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Kanishka Perera

Publications and source records attributed to Kanishka Perera.

At least 73 records · Page 4Linked to original sources
arXiv2016

$N$-Laplacian problems with critical Trudinger-Moser nonlinearities

We prove existence and multiplicity results for a $N$-Laplacian problem with a critical exponential nonlinearity that is a natural analog of the Brezis-Nirenberg problem for the borderline case of the Sobolev inequality. This extends results in the literature for the semilinear case $N = 2$ to all $N \ge 2$. When $N > 2$ the nonlinear operator $- Δ_N$ has no linear eigenspaces and hence this extension requires new abstract critical point theorems that are not based on linear subspaces. We prove new abstract results based on the ${\mathbb Z}_2$-cohomological index and a related pseudo-index that are applicable here.

math.AP↗
arXiv2015

$(N,q)$-Laplacian problems with critical Trudinger-Moser nonlinearities

We obtain nontrivial solutions of a $(N,q)$-Laplacian problem with a critical Trudinger-Moser nonlinearity in a bounded domain. In addition to the usual difficulty of the loss of compactness associated with problems involving critical nonlinearities, this problem lacks a direct sum decomposition suitable for applying the classical linking theorem. We show that every Palais-Smale sequence at a level below a certain energy threshold admits a subsequence that converges weakly to a nontrivial critical point of the variational functional. Then we prove an abstract critical point theorem based on a cohomological index and use it to construct a minimax level below this threshold.

math.AP↗
arXiv2015

On critical $p$-Laplacian systems

We consider the critical $p$-Laplacian system \begin{equation}\label{92} \begin{cases}-Δ_p u-\frac{λa}{p}|u|^{a-2}u|v|^b =μ_1|u|^{p^\ast-2}u+\frac{αγ}{p^\ast}|u|^{α-2}u|v|^β, &x\inΩ,\\ -Δ_p v-\frac{λb}{p}|u|^a|v|^{b-2}v =μ_2|v|^{p^\ast-2}v+\frac{βγ}{p^\ast}|u|^α|v|^{β-2}v, &x\inΩ,\\ u,v\ \text{in } D_0^{1,p}(Ω), \end{cases} \end{equation} where $Δ_p:=\text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator defined on $D^{1,p}(\mathbb{R}^N):=\{u\in L^{p^\ast}(\mathbb{R}^N):|\nabla u|\in L^p(\mathbb{R}^N)\}$, endowed with norm $\|u\|_{D^{1,p}}:=\big(\int_{\mathbb{R}^N}|\nabla u|^p\text{d}x\big)^{\frac{1}{p}}$, $N\ge3$, $1 1$ satisfy $a + b = p, α+ β= p^\ast:=\frac{Np}{N-p}$, the critical Sobolev exponent, $Ω$ is $\mathbb{R}^N$ or a bounded domain in $\mathbb{R}^N$, $D_0^{1,p}(Ω)$ is the closure of $C_0^\infty(Ω)$ in $D^{1,p}(\mathbb{R}^N)$. Under suitable assumptions, we establish the existence and nonexistence of a positive least energy solution. We also consider the existence and multiplicity of nontrivial nonnegative solutions.

math.AP↗
arXiv2015

The Brezis-Nirenberg problem for the fractional $p$-Laplacian

We obtain nontrivial solutions to the Brezis-Nirenberg problem for the fractional $p$-Laplacian operator, extending some results in the literature for the fractional Laplacian. The quasilinear case presents two serious new difficulties. First an explicit formula for a minimizer in the fractional Sobolev inequality is not available when $p \ne 2$. We get around this difficulty by working with certain asymptotic estimates for minimizers recently obtained by Brasco, Mosconi and Squassina. The second difficulty is the lack of a direct sum decomposition suitable for applying the classical linking theorem. We use an abstract linking theorem based on the cohomological index proved by Perera and Yang to overcome this difficulty.

math.AP↗
arXiv2015

Critical fractional $p$-Laplacian problems with possibly vanishing potentials

We obtain nontrivial solutions of a critical fractional $p$-Laplacian equation in the whole space and with possibly vanishing potentials. In addition to the usual difficulty of the lack of compactness associated with problems involving critical Sobolev exponents, the problem is further complicated by the absence of a direct sum decomposition suitable for applying classical linking arguments. We overcome this difficulty using a generalized linking construction based on the ${\mathbb Z}_2$-cohomological index.

math.AP↗
arXiv2015

Existence and nondegeneracy of ground states in critical free boundary problems

Existence and regularity of minimizers in elliptic free boundary problems have been extensively studied in the literature. The corresponding study of higher critical points was recently initiated in Jerison and Perera [30, 31]. In particular, the existence and nondegeneracy of a mountain pass point in a superlinear and subcritical free boundary problem related to plasma confinement was proved in [30]. In this paper we study ground states of a critical free boundary problem related to the Brezis-Nirenberg problem [5]. We extend the results of [30] to this problem by combining the method introduced there with the concentration compactness principle to overcome the difficulties arising from lack of compactness.

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arXiv2014

Existence and regularity of higher critical points for elliptic free boundary problems

Existence and regularity of minimizers in elliptic free boundary problems have been extensively studied in the literature. We initiate the corresponding study of higher critical points by considering a superlinear free boundary problem related to plasma confinement. The associated energy functional is nondifferentiable, and therefore standard variational methods cannot be used directly to prove the existence of critical points. Here we obtain a nontrivial generalized solution $u$ of mountain pass type as the limit of mountain pass points of a suitable sequence of $C^1$-functionals approximating the energy. We show that $u$ minimizes the energy on the associated Nehari manifold and use this fact to prove that it is nondegenerate. We use the nondegeneracy of $u$ to show that it satisfies the free boundary condition in the viscosity sense. Moreover, near any free boundary point that has a measure-theoretic normal, the free boundary is a smooth surface, and hence the free boundary condition holds in the classical sense.

math.AP↗
arXiv2014

A note on the Dancer-Fucik spectra of the fractional p-Laplacian and Laplacian operators

We study the Dancer-Fucik spectrum of the fractional p-Laplacian operator. We construct an unbounded sequence of decreasing curves in the spectrum using a suitable minimax scheme. For p=2, we present a very accurate local analysis. We construct the minimal and maximal curves of the spectrum locally near the points where it intersects the main diagonal of the plane. We give a sufficient condition for the region between them to be nonempty, and show that it is free of the spectrum in the case of a simple eigenvalue. Finally we compute the critical groups in various regions separated by these curves. We compute them precisely in certain regions, and prove a shifting theorem that gives a finite-dimensional reduction in certain other regions. This allows us to obtain nontrivial solutions of perturbed problems with nonlinearities crossing a curve of the spectrum via a comparison of the critical groups at zero and infinity.

math.AP↗
arXiv2014

On a class of critical $(p,q)$-Laplacian problems

We obtain nontrivial solutions of a critical $(p,q)$-Laplacian problem in a bounded domain. In addition to the usual difficulty of the loss of compactness associated with problems involving critical Sobolev exponents, this problem lacks a direct sum decomposition suitable for applying the classical linking theorem. We show that every Palais-Smale sequence at a level below a certain energy threshold admits a subsequence that converges weakly to a nontrivial critical point of the variational functional. Then we prove an abstract critical point theorem based on a cohomological index and use it to construct a minimax level below this threshold.

math.AP↗
arXiv2014

Existence results for fractional p-Laplacian problems via Morse theory

We investigate a class of quasi-linear nonlocal problems, including as a particular case semi-linear problems involving the fractional Laplacian and arising in the framework of continuum mechanics, phase transition phenomena, population dynamics and game theory. Under different growth assumptions on the reaction term, we obtain various existence as well as finite multiplicity results by means of variational and topological methods and, in particular, arguments from Morse theory.

math.AP↗
arXiv2014

A nodal solution of the scalar field equation at the second minimax level

We prove the existence of a sign-changing eigenfunction at the second minimax level of the eigenvalue problem for the scalar field equation under a slow decay condition on the potential near infinity. The proof involves constructing a set consisting of sign-changing functions that is dual to the second minimax class. We also obtain a nonradial sign-changing eigenfunction at this level when the potential is radial.

math.AP↗
arXiv2014

Ground states for scalar field equations with anisotropic nonlocal nonlinearities

We consider a class of scalar field equations with anisotropic nonlocal nonlinearities. We obtain a suitable extension of the well-known compactness lemma of Benci and Cerami to this variable exponent setting, and use it to prove that the Palais-Smale condition holds at all level below a certain threshold. We deduce the existence of a ground state when the variable exponent slowly approaches the limit at infinity from below.

math.AP↗
arXiv2013

A multiplicity result for the scalar field equation

We prove the existence of $N - 1$ distinct pairs of nontrivial solutions of the scalar field equation in ${\mathbb R}^N$ under a slow decay condition on the potential near infinity, without any symmetry assumptions. Our result gives more solutions than the existing results in the literature when $N \ge 6$. When the ground state is the only positive solution, we also obtain the stronger result that at least $N - 1$ of the first $N$ minimax levels are critical, i.e., we locate our solutions on particular energy levels with variational characterizations. Finally we prove a symmetry breaking result when the potential is radial. To overcome the difficulties arising from the lack of compactness we use the concentration compactness principle of Lions, expressed as a suitable profile decomposition for critical sequences.

math.AP↗
arXiv2013

$p$-Laplace equations with singular weights

We study a class of $p$-Laplacian Dirichlet problems with weights that are possibly singular on the boundary of the domain, and obtain nontrivial solutions using Morse theory. In the absence of a direct sum decomposition, we use a cohomological local splitting to get an estimate of the critical groups.

math.AP↗
arXiv2013

On the second minimax level for the scalar field equation

The paper studies eigenfunctions for the scalar field equation on $\R^N$ at the second minimax level $λ_2$. Similarly to the well-studied case of the ground state, there is a threshold level $λ^#$ such that $λ_2\le λ^#$, and a critical point at the level $λ_2$ exists if the inequality is strict. Unlike the case of the ground state, the level $λ_2$ is not attained in autonomous problems, and the existence is shown when the potential near infinity approaches the constant level from below not faster than $e^{- \varepsilon |x|}$. The paper also considers questions about the nodal character and the symmetry breaking for solutions at the level $λ_2$.

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