130 research outputs found

### Sequences of weak solutions for fractional equations

This work is devoted to study the existence of infinitely many weak solutions
to nonlocal equations involving a general integrodifferential operator of
fractional type. These equations have a variational structure and we find a
sequence of nontrivial weak solutions for them exploiting the
${\mathbb{Z}}_2$-symmetric version of the Mountain Pass Theorem. To make the
nonlinear methods work, some careful analysis of the fractional spaces involved
is necessary. As a particular case, we derive an existence theorem for the
fractional Laplacian, finding nontrivial solutions of the equation
\left\{\begin{array}{ll} (-\Delta)^s u=f(x,u) & {\mbox{in}} \Omega\\ u=0 &
{\mbox{in}} \erre^n\setminus \Omega. \end{array} \right. As far as we know,
all these results are new and represent a fractional version of classical
theorems obtained working with Laplacian equations

### Yamabe-type equations on Carnot groups

This article is concerned with a class of elliptic equations on Carnot groups
depending of one real positive parameter and involving a critical nonlinearity.
As a special case of our results we prove the existence of at least one
nontrivial solution for a subelliptic critical equation defined on a smooth and
bounded domain $D$ of the {Heisenberg group} $\mathbb{H}^n=\mathbb{C}^n\times
\mathbb{R}$. Our approach is based on pure variational methods and locally
sequentially weakly lower semicontinuous arguments

### Existence Results for a critical fractional equation

We are concerned with existence results for a critical problem of
Brezis-Nirenberg Type involving an integro-differential operator. Our study
includes the fractional Laplacian. Our approach still applies when adding small
singular terms. It hinges on appropriate choices of parameters in the
mountain-pass theore

### Existence results for nonlinear elliptic problems on fractal domains

Some existence results for a parametric Dirichlet problem defined on the
Sierpi\'nski fractal are proved. More precisely, a critical point result for
differentiable functionals is exploited in order to prove the existence of a
well determined open interval of positive eigenvalues for which the problem
admits at least one non-trivial weak solution

### An Eigenvalue Problem for Nonlocal Equations

In this paper we study the existence of a positive weak solution for a class of nonlocal equations under Dirichlet boundary conditions and involving the regional fractional Laplacian operator...Our result extends to the fractional setting some theorems obtained recently for ordinary and classical elliptic equations, as well as some characterization properties proved for differential problems involving different elliptic operators. With respect to these cases studied in literature, the nonlocal one considered here presents some additional difficulties, so that a careful analysis of the fractional spaces involved is necessary, as well as some nonlocal L^q estimates, recently proved in the nonlocal framework

### Multiple solutions of nonlinear equations involving the square root of the Laplacian

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 zero-boundary conditions, i.e. \begin{equation*} \left\{
\begin{array}{ll} A_{1/2}u=\lambda f(u) & \mbox{ in } \Omega\\ u=0 & \mbox{ on
} \partial\Omega. \end{array}\right. \end{equation*} The existence of at least
three $L^{\infty}$-bounded weak solutions is established for certain values of
the parameter $\lambda$ requiring that the nonlinear term $f$ is continuous and
with a suitable growth. Our approach is based on variational arguments and a
variant of Caffarelli-Silvestre's extension method

### Elliptic problems on complete non-compact Riemannian manifolds with asymptotically non-negative Ricci curvature

In this paper we discuss the existence and non--existence of weak solutions
to parametric equations involving the Laplace-Beltrami operator $\Delta_g$ in a
complete non-compact $d$--dimensional ($d\geq 3$) Riemannian manifold
$(\mathcal{M},g)$ with asymptotically non--negative Ricci curvature and
intrinsic metric $d_g$. Namely, our simple model is the following problem
\left\{ \begin{array}{ll} -\Delta_gw+V(\sigma)w=\lambda \alpha(\sigma)f(w) &
\mbox{ in } \mathcal{M}\\ w\geq 0 & \mbox{ in } \mathcal{M} \end{array}\right.
where $V$ is a positive coercive potential, $\alpha$ is a positive bounded
function, $\lambda$ is a real parameter and $f$ is a suitable continuous
nonlinear term. The existence of at least two non--trivial bounded weak
solutions is established for large value of the parameter $\lambda$ requiring
that the nonlinear term $f$ is non--trivial, continuous, superlinear at zero
and sublinear at infinity. Our approach is based on variational methods. No
assumptions on the sectional curvature, as well as symmetry theoretical
arguments, are requested in our approach

### An Eigenvalue Problem for Nonlocal Equations

In this paper we study the existence of a positive weak solution for a class of nonlocal equations under Dirichlet boundary conditions and involving the regional fractional Laplacian operator...Our result extends to the fractional setting some theorems obtained recently for ordinary and classical elliptic equations, as well as some characterization properties proved for differential problems involving different elliptic operators. With respect to these cases studied in literature, the nonlocal one considered here presents some additional difficulties, so that a careful analysis of the fractional spaces involved is necessary, as well as some nonlocal L^q estimates, recently proved in the nonlocal framework

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