On $$p(\cdot )$$-Laplacian problem with singular nonlinearity having variable exponent

Elharrar, N.; Igbida, Jaouad; Bouhlal, Ahmed

doi:10.1007/s41808-021-00112-1

Cited by 4 publications

(8 citation statements)

References 23 publications

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“…Remark 3.10. Note that the condition on p(•) in (1.6) is stronger than in [14,24], but is an outcome of the condition on q(•) in (3.3). This condition is important to prove Lemma 5.3 below.…”

Section: Main Results and Some Remarksmentioning

confidence: 99%

“…We will further assume greater regularity on f in problem (1.2) to guarantee the existence of a very weak solution. Here we use a new fresh ideas different from that used in [8,24,41] to generalize the previously obtained some results in [6,19,10,11,14,38,36] to the setting of variable exponents p, m and γ which appear in (1.6)-(1.7)-(1.8) respectively depends on the variable x. Obviously, the nonlinearity of (1.1) (resp.…”

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confidence: 95%

“…To analyse our problem more clearly, we point out that our estimates on the gradients of u (see Lemma 5.3) are new compared to [14,24]. In fact, it follows from Remarks 3.8-3.10 that the regularity given in Theorem 3.6 is better than ( [14, Theorem 3.2]; [24,Theorem 3.2]). To prove Theorem 3.6, a key result (Lemma 5.3) about an L p(•) estimate for the gradients of solution to problem (1.2).…”

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confidence: 96%

“…In recent years, boundary value problems with variable exponents has received a lot of attention, reader can look at the nice surveys books [16,18,22,39] and the references therein, while the study of the elliptic and parabolic problems involving singular nonlinearities with variable exponents is still developing with slowness, because there are a very limited number of results exist on this topic, reader can have the opportunity to look at the new marvellous works [8,24,28,41]. We mention also that much attention has been devoted to nonlinear elliptic equations with singularities because of their wide application to physical models such as Non-Newtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogenous, etc.…”

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confidence: 99%

“…Recently, in the variable exponents cases, the authors in [24] improved the results of [6,30] and studied the problem (1.2) in general form depending on γ(•) > 0, µ = 0, m(•) = m ≥ 1, 2 < p(•) ≤ N and p(•)-Laplace operator. They proved the existence and regularity results of problem (1.2) by adapting some techniques used in [14], nevertheless it was a new spirit of idea different from that used in [12], ( when γ(•) = γ ≥ 1, see [25] for a similar paper ).…”

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confidence: 99%

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Existence and Uniqueness Results for a Singular Nonlinear Elliptic Equations with Variable Exponents

Abdelaziz

2024

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This paper deals with study of the nonlinear singular elliptic equations in a bounded domain $\Omega\subset\mathbb{R}^N$, $(N\geq2)$ with Lipschitz boundary $\partial\Omega$, $$ Au=\frac{f}{u^{\gamma(\cdot)}}+\mu, $$ Where $A:=-\mathrm{div}\left(\widehat{a}(\cdot,Du)\right)$ is a Leray-Lions type operator which maps continuously $W^{1,p(\cdot)}_0(\Omega)$ into its dual $W^{-1,p’(\cdot)}(\Omega)$, whose simplest model is the $p(\cdot)$-laplacian type operator ( i.e. $\widehat{a}(\cdot,\xi)=|\xi|^{p(\cdot)-2}\xi$ ), such taht $f$ is a nonnegative function belonging to the Lebesgue space with variable exponents $L^{m(\cdot)}(\Omega)$, with $m(\cdot)$ being small ( or $L^{1}(\Omega)$ ) and $\mu$ is a nonnegative bounded Radon measure, while $m:\overline{\Omega}\to (1,+\infty)$, $\gamma:\overline{\Omega}\to (0,1)$ are continuous functions satisfying certain conditions depend on $p(\cdot)$. We prove the existence, uniqueness and regularity of nonnegative weak solutions or this class of problems with $p(\cdot)$-growth conditions. More precisely, we will discuss that the nonlinear singular term has some regularizing effects on the solutions of our problem which depends on the summability of $f$, $m(\cdot)$ and the value of $\gamma(\cdot)$. The functional framework involves Sobolev spaces with variable exponents as well as Lebesgue spaces with variable exponents. Our results can be seen as a generalization of some results given in the constant exponents case.

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Section: Main Results and Some Remarksmentioning

confidence: 99%