arXiv subjects

Giovanni Molica Bisci

Publications and source records attributed to Giovanni Molica Bisci.

At least 37 records · Page 2Linked to original sources

arXiv2023

A note on the NLS equation on Cartan-Hadamard manifolds with unbounded and vanishing potentials

We study the semilinear equation $-Δ_g u + V(σ) u = f(u)$ on a Cartan-Hadamard manifold ${\cal M}$ of dimension $N \geq 3$, and we prove the existence of a nontrivial solution under suitable assumptions on the potential function $V \in C({\cal M})$. In particular, the decay of $V$ at infinity is allowed, with some restrictions related to the geometry of ${\cal M}$. We generalize some results proved in $\mathbb{R}^N$ by Alves \emph{et al.}

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arXiv2022

Schrödinger equation on Cartan-Hadamard manifolds with oscillating nonlinearities

We study the equation $-Δ_g w+w=λα(σ) f(w)$ on a $d$-dimensional homogeneous Cartan-Hadamard Manifold $\mathcal{M}$ with $d \geq 3$. Without using the theory of topological indices, we prove the existence of infinitely many solutions for a class of nonlinearities $f$ which have an oscillating behavior either at zero or at infinity.

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arXiv2022

Multiple solutions for Schrödinger equations on Riemannian manifolds via $\nabla$-theorems

We consider a smooth, complete and non-compact Riemannian manifold $(\mathcal{M},g)$ of dimension $d \geq 3$, and we look for positive solutions to the semilinear elliptic equation $$ -Δ_g w + V w = αf(w) + λw \quad\hbox{in $\mathcal{M}$}. $$ The potential $V \colon \mathcal{M} \to \mathbb{R}$ is a continuous function which is coercive in a suitable sense, while the nonlinearity $f$ has a subcritical growth in the sense of Sobolev embeddings. By means of $\nabla$-Theorems introduced by Marino and Saccon, we prove that at least three solution exists as soon as the parameter $λ$ is sufficiently close to an eigenvalue of the operator $-Δ_g$.

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arXiv2021

On Critical Kirchhoff problems driven by the fractional Laplacian

We study a nonlocal parametric problem driven by the fractional Laplacian operator combined with a Kirchhoff-type coefficient and involving a critical nonlinearity term in the sense of Sobolev embeddings. Our approach is of variational and topological nature. The obtained results can be viewed as a nontrivial extension to the nonlocal setting of some recent contributions already present in the literature.

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arXiv2020

Existence results for some problems on Riemannian manifolds

By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional ($d\geq 3$) Riemannian manifold without boundary. As a direct consequence of our main theorems, we prove the existence of at least one solution to the following singular Yamabe-type problem $$ \left\lbrace \begin{array}{ll} -Δ_g w + α(σ)w = μK(σ) w^\frac{d+2}{d-2} +λ\left( w^{r-1} + f(w)\right), \quad σ\in\mathcal{M} &\\ &\\ w\in H^2_α(\mathcal{M}), \quad w>0 \ \ \mbox{in} \ \ \mathcal{M} & \end{array} \right.$$ where, as usual, $Δ_g$ denotes the Laplace-Beltrami operator on $(\mathcal{M},g)$, $α, K:\mathcal{M}\to\mathbb{R}$ are positive (essentially) bounded functions, $r\in(0,1)$, and $f:[0,+\infty)\to[0,+\infty)$ is a subcritical continuous function. Restricting ourselves to the unit sphere ${\mathbb{S}}^d$ via the stereographic projection, we also solve some parametrized Emden-Fowler equations in the Euclidean case.

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arXiv2020

Existence and localization of solutions for nonlocal fractional equations

This work is devoted to the study of the existence of at least one weak solution to nonlocal equations involving a general integro-differential operator of fractional type. As a special case, we derive an existence theorem for the fractional Laplacian, finding a nontrivial weak solution of the equation \begin{eqnarray*} \begin{cases} (-Δ)^s u=h(x)f(u) & {\mbox{ in }} Ω\\ u=0 & {\mbox{ in }} \mathbb{R}^n\setminus Ω, \end{cases} \end{eqnarray*} where $h\in L^{\infty}_+(Ω)\setminus\{0\}$ and $f:\mathbb{R}\rightarrow\mathbb{R}$ is a suitable continuous function. These problems have a variational structure and we find a nontrivial weak solution for them by exploiting a recent local minimum result for smooth functionals defined on a reflexive Banach space. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary.

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arXiv2019

Gradient-type systems on unbounded domains of the Heisenberg group

The purpose of this paper is to study the existence of weak solutions for some classes of one-parameter subelliptic gradient-type systems involving a Sobolev-Hardy potential defined on an unbounded domain $Ω_ψ$ of the Heisenberg group $\mathbb{H}^n=\mathbb{C}^n\times \mathbb{R}$ ($n\geq 1$) whose geometrical profile is determined by two real positive functions $ψ_1$ and $ψ_2$ that are bounded on bounded sets. The treated problems have a variational structure and thanks to this, we are able to prove the existence of an open interval $Λ\subset (0,\infty)$ such that, for every parameter $λ\in Λ$, the system has at least two nontrivial symmetric weak solutions that are uniformly bounded with respect to the Sobolev $HW^{1,2}_0$-norm. Moreover, the existence is stable under certain small subcritical perturbations of the nonlinear term. The main proof, crucially based on the Palais principle of symmetric criticality, is obtained by developing a group-theoretical procedure on the unitary group $\mathbb{U}(n)=U(n)\times\{1\}$ and by exploiting some compactness embedding results into Lebesgue spaces, recently proved for suitable $\mathbb{U}(n)$-invariant subspaces of the Folland-Stein space $HW^{1,2}_0(Ω_ψ)$. A key ingredient for our variational approach is a very general min-max argument valid for sufficiently smooth functionals defined on reflexive Banach spaces.

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arXiv2019

Some hemivariational inequalities in the Euclidean space

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, thanks to this, we are able to find a non-trivial weak solution for them by using variational methods and a non-smooth version of the Palais principle of symmetric criticality for locally Lipschitz continuous functionals, due to Krawcewicz and Marzantowicz. The main tools in our approach are based on appropriate theoretical arguments on suitable subgroups of the orthogonal group $O(d)$ and their actions on the Sobolev space $H^1(\mathbb{R}^d)$. Moreover, under an additional hypotheses on the dimension $d$ and in the presence of symmetry on the nonlinear datum, the existence of multiple pairs of sign-changing solutions with different symmetries structure has been proved. In connection to classical Schrödinger equations a concrete and meaningful example of an application is presented.

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arXiv2018

Existence of solutions for a class of fractional elliptic problems on exterior domains

This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{equation*}\label{00} \left\{ \begin{array}{l} (-Δ)^{s}u + u = |u|^{p-2}u\;\;\mbox{in $Ω$},\\ u \geq 0 \quad \mbox{in} \quad Ω\quad \mbox{and} \quad u \not\equiv 0, \\ u=0 \quad \mathbb{R}^N \setminus Ω, \end{array} \right. \end{equation*} involving the fractional Laplacian operator $(-Δ)^{s}$, where $s\in (0,1)$, $N> 2s$, $Ω\subset \R^N$ is an exterior domain with (non-empty) smooth boundary $\partial Ω$ and $p\in (2, 2_{s}^{*})$. The main technical approach is based on variational and topological methods. The variational analysis that we perform in this paper dealing with exterior domains is quite general and may be suitable for other goals too.

math.AP↗

arXiv2018

Nonlinear equations involving the square root of the Laplacian

In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian $A_{1/2}$ in a smooth bounded domain $Ω\subset \mathbb{R}^n$ ($n\geq 2$) and with zero Dirichlet boundary conditions. Namely, our simple model is the following equation \begin{equation*} \left\{ \begin{array}{ll} A_{1/2}u=λf(u) & \mbox{ in } Ω\\ u=0 & \mbox{ on } \partialΩ. \end{array}\right. \end{equation*} The existence of at least two non-trivial $L^{\infty}$-bounded weak solutions is established for large value of the parameter $λ$ requiring that the nonlinear term $f$ is continuous, superlinear at zero and sublinear at infinity. Our approach is based on variational arguments and a suitable variant of the Caffarelli-Silvestre extension method.

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arXiv2018

Sign-changing solutions for a class of zero mass nonlocal Schrödinger equations

We consider the following class of fractional Schrödinger equations $$ (-Δ)^α u + V(x)u = K(x) f(u) \mbox{in} \mathbb{R}^{N} $$ where $α\in (0, 1)$, $N>2α$, $(-Δ)^α$ is the fractional Laplacian, $V$ and $K$ are positive continuous functions which vanish at infinity, and $f$ is a continuous function. By using a minimization argument and a quantitative deformation lemma, we obtain the existence of a sign-changing solution. Furthermore, when $f$ is odd, we prove that the above problem admits infinitely many nontrivial solutions. Our result extends to the fractional framework some well-known theorems proved for elliptic equations in the classical setting. With respect to these cases studied in the literature, the nonlocal one considered here presents some additional difficulties, such as the lack of decompositions involving positive and negative parts, and the non-differentiability of the Nehari Manifold, so that a careful analysis of the fractional spaces involved is necessary.

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arXiv2018

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 $Δ_g$ 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 $$ \left\{ \begin{array}{ll} -Δ_gw+V(σ)w=λα(σ)f(w) & \mbox{ in } \mathcal{M}\\ w\geq 0 & \mbox{ in } \mathcal{M} \end{array}\right. $$ where $V$ is a positive coercive potential, $α$ is a positive bounded function, $λ$ 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 $λ$ 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.

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arXiv2017

Nonlinear elliptic equations on Carnot groups

This article concerns a class of elliptic equations on Carnot groups depending on one real positive parameter and involving a subcritical nonlinearity (for the critical case we refer to G. Molica Bisci and D. Repovš, Yamabe-type equations on Carnot groups, Potential Anal. 46:2 (2017), 369-383; arXiv:1705.10100 [math.AP]). As a special case of our results we prove the existence of at least one nontrivial solution for a subelliptic equation defined on a smooth and bounded domain $D$ of the Heisenberg group $\mathbb{H}^n=\mathbb{C}^n\times \mathbb{R}$. The main approach is based on variational methods.

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arXiv2017

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.

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arXiv2017

Nonlinear problems on the Sierpiński gasket

This paper concerns with a class of elliptic equations on fractal domains depending on a real parameter. Our approach is based on variational methods. More precisely, the existence of at least two non-trivial weak (strong) solutions for the treated problem is obtained exploiting a local minimum theorem for differentiable functionals defined on reflexive Banach spaces. A special case of the main result improves a classical application of the Mountain Pass Theorem in the fractal setting, given by Falconer and Hu (1999).

math.AP↗

arXiv2017

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 $Ω\subset \mathbb{R}^n$ ($n\geq 2$) and with Dirichlet zero-boundary conditions, i.e. \begin{equation*} \left\{ \begin{array}{ll} A_{1/2}u=λf(u) & \mbox{ in } Ω\\ u=0 & \mbox{ on } \partialΩ. \end{array}\right. \end{equation*} The existence of at least three $L^{\infty}$-bounded weak solutions is established for certain values of the parameter $λ$ 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.

math.AP↗

arXiv2017

Periodic solutions for a fractional asymptotically linear problem

We study the existence and multiplicity of periodic weak solutions for a non-local equation involving an odd subcritical nonlinearity which is asymptotically linear at infinity. We investigate such problem by applying the the pseudo-index theory developed by Bartolo, Benci and Fortunato \cite{bbf} after transforming the problem to a degenerate elliptic problem in a half-cylinder with a Neumann boundary condition, via a Caffarelli-Silvestre type extension in periodic setting. The periodic nonlocal case, considered here, presents, respect to the cases studied in literature, some new additional difficulties and a careful analysis of the fractional spaces involved is necessary.

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arXiv2016

Nontrivial solutions of superlinear nonlocal problems

We study the question of the existence of infinitely many weak solutions for nonlocal equations of fractional Laplacian type with homogeneous Dirichlet boundary data, in presence of a superlinear term. Starting from the well-known Ambrosetti-Rabinowitz condition, we consider different growth assumptions on the nonlinearity, all of superlinear type. We obtain three different existence results in this setting by using the Fountain Theorem, which extend some classical results for semilinear Laplacian equations to the nonlocal fractional setting.

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