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On the connection between two quasilinear elliptic problems with source terms of order 0 or 1
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In this paper, the authors established a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of N. The correlation gave new results of existence, nonexistence, regularity and multiplicity of the solutions for the two problems, without or with measures.Abstract:
We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of $\mathbb{R}^{N}.$ The first one, of the form \[ -\Delta_{p}u=\beta(u)| \nabla u| ^{p}+\lambda f(x)+\alpha, \] involves a source gradient term with natural growth, where $\beta$ is nonnegative, $\lambda>0,f(x)\geqq0$, and $\alpha$ is a nonnegative measure. The second one, of the form \[ -\Delta_{p}v=\lambda f(x)(1+g(v))^{p-1}+\mu, \] presents a source term of order $0, $where $g$ is nondecreasing, and $\mu$ is a nonnegative measure. Here $\beta$ and $g$ can present an asymptote. The correlation gives new results of existence, nonexistence, regularity and multiplicity of the solutions for the two problems, without or with measures. New informations on the extremal solutions are given when $g$ is superlinear.read more
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References
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Combined Effects of Concave and Convex Nonlinearities in Some Elliptic Problems
TL;DR: In this article, a class of semilinear elliptic Dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results is considered.
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Non-linear elliptic and parabolic equations involving measure data
Lucio Boccardo,Thierry Gallouët +1 more
TL;DR: In this paper, the existence of solutions for equations of the type −div(a(·, Du)) = f in a bounded open set Ω, u = 0 on ∂Ω, where a is a possibly non-linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure.
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On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN
TL;DR: In this paper, the authors derive a generic theorem for a wide class of functionals, having a mountain pass geometry, and show how to obtain, for a given functional, a special Palais-Smale sequence possessing extra properties that help to ensure its convergence.
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Blow-up solutions of some nonlinear elliptic problems
Haim Brezis,Juan Luis Vázquez +1 more
TL;DR: In this article, a generalization of Hardy's and Poincaré's inequalities is proposed to deal with unbaunded exiremal solution problems in a continuous, positive, increasing and convex funetion setting.
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Renormalized solutions of elliptic equations with general measure data
TL;DR: In this article, the authors studied the nonlinear monotone elliptic problem and proved the existence of a renormalized solution by an approximation procedure, where the key point is a stability result (the strong convergence in W 1,p 0 (Ω) of the truncates).