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For p∈ (1 , N) and a domain Ω in RN, we study the following quasi-linear problem involving the critical growth: -Δpu-μg|u|p-2u=|u|p∗-2uinDp(Ω),where Δ p is the p-Laplace operator defined as Δ p(u) = div (| ∇ u| p-2∇ u) , p∗=NpN-p is the critical Sobolev exponent and Dp(Ω) is the Beppo-Levi space defined as the completion of Cc∞(Ω) with respect to the norm ‖u‖Dp:=[∫Ω|∇u|pdx]1p. In this article, we provide various sufficient conditions on g and Ω so that the above problem admits a positive solution for certain range of μ. As a consequence, for N≥ p2, if g is such that g+≠ 0 and the map u↦ ∫ Ω| g| | u| pd x is compact on Dp(Ω) , we show that the problem under consideration has a positive solution for certain range of μ. Further, for Ω = RN, we give a necessary condition for the existence of positive solution. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

Department of Mathematics

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Nonlinear Differential Equations and Applications

On the generalised Brézis–Nirenberg problem

For p∈ (1 , N) and Ω ⊆ RN open, the Beppo-Levi space D01,p(Ω) is the completion of Cc∞(Ω) with respect to the norm [∫Ω|∇u|pdx]1p. Using the p-capacity, we define a norm and then identify the Banach function space H(Ω) with the set of all g in Lloc1(Ω) that admits the following Hardy–Sobolev type inequality: ∫Ω|g||u|pdx≤C∫Ω|∇u|pdx,∀u∈D01,p(Ω),for some C&gt; 0. Further, we characterize the set of all g in H(Ω) for which the map G(u)=∫Ωg|u|pdx is compact on D01,p(Ω). We use a variation of the concentration compactness lemma to give a sufficient condition on g∈ H(Ω) so that the best constant in the above inequality is attained in D01,p(Ω). © 2021, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.

Department of Humanities and Social Sciences

Annali di Matematica Pura ed Applicata

The compactness and the concentration compactness via p-capacity

In 1961, Payne and Weinberger showed that ‘among the class of membranes with given area, free along the interior boundaries and fixed along the outer boundary of given length, the concentric annulus has the highest fundamental frequency.’ In the present study, we extend this result for the first eigenvalue of the p-Laplacian, for p∈(1,∞), in higher dimensional domains where the outer boundary is a fixed sphere. As an application, we prove that the nodal set of the second eigenfunctions of the p-Laplacian (with mixed boundary conditions) on a ball cannot be a concentric sphere. © 2019 Elsevier Inc.

Journal of Mathematical Analysis and Applications

On reverse Faber-Krahn inequalities

In this paper, we look for the weight functions (say g) that admit the following generalized Hardy-Rellich type inequality: for some constant C &gt; 0, where ω is an open set in RN with N ≥ 1We find various classes of such weight functions, depending on the dimension N and the geometry of ω Firstly, we use the Muckenhoupt condition for the one-dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of D0 2,2(ω) into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger. © 2019 Royal Society of Edinburgh.

Proceedings of the Royal Society of Edinburgh Section A: Mathematics

On the generalized Hardy-Rellich inequalities

For p∈(1,∞), we consider the following weighted Neumann eigenvalue problem on B1 c, the exterior of the closed unit ball in RN: −Δpϕ=λg|ϕ|p−2ϕinB1 c, [Formula presented]=0on∂B1,where Δp is the p-Laplace operator and g∈Lloc 1(B1 c)is an indefinite weight function. Depending on the values of p and the dimension N, we take g in certain Lorentz spaces or weighted Lebesgue spaces and show that (0.1)admits an unbounded sequence of positive eigenvalues that includes a unique principal eigenvalue. For this purpose, we establish the compact embeddings of W1,p(B1 c)into Lp(B1 c,|g|)for g in certain weighted Lebesgue spaces. For N&gt;p, we also provide an alternate proof for the embedding of W1,p(B1 c)into the Lorentz space Lp∗,p(B1 c). Further, we show that the set of all eigenvalues of (0.1)is closed. © 2019 Elsevier Ltd

Nonlinear Analysis, Theory, Methods and Applications

Neumann eigenvalue problems on the exterior domains

Let B1 be a ball in ℝN centred at the origin and let B0 be a smaller ball compactly contained in B1. For p ∈ (1, ∞), using the shape derivative method, we show that the first eigenvalue of the p-Laplacian in annulus B1 \B0 strictly decreases as the inner ball moves towards the boundary of the outer ball. The analogous results for the limit cases as p → 1 and p →∞ are also discussed. Using our main result, further we prove the nonradiality of the eigenfunctions associated with the points on the first nontrivial curve of the Fučik spectrum of the p-Laplacian on bounded radial domains. ©2018 American Mathematical Society.

Transactions of the American Mathematical Society

On the strict monotonicity of the first eigenvalue of the p-Laplacian on annuli

In this paper, we prove that the second eigenfunctions of the p- Laplacian, p &gt; 1, are not radial on the unit ball in RN, for any N ≥ 2. Our proof relies on the variational characterization of the second eigenvalue and a variant of the deformation lemma. We also construct an infinite sequence of eigenpairs {τn,Ψn} such that Ψn is nonradial and has exactly 2n nodal domains. A few related open problems are also stated. © 2015 American Mathematical Society.

Proceedings of the American Mathematical Society

On the structure of the second eigenfunctions of the p-laplacian on a ball

We consider the antimaximum principle for the p-Laplacian in the exterior domain: {-Δpu=λK(x)|u|p-2u+h(x)in B1c,u=0on ∂B1, where Δp is the p-Laplace operator with p&gt;1,λ is the spectral parameter and B1c is the exterior of the closed unit ball in RN with N ≥ 1. The function h is assumed to be nonnegative and nonzero, however the weight function K is allowed to change its sign. For K in a certain weighted Lebesgue space, we prove that the antimaximum principle holds locally. A global antimaximum principle is obtained for h with compact support. For a compactly supported K, with N=1 and p=2, we provide a necessary and sufficient condition on h for the global antimaximum principle. In the course of proving our results we also establish the boundary regularity of solutions of certain boundary value problems. © 2015 Elsevier Ltd.

Antimaximum principle in exterior domains

We consider the following weighted eigenvalue problem in the exterior domain:-Apu = K(x)|u|P-2 u in B,โ u = 0 on 3B1,where A, is the p-Laplace operator with p &gt; 1, and B is the exterior of the closed unit ball in RN with N &gt; 1. There is no restriction on the dimension N in terms of p, i.e., we allow both 1&lt;p&lt; N and p &gt; N. The weight function K is locally integrable on B and is allowed to change its sign. For some appropriate choice of w, a positive weight function on the interval (1,∞), we prove that the Beppo-Levi space D (B) is compactly embedded into the weighted Lebesgue space LP (B; w(x)). The existence of the positive eigenvalue for the above problem is proved for K such that suppK is of non-zero measure and K &lt;w. Further, we discuss the positivity, the regularity and the asymptotic behaviour at infinity of the first eigenfunctions.

Calculus of Variations and Partial Differential Equations

Weighted quasilinear eigenvalue problems in exterior domains

We consider the nonlinear eigenvalue problem where Δp is the p-Laplacian operator, Ω is a connected domain in RN with N &gt; p and the weight function g is locally integrable. We obtain the existence of a unique positive principal eigenvalue for g such that g+ lies in certain subspace of weak-LN/p(Ω). The radial symmetry of the first eigenfunctions are obtained for radial g, when Ω is a ball centered at the origin or RN. The existence of an infinite set of eigenvalues is proved using the Ljusternik-Schnirelmann theory on C1 manifolds. © 2011 Texas State University - San Marcos.

Electronic Journal of Differential Equations

Weighted eigenvalue problems for the p-Laplacian with weights in weak lebesgue spaces

In this note, we give sufficient conditions for the existence of a positive solution branch for the problem-Au = \a (x) f (u) in RN,u(x)→ 0 as x → ∞,with sign changing weight a, where N &gt; 3 and ƒ is a smooth nonlinearity with ƒ (0) 0

Nonlinear Analysis: Theory, Methods & Applications

A positive solution branch for nonlinear eigenvalue problems in

Given V, w locally integrable functions on a general domain Ω with V ≥ 0 but w allowed to change sign, we study the existence of ground states for the nonlinear eigenvalue problem: -Δu + Vu= p-2u, uΩ =0, with p subcritical. These are minimizers of the associated Rayleigh quotient whose existence is ensured under suitable assumptions on the weight w. In the present paper we show that an admissible space of weight functions is provided by the closure of smooth functions with compact support in the Lorentz space with L(p̃ ∞). This generalizes previous results and gives new sufficient conditions ensuring existence of extremals for generalized Hardy-Sobolev inequalities. The existence in such a generality of a principal eigenfunction in the linear case p = 2 is applied to study the bifurcation for semilinear problems of the type Δu= λ (a(x)u + b(x) r(u)), where a, b are indefinite weights belonging to some Lorentz spaces, and the function r has subcritical growth at infinity. © Springer-Verlag 2009.