Related papers: Concentration Phenomena for $(p,N)$-Laplace Equati…
In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} -\Delta u + V(x) u = \mu_q u + a|u|^q u \quad {\rm…
We study a Dirichlet problem driven by the (degenerate or singular) fractional $p$-Laplacian and involving a $(p-1)$-superlinear reaction at infinity, not necessarily satisfying the Ambrosetti-Rabinowitz condition. Using critical point…
The present paper studies concentration phenomena of semiclassical approximation of a massive Dirac equation with general nonlinear self-coupling: \[ -i\hbar\alpha\cdot\nabla w+a\beta w+V(x)w=g(|w|)w \,. \] Compared with some existing…
In this paper we extend the well-known concentration -- compactness principle of P.L. Lions to the variable exponent case. We also give some applications to the existence problem for the $p(x)-$Laplacian with critical growth.
We consider homogenization problems for linear elliptic equations in divergence form. The coecients are assumed to be a local perturbation of some periodic background. We prove $W^{1,p}$ and Lipschitz convergence of the two-scale expansion,…
We study the behavior of solutions for the parametric equation $$-\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z)=\lambda |u(z)|^{q-2} u(z)+f(z,u(z)) \quad \mbox{in } \Omega,\, \lambda >0,$$ under Dirichlet condition, where $\Omega \subseteq…
We prove the existence of one positive, one negative, and one sign-changing solution of a $p$-Laplacian equation on $\mathbb{R}^N$, with a $p$-superlinear subcritical term. Sign-changing solutions of quasilinear elliptic equations set on…
We consider a nonlinear parametric Neumann problem driven by the anisotropic $(p,q)$-Laplacian and a reaction which exhibits the combined effects of a singular term and of a parametric superlinear perturbation. We are looking for positive…
In this paper we study the nonlinear elliptic problem involving p(x)-Laplacian with nonsmooth potential, where the weighted function may change sign. By using critical point theory for locally Lipschitz functionals due to Chang, we obtain…
We prove the existence of solutions $(\lambda, v)\in \mathbb{R}\times H^{1}(\Omega)$ of the elliptic problem \[ \begin{cases} -\Delta v+(V(x)+\lambda) v =v^{p}\ &\text{ in $ \Omega, $} \ v>0,\qquad \int_\Omega v^2\,dx =\rho. \end{cases} \]…
We study a weighted $\frac{N}{2}$ biharmonic equation involving a positive continuous potential in $\overline{B}$. The non-linearity is assumed to have critical exponential growth in view of logarithmic weighted Adams' type inequalities in…
In this paper, we study the existence of ground state solutions to the following p-Laplacian equation in some dimension $N\geq3$ with an $L^2$ constraint: \begin{equation*} \begin{cases} -\Delta_{p}u+{\vert u\vert}^{p-2}u=f(u)-\mu u \quad…
Our purpose is to find positive solutions $u \in D^{1,2}(\rz^N)$ of the semilinear elliptic problem $-\laplace u = h(x) u^{p-1}$ for $2<p$. The function $h$ may have an indefinite sign. Key ingredients are a $h$-dependent…
In this article we develop and analyze novel iterative regularization techniques for the solution of systems of nonlinear ill--posed operator equations. The basic idea consists in considering separately each equation of this system and…
We investigate a class of quasi-linear nonlocal problems, including as a particular case semi-linear problems involving the fractional Laplacian and arising in the framework of continuum mechanics, phase transition phenomena, population…
In this paper, we establish global $C^{1,\alpha}$-regularity for bounded generalized solutions of elliptic equations in divergence form with Musielak-Orlicz growth and subject to Dirichlet or Neumann boundary conditions. In fact, our…
We consider non-autonomous $N$-body-type problems with strong force type potentials at the origin and sub-quadratic growth at infinity, and using Ljusternik-Schnirelmann theory, we prove the existence of unbounded sequences of critical…
In this manuscript we deal with existence/uniqueness and regularity issues of suitable weak solutions to nonlocal problems driven by fractional Laplace type operators. Different from previous researches, in our approach we consider gradient…
In this paper, we consider a class of quasilinear stationary Kirchhoff type potential systems with Neumann Boundary conditions, which involves a general variable exponent elliptic operator with critical growth. Under some suitable…
In this paper, we prove the discrete Caffarelli-Kohn-Nirenberg inequalities on the lattice $\mathbb{Z}^{N}$ ($N\geq 1$) in a broader range of parameters than the classical continuous version [8]: \[ \parallel u\parallel_{\ell_{b}^{q}}\leq…