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In this article we study the existence of weak solution, existence of ground state solution using Nehari manifold and existence of infinitely many solutions using Fountain theorem and Dual fountain theorem for a class of doubly nonlocal…

We study the quasilinear elliptic system \[ -\textbf{div}(A(x,\boldsymbol u)|D\boldsymbol u|^{p-2}D\boldsymbol u) +\frac{1}{p}\nabla_{\boldsymbol s}A(x,\boldsymbol u)|D\boldsymbol u|^p = \boldsymbol g(x,\boldsymbol u) \quad \text{in }…

In this article, we establish a $L^1$ estimate for solutions to Poisson equation with mixed boundary condition, on complete noncompact manifolds with nonnegative Ricci curvature and compact manifolds with positive Ricci curvature…

In this paper, we obtain the existence of weak solutions to the Choquard-Kirchhoff type critical multiphase problem: \begin{equation*} \left\{\begin{array}{cc} &-M(\varphi_{\h}(\lvert{\nabla u}\rvert))div(\lvert{\nabla…

We study boundary value problems for bounded uniform domains in $\mathbb{R}^n$, $n\geq 2$, with non-Lipschitz (and possibly fractal) boundaries. We prove Poincar\'e inequalities with trace terms and uniform constants for uniform…

We study the quasilinear equation \[(P)\qquad - {\rm div} (A(x,u) |\nabla u|^{p-2} \nabla u) + \frac1p\ A_t(x,u) |\nabla u|^p + |u|^{p-2}u\ =\ g(x,u) \qquad \hbox{in ${\mathbb R}^N$,} \] with $N\ge 3$, $p > 1$, where $A(x,t)$, $A_t(x,t) =…

This paper studies the existence and uniqueness of a classical solution to a type of Robin boundary problems on the nonnegative orthant. We propose a new decomposition-homogenization method for the Robin boundary problem based on…

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite potential. The reaction term involves competing nonlinearities. More precisely, it is the sum of a parametric sublinear (concave) term and a…

In the present paper, we deal with a new continuous and compact embedding theorems for the fractional Orlicz-Sobolev spaces, also, we study the existence of infinitely many nontrivial solutions for a class of non-local fractional…

In this paper we study the existence of multiple nontrivial positive weak solutions to the following system of problems. \begin{align*} \begin{split} -\Delta_{p}u-\Delta_q u &= \lambda f(x)|u|^{r-2}u+\nu\frac{1-\alpha}{2-\alpha-\beta}h(x)…

The purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space $\mathbb{R}^d$ ($d\geq 3$). These hemivariational inequalities have a variational structure and,…

In this article, we study the following problem $$-{\rm div} (\omega(x)|\nabla u|^{N-2} \nabla u) = \lambda\ f(x,u) \quad\mbox{ in }\quad B, \quad u=0 \quad\mbox{ on } \quad\partial B,$$ where $B$ is the unit ball of $\mathbb{R^{N}}$,…

The Aubin-Lions lemma and its variants play crucial roles for the existence of weak solutions of nonlinear evolutionary PDEs. In this paper, we aim to develop some compactness criteria that are analogies of the Aubin--Lions lemma for the…

In this work, we establish a continuous version of the Fountain theorem by using the framework of the weak slope for continuous functionals, which generalizes Theorem 3.6 of Willem \cite{Wi}. Then we present an application to a semilinear…

We consider a fluid-structure interaction problem with Navier-slip boundary conditions in which the fluid is considered as a non-Newtonian fluid and the structure is described by a nonlinear multi-layered model. The fluid domain is driven…

We deal with a weighted biharmonic problem in the unit ball of $\mathbb{R}^{4}$. The non-linearity is assumed to have critical exponential growth in view of Adam's type inequalities. The weight $w(x)$ is of logarithm type and the potential…

Using the Mountain--Pass Theorem of Ambrosetti and Rabinowitz we prove that $-\Delta_p u-\mu|x|^{-p}{u^{p-1}}=|x|^{-s}{u^{\crits-1}}+u^{\crit-1}$ admits a positive weak solution in $\rn$ of class $\dunp\cap C^1(\rn\setminus\{0\})$, whenever…

We consider a nonlinear Robin problems driven by the $p$-Laplacian plus an indefinite potential. The reaction is resonant with respect to a variational eigenvalue. For the principal eigenvalue we assume strong resonance. Using variational…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a bounded smooth domain and $\delta(x)=\text{dist}(x,\partial \Omega)$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to $$…

Robin's Inequality posits $G(n)<e^{\gamma}$ for $n>5040$. Robin also showed that if the Riemann Hypothesis (RH) is false, then $G(n)>e^{\gamma}\left(1+\displaystyle\frac{c}{(\log n)^{b}}\right)$ for infinitely many values of $n$. By…