“…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.…”
We are concerned with the existence and multiplicity of weak solutions for a general form of a $$(p_1, \ldots ,p_n)$$
(
p
1
,
…
,
p
n
)
-Laplacian elliptic problem including singular terms. Our approaches are mainly based on critical points theory.
“…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.…”
We are concerned with the existence and multiplicity of weak solutions for a general form of a $$(p_1, \ldots ,p_n)$$
(
p
1
,
…
,
p
n
)
-Laplacian elliptic problem including singular terms. Our approaches are mainly based on critical points theory.
“…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).…”
We make use of variational methods to prove the existence of at least one positive radial increasing weak solution to a Leray–Lions type problem under Steklov boundary conditions.
“…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.…”
The game-theoretic p-Laplacian operator is a version of classical variational p-Laplacian which is in connection with stochastic games called Tug-of-War with noise. The existence of positive singular and Holder continuous solutions of the game-theoretic p-Laplace operator involving the gradient in a small C2 perturbation of the unit ball in Rn are proved. Finally, a more case problem is introduced.
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