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In this work we establish existence and multiplicity of solutions for elliptic problem with nonlinear boundary conditions under strong resonance conditions at infinity. The nonlinearity is resonance at infinity and the reso- nance phenomena…
In this work, we use monotonicity-based methods for the fractional Schr\"odinger equation with general potentials $q\in L^\infty(\Omega)$ in a Lipschitz bounded open set $\Omega\subset \mathbb R^n$ in any dimension $n\in \mathbb N$. We…
We consider the Neumann problem for the equation $u_{xx}+\lambda f(u)=0$ in the punctured interval $(-1,1) \setminus \{0\}$, where $\lambda>0$ is a bifurcation parameter and $f(u)=u-u^3$. At $x=0$, we impose the conditions…
In this paper we consider the overdetermined boundary problem for a general second order semilinear elliptic equation on bounded domains of $\mathbf{R}^n$, where one prescribes both the Dirichlet and Neumann data of the solution. We are…
In this paper we study a class of elliptic boundary hemivariational inequalities which originates in the steady-state heat conduction problem with nonmonotone multivalued subdifferential boundary condition on a portion of the boundary…
We prove semi-classical resolvent estimates for the Schr{\"o}dinger operator with a real-valued L $\infty$ potential on non-compact, connected Riemannian manifolds which may have a compact smooth boundary. We show that the resolvent bound…
We prove the existence of radial self-similar singular solutions for the mass supercritical Nonlinear Schr\"odinger Equation far from the critical regime and, more generally, branches of such solutions for the Complex Ginzburg-Landau…
Some new global results are given about solutions to the boundary value problem for the Euler-Lagrange equations for the Ginzburg-Landau model of a one-dimensional superconductor. The main advance is a proof that in some parameter range…
The paper deals with standing wave solutions of the dimensionless nonlinear Schr\"odinger equation \label{eq:abs1} i\Phi_t(x,t) = -\Delta_x\Phi +V_\la(x)\Phi + f(x,\Phi), \quad x\in\R^N,\ t\in\R,\tag{$NLS_\la$} where the potential…
The purpose of this article is to construct global solutions, in a probabilistic sense, for the nonlinear Schr{\"o}dinger equation posed on $\mathbb{R}^d$, in a supercritical regime. Firstly, we establish Bourgain type bilinear estimates…
We examine the conditions under which the solution of the radial stationary Schr\"odinger equation for the sextic anharmonic oscillator can be expanded in terms of Hermite functions. We find that this is possible for an infinite hierarchy…
In this paper we study the positive solutions of sub linear elliptic equations with a Hardy potential which is singular at the boundary. By means of ODE techniques a fairly complete picture of the class of radial solutions is given. Local…
We derive and justify a normal form reduction of the nonlinear Schrodinger equation for a general pitchfork bifurcation of the symmetric bound state that occurs in a double-well symmetric potential. We prove persistence of normal form…
In this paper, we study the existence of solution for the following class of nonlocal problem, $$ \left\{ \begin{array}{lcl} -\Delta u=\left(\lambda f(x)-\int_{\R^N}K(x,y)|u(y)|^{\gamma}dy\right)u,\quad \mbox{in} \quad \R^{N}, \\…
We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in domains with conic singularities and establish asymptotic expansions with respect to two normal directions simultaneously. The spherical domains over which cones…
We investigate the problem of entire solutions for a class of fourth order, dilation invariant, semilinear elliptic equations with power-type weights and with subcritical or critical growth in the nonlinear term. These equations define non…
We study a nonlinear equation in the half-space $\{x_1>0\}$ with a Hardy potential, specifically \[-\Delta u -\frac{\mu}{x_1^2}u+u^p=0\quad\text{in}\quad \mathbb R^n_+,\] where $p>1$ and $-\infty<\mu<1/4$. The admissible boundary behavior…
We review some results concerning the semi-classical limit for the nonlinear Schrodinger equation, with or without an external potential. We consider initial data which are either of the WKB type, or very concentrated as the semi-classical…
In this paper, we study the nonhomogeneous Dirichlet problem concerning general semilinear elliptic equations in divergence form. We establish that the boundary Lipschitz regularity of solutions under some more weaker conditions on the…
We present new $L^\infty$ a priori estimates for weak solutions of a wide class of subcritical elliptic equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is…