Mukherjee, Tuhina

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+u2s⁎−1,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 multiplicity results for double phase problems of Kirchhoff type with right-hand sides that include a parametric singular term and a nonlinear term of subcritical growth. Under very general assumptions on the data, we prove the existence of at least two weak solutions that have different energy sign. Our treatment is based on the fibering method in form of the Nehari manifold. We point out that we cover both the nondegenerate as well as the degenerate Kirchhoff case in our setting.

The paper deals with the following double phase problem -m[∫Ω(|∇u|pp+a(x)|∇u|qq)dx]div(|∇u|p-2∇u+a(x)|∇u|q-2∇u)=λu-γ+up∗-1inΩ,u>0inΩ,u=0on∂Ω,where Ω ⊂ RN is a bounded domain with Lipschitz boundary ∂Ω , N≥ 2 , m represents a Kirchhoff coefficient, 1 < p< q< p∗ with p∗= Np/ (N- p) being the critical Sobolev exponent to p, a bounded weight a(·) ≥ 0 , λ> 0 and γ∈ (0 , 1). By the Nehari manifold approach, we establish the existence of at least one weak solution.

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 article, we study the following anisotropic p-Laplacian equation with variable exponent given by (formula Presented) under the assumption is a bounded smooth domain in RN with p;N ≥ 2,λ > 0 and 0 < q ε C(Ω). For the purely singular case that is g ≡ 0, we proved existence and uniqueness of solution. We also demonstrate the existence of multiple solution to (P) provided f≡1 and g(u) = ur for r ε (p-1, p*-1).

This article consists of study of anisotropic double phase problems with singular term and sign changing subcritical as well as critical nonlinearity. Seeking the help of well known Nehari manifold technique, we establish existence of at least two opposite sign energy solutions in the subcritical case and one negative energy solution in the critical case. The results in the critical case are new also in the classical p-Laplacian case.

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.

In this article, we deal with the existence of non-negative solutions of the class of following non local problem {−M(∫Rn∫Rn|u(x)−u(y)|ns|x−y|2ndxdy)(−Δ)n/ssu=(∫ΩG(y,u)|x−y|μdy)g(x,u)inΩ,u=0inRn∖Ω, where (−Δ)n/ss is the n/ s-fractional Laplace operator, n≥ 1 , s∈ (0 , 1) such that n/ s≥ 2 , Ω ⊂ Rn is a bounded domain with Lipschitz boundary, M: R+→ R+ and g: Ω × R→ R are continuous functions, where g behaves like exp(|u|nn−s) as | u| → ∞. The key feature of this article is the presence of Kirchhoff model along with convolution type nonlinearity having exponential growth which appears in several physical and biological models.