Mukherjee, Tuhina

In this paper, we deal with the existence of nontrivial nonnegative solutions for a (p, N)-Laplacian Schrödinger-Kirchhoff problem in ℝN with singular exponential nonlinearity. The main features of the paper are the (p, N) growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger-Moser inequality, and a completely new Brézis-Lieb-type lemma for singular exponential nonlinearity. © 2024 Walter de Gruyter GmbH, Berlin/Boston.

In this paper, we deal with the existence of nontrivial nonnegative solutions for a (p, N)-Laplacian Schrödinger–Kirchhoff problem in ℝN with singular exponential nonlinearity. The main features of the paper are the (p, N) growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity.

In this paper we establish the existence of at least two (weak) solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearities (equation presented) where is a smooth bounded domain of Rn, n ≥1, s 2 (0; 1), μ > 0 is a real parameter, β < n=(n - s) and q 2 (0; 1). The paper covers the so called degenerate Kirchhoff case and the existence proofs rely on the Nehari manifold techniques.

This article contains the study of the following problem with critical growth that involves the classical Laplacian and fractional Laplacian operators precisely (Formula presented.) where Ω⊆Rn, n≥3 is a bounded domain with smooth boundary ∂Ω, u+=max{u,0}, λ>0 is a real parameter, 2∗=2nn-2 and L=-Δ+(-Δ)s,fors∈(0,1). Here φ1 is the first eigenfunction of L with homogeneous Dirichlet boundary condition, t∈R and h∈L∞(Ω) satisfies ∫Ωhφ1dx=0. We establish existence and multiplicity results for the above problem, based on different ranges of the spectrum of L, using the Linking Theorem.

In this paper, we study the following singular problem, under mixed Dirichlet-Neumann boundary conditions, and involving the fractional Laplacian (Pλ) {(−Δ)su=λu−q+u2,u>0in Ω,A(u)=0on∂Ω=∑D∪∑N, where Ω⊂RN is a bounded domain with smooth boundary ∂Ω, 1/20 is a real parameter, 02s, 2s⁎=2N/(N−2s) and [Formula presented] Here ∑D, ∑N are smooth (N−1) dimensional submanifolds of ∂Ω such that ∑D∪∑N=∂Ω, ∑D∩∑N=∅ and ∑D∩∑N‾=τ′ is a smooth (N−2) dimensional submanifold of ∂Ω. Within a suitable range of λ, we establish existence of at least two opposite energy solutions for (Pλ) using the standard Nehari manifold technique.

In this paper, we study the following singular problem associated with mixed operators (the combination of the classical Laplace operator and the fractional Laplace operator) under mixed boundary conditions (Formula presented.) where U=(Ω∪N∪(∂Ω∩N¯)), Ω⊆RN is a non empty open set, D, N are open subsets of RN\Ω¯ such that D∪N=RN\Ω¯, D∩N=∅ and Ω∪N is a bounded set with smooth boundary, λ>0 is a real parameter and L=-Δ+(-Δ)s,fors∈(0,1). Here g(u)=u-q or g(u)=λu-q+up with 0

In this paper, we study the existence, nonexistence and multiplicity of positive solutions to the problem given by (Figure presented.) where D=Ω∪Π2∪(∂Ω∩Π2¯) and Dc is the complement of D, Ω⊆Rn is a non empty bounded open set with sufficiently smooth boundary ∂Ω, say of class C1. Π1, Π2 are open subsets of Rn\Ω¯ such that Π1∪Π2¯=Rn\Ω, Π1∩Π2=∅, ∂Ω∩Π2¯≠∅ and Ω∪Π2 is a bounded set with sufficiently smooth boundary, λ>0 is a real parameter, 02 and L=-Δ+(-Δ)s,fors∈(0,1). We first present a functional setting to study any problem involving L under mixed boundary conditions in the presence of concave-convex power nonlinearity, for a suitable range of λ, q and p. Our article also contains results related to Picone’s identity, strong maximum principles and comparison principles. We have extended the results of [1] to problems admitting mixed type operator as well as mixed boundary conditions. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.

This article focuses on the study of the existence, multiplicity and concentration behavior of ground states as well as the qualitative aspects of positive solutions for a (p, N)-Laplace Schrödinger equation with logarithmic nonlinearity and critical exponential nonlinearity in the sense of Trudinger-Moser in the whole Euclidean space ℝN. Through the use of smooth variational methods, penalization techniques, and the application of the Lusternik–Schnirelmann category theory, we establish a connection between the number of positive solutions and the topological properties of a set in which the potential function achieves its minimum values. © 2025 Elsevier B.V., All rights reserved.

This article focuses on the existence and non-existence of solutions for the following system of local and nonlocal type{−∂xxu+(−Δ)ys1u+u−u2s1−1=καh(x,y)uα−1vβinR2,−∂xxv+(−Δ)ys2v+v−v2s2−1=κβh(x,y)uαvβ−1inR2,u,v⩾0inR2, where s1,s2∈(0,1),α[jls-end-space/], β>1[jls-end-space/], α+β⩽min{2s1,2s2}[jls-end-space/], and 2si=2(1+si)1−si[jls-end-space/], i=1,2[jls-end-space/]. The existence of a ground state solution entirely depends on the behaviour of the parameter κ>0 and on the function h. In this article, we prove that a ground state solution exists in the subcritical case if κ is large enough and h satisfies (H). Further, if κ becomes very small, then there is no solution to our system. The study of the critical case, i.e., s1=s2=s[jls-end-space/], α+β=2s[jls-end-space/], is more complex, and the solution exists only for large κ and radial h satisfying (H1). Finally, we establish a Pohozaev identity which enables us to prove the non-existence results under some smooth assumptions on h. © 2025 Elsevier B.V., All rights reserved.

In this paper, we study a class of eigenvalue problems involving both local as well as nonlocal operators, precisely the classical Laplace operator and the fractional Laplace operator in the presence of mixed boundary conditions, that is {́¦́¦'''''''' Ns(∂u∂νL u u u ) = === 0 00 λu, in inin u∂U N Ωc >∩0_N _in _Ω, (Pλ) where U = (Ω ∪ N ∪ (∂Ω ∩ N)), Ω ⊆ Rn is a non empty bounded open set with smooth boundary ∂Ω, say of class C1 for n ≥ 3, D, N are open subsets of Rn \ Ω̄ such that D ∪ N = Rn \ Ω, D ∩ N = ∅ and Ω ∪ N is a bounded set with sufficiently smooth boundary, λ > 0 is a real parameter, and L = −∆ + (−∆)s, for s ∈ (0, 1). We establish the existence and some characteristics of the first eigenvalue and associated eigenfunctions to the above problem, based on the topology of the sets D and N. Next, we apply these results to establish bifurcation type results, both from zero and infinity for the problem (Qλ) which is an asymptotically linear problem inclined with (Pλ). © 2026 American Institute of Mathematical Sciences. All rights reserved.