Existence of positive radial solution for Neumann problem on the Heisenberg group

Abstract

The existence of at least one positive radial solution of the Neumann problem $$ -\Delta _{\mathbb{H}^{n}} u+R(\xi ) u=a \bigl( \vert \xi \vert _{\mathbb{H}^{n}} \bigr) \vert u \vert ^{p-2} u - b\bigl( \vert \xi \vert _{\mathbb{H}^{n}}\bigr) \vert u \vert ^{q-2}u, $$−ΔHnu+R(ξ)u=a(|ξ|Hn)|u|p−2u−b(|ξ|Hn)|u|q−2u, is proved on the Heisenberg group $\mathbb{H}^{n}$Hn, via the variational principle, where $a(|\xi |_{\mathbb{H}^{n}})$a(|ξ|Hn), $b(|\xi |_{\mathbb{H}^{n}})$b(|ξ|Hn) are nonnegative radial functions and $R(\xi )$R(ξ) satisfies suitable conditions.

Solutions to a $$(p_1, \ldots ,p_n)$$-Laplacian Problem with Hardy Potentials

“…We point out that our approaches also fit with slightly different versions of the problem ( P ), e.g., with p(x)-Laplacian operator or the Heisenberg p-Laplacian operator or even the weighted Heisenberg p-Laplacian operator on the left hand side. Interested reader can see more details in [18,19,[26][27][28][29][30][31][32] and the references therein.…”

Existence of radial weak solutions to Steklov problem involving Leray–Lions type operator

“…The purpose of this paper is to establish the existence of at least one positive radial increasing weak solution of the problem (1.1) in the first order Sobolev space with variable exponent. We point out the authors have proved the existence of solutions to the problems in some special cases of f and g for a(x, t) = |t| p(x)−2 t on the Heisenberg groups (see [15,[19][20][21][22][23][24] for more details).…”

Game-theoretic p-Laplace operator involving the gradient

“…The standard mathematical techniques are not adequate to study these problems and they need new techniques. This may be the central development of mathematical ideas in active areas of pure mathematics which have had a decisive interaction with PDE's (see [1,[10][11][12][13]15,18,20,21,[24][25][26][27][28][29][30][31][32][33][34][36][37][38] for more relevant problems). It is very remarkable to write that in the classical theory of the p-Laplace equation (as well as Laplace equation) several main parts of mathematics such as Calculus of Variations, Partial Differential Equations, Potential Theory, Function Theory are joined.…”