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Register a new userOn symmetry results for elliptic equations with convex nonlinearities
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We investigate partial symmetry of solutions to semi-linear and quasi-linear elliptic problems with convex nonlinearities, in domains that are either axially symmetric or radially symmetric.
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In this paper we prove regularity results for a class of nonlinear degenerate elliptic equations of the form $displaystyle -operatorname{div}(A(| abla u|) abla u)+Bleft( | abla u|right) =f(u)$; in particular, we investigate the second order regularity of the solutions. As a consequence of these results, we obtain symmetry and monotonicity properties of positive solutions for this class of degenerate problems in convex symmetric domains via a suitable adaption of the celebrated moving plane method of Alexandrov-Serrin.
We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. By using a fractional order adaptation of this method, we show that the results of [LLLS20a, LLLS20b] remain valid for general power type nonlinearities.
Given $n geq 2$ and $1<p<n$, we consider the critical $p$-Laplacian equation $Delta_p u + u^{p^*-1}=0$, which corresponds to critical points of the Sobolev inequality. Exploiting the moving planes method, it has been recently shown that positive solutions in the whole space are classified. Since the moving plane method strongly relies on the symmetries of the equation and the domain, in this paper we provide a new approach to this Liouville-type problem that allows us to give a complete classification of solutions in an anisotropic setting. More precisely, we characterize solutions to the critical $p$-Laplacian equation induced by a smooth norm inside any convex cone. In addition, using optimal transport, we prove a general class of (weighted) anisotropic Sobolev inequalities inside arbitrary convex cones.
It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form $$ begin{array}{rcl} -Delta u +V(x) u &=& (I_alpha* |u|^p)|u|^{p-2}u+ lambda |u|^{q-2}u, , u in H^1(mathbb{R}^{N}), end{array} $$ where $lambda > 0, N geq 3, alpha in (0, N)$. The potential $V$ is a continuous function and $I_alpha$ denotes the standard Riesz potential. Assume also that $1 < q < 2,~2_{alpha} < p < 2^*_alpha$ where $2_alpha=(N+alpha)/N$, $2_alpha=(N+alpha)/(N-2)$. Our main contribution is to consider a specific condition on the parameter $lambda > 0$ taking into account the nonlinear Rayleigh quotient. More precisely, there exists $lambda_n > 0$ such that our main problem admits at least two positive solutions for each $lambda in (0, lambda_n]$. In order to do that we combine Nehari method with a fine analysis on the nonlinear Rayleigh quotient. The parameter $lambda_n > 0$ is optimal in some sense which allow us to apply the Nehari method.
We give pointwise gradient bounds for solutions of (possibly non-uniformly) elliptic partial differential equations in the entire Euclidean space. The operator taken into account is very general and comprises also the singular and degenerate nonlinear case with non-standard growth conditions. The sourcing term is also allowed to have a very general form, depending on the space variables, on the solution itself, on its gradient, and possibly on higher order derivatives if additional structural conditions are satisfied.
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