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Giovanni Molica Bisci

Publications and source records attributed to Giovanni Molica Bisci.

At least 19 recordsLinked to original sources
arXiv2026

A sharp eigenvalue theorem for mixed elliptic problems under mixed boundary conditions

In this paper, we study a class of eigenvalue problems involving both local and nonlocal operators, namely the classical Laplacian and the fractional Laplacian, under mixed boundary conditions. More precisely, we consider the problem \begin{equation}\label{1} \left\{ \begin{aligned} \mathcal{L}u &= λf(u), \quad u>0 &&\text{in }Ω,\\ u&=0 &&\text{in }U^c,\\ \mathcal{N}_s(u)&=0 &&\text{in }\mathcal{N},\\ \frac{\partial u}{\partialν}&=0 &&\text{on }\partialΩ\cap\overline{\mathcal{N}}, \end{aligned} \right. \tag{$P_λ$} \end{equation} where \( U=Ω\cup\mathcal{N}\cup \bigl(\partialΩ\cap\overline{\mathcal{N}}\bigr), \) \(Ω\subseteq\mathbb{R}^n\) is a bounded open set with smooth boundary, \(λ>0\) is a real parameter, \(f\) is continuous function with \(f(0)=0\), and \[ \mathcal{L}=-Δ+(-Δ)^s, \qquad s\in(0,1). \] We establish a characterization theorem for the existence of positive weak solutions to problem (P_λ). Motivated by the classical elliptic framework developed by Molica Bisci and Rădulescu \cite{MolicaBisciRadulescu2017}, we establish a corresponding characterization result for mixed local-nonlocal operators under mixed boundary conditions.

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arXiv2026

Bifurcation and multiplicity results for critical Grushin-Choquard problems

We consider the following nonlocal Brézis-Nirenberg type critical Choquard problem involving the Grushin operator \begin{equation*} \left\{ \begin{aligned} -Δ_γ& u =λu + \left(\displaystyle\int_Ω\frac{|u(w)|^{2^*_{γ,μ}}}{d(z-w)^μ}dw\right) |u|^{2^*_{γ,μ}-2}u \quad &&\text{in} \ Ω, u &= 0 \quad &&\text{on} \, \partial Ω, \end{aligned} \right. \end{equation*} where $Ω$ is an open bounded domain in $\mathbb{R}^N$, $N \geq 3$, with $Ω\cap \{ x=0\} \neq \emptyset$, and $λ>0$ is a parameter. Here, $Δ_γ$ represents the Grushin operator, defined as \[ Δ_γu(z) = Δ_x u(z) +(1+γ)^2 |x|^{2γ} Δ_y u(z), \quad γ\geq 0, \] where $z=(x,y)\in Ω\subset \mathbb{R}^m\times \mathbb{R}^n$, $m+n=N \geq 3$ and $2^*_{γ,μ}= \frac{2N_γ-μ}{N_γ-2}$ is the Sobolev critical exponent in the Hardy-Littlewood context with $N_γ= m+(1+γ)n$ is the homogeneous dimension associated to the Grushin operator and $0<μ<N_γ$. The homogeneous norm related to the Grushin operator is denoted by $d(\cdot)$. In this article, we prove the existence of bifurcation from any eigenvalue $λ^*$ of $-Δ_γ$ under Dirichlet boundary conditions. Furthermore, we show that in a suitable left neighborhood of $λ^*$, the number of nontrivial solutions to the problem is at least twice the multiplicity of $λ^*$.

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arXiv2026

Asymptotically linear fractional problems with mixed boundary conditions

We derive the existence of solutions for an asymptotically linear equation driven by the spectral fractional Laplacian operator with mixed Dirichlet-Neumann boundary conditions. When the nonlinear term $f$ is odd and a suitable relation between the perturbation parameter, the limit of $f(\cdot,t)/t$ as $t\to 0$ and the eigenvalues occurs, we establish also a multiplicity result via the pseudo-index theory related to the genus.

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arXiv2026

A note on critical problems involving the $p$-Grushin Operator: existence of infinitely many solutions

We consider a critical problem in a bounded domain involving the $p$-Grushin operator $Δ_α^p$. After a truncation argument, we obtain infinitely many solutions to our problem via Krasnoselskii's genus, extending a previous result of García Azorero and Peral Alonso to the $p$-Grushin operator. A central part of our analysis is the verification of the Palais-Smale condition of the associated functional under a certain level.

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arXiv2025

Regularizing effect of the interplay between coefficients in linear and semilinear $X$-elliptic equations

We study the regularizing effect arising from the interaction between the coefficient \(a\) of the zero order term and the datum \(f\) in the problem $$ \left\lbrace \begin{array}{ll} -\mathcal{L}u + a(x) g(u) = f(x) \quad &\mbox{in} \;\; Ω, u = 0 \quad &\mbox{on} \;\; \partialΩ, \end{array} \right. $$ where $Ω\subseteq\mathbb{R}^N$ is a bounded domain and $\mathcal{L}$ is an $X$-elliptic operator introduced by Lanconelli and Kogoj. If $f \in L^1(Ω)$, we prove that the \(Q\)-condition introduced by Arcoya and Boccardo is sufficient to ensure the existence and boundedness of solutions in the framework of $X$-elliptic operators as well. Finally, we prove the existence of a bounded solution for linear problems under a more general condition between $f$ and $a$.

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arXiv2025

Multiple solutions to asymptotically linear problems driven by superposition operators

In this paper, we investigate the existence and multiplicity of weak solutions to problems involving a superposition operator of the type $$\int_{[0, 1]}(- Δ)^s u d μ(s),$$ for a signed measure $μ$ on the interval of fractional exponents $[0,1]$, when the nonlinearity is subcritical and asymptotically linear at infinity; thus, we deal with a perturbation of the eigenvalue problem for the superposition operator. We use variational tools, extending to this setting well-known results for the classical and the fractional Laplace operators.

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arXiv2025

Bifurcation and multiplicity results for critical problems involving the $p$-Grushin operator

In this article we prove a bifurcation and multiplicity result for a critical problem involving a degenerate nonlinear operator $Δ_γ^p$. We extend to a generic $p>1$ a result which was proved only when $p=2$. When $p\neq 2$, the nonlinear operator $-Δ_γ^p$ has no linear eigenspaces, so our extension is nontrivial and requires an abstract critical theorem which is not based on linear subspaces. We also prove a new abstract result based on a pseudo-index related to the $\mathbf{Z}_2$-cohomological index that is applicable here. We provide a version of the Lions' Concentration-Compactness Principle for our operator.

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arXiv2024

Gradient regularity for $(s,p)$-harmonic functions

We study the local regularity properties of $(s,p)$-harmonic functions, i.e. local weak solutions to the fractional $p$-Laplace equation of order $s\in (0,1)$ in the case $p\in (1,2]$. It is shown that $(s,p)$-harmonic functions are weakly differentiable and that the weak gradient is locally integrable to any power $q\geq 1$. As a result, $(s,p)$-harmonic functions are Hölder continuous to arbitrary Hölder exponent in $(0,1)$. In addition, the weak gradient of $(s,p)$-harmonic functions has certain fractional differentiability. All estimates are stable when $s$ reaches $1$, and the known regularity properties of $p$-harmonic functions are formally recovered, in particular the local $W^{2,2}$-estimate.

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arXiv2024

Mountain Pass Solutions for an entire semipositone problem involving the Grushin Subelliptic Operator

For $N\ge 3$ we study the following semipositone problem $$ -Δ_γu = g(z) f_a(u) \quad \hbox{in $\mathbb{R}^N$}, $$ where $Δ_γ$ is the Grushin operator $$ Δ_ γu(z) = Δ_x u(z) + \vert x \vert^{2γ} Δ_y u (z) \quad (γ\ge 0), $$ $g\in L^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ is a positive function, $a>0$ is a parameter and $f_a$ is a continuous function on $\mathbb{R}$ that coincides with $f(t) -a$ for $t\in\mathbb{R}^+$, where $f$ is a continuous function with subcritical and Ambrosetti-Rabinowitz type growth and which satisfies $f(0) = 0$. Depending on the range of $a$, we obtain the existence of positive mountain pass solutions in $D_γ(\mathbb{R}^N)$

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arXiv2024

Regularity for the fractional $p$-Laplace equation

Higher Sobolev and Hölder regularity is studied for local weak solutions of the fractional $p$-Laplace equation of order $s$ in the case $p\ge 2$. Depending on the regime considered, i.e. $$0<s\le\tfrac{p-2}{p}\quad \text{or} \quad\tfrac{p-2}{p}<s<1,$$ precise local estimates are proven. The relevant estimates are stable if the fractional order $s$ reaches $1$; the known Sobolev regularity estimates for the local $p$-Laplace are recovered. The case $p=2$ reproduces the almost $W^{1+s,2}_{\rm loc}$-regularity for the fractional Laplace equation of any order $s\in(0,1)$.

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arXiv2024

A note on nonlinear critical problems involving the Grushin Subelliptic Operator: bifurcation and multiplicity results

We consider the boundary value problem $$ \cases{ -Δ_γu = λu + \left\vert u \right\vert^{2^*_γ-2}u &in $Ω$\cr u = 0 &on $\partialΩ$,\cr } $$ where $Ω$ is an open bounded domain in $\mathbb{R}^N$, $N \geq 3$, while $Δ_γ$ is the Grushin operator $$ Δ_ γu(z) = Δ_x u(z) + \vert x \vert^{2γ} Δ_y u (z) \quad (γ\ge 0). $$ We prove a multiplicity and bifurcation result for this problem, extending the results of Cerami, Fortunato and Struwe and of Fiscella, Molica Bisci and Servadei.

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arXiv2023

Asymptotically linear magnetic fractional problems

The aim of this paper is investigating the existence and multiplicity of weak solutions to non--local equations involving the {\em magnetic fractional Laplacian}, when the nonlinearity is subcritical and asymptotically linear at infinity. We prove existence and multiplicity results by using variational tools, extending to the magnetic local and non--local setting some known results for the classical and the fractional Laplace operators.

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arXiv2023

Nonlocal critical growth elliptic problems with jumping nonlinearities

In this paper we study a nonlocal critical growth elliptic problem driven by the fractional Laplacian in presence of jumping nonlinearities. In the main results of the paper we prove the existence of a nontrivial solution for the problem under consideration, using variational and topological methods and applying a new linking theorems recently got by Perera and Sportelli in [10]. The existence results provided in this paper can be seen as the nonlocal counterpart of the ones obtained in [10] in the context of the Laplacian equations. In the nonlocal framework the arguments used in the classical setting have to be refined. Indeed the presence of the fractional Laplacian operator gives rise to some additional difficulties, that we are able to overcome proving new regularity results for weak solutions of nonlocal problems, which are of independent interest.

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arXiv2023

New minimax theorems for lower semicontinuous functions and applications

In this paper we prove a version of the Fountain Theorem for a class of nonsmooth functionals that are sum of a $C^1$ functional and a convex lower semicontinuous functional, and also a version of a theorem due to Heinz for this class of functionals. The new abstract theorems will be used to prove the existence of infinitely many solutions for some elliptic problems whose the associated energy functional is of the above mentioned type. We study problems with logarithmic nonlinearity and a problem involving the 1-Laplacian operator.

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arXiv2023

On nonlocal Dirichlet problems with oscillating term

In this paper, a class of nonlocal fractional Dirichlet problems is studied. By using a variational principle due to Ricceri (whose original version was given in J. Comput. Appl. Math. 113 (2000), 401-410), the existence of infinitely many weak solutions for these problems is established by requiring that the nonlinear term $f$ has a suitable oscillating behaviour either at the origin or at infinity.

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arXiv2023

Subcritical nonlocal problems with mixed boundary conditions

In this paper, by variational and topological arguments based on linking and $\nabla$-theorems, we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet-Neumann boundary data, $$ \left\{ \begin{array}{lcl} (-Δ)^su=λu+f(x,u) & &\text{in } Ω, \\[2pt] \mkern+39mu u=0& &\text{on } Σ_{\mathcal{D}}, \\[2pt] \mkern+26mu \displaystyle \frac{\partial u}{\partial ν}=0& &\text{on } Σ_{\mathcal{N}}, \end{array} \right. $$ where $(-Δ)^s$, $s\in (1/2,1)$, is the spectral fractional Laplacian operator, $Ω\subset\mathbb{R}^N$, $N>2s$, is a smooth bounded domain, $λ>0$ is a real parameter, $ν$ is the outward normal to $\partialΩ$, $Σ_{\mathcal{D}}$, $Σ_{\mathcal{N}}$ are smooth $(N-1)$-dimensional submanifolds of $\partialΩ$ such that $Σ_{\mathcal{D}}\cupΣ_{\mathcal{N}}=\partialΩ$, $Σ_{\mathcal{D}}\capΣ_{\mathcal{N}}=\emptyset$ and $Σ_{\mathcal{D}}\cap\overlineΣ_{\mathcal{N}}=Γ$ is a smooth $(N-2)$-dimensional submanifold of $\partialΩ$.

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arXiv2023

Elliptic problems on weighted locally finite graphs

Let $\mathscr G:= (V,E)$ be a weighted locally finite graph whose finite measure $μ$ has a positive lower bound. Motivated by wide interest in the current literature, in this paper we study the existence of classical solutions for a class of elliptic equations involving the $μ$-Laplacian operator on the graph $\mathscr G$, whose analytic expression is given by \begin{equation*} Δ_μ u(x) := \frac{1}{μ(x)} \sum_{y\sim x} w(x,y) (u(y)-u(x)),\ \hbox{for all} \ x\in V, \end{equation*} where $w : V\times V \rightarrow [0,+\infty)$ is a weight symmetric function and the sum on the right-hand side of the above expression is taken on the neighbors vertices $x,y\in V$, that is $x\sim y$ whenever $w(x,y) > 0$. More precisely, by exploiting direct variational methods, we study problems whose simple prototype has the following form $$-Δ_μ u(x)=λf(x,u(x)),\ x \in \mathop D\limits^ \circ , \ \hbox{and} \ u|_{\partial D}=0,$$ where $D$ is a bounded domain of $V$ such that $\mathop D\limits^ \circ\neq \emptyset$ and $\partial D\neq \emptyset$, the nonlinear term $f : D \times \mathbb{R} \rightarrow \mathbb{R}$ satisfy suitable structure conditions and $λ$ is a positive real parameter. By applying a critical point result coming out from a classical Pucci-Serrin theorem in addition to a local minimum result for differentiable functionals due to Ricceri, we are able to prove the existence of at least two solutions for the treated problems. We emphasize the crucial role played by the famous Ambrosetti-Rabinowitz growth condition along the proof of the main theorem and its consequences. Our results improve the general results obtained by Grigor'yan, Lin, and Yang (J. Differential Equations 261(9) (2016), 4924-4943).

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