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Kanishka Perera

Publications and source records attributed to Kanishka Perera.

At least 19 recordsLinked to original sources
arXiv2026

Doubly critical mixed-order quasilinear equations: existence, multiplicity and regularity

We investigate a class of mixed-order quasilinear elliptic equations in \(\mathbb R^N\) driven by the \(q\)-biharmonic and \(p\)-Laplacian operators, \[ Δ_q^2u-Δ_pu = λ\frac{|u|^{r_σ-2}u}{|x|^σ} +|u|^{p^*-2}u +|u|^{q^{**}-2}u, \] where \(1<q<N/2\), \(1<p<q^*\), \(p\ne q\), \(q\ne p^*\), and the weighted exponent is determined by the natural scaling of the equation. We establish compact weighted embeddings compatible with this mixed scaling and develop the corresponding variational framework. As an application, we prove existence and multiplicity of nontrivial solutions under suitable assumptions on the parameters. We also establish a local regularity result for the more general equation $Δ_q^2u-Δ_pu=f(x,u),$ where \(f\) is a Carathéodory function with local critical growth. By combining nonlinear potential estimates, a regularity-lifting argument, and a finite bootstrap for an associated second-order system, we obtain higher local regularity of both \(u\) and the nonlinear flux \(|Δu|^{q-2}Δu\). This regularity result, which is also of independent interest, enables us to derive a Pohozaev-type identity for the equations under consideration.

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arXiv2026

Critical quasilinear elliptic systems with convex subcritical perturbations

We study a class of Dirichlet problems for coupled quasilinear elliptic systems driven by the $p$--Laplacian in a bounded domain, where the nonlinear terms split into a critical homogeneous part and a convex subcritical perturbation. Using variational methods, a nonlinear eigenvalue theory based on the Fadell--Rabinowitz $\mathbb Z_2$--cohomological index, and a refined linking construction, we prove the existence of a nontrivial solution for every parameter value under suitable dimensional assumptions. We also treat a nonhomogeneous version with small forcing terms by applying a recent abstract perturbation theorem, yielding the existence of two distinct nontrivial solutions.

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arXiv2026

A unifying zero-mass equation

We introduce the nonlocal zero-mass equation \[ - Δu + Φ_u(|x|)\, |u|^{p-2}\, u = f(u) \quad \text{in } {\mathbb R}^N, \qquad Φ_u(r) = a \int_r^\infty ρ^{-b}\, h_u^{q-1}(ρ)\, dρ, \] where $h_u(ρ)$ is the mass of $|u|^p$ in the ball of radius $ρ$. For radial functions, this equation includes the defocusing inverse-power Schrödinger equation, the Chern-Simons-Schrödinger equation, and the Schrödinger-Poisson-Slater equation as special cases. We develop the associated ball-mass Lebesgue and Sobolev spaces, which are uniformly convex, and prove sharp compact radial embeddings above a new critical exponent, a Brézis-Lieb type splitting, and a Pohožaev identity. Exploiting a scaling invariance of the operator, we construct an unbounded sequence of eigenvalues of a scaled eigenvalue problem using the Fadell-Rabinowitz cohomological index, and obtain existence and multiplicity results in the subscaled, superscaled, and critical regimes. Our main result is a Brézis-Nirenberg type theorem, proved by means of a new scaled linking theorem, which gives a nontrivial radial solution for every $λ> 0$ that is not an eigenvalue. Specializing to the three models recovers several known results in a unified way and yields new ones, including a Brézis-Nirenberg type result for the Schrödinger-Poisson-Slater equation in dimensions $N \ge 4$.

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arXiv2026

On biharmonic equations with $p$-Laplacian and indefinite potentials or critical nonlinearity

In this paper we consider nonlinear biharmonic equations with $p$-Laplacian ($p\ge2$) of the form $$ \left\{ \begin{array}{l} Δ^2 u - Δ_p u + V (x) u = f (x, u) \text{,} u \in H^2 (\mathbb{R}^N) \text{,} \end{array} \right. $$ where the potential $V(x)$ may be indefinite. Using local linking and Morse theory, nontrivial solutions are obtained. In case the nonlinearity $f(x,\cdot)$ is odd, we obtain a sequence of large energy solutions. In the second part of the paper, for bounded positive potential, we get multiple solutions for the case that $$f(x,u)=λg (x) | u |^{q - 2} u + | u |^{m - 2} u$$ with exponent $m$ critical or subcritical.

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arXiv2026

Solutions to critical equations with a superposition of nonlocal Hartree-type nonlinearities

We study a class of nonlinear nonlocal elliptic equations in $\mathbb{R}^N$ involving superpositions of Hartree-type nonlinearities. Motivated by the Schrödinger-Poisson-Slater system, these equations arise as natural generalizations of problems with a single nonlocal interaction term. More precisely, we consider equations driven by a family of Riesz potentials weighted by a positive Borel measure, which gives rise to a superposed nonlocal operator. To treat this problem variationally, we introduce suitable functional settings, namely the superposed Coulomb space and the associated superposed Coulomb-Sobolev space, and study their main properties. Combining variational methods with a recently developed scaling-based critical point theory, we prove existence and multiplicity results for radial solutions. We also investigate a Brezis-Nirenberg-type problem and obtain multiplicity results near eigenvalues of an associated nonlinear eigenvalue problem. Our results extend previous works on single Hartree-type equations and provide a unified framework for treating superpositions of nonlocal interactions of Hartree type.

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arXiv2026

Inhomogeneous nonlinear Schrödinger equations with competing singular nonlinearities

We study nonlinear elliptic equations that arise as stationary states of inhomogeneous nonlinear Schrödinger equations with competing singular nonlinearities. The model involves the Laplacian combined with weighted power-type terms and naturally leads to a variational formulation in a weighted Sobolev space obtained from the intersection of the homogeneous Sobolev space with a weighted Lebesgue space. Using sharp weighted Sobolev and Caffarelli--Kohn--Nirenberg type inequalities, we establish continuous and compact embeddings of this space into suitable weighted Lebesgue spaces. These embedding results, together with a natural scaling structure of the model, allow us to apply the abstract critical point framework of Mercuri and Perera (2026), yielding a sequence of nonlinear eigenvalues for the associated problem via a min--max scheme based on the Fadell--Rabinowitz cohomological index. Within this framework we obtain a broad collection of existence and multiplicity results for equations driven by sums of weighted power nonlinearities, covering interactions in both subcritical and critical cases. We also establish a nonexistence result derived from a Pohozaev-type identity. Finally, we analyze the radial setting, where improved radial Caffarelli--Kohn--Nirenberg inequalities allow us to enlarge some of the admissible embedding ranges. This leads to strengthened radial versions of our main results.

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arXiv2026

New solutions to Schrödinger-Poisson-Slater equations in Coulomb-Sobolev spaces

We prove existence and multiplicity results for the nonlinear and nonlocal PDE $$ - Δu + (I_α\star |u|^p)\, |u|^{p-2}\, u = f(|x|,u) \quad \textrm{in} \,\,\mathbb {R}^N, $$ where $N \geq 2$, $I_α: \mathbb{R}^N \setminus \{0\} \rightarrow \mathbb{R}$ is the Riesz potential of order $α\in (1,N),$ $p>1,$ and the local nonlinearity $f: [0,\infty) \times \mathbb{R} \rightarrow \mathbb R$ is subject to a new class of assumptions. We find solutions to this zero-mass problem in a Coulomb-Sobolev space using a new scaling based approach in critical point theory, by which we classify the possibly different behaviour of the nonlinearity $f$ at zero and at infinity in terms of the scaling properties of the left hand side of the equation. This is accomplished identifying a scaling invariant PDE which can be interpreted as a nonlinear eigenvalue problem, for which a sequence of eigenvalues $\{λ_k\}$ is conveniently defined via the ${\mathbb{Z}}_2$-cohomological index of Fadell and Rabinowitz. This index allows us to use new critical group estimates (and scaling-based linking sets) which might not be possible via the classical genus. Within a fairly broad set of parameters $N,α, p$ and class of assumptions on the local nonlinearity $f,$ we establish compactness results for an associated action functional and find multiple solutions as critical points, whose existence and number is sensitive to the ''resonance'' of $f$ with the sequence of eigenvalues for the scaling invariant problem, a construction which is at places reminiscent, in the present nonlinear setting, of the classical Fredholm alternative. As a byproduct of our analysis, letting $p\neq 2$ allows us to capture general nonlinearities $f$ of Sobolev-subcritical, critical, or supercritical growth.

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arXiv2025

Nonlocal Dirichlet problems involving the Logarithmic $p$-Laplacian

In this work, we show the existence of an unbounded sequence of minimax eigenvalues for the logarithmic $p$-Laplacian via the $\mathbb{Z}_2$-cohomological index of Fadell and Rabinowitz. As an application of these minimax eigenvalues and $p$-logarithmic Sobolev inequality proved in [4], we prove new existence results for nonlocal Dirichlet problems involving logarithmic $p$-Laplacian and nonlinearities with $p$-superlinear and subcritical growth at infinity.

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arXiv2025

Prescribed energy solutions of concave-convex type problems involving sign-changing or vanishing weights

We provide an abstract approach to find couples $(λ,u) \in \mathbb{R} \times X$ satisfying $$Φ_λ(u)=c \quad \mbox{and} \quad Φ'_λ(u)=0,$$ for some suitable values of $c \in \mathbb{R}$. Here $Φ_λ$ is a $C^1$ functional (set on a Banach space $X$) whose main prototype is the energy functional associated to a concave-convex problem with sign-changing or vanishing weights. This approach allows us to derive several existence, multiplicity and bifurcation type results for the equation $Φ'_λ(u)=0$ with $λ$ fixed.

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arXiv2025

Fractional Schrödinger-Poisson-Slater equations in Coulomb-Sobolev spaces

We prove existence and multiplicity results for the fractional Schroedinger--Poisson--Slater equation $(-Δ)^s u + (I_α* u^2)u = f(|x|,u)$ in $\mathbb{R}^N$, where $0<s<1$ and $α\in (1,N)$. We seek solutions in a fractional Coulomb-Sobolev space and employ new tools in critical point theory that link the behavior of $f$ at zero and at infinity to the scaling properties of the left-hand side. For several regimes of $f$, we establish compactness for an associated action functional and obtain multiple solutions as critical points, with the number governed by the interaction of $f$ with a sequence of eigenvalues $\{λ_k\}$ defined via the $\mathbb{Z}_2$ cohomological index of Fadell and Rabinowitz (rather than the classical Krasnosel'skii genus). In this fractional setting we also prove new regularity results and necessary conditions for the existence of solutions.

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arXiv2025

Schrodinger-Poisson-Slater equations with nonlinearity subscaled near zero

We study the following zero-mass Schr{ö}dinger-Poisson-Slater equation \[ - Δu + \left( \frac{1}{4 π| x |} \ast u^2 \right) u = f (| x |, u) \text{,} \qquad u \in \mathcal{D}^{1, 2} (\mathbb{R}^3) \text{} \] with nonlinearity subscaled near zero in the sense that $f (| x |, t) \approx a | t |^{p - 2} t$ as $| t | \rightarrow 0$ for some $p\in\big(\frac{18}{7},3\big)$. A nonzero solution is obtained via Morse theory when the nonlinearity is asymptotically scaled at infinity. For this purpose we prove an abstract result on the critical groups at infinity for functionals satisfying the geometric assumptions of the scaled saddle point theorem of Mercuri \& Perera [arXiv:2411.15887]. For the case that $f (| x |, \cdot)$ is odd, a sequence of solutions are obtained via a version of Clark's theorem due to Kajikiya [J.\ Funct.\ Anal.\ 225 (2005) 352--370].

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arXiv2025

Critical p-biharmonic problems and applications to Hamiltonian systems

We study fourth-order quasilinear elliptic problems that involve the p-biharmonic operator and Navier boundary conditions. The nonlinear term grows at the critical Sobolev rate. Starting from a Hamiltonian system of two second-order equations, we use an inversion step to reduce it to a single p-biharmonic equation with a lower-order perturbation. We handle both non-resonant and resonant cases and show that the problem admits non-trivial solutions when the forcing term and the superscaled perturbation are small enough. The proof combines concentration-compactness with an abstract critical point method based on the cohomological index. Our theorems cover both homogeneous and nonhomogeneous settings and extend Tarantello's classical results for the Laplacian, improving earlier work on p-biharmonic equations (including the case p = 2) and on critical Hamiltonian systems.

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arXiv2025

Nonlocal critical elliptic equations in homogeneous fractional Sobolev spaces

We prove new multiplicity results for some nonlocal critical growth elliptic equations in homogeneous fractional Sobolev spaces. The proofs are based on an abstract critical point theorem based on the ${\mathbb Z}_2$-cohomological index and on a novel regularity result for fractional $p$-Laplacian equations as well as on some compact embeddings.

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arXiv2025

Global $L^\infty$ and decay estimate for fractional $p$-Laplacian equations in $D^{s,p}(\R^N)$

In this paper we present a new global $L^\infty$-estimate for solutions $u\in D^{s,p}(\R^N)$ of the fractional $p$-Laplacian equation % $$ u\in D^{s,p}(\R^N): (-Δ_p)^s u=f(x,u) \quad\mbox{in }\R^N, $$ % of the form % $$ \|u\|_{\infty}\le C Φ(\|u\|_β) $$ % for some $β> p$, where $Φ: \R^+\to \R^+$ is a data independent function with $\lim_{s\to 0^+}Φ(s)=0$. The obtained $L^\infty$-estimate is used to prove a decay estimate based on pointwise estimates in terms of nonlinear Wolff potentials. Taking advantage of both the $L^\infty$ and decay estimate we prove a Brezis-Nirenberg type result regarding $D^{s,2}(\R^N)$ versus $C_b\left(\R^N, 1+|x|^{N-2s}\right)$ local minimizers.

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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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