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Kanishka Perera

Publications and source records attributed to Kanishka Perera.

At least 37 records · Page 2Linked to original sources
arXiv2024

Variational methods for scaled functionals with applications to the Schrödinger-Poisson-Slater equation

We develop novel variational methods for solving scaled equations that do not have the mountain pass geometry, classical linking geometry based on linear subspaces, or $\mathbb Z_2$ symmetry, and therefore cannot be solved using classical variational arguments. Our contributions here include new critical group estimates for scaled functionals, nonlinear saddle point and linking geometries based on scaling, a notion of local linking based on scaling, and scaling-based multiplicity results for symmetric functionals. We develop these methods in an abstract setting involving scaled operators and scaled eigenvalue problems. Applications to subcritical and critical Schrödinger-Poisson-Slater equations are given.

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arXiv2024

Local and nonlocal critical growth anisotropic quasilinear elliptic systems

In this paper we prove new multiplicity results for a critical growth anisotropic quasilinear elliptic system that is coupled through a subcritical perturbation term. We identify a certain scaling for the system and a parameter γ related to this scaling that determines the geometry of the associated variational functional. This leads to a natural classification of different nonlinear regimes for the system in terms of scaling properties of the perturbation term. We give three different types of multiplicity results in the three regimes γ = 1, γ > 1, and γ < 1. Proofs of our multiplicity results are based on a new abstract critical point theorem for symmetric functionals on product spaces, which we prove using the piercing property of the Z2-cohomological index of Fadell and Rabinowitz. This abstract result only requires a local (PS) condition and is therefore applicable to systems with critical growth. It is of independent interest as it has wide applicability to many different types of critical elliptic systems. We also indicate how it can be applied to obtain similar multiplicity results for nonlocal systems.

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arXiv2024

A bifurcation and multiplicity result for a critical growth elliptic problem

We consider a Brézis-Nirenberg type critical growth $p$-Laplacian problem involving a parameter $μ> 0$ in a smooth bounded domain $Ω$. We prove the existence of multiple nontrivial solutions if either $μ$ or the volume of $Ω$ is sufficiently small. The proof is based on an abstract critical point theorem that only assumes a local $(\text{PS})$ condition. Our results are new even in the semilinear case $p = 2$.

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arXiv2024

Abstract multiplicity theorems and applications to critical growth problems

We prove some abstract multiplicity theorems that can be used to obtain multiple nontrivial solutions of critical growth $p$-Laplacian and $(p,q)$-Laplacian type problems. We show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter $λ> 0$. In particular, the number of solutions goes to infinity as $λ\to \infty$. Moreover, we give an explicit lower bound on $λ$ in order to have a given number of solutions. This lower bound is in terms of a sequence of eigenvalues constructed using the ${\mathbb Z}_2$-cohomological index. This is a consequence of the fact that our abstract multiplicity results make essential use of the piercing property of the cohomological index, which is not shared by the genus. Our result for the $p$-Laplacian is new even in the semilinear case $p = 2$.

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arXiv2024

Existence results for a borderline case of a class of p-Laplacian problems

The aim of this paper is investigating the existence of at least one nontrivial bounded solution of the new asymptotically ``linear'' problem \[ \left\{ \begin{array}{ll} - {\rm div} \left[\left(A_0(x) + A(x) |u|^{ps}\right) |\nabla u|^{p-2} \nabla u\right] + s\ A(x) |u|^{ps-2} u\ |\nabla u|^p &\\ \qquad\qquad\qquad =\ μ|u|^{p (s + 1) -2} u + g(x,u) & \hbox{in $Ω$,}\\ u = 0 & \hbox{on $\partialΩ$,} \end{array}\right.\] where $Ω$ is a bounded domain in $\mathbb{R}^N$, $N \ge 2$, $1 < p < N$, $s > 1/p$, both the coefficients $A_0(x)$ and $A(x)$ are in $L^\infty(Ω)$ and far away from 0, $μ\in \mathbb{R}$, and the ``perturbation'' term $g(x,t)$ is a Carathéodory function on $Ω\times \mathbb{R}$ which grows as $|t|^{r-1}$ with $1\le r < p (s + 1)$ and is such that $g(x,t) \approx ν|t|^{p-2} t$ as $t \to 0$. By introducing suitable thresholds for the parameters $ν$ and $μ$, which are related to the coefficients $A_0(x)$, respectively $A(x)$, under suitable hypotheses on $g(x,t)$, the existence of a nontrivial weak solution is proved if either $ν$ is large enough with $μ$ small enough or $ν$ is small enough with $μ$ large enough. Variational methods are used and in the first case a minimization argument applies while in the second case a suitable Mountain Pass Theorem is used.

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arXiv2024

Critical growth double phase problems: the local case and a Kirchhoff type case

We study Brezis-Nirenberg type problems, governed by the double phase operator $- \mathrm{div}\left(|\nabla u|^{p-2}\, \nabla u + a(x)\, |\nabla u|^{q-2}\, \nabla u\right)$, that involve a critical nonlinearity of the form $|u|^{p^\ast - 2}\, u + b(x)\, |u|^{q^\ast - 2}\, u$. Both for the local case and for related nonlocal Kirchhoff type problems, we prove new compactness and existence results using variational methods in suitable Musielak-Orlicz Sobolev spaces. For these functional spaces, we prove some continuous and compact embeddings that are of independent interest. The study of the local problem is complemented by some nonexistence results of Pohožaev type.

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arXiv2024

Multiplicity results for elliptic problems with critical exponential growth

We prove new multiplicity results for some elliptic problems with critical exponential growth. More specifically, we show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter $μ> 0$. In particular, the number of solutions goes to infinity as $μ\to \infty$. The proof is based on an abstract critical point theorem.

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arXiv2023

An existence theory for nonlinear superposition operators of mixed fractional order

We establish the existence of multiple solutions for a nonlinear problem of critical type. The problem considered is fractional in nature, since it is obtained by the superposition of $(s,p)$-fractional Laplacians of different orders. The results obtained are new even in the case of the sum of two different fractional $p$-Laplacians, or the sum of a fractional $p$-Laplacian and a classical $p$-Laplacian, but our framework is general enough to address also the sum of finitely, or even infinitely many, operators. In fact, we can also consider the superposition of a continuum of operators, modulated by a general signed measure on the fractional exponents. When this measure is not positive, the contributions of the individual operators to the whole superposition operator is allowed to change sign. In this situation, our structural assumption is that the positive measure on the higher fractional exponents dominates the rest of the signed measure.

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arXiv2023

An existence theory for superposition operators of mixed order subject to jumping nonlinearities

We consider a superposition operator of the form $$ \int_{[0, 1]} (-Δ)^s u\, dμ(s),$$ for a signed measure $μ$ on the interval of fractional exponents $[0,1]$, joined to a nonlinearity whose term of homogeneity equal to one is "jumping", i.e. it may present different coefficients in front of the negative and positive parts. The signed measure is supposed to possess a positive contribution coming from the higher exponents that overcomes its negative contribution (if any). The problem taken into account is also of "critical" type, though in this case the critical exponent needs to be carefully selected in terms of the signed measure $μ$. Not only the operator and the nonlinearity considered here are very general, but our results are new even in special cases of interest and include known results as particular subcases. The possibility of considering operators "with the wrong sign" is also a complete novelty in this setting.

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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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arXiv2022

New linking theorems with applications to critical growth elliptic problems with jumping nonlinearities

We study critical growth elliptic problems with jumping nonlinearities. Standard linking arguments based on decompositions of $H^1_0(Ω)$ into eigenspaces of $- Δ$ cannot be used to obtain nontrivial solutions to such problems. We show that the associated variational functional admits certain linking structures based on splittings of $H^1_0(Ω)$ into nonlinear submanifolds. In order to capture these linking geometries, we prove several generalizations of the classical linking theorem of Rabinowitz that are not based on linear subspaces. We then use these new linking theorems to obtain nontrivial solutions of our problems. Our abstract results are of independent interest and can be used to obtain nontrivial solutions of other types of problems with jumping nonlinearities as well.

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arXiv2022

A system of equations involving the fractional $p$-Laplacian and doubly critical nonlinearities

This paper deals with existence of solutions to the following fractional $p$-Laplacian system of equations \begin{equation*} %\tag{$\mathcal P$}\label{MAT1} \begin{cases} (-Δ_p)^s u =|u|^{p^*_s-2}u+ \frac{γα}{p_s^*}|u|^{α-2}u|v|^β\;\;\text{in}\;Ω, (-Δ_p)^s v =|v|^{p^*_s-2}v+ \frac{γβ}{p_s^*}|v|^{β-2}v|u|^α\;\;\text{in}\;Ω, % % u,\;v\in\wsp, \end{cases} \end{equation*} where $s\in(0,1)$, $p\in(1,\infty)$ with $N>sp$, $α,\,β>1$ such that $α+β= p^*_s:=\frac{Np}{N-sp}$ and $Ω=\mathbb{R}^N$ or smooth bounded domains in $\mathbb{R}^N$. For $Ω=\mathbb{R}^N$ and $γ=1$, we show that any ground state solution of the above system has the form $(λU, τλV)$ for certain $τ>0$ and $U,\;V$ are two positive ground state solutions of $(-Δ_p)^s u =|u|^{p^*_s-2}u$ in $\mathbb{R}^N$. For all $γ>0$, we establish existence of a positive radial solution to the above system in balls. For $Ω=\mathbb{R}^N$, we also establish existence of positive radial solutions to the above system in various ranges of $γ$.

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arXiv2022

A general perturbation theorem with applications to nonhomogeneous critical growth elliptic problems

We prove a general perturbation theorem that can be used to obtain pairs of nontrivial solutions of a wide range of local and nonlocal nonhomogeneous elliptic problems. Applications to critical $p$-Laplacian problems, $p$-Laplacian problems with critical Hardy-Sobolev exponents, critical fractional $p$-Laplacian problems, and critical $(p,q)$-Laplacian problems are given. Our results are new even in the semilinear case $p = 2$.

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arXiv2022

A multiplicity result for critical elliptic problems involving differences of local and nonlocal operators

We study some critical elliptic problems involving the difference of two nonlocal operators, or the difference of a local operator and a nonlocal operator. The main result is the existence of two nontrivial weak solutions, one with negative energy and the other with positive energy, for all sufficiently small values of a parameter. The proof is based on an abstract result recently obtained in [20].

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arXiv2022

On a class of critical $N$-Laplacian problems

We establish some existence results for a class of critical $N$-Laplacian problems in a bounded domain in ${\mathbb R}^N$. In the absence of a suitable direct sum decomposition, we use an abstract linking theorem based on the ${\mathbb Z}_2$-cohomological index to obtain a nontrivial critical point.

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arXiv2021

On the critical $p$-Kirchhoff equation

We study a nonlocal elliptic equation of $p$-Kirchhoff type involving the critical Sobolev exponent. First we give sufficient conditions for the (PS) condition to hold. Then we prove some existence and multiplicity results using tools from Morse theory, in particular, the notion of a cohomological local splitting and eigenvalues based on the Fadell-Rabinowitz cohomological index.

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