Existence of radial solutions for a $p(x)$-Laplacian Dirichlet problem

Abstract

In this paper, using variational methods, we prove the existence of at least one positive radial solution for the generalized $p(x)$ p ( x ) -Laplacian problem $$ -\Delta _{p(x)} u + R(x) u^{p(x)-2}u=a (x) \vert u \vert ^{q(x)-2} u- b(x) \vert u \vert ^{r(x)-2} u $$ − Δ p ( x ) u + R ( x ) u p ( x ) − 2 u = a ( x ) | u | q ( x ) − 2 u − b ( x ) | u | r ( x ) − 2 u with Dirichlet boundary condition in the unit ball in $\mathbb{R}^{N}$ R N (for $N \geq 3$ N ≥ 3 ), where a, b, R are radial functions.

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

“…The next is a fact [12, problem 127, P. 81] established in [15]. Proof Clearly ū is a positive radial function.…”

“…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).…”

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.…”

Global existence and exponential decay of solutions for higher-order parabolic equation with logarithmic nonlinearity

“…Also, He et al [5] proved the decay and the finite time blow-up for weak solutions of the equation, by using the potential well technique and concave technique. Recently many other authors investigated higher-order hyperbolic and parabolic type equation [2,3,6,[11][12][13][14][15]. Ishige et al [6] studied the Cauchy problem for nonlinear higher-order heat equation as follows…”