu|u|^{-\gamma-1}u,~\text{in}~\mathbb{R},…

In this paper we investigate the existence of solution for the following nonlocal problem with anisotropic Stein-Weiss convolution term $$ -\Delta_{\Phi} u+V(x)\phi(|u|)u=\dfrac{1}{|x|^\alpha}\left(\int_{\mathbb{R}^{N}}…

In this paper, we investigate the existence of normalized solutions for the following nonlinear Kirchhoff type problem \begin{equation*} \begin{cases} -(a+b\int_{\Omega}\vert\nabla u\vert^2dx)\Delta u+\lambda u=\vert u\vert^{p-2}u & \text{…

The aim of this paper is to obtain the existence of solution for the fractional p-Laplacian Dirichlet problem with mixed derivatives \begin{eqnarray*} &{_{t}}D_{T}^{\alpha}\left(|_{0}D_{t}^{\alpha}u(t))|^{p-2}{_{0}}D_{t}^{\alpha}u(t)\right)…

In this paper a quasi-linear elliptic equation in the whole Euclidean space is considered. The nonlinearity of the equation is assumed to have exponential growth or have critical growth in view of Trudinger-Moser type inequality. Under some…

We study the existence of positive solutions with prescribed $L^2$-norm for the Schr\"odinger equation \[ -\Delta u-V(x)u+\lambda u=|u|^{p-2}u\qquad\lambda\in \mathbb{R},\quad u\in H^1(\mathbb{R}^N), \] where $V\ge 0$, $N\ge 1$ and…

In this paper, we show the existence of non-trivial solutions to very general elliptic systems with critical non-linearities in the sense of embeddings in Orlicz-Sobolev spaces. This allows to consider non-linearities which do not have…

This work extends the study of mean field equations arising in two-dimensional (2D) turbulence by introducing generalized weighted Sobolev operators. Employing variational methods, particularly the mountain pass theorem and a refined…

We study a general nonlinear parabolic equation on a Lipschitz bounded domain in $\mathbb{R}^N$, \begin{equation*} \left\{\begin{array}{l l} \partial_t u-\mathrm{div} A(t,x,\nabla u)= f(t,x)&\text{in}\ \ \Omega_T,\\ u(t,x)=0 &\ \mathrm{ on}…

We study the existence and multiplicity of positive solutions in $H^1(\mathbb{R}^N)$, $N\ge3$, with prescribed $L^2$-norm, for the (stationary) nonlinear Schr\"odinger equation with Sobolev critical power nonlinearity. It is well known…