{"title":"具有双指数增长的非自治加权椭圆方程","authors":"S. Baraket, Rached Jaidane","doi":"10.2478/auom-2021-0033","DOIUrl":null,"url":null,"abstract":"Abstract We consider the existence of solutions of the following weighted problem: {L:=-div(ρ(x)|∇u|N-2∇u)+ξ(x)|u|N-2u=f(x,u)inBu>0inBu=0on∂B, \\left\\{ {\\matrix{{L: = - div\\left( {\\rho \\left( x \\right){{\\left| {\\nabla u} \\right|}^{N - 2}}\\nabla u} \\right) + \\xi \\left( x \\right){{\\left| u \\right|}^{N - 2}}} \\hfill & {u = f\\left( {x,u} \\right)} \\hfill & {in} \\hfill & B \\hfill \\cr {} \\hfill & {u > 0} \\hfill & {in} \\hfill & B \\hfill \\cr {} \\hfill & {u = 0} \\hfill & {on} \\hfill & {\\partial B,} \\hfill \\cr } } \\right. where B is the unit ball of ℝN, N #62; 2, ρ(x)=(loge|x|)N-1 \\rho \\left( x \\right) = {\\left( {\\log {e \\over {\\left| x \\right|}}} \\right)^{N - 1}} the singular logarithm weight with the limiting exponent N − 1 in the Trudinger-Moser embedding, and ξ(x) is a positif continuous potential. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of double exponential type. We prove the existence of positive solution by using Mountain Pass theorem. In the critical case, the function of Euler Lagrange does not fulfil the requirements of Palais-Smale conditions at all levels. We dodge this problem by using adapted test functions to identify this level of compactness.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"14","resultStr":"{\"title\":\"Non-autonomous weighted elliptic equations with double exponential growth\",\"authors\":\"S. Baraket, Rached Jaidane\",\"doi\":\"10.2478/auom-2021-0033\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract We consider the existence of solutions of the following weighted problem: {L:=-div(ρ(x)|∇u|N-2∇u)+ξ(x)|u|N-2u=f(x,u)inBu>0inBu=0on∂B, \\\\left\\\\{ {\\\\matrix{{L: = - div\\\\left( {\\\\rho \\\\left( x \\\\right){{\\\\left| {\\\\nabla u} \\\\right|}^{N - 2}}\\\\nabla u} \\\\right) + \\\\xi \\\\left( x \\\\right){{\\\\left| u \\\\right|}^{N - 2}}} \\\\hfill & {u = f\\\\left( {x,u} \\\\right)} \\\\hfill & {in} \\\\hfill & B \\\\hfill \\\\cr {} \\\\hfill & {u > 0} \\\\hfill & {in} \\\\hfill & B \\\\hfill \\\\cr {} \\\\hfill & {u = 0} \\\\hfill & {on} \\\\hfill & {\\\\partial B,} \\\\hfill \\\\cr } } \\\\right. where B is the unit ball of ℝN, N #62; 2, ρ(x)=(loge|x|)N-1 \\\\rho \\\\left( x \\\\right) = {\\\\left( {\\\\log {e \\\\over {\\\\left| x \\\\right|}}} \\\\right)^{N - 1}} the singular logarithm weight with the limiting exponent N − 1 in the Trudinger-Moser embedding, and ξ(x) is a positif continuous potential. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of double exponential type. We prove the existence of positive solution by using Mountain Pass theorem. In the critical case, the function of Euler Lagrange does not fulfil the requirements of Palais-Smale conditions at all levels. We dodge this problem by using adapted test functions to identify this level of compactness.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2021-11-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"14\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.2478/auom-2021-0033\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2478/auom-2021-0033","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 14
摘要
摘要考虑以下加权问题解的存在性:{L:=-div(ρ(x)|∇u|N-2∇u)+ξ(x)|u|N-2u=f(x,u)inBu>0inBu=0on∂B, \left\{\矩阵{{L:= - div \离开({\ρ\离开(x \右){{\左|{\微分算符u} \右|}^ {N - 2}} \微分算符u} \右)+ \ xi \离开(x \右){{\左| u \右|}^ {N - 2}}} \ hfill & {u = f \离开({x, u} \右)}\ hfill &的{}\ hfill & B \ hfill \ cr {} \ hfill & u > {0} \ hfill &的{}\ hfill & B \ hfill \ cr {} \ hfill & u = {0} \ hfill &上{}\ hfill & {\ B部分}\ hfill \ cr}} \。式中B为单位球,N #62;2、ρ(x)=(loge|x|)N-1 \rho \left(x \right) = {\left({\log {e \ / {\left| x \right}}} \right)^{N -1}}是Trudinger-Moser嵌入中极限指数为N−1的奇异对数权值,ξ(x)是一个正连续势。对于双指数型Trudinger-Moser不等式,非线性是临界或亚临界增长。利用山口定理证明了正解的存在性。在临界情况下,欧拉-拉格朗日函数在各个层次上都不满足Palais-Smale条件的要求。我们通过使用适应的测试函数来识别这种紧凑程度来避免这个问题。
Non-autonomous weighted elliptic equations with double exponential growth
Abstract We consider the existence of solutions of the following weighted problem: {L:=-div(ρ(x)|∇u|N-2∇u)+ξ(x)|u|N-2u=f(x,u)inBu>0inBu=0on∂B, \left\{ {\matrix{{L: = - div\left( {\rho \left( x \right){{\left| {\nabla u} \right|}^{N - 2}}\nabla u} \right) + \xi \left( x \right){{\left| u \right|}^{N - 2}}} \hfill & {u = f\left( {x,u} \right)} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u > 0} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u = 0} \hfill & {on} \hfill & {\partial B,} \hfill \cr } } \right. where B is the unit ball of ℝN, N #62; 2, ρ(x)=(loge|x|)N-1 \rho \left( x \right) = {\left( {\log {e \over {\left| x \right|}}} \right)^{N - 1}} the singular logarithm weight with the limiting exponent N − 1 in the Trudinger-Moser embedding, and ξ(x) is a positif continuous potential. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of double exponential type. We prove the existence of positive solution by using Mountain Pass theorem. In the critical case, the function of Euler Lagrange does not fulfil the requirements of Palais-Smale conditions at all levels. We dodge this problem by using adapted test functions to identify this level of compactness.