Combined effects of concave–convex nonlinearities and indefinite potential in some elliptic problems

Papageorgiou, Nikolaos S.; Rădulescu, Vicenţiu D.

doi:10.3233/asy-151292

Cited by 4 publications

(1 citation statement)

References 17 publications

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“…The work was enhanced to nonlinear problems for p-Laplacian by Azorero, Manfredi and Alonso [3], Guo and Zhang [20]. In general cases, the problem were studied in [4], [9], [19], [24], [28], [29]. In [1] Azorero and Alonso had considered the problem…”

Section: Introductionmentioning

confidence: 99%

On strong solvability of the Dirichlet problem for a class of semilinear elliptic equations with discontinuous coefficients

Mamedov¹,

Salmanova²

2021

Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan

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In this paper, we study a solvability result for the nonlinear problemassuming for the weight functions v ∈ A ∞ , ω ∈ A p to belong the Muckenhoupt class and a balance condition of Chanillo-Wheeden's type, with degenerate gradient ∇ ω u = ω 1/p ∇ x , ∇ y and its mod-The range conditions q ∈ (p, pN/(N − p)) and γ ∈ (1, N/(N − 1)) (or γ ∈ (1, p) and v −γ/(q−γ) ∈ L 1,loc (Ω) additionally) and µ ∈ (0, Λ) with sufficiently small Λ are assumed also.

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Section: Introductionmentioning

confidence: 99%

On strong solvability of the Dirichlet problem for a class of semilinear elliptic equations with discontinuous coefficients

Mamedov¹,

Salmanova²

2021

Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan

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An abstract critical point theorem with applications to elliptic problems with combined nonlinearities

Perera

2021

Calc. Var.

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Positive Solutions of Nonuniformly Elliptic Equations with Weighted Convex-Concave Nonlinearity

Mamedov

Гасымов

Gasymov

2022

Matematicheskie Zametki

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В данной статье мы доказываем результаты о существовании двух различных положительных решений задачи $$ \frac{\partial}{\partial z_i}(a_{ij}(z) \frac{\partial u}{\partial z_j})+v(x)u^{q-1}+ \mu u^{p-1}=0, \qquad z\in \Omega, \quad u|_{\partial\Omega}=0, $$ содержащей выпуклую и вогнутую нелинейности, параметр $\mu=\operatorname{const}$; переменные $z=(x,y) \in \mathbb{R}^n \times \mathbb{R}^{N-n}$. Матрица коэффициентов $A=\{a_{ij}(z)\}_{i,j=1}^N$ удовлетворяет условию неравномерной эллиптичности $$ C_1(\omega(x)|\xi|^2+|\eta|^2)\leqslant A(z) \zeta \cdot \zeta \leqslant C_2(\omega(x)|\xi|^2+|\eta|^2) $$ в ограниченной области $\Omega \subset \mathbb{R}^N$, $\zeta=(\xi,\eta) \in \mathbb{R}^n \times \mathbb{R}^{N-n}$, $\zeta \ne 0$. Для достижения своих целей мы рассматриваем условия на диапазон изменения показателей нелинейности $q \in (2,2N/(N-2))$ и $p\in (1,N/(N-1))$ (или $p\in (1,2)$ и дополнительное условие $v^{-p/(q-p)}\in L_1(\Omega)$) и $\mu \in (0,\Lambda)$ при достаточно малой $\Lambda$; положительные весовые функции $v \in A_\infty$, $\omega \in A_2$ принадлежат соответствующим классам Маккенхаупта в метрике $n$-мерного евклидового пространства, а также выполняется условие баланса типа Джанило-Видена. Библиография: 25 названий.

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Combined effects of concave–convex nonlinearities and indefinite potential in some elliptic problems

Cited by 4 publications

References 17 publications

On strong solvability of the Dirichlet problem for a class of semilinear elliptic equations with discontinuous coefficients

On strong solvability of the Dirichlet problem for a class of semilinear elliptic equations with discontinuous coefficients

An abstract critical point theorem with applications to elliptic problems with combined nonlinearities

Positive Solutions of Nonuniformly Elliptic Equations with Weighted Convex-Concave Nonlinearity

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